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Transcription

Thesis
Spatio-temporal sampling strategies and spiral imaging for
translational cardiac MRI
DELATTRE, Bénédicte
Abstract
Magnetic Resonance Imaging is a reference tool to assess myocardial function and viability,
the two key measurements in clinics. However, several technical challenges remain. This
thesis focuses on the development of new strategies to provide an efficient characterization of
the myocardium. Using tools provided by MR physics and image processing a translational
"bench-to-bedside" approach was adopted. Concerning the "bench", Manganese was studied
as a contrast agent for myocardial viability assessment. A new cine sequence, "interleaved
cine", was also developed to increase the time resolution and opens up the possibility of
stress studies in mice on clinical scanners. In parallel, spiral imaging was applied to the
"bedside". In the context of real-time imaging, the proposed reconstruction method, k-t SPIRE,
takes into account the temporal information of data which helps to resolve the undersampling
artifacts and showed important improvements compared to the classical method both in
numerical simulations and in-vivo.
Reference
DELATTRE, Bénédicte. Spatio-temporal sampling strategies and spiral imaging for
translational cardiac MRI. Thèse de doctorat : Univ. Genève, 2011, no. Sc. 4302
URN : urn:nbn:ch:unige-150342
Available at:
http://archive-ouverte.unige.ch/unige:15034
Disclaimer: layout of this document may differ from the published version.
[ Downloaded 25/10/2016 at 18:13:48 ]
UNIVERSITÉ DE GENÈVE
Service de radiologie
FACULTÉ DE MÉDECINE
Professeur J.-P. Vallée
Groupe de Physique Appliquée
FACULTÉ DES SCIENCES
Professeur J.-P. Wolf
Spatio-temporal Sampling Strategies and Spiral
Imaging for Translational Cardiac MRI
THÈSE
présentée à la Faculté des sciences de l’Université de Genève
pour obtenir le grade de Docteur ès sciences, mention Physique
par
Bénédicte Delattre
de
Prévessin-Moëns (France)
Thèse No 4302
GENÈVE
Atelier de reproduction ReproMail
2011
Ne désespérez jamais. Faites infuser davantage.
Henri Michaux
Contents
Contents
v
Acknowledgements
ix
Abstracts
xi
List of Figures
xv
List of Tables
xix
1 Introduction
1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 MRI basis . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Signal detection . . . . . . . . . . . . . . . . . .
1.2.2 Relaxation phenomenon . . . . . . . . . . . . . .
1.2.3 Spatial encoding of information . . . . . . . . . .
1.3 Cardiac MRI . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Sequences for function measurement . . . . . . .
1.3.2 Acquisition and reconstruction methods for rapid
1.3.3 Viability measurement . . . . . . . . . . . . . . .
1.3.4 Translational research . . . . . . . . . . . . . . .
1.4 Aim of the project . . . . . . . . . . . . . . . . . . . . .
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Small animal imaging (from bench...)
2 Manganese-enhanced MRI in mice
2.1 Manganese as a contrast agent . . . . . . . . .
2.2 Imaging myocardial viability in mice . . . . . .
2.2.1 Material and methods . . . . . . . . . .
2.2.2 Manganese optimal dose determination
2.2.3 Myocardial function quantification . . .
2.2.4 Infarction quantification . . . . . . . . .
2.3 Manganese kinetics . . . . . . . . . . . . . . . .
2.3.1 Animal groups . . . . . . . . . . . . . .
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Contents
2.4
2.3.2 Myocardial function quantification . . .
2.3.3 Kinetic curves . . . . . . . . . . . . . .
2.3.4 Measurements of infarction extension .
Discussion . . . . . . . . . . . . . . . . . . . . .
2.4.1 Acute infarction . . . . . . . . . . . . .
2.4.2 Chronic infarction . . . . . . . . . . . .
2.4.3 Manganese as a marker of cell viability .
3 Highly time-resolved functional imaging
3.1 Sequence presentation . . . . . . . . . . . . .
3.1.1 Sequence parameters . . . . . . . . . .
3.1.2 Temporal regularization . . . . . . . .
3.1.3 Validation experiments . . . . . . . . .
3.2 Mass and function measurements . . . . . . .
3.2.1 Animals . . . . . . . . . . . . . . . . .
3.2.2 Image analysis . . . . . . . . . . . . .
3.2.3 Results . . . . . . . . . . . . . . . . .
3.3 Going further with the image enhancement...
3.3.1 Model presentation . . . . . . . . . . .
3.3.2 Validation experiments . . . . . . . . .
3.3.3 Function and mass measurements . . .
3.4 Discussion . . . . . . . . . . . . . . . . . . . .
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Spiral imaging (...to bedside)
4 Spiral Sequence
4.1 What is spiral ? . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Introduction . . . . . . . . . . . . . . . . . . .
4.1.2 What is spiral ? . . . . . . . . . . . . . . . . .
4.1.3 Spiral trajectory . . . . . . . . . . . . . . . . .
4.1.4 Specific advantages of spiral trajectory . . . . .
4.1.5 Eddy currents . . . . . . . . . . . . . . . . . . .
4.1.6 Sensibility to inhomogeneities . . . . . . . . . .
4.1.7 Concomitant fields . . . . . . . . . . . . . . . .
4.2 Designing the trajectory . . . . . . . . . . . . . . . . .
4.2.1 General solution . . . . . . . . . . . . . . . . .
4.2.2 Glover’s proposition to manage k-space center
4.2.3 Zhao’s adaptation to variable density spiral . .
4.2.4 Comparison of the 3 propositions . . . . . . . .
4.3 Coping with blurring in spiral images . . . . . . . . . .
4.3.1 Measuring gradient deviations . . . . . . . . . .
4.3.2 Correcting off-resonance effects . . . . . . . . .
4.3.3 Managing with concomitant fields . . . . . . .
4.3.4 Summary . . . . . . . . . . . . . . . . . . . . .
4.4 Spiral image reconstruction . . . . . . . . . . . . . . .
4.4.1 Gridding algorithm . . . . . . . . . . . . . . . .
4.4.2 Spiral imaging and parallel imaging . . . . . .
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Contents
5 Spline-based image model for spiral reconstruction: SPIRE
5.1 Model assumptions and justification . . . . . . . . . . . . . . .
5.1.1 Spline-based image model . . . . . . . . . . . . . . . . .
5.1.2 Spline-based Image REconstruction: SPIRE . . . . . . .
5.2 Algorithm implementation . . . . . . . . . . . . . . . . . . . . .
5.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Automatic parameter adjustment . . . . . . . . . . . . .
5.4 Evaluation on numercal Shepp-Logan phantom . . . . . . . . .
5.4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 MRI experiments . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 k − t SPIRE: time extension of spline-based image model
6.1 Spatio-temporal model . . . . . . . . . . . . . . . . . . . . .
6.2 Model fitting & implementation . . . . . . . . . . . . . . . .
6.3 Evaluation on numerical phantom . . . . . . . . . . . . . .
6.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 k − t SPIRE for real-time cardiac imaging . . . . . . . . . .
6.4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Reconstruction artifacts . . . . . . . . . . . . . . . .
6.5 Comparison with existing reconstruction methods . . . . . .
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7 Conclusions and perspectives
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7.1 MEMRI and highly time-resolved cine . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Spiral imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Bibliography
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Appendix
151
viii
Contents
Remerciements
Ce travail de thèse, qui aura duré 3 ans au sein du service de radiologie de l’Hôpital Cantonal Universitaire de Genève, a bénéficié de l’interaction de nombreuses personnes, qui toutes ont eu une
grande importance dans la réalisation de ce travail. J’aimerais ici leur adresser mes remerciements.
Je souhaite tout d’abord remercier le Prof. Jean-Paul Vallée qui m’a permis d’effectuer ce travail
de thèse sur un sujet passionnant, en me donnant la possibilité d’être encadrée par des personnes
très compétentes dans leur domaine. J’aimerais également remercier le Prof. Jean-Pierre Wolf qui a
sans hésité accepté de co-diriger cette thèse, en m’accordant une grande confiance lors de ce travail.
Je remercie le Prof. Christophe Becker pour m’avoir accueillie au sein de son service. Je tiens aussi
à remercier le Prof. Dimitri Van De Ville, le Prof. Matthias Stüber et le Prof. Sebastian Kozerke
pour avoir accepté de faire partie de mon jury de thèse.
Il y a ensuite des personnes sans qui ce travail ne serait certainement pas ce qu’il est. Tout
d’abord, je veux remercier le Dr Jean-Noël Hyacinthe qui m’a suivie, épaulée, soutenue tout au long
de ces trois ans. Son enthousiasme communicatif, sa disponibilité, ses remarques riches en questions
pertinentes font de lui un excellent mentor grâce à qui j’ai beaucoup progressé. Une grande partie
de ce travail n’aurait également pas vue le jour sans les précieux conseils du Prof. Dimitri Van De
Ville. Son expertise en traitement d’images a été un véritable atout. Très pédagogue, il a toujours
su expliquer des concepts tout d’abord assez abstraits pour moi avec des images simples. Toujours
de bonne humeur, il s’est rendu disponible à chaque fois que j’en ai eu besoin, malgré un emploi du
temps assez serré, et je lui en suis très reconnaissante.
Je voudrais également remercier le Prof. François Mach pour m’avoir permis de collaborer avec
son groupe pour toute la partie concernant les modèles murins, mes remerciements vont notamment
au Dr Vincent Braunersreuther pour sa disponibilité lors de nos nombreuses expériences.
Concernant la programmation de séquence, je remercie le Dr Gunnar Krüger pour m’avoir donné
accès à la séquence spirale, ainsi que pour son aide sur quelques points sensibles du debuggage de
la séquence.
Je souhaiterais également remercier le Dr Magalie Viallon pour sa disponibilité et son efficacité
concernant la résolution des problèmes relatifs à l’IRM ainsi que pour sa générosité concernant le
partage de son expérience en IRM cardiaque.
ix
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Remerciements
Mes prochains remerciements vont à mes collègues de groupe. Un grand merci à Lindsey qui
a patiemment relu l’anglais de mes nombreuses “proses” et avec qui j’ai partagé une cohabitation
très agréable dans le même espace de bureau. Je remercie Stéphany pour son avis pertinent durant les nombreuses discussions concernant les aspects biologiques de mon travail qui m’ont évité
les dangereux raccourcis que j’avais tendance à faire, mais aussi pour son regard éclairé qui a bien
souvent remis les morceaux de mon travail dans le bon ordre dans mon esprit. Merci également à
Jean-Luc, qui lors de mon arrivée dans le groupe, m’a appris la manipulation de l’IRM et formée aux
examens cardiaques de souris ainsi qu’à toutes les techniques relatives à cette partie de mon travail.
Merci également à Xavier, Karin et Frank pour les discussions enrichissantes que nous avons pu avoir.
Enfin, j’ai pu bénéficier d’un excellent cadre de travail grâce au Dr François Lazeyras, qui m’a
accueillie au sein des locaux du CIBM. Cet environnement où cohabitent des personnes de spécialités, mais aussi de personnalités, très différentes a souvent été le siège d’émulations scientifiques, ou
non, très enrichissantes. Merci à Suzanne, Stéphane, Tamara, Laura, Elda, Djano, Isik, César, Jeff,
Lorena, Vincent, Thomas, Rares, Michel, et à tous ceux qui ont passé du temps dans l’open-space
pour ces bons moments. Des remerciements particuliers vont à Jonas pour son aide avec certains
tests statistiques ainsi qu’avec quelques subtilités de LATEX.
Mais une thèse est également une grande aventure humaine. Elle m’a permis de rencontrer des
personnes pleines de coeur (ce qui a pu être vérifié à l’IRM !) qui ont été présentes dans tous les
moments importants, de la thèse, mais aussi de la vie: Steph, Jean-No, Lindsey, Thomas, Frank,
les mots sont un peu faibles pour exprimer mes sentiments, donc je me limiterai à l’essentiel: merci
d’être là.
Enfin, mes derniers remerciements, et non des moindres, vont à mes parents. À ma maman
qui a supporté beaucoup trop de choses durant ces trois ans, mais qui a tout de même trouvé les
ressources pour me soutenir sur tous les plans durant ce marathon. À mon papa, qui n’aura pas vu
la fin de l’aventure, mais qui était déjà très fier de moi lorsqu’elle a commencé... Ils m’ont toujours
encouragée à faire ce dont j’avais envie en me donnant la possibilité et les moyens de le faire. C’est
une grande chance.
Genève, janvier 2011
Bénédicte Delattre
Abstract
Currently, cardiovascular diseases are the leading cause of mortality worldwide. Overall, the majority of deaths are due to coronary heart disease. Cardiac imaging has thus an important role in
early diagnosis of the disease but also for the development of new treatments. Magnetic resonance
imaging (MRI) is a reference tool to assess myocardial function and viability, the two key measurements in clinics. Despite the important developments in this domain during the last two decades,
several technical challenges remain. This work focuses on the development of new techniques to
provide an efficient characterization of the myocardium. Using tools provided by MR physics and
image processing we adopted a translational “bench-to-bedside” approach, from mice to patients.
The first part of the thesis starts from the “bench” with the development of cardiac imaging in
mice. Translational research can greatly benefit from mouse imaging on clinical scanners, nevertheless it remains a challenge. In order to assess the myocardial viability of mice, we chose to use
Manganese (Mn2+ ) as a contrast agent. Since this is an analog of Calcium (Ca2+ ), it constitutes
a powerful tool for this application. A robust protocol to quantify myocardial infarction in mice
was thus set up and we showed a high correlation between infarct volume evaluated with Mn2+ enhanced MRI and with the histologic reference method. The study of Mn2+ kinetics in infarction
demonstrated a faster accumulation of the contrast agent in infarction in the acute phase compared
to the chronic phase. Mn2+ kinetics provided thus an interesting tool to differentiate acute from
chronic infarction. Moreover, important information concerning cardiac function can be derived
from moving cine images, the tradeoff between spatial and temporal resolution is however strong
on clinical scanners. We developed a new cine sequence, “interleaved cine”, to increase the final
time resolution and reach values of the same order as dedicated scanners. The images provided by
this sequence were then enhanced with a post processing algorithm that allowed the reduction of
artifacts produced by the sequence itself. Finally, interleaved cine was successfully validated with
mass and function measurements. This sequence opens up the possibility of stress studies in mice
on clinical scanners.
The second part of the thesis is concerned with spiral imaging applied to the “bedside”. Spiral
is probably the most relevant sampling scheme for cardiac imaging due to its inherent advantages
such as an efficient coverage of k-space and inherent flow compensation. Several applications can
thus benefit from it, a clinically relevant one being real-time imaging. However, the classical FFT
(Fast Fourier Transform) image reconstruction method is not applicable to this trajectory and the
strategy usually used in this case (gridding algorithm) does not perform well when k-space is highly
undersampled. To face this problem, we first developed an innovative spline-based reconstruction
method, SPIRE, that was shown to be more robust to undersampling artifacts than gridding. This
method was then extended to the temporal domain (k − t SPIRE) and applied to real-time imaging.
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Abstract
Instead of considering that a set of data is acquired at the same time point as is the case in existing
reconstruction methods, k − t SPIRE takes into account the temporal acquisition information of
data which helps to resolve the undersampling artifacts. Compared to gridding associated with
sliding window (the reference method), we demonstrated an important improvement of signal-tonoise ratio as well as a better preservation of edge sharpness in numerical simulations with k − t
SPIRE. We also observed a better temporal definition of the heart motion in volunteer experiments.
As a future perspective, this method can be integrated with existing ones to further accelerate the
acquisition. In particular, the combination with parallel imaging has to be investigated, but trendy
regularization methods such as “compressed sensing” are also of great interest. k − t SPIRE is also
suited to other applications such as perfusion imaging or hyperpolarized studies.
Résumé
Les maladies cardiovasculaires sont actuellement la première cause de mortalité dans le monde, et
les cardiopathies coronariennes représentent la majeure cause de ces décès. L’imagerie cardiaque a
donc un rôle très important dans le diagnostic précoce des cardiopathies ainsi que dans l’évaluation
et le développement de nouveaux traitements. L’imagerie par résonance magnétique (IRM) est un
outil de référence dans la caractérisation de la fonction et de la viabilité myocardique, deux indices
clés en pratique clinique. Malgré d’importants développements dans le domaine de l’IRM ces vingt
dernières années, plusieurs défis techniques restent à relever. Ce travail est consacré au développement de nouvelles techniques permettant une meilleure compréhension de la pathophysiologie cardiaque. Pour ce faire, une approche translationnelle a été adoptée, utilisant les outils de la Physique
de la Résonance Magnétique ainsi que du traitement d’images appliqués de la souris jusqu’au patient.
La première partie de cette thèse a été dédiée au développement de l’imagerie cardiaque chez
la souris. L’imagerie cardiaque des modèles murins sur des IRM cliniques peut apporter un grand
bénéfice à la recherche translationnelle mais reste un challenge technique important. Concernant la
mesure de la viabilité myocardique, nous avons choisi d’utiliser le Manganese (Mn2+ ) comme agent
de contraste. En tant qu’analogue du Calcium (Ca2+ ) il constitue un puissant outil pour cette application. Un protocole robuste mis en place pour la quantification de l’infarctus chez la souris avec
le Mn2+ a ainsi permis de montrer une excellente corrélation de la mesure du volume infarci avec la
méthode histologique de référence. L’étude de la cinétique du Mn2+ dans l’infarctus a montré une
accumulation plus rapide de cet agent de contraste dans la phase aigue que dans la phase chronique.
La cinétique du Mn2+ constitue donc un outil intéressant pour la discrimination des infarctus aigus
et chroniques. Par ailleurs, en plus des mesures de viabilité, d’importantes informations concernant l’évaluation de la pathologie cardiaque sont données par la mesure de la fonction, dérivée des
images cine. Cependant, le compromis entre résolution spatiale et temporelle est assez important
sur les systèmes cliniques. Nous avons donc développé une nouvelle séquence appelée “interleaved
cine” afin d’arriver à une résolution temporelle dans le même ordre de grandeur que celle obtenue
sur des systèmes dédiés. Les images produites par cette séquence ont également pu être améliorées
avec un algorithme de post-processing qui a permis de réduire les artéfacts produits par la séquence
elle-même. Enfin, la séquence ”interleaved cine” a été validée avec succès avec des mesures de masses
et de fonction. Cette séquence ouvre la voie à des études de stress chez la souris sur scanners cliniques.
La seconde partie de cette thèse a été consacrée à l’imagerie spirale appliquée au patient. La
spirale est probablement le schéma d’acquisition le plus pertinent grâce à ses avantages intrinsèques
que sont, entre autres, une couverture efficace de l’espace de Fourier, et une auto-compensation des
flux. De nombreuses applications peuvent donc bénéficier de cette trajectoire, l’une d’entre elles
étant l’imagerie cardiaque en temps réel qui est très pertinente au niveau clinique. Cependant, la
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Résumé
méthode de reconstruction classique, la FFT (transformée de Fourier rapide), ne s’applique plus avec
cette trajectoire et la stratégie habituellement utilisée (algorithme de gridding) ne donne pas de bons
résultats lorsque l’espace de Fourier est sous-échantillonné de manière importante. Pour faire face à
ce problème, nous avons tout d’abord développé une méthode de reconstruction basée sur un modèle
d’image interpolée avec des fonctions spline: SPIRE. Cette méthode s’est révélée plus robuste aux
artéfacts de sous-échantillonnage que le gridding. SPIRE a ensuite été étendue dans le domaine
temporel (k − t SPIRE) et appliquée à l’imagerie temps réel. Au lieu de considérer qu’un ensemble
de données a été acquise au même point dans le temps, comme cela est fait dans les méthodes
de reconstruction existantes, k − t SPIRE prend en compte l’information temporelle concernant
l’acquisition de chaque échantillon de signal, ce qui résout les artéfacts de sous-échantillonnage.
Comparé au gridding, associé avec la méthode de “sliding window” (méthode de référence), nous
avons démontré, lors des simulations numériques, une importante amélioration du rapport signalsur-bruit, mais également une meilleure conservation des bords des objets contenus dans l’image.
Nous avons également observé une meilleure définition temporelle du mouvement cardiaque lors
d’expériences sur volontaires. Une perspective intéressante serait l’intégration de cette méthode
avec les algorithmes existants pour accélérer encore l’acquisition. En particulier, la combinaison
avec l’imagerie parallèle doit être investiguée mais aussi avec des méthodes de régularisation actuelles
comme le “compressed sensing”. k − t SPIRE peut également s’avérer particulièrement adaptée à
d’autres applications comme l’imagerie de la perfusion cardiaque ainsi que dans des études utilisant
l’hyperpolarisation.
List of Figures
1.1
1.2
1.3
Illustration of nuclear spins orientation in a magnetic field . . . . . . . . . . . . . . .
Illustration of radiofrequency excitation of the magnetization vector . . . . . . . . .
Illustration of longitudinal and transversal relaxation for myocardial tissue and blood
at 3T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Illustration of spin echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Illustration of slice selection with application of a gradient . . . . . . . . . . . . . . .
1.6 Illustration of relative importance of low and high frequencies in k-space sampling .
1.7 Scheme of the cine sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Schematic representation of phased array coil . . . . . . . . . . . . . . . . . . . . . .
1.9 Scheme of the Keyhole and BRISK methods. . . . . . . . . . . . . . . . . . . . . . .
1.10 Illustration of the UNFOLD and k − t BLAST methods . . . . . . . . . . . . . . . .
1.11 Longitudinal magnetization relaxation in inversion recovery sequence. . . . . . . . .
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
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Scheme of the experimental setup for MRI exams of mice . . . . . . . . . . . . . . .
Scheme of EDV and ESV evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . .
Myocardial segmentation as defined by the American Heart Association . . . . . . .
Examples of dose response curves for low and high Mn2+ concentrations . . . . . . .
Synthesis of dose response curves for lvarying Mn2+ concentrations . . . . . . . . . .
Scheme of SI after inversion pulse for different T1 . . . . . . . . . . . . . . . . . . . .
Example of cine images of systolic and diastolic phases in middle slice of the heart .
Examples of cine images before and after Mn2+ injection . . . . . . . . . . . . . . . .
Wall thickening between diastolic and systolic phase for the different myocardial sectors
Left ventricular cavity and wall areas before and after Mn2+ injection . . . . . . . .
T1-weighted PSIR short axis images and corresponding TTC staining for IR60 and
sham mouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contrast to noise ratio for control, sham and IR60 groups . . . . . . . . . . . . . . .
Segmentation method for infarction volume quantification . . . . . . . . . . . . . . .
Infarction volume quantification with MEMRI versus TTC staining . . . . . . . . . .
Mean Mn2+ kinetics 24 hours and 8 days after reperfusion . . . . . . . . . . . . . . .
Slopes of Mn2+ wash-in kinetics 24 hours and 8 days after reperfusion . . . . . . . .
Examples of PSIR images taken at representative time points during the MRI exam
24h and 8 days after surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic representation of infarct extension measurement . . . . . . . . . . . . . .
Comparison of MR images and ex-vivo colorations . . . . . . . . . . . . . . . . . . .
Correlation between MEMRI and ex-vivo infarction extent measurements . . . . . .
xv
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List of Figures
3.1
3.2
3.3
3.4
3.5
3.6
3.13
3.14
Schematic representation of sequence design for interleaved cine . . . . . . . . . . . .
Illustration of the ghosting artifacts corrupting the interleaved cine . . . . . . . . . .
Validation experiment for interleaved cine . . . . . . . . . . . . . . . . . . . . . . . .
Effect of threshold value on data term . . . . . . . . . . . . . . . . . . . . . . . . . .
Cine image example with corresponding k-space . . . . . . . . . . . . . . . . . . . . .
Time course of the signal if one representative k-space line for basic and interleaved
cine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of interleaved cine for mouse with an infarction . . . . . . . . . . . . . . . .
Example of the temporal regularization . . . . . . . . . . . . . . . . . . . . . . . . .
Correlation between MRI and ex-vivo mass measurements . . . . . . . . . . . . . . .
Ejection fraction for control mice and mice with an infarction . . . . . . . . . . . . .
Example of temporal profile of interleaved cine for several threshold values . . . . . .
llustration of the proposed algorithm to remove the flickering artifact on numerical
phantom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Results of the denoising algorithm on the numerical phantom . . . . . . . . . . . . .
Example of temporal profile of interleaved cine for several threshold values . . . . . .
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Example of variable density spiral trajectory on a cartesian grid . . . . . . . . .
Constant-linear-velocity versus constant-angular-velocity spiral trajectory . . .
Examples of spiral trajectories, k-space values and gradient wavefroms . . . . .
Comparison of different trajectory designs, effect on the slew-rate overshooting
Synthetic scheme of the time-segmented reconstruction algorithm . . . . . . . .
Synthetic scheme of the frequency-segmented reconstruction algorithm . . . . .
Voronoi diagram for density compensation . . . . . . . . . . . . . . . . . . . . .
Illustration of the gridding method in image domain . . . . . . . . . . . . . . .
Simulated aliasing artifact behavior of Cartesian and spiral imaging . . . . . .
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Illustration of weaknesses of gridding reconstruction . . . . . . . . . . . .
B-spline functions of different degree . . . . . . . . . . . . . . . . . . . . .
Spline interpolation of 1-D signal . . . . . . . . . . . . . . . . . . . . . . .
Scheme of the automatic setting of parameter µ . . . . . . . . . . . . . . .
Shepp-Logan analytic phantom . . . . . . . . . . . . . . . . . . . . . . . .
Trajectories used in Shepp-Logan experiments . . . . . . . . . . . . . . . .
Normalization areas on Shepp-Logan phantom . . . . . . . . . . . . . . .
Shepp-Logan reconstruction with gridding and SPIRE . . . . . . . . . . .
Shepp-Logan reconstruction with gridding and SPIRE in the case of noisy
Comparison of SPIRE and gridding reconstruction of a phantom . . . . .
Comparison of SPIRE and gridding reconstruction of a volunteer heart . .
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k − t space representation of spiral trajectory . . . . . . . . . . . . . . .
Temporal interpolation with spline functions . . . . . . . . . . . . . . .
Numerical phantom reference for k-t reconstruction . . . . . . . . . . . .
Spiral trajectory with golden ratio angle rotation . . . . . . . . . . . . .
Scheme of the sliding window technique . . . . . . . . . . . . . . . . . .
Temporal profiles of pixels on diagonal for all methods . . . . . . . . . .
SNR for the different reconstruction methods . . . . . . . . . . . . . . .
Sequence of steps performed to determine the edge SNR . . . . . . . . .
Gradient of the angular average profile for the different reconstructions .
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List of Figures
6.10 Edge SNR for the different reconstructions . . . . . . . . . . . . . . . . . . . . . . . .
6.11 Edge SNR for the different reconstructions . . . . . . . . . . . . . . . . . . . . . . . .
6.12 Images of apex and basis slices reconstructed with gridding and k−t SPIRE compared
with cine sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.13 Gridding and k − t SPIRE images for apex slice . . . . . . . . . . . . . . . . . . . . .
6.14 Temporal profiles of apex slice for all reconstructions . . . . . . . . . . . . . . . . . .
6.15 Selection of temporal profiles of apex slices. . . . . . . . . . . . . . . . . . . . . . .
6.16 Reconstructed frames around the systole for reference sequence and k − t SPIRE . .
6.17 Comparison of temporal profiles and gradient of CINE, gridding and k − t SPIRE for
apex slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.18 Comparison of temporal profiles and gradient of CINE, gridding and k − t SPIRE for
basis slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.19 PSF for gridding and k − t SPIRE . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.20 Temporal profile of the PSF for gridding and k − t SPIRE . . . . . . . . . . . . . . .
6.21 Effect of the matrix size and sampling strategy on the grid-like artifact. . . . . . . .
6.22 Effect of the matrix size on the grid-like artifact on real data . . . . . . . . . . . . .
6.23 Effect of temporal assumption on the sample acquisition . . . . . . . . . . . . . . . .
xvii
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List of Figures
List of Tables
2.1
2.2
2.3
2.4
2.5
2.6
3.1
Mn2+ injection parameters for dose determination study . . . . . . . . . . . . . . . .
EDV, ESV, EF and heart rate for IR60, sham and control groups measured before
and after Mn injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wall thickening between diastolic and systolic phase for IR60, sham and control
groups, before and after Mn2+ injection . . . . . . . . . . . . . . . . . . . . . . . . .
Global function parameters measured 24 hours and 8 days after reperfusion . . . . .
Manganese entry slopes and mean correlation coefficient for acute and chronic time
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linear regression between infarction extent measurements with MEMRI and ex-vivo
methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
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Evaluation of the flickering artifact reduction with the two proposed denoising algroithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.1
Comparison of proposed methods efficiency for deblurring images . . . . . . . . . . .
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5.2
5.3
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Gradient of the regularization term for different spline degree α and derivative order γ 94
Parameters used for the different experiment on Shepp-Logan numerical phantom . . 95
SNR for reconstruction of Shepp-Logan phantom with gridding and SPIRE . . . . . 98
SNR for reconstruction of Shepp-Logan phantom in the case of noisy data, with
gridding and SPIRE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1
Detailed calculation times for sp5 on 3 interleaves on matrix size N=64 and N=128
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List of Tables
1
Introduction
1.1
Context
Cardiovascular diseases include disorders of the heart and the blood vessels. This is currently the
leading cause of mortality, representing one third of deaths worldwide. Overall, an estimated 42%
of deaths were due to coronary heart disease (CHD) in 2004 [1]. Governments are increasing public
awareness about the risk factors of CHD, such as smoking, diabetes, high blood pressure and inactivity [2], however, more than ever before, there is a real need to face this pandemic with efficient
tools and treatments.
In this context, imaging tools are of particular importance for diagnostic purposes, since an early
an accurate assessment of the disease is essential to optimize patient management and treatment
decisions. Imaging has also a major role in fundamental research as well as treatment development
[3]. Magnetic Resonance Imaging (MRI) has emerged as an important imaging technique to assess
patients with CHD, with the advantage of using non-ionizing radiation [4]. It has already demonstrated relevant diagnostic and prognostic information in many forms of heart disease [5] but still
holds challenges.
1.2
MRI basis
This introductory part succinctly presents the theoretical notions of Nuclear Magnetic Resonance
(NMR) and MRI that are needed for the comprehension of the following chapters. The interested
reader can find more detailed information in references [6–8].
Before MRI emerged, it was already known from NMR that the angular momentum, or spin, of
a hydrogen nucleus placed in a magnetic field precesses about that field at the Larmor (or resonant)
frequency [9]:
ω = γB0 ,
(1.1)
1
2
Chapter 1. Introduction
where γ is the gyromagnetic ratio specific for a given nucleus, for Hydrogen 1 H in water γ = 2.68·108
rad/(s·T). The resonance frequency may be shifted from the Larmor frequency depending on the
nuclei environment, which makes NMR specifically sensitive to the different components of a sample.
This is referred to as chemical shift. The idea of Lauterbur [10] and Mansfield [11] was to add to
the static magnetic field B0 , spatially varying magnetic fields (commonly referred to as gradients)
to encode the position of the object of interest. Resonant frequency will then vary proportionally
to the gradient added and thus contain its location information. This was the key concept leading
to MRI. We will see in the next sections how signal frequency measurement can lead to an image of
tissues.
In biological tissues, there is a natural abundance of hydrogen. This is the reason why clinical
MRI focuses on the signal provided by this nucleus. However, other nuclei, present in smaller
quantity in the human body, also have magnetic properties, such as 13 C, 19 F, 31 P and 23 Na. When
a biological tissue is placed in a magnetic field, the proton spins will rotate around an axis aligned
along the field direction. The signal measured in MRI comes from the difference between the number
of spins that are in the low energy level (with a parallel alignment to the magnetic field) and those
that are in the high energy level (with an anti-parallel alignment). The portion of spins in the
low energy state is slightly higher than the one in high energy state and is given by Boltzmann
equilibrium. This depends on the factor [7]:
~ω0
,
(1.2)
2kT
where N is the number of protons in the sample, ω0 is the Larmor frequency, k the Boltzmann
h
constant and ~ = 2π
with h Planck’s constant. The spin excess will be more important with an
increasing magnetic field, see figure 1.1. For example at 3T, the MR signal is given by only 10
nuclear spins on 106 protons.
spin excess ≈ N
Figure 1.1 — Illustration of nuclear spin orientation in a magnetic field. Spins with a parallel alignment
to the magnetic field are in the low energy level and are slightly in excess compared to the one in the high
energy level, with an anti-parallel alignment to the field. This fraction excess is greater for a higher magnetic
field.
1.2.1
Signal detection
−→
In MRI the signal is produced by tipping the magnetization vector M0 (the resultant sum of all
angular momentum vectors) away from the static magnetic field B0 with a radiofrequency field B1
3
1.2. MRI basis
−→
(rf pulse) set at the Larmor frequency. If M0 is aligned along z and the B1 field is applied along
x, the magnetization vector then rotates around x axis of an angle α (the flip angle). The varying
−
→
magnetic flux produced by the transverse magnetization M through a nearby conductor loop induces
a current in this conductor that can be measured. Figure 1.2 illustrate this process in a frame that
is rotating at the Larmor frequency (referred to as the rotating frame). The corresponding induced
elecrtomotive force (emf ) is given by:
Z
d
~ (~r, t) · B
~ receive (~r)d~r,
M
(1.3)
emf = −
dt
where B receive (~r) is the received field produced by the detection coil at all point where magnetization
is non-zero. The dependance of the emf on the applied B1 field is implicitly contained in the
~ . Signal is proportional to the magnetization which is in turn proportional to the
magnetization M
spin density ρ. After some simplifications it can thus be expressed as (see [7]):
Z
s(t) = ρ(~r)ei(ω0 t+φ(~r,t)) d~r,
(1.4)
where φ(t) is the accumulated phase expressed as:
φ(~r, t) =
Z
t
ω(~r, t′ )dt′ .
(1.5)
0
In the presence of only a static magnetic field B0 , ω = ω0 .
−→
Figure 1.2 — Illustration of radiofrequency excitation of the magnetization vector M0 in the rotating
−→
frame. An rf pulse (B1 field) is applied along x axis and tips M0 at π/2 angle from z axis. A conductor
loop measures the signal produced.
Noise in MRI
Noise in MRI has several origins, however in an ideal experiment some sources of noise such as
digitization noise or pseudo-random ghosting due to moving spins can be neglected. The main
source of noise derives thus from random fluctuations in the receive coil electronics and the sample.
4
The variance of this noise can be expressed as:
2
var(emfnoise ) ≡ σthermal
(~k) = 4kT · R · BW,
(1.6)
where R is the effective resistance of the coil load by the body, and BW is the bandwidth of
the detecting system. The bandwidth is the main noise contribution since the temperature and
resistance of the coils and bodies are not variable. The effective resistance R can also be expressed
as the sum of the contributions of the body and coil load as well as electronic noise:
Ref f ective = Rbody + Rcoil + Relectronics .
(1.7)
The noise expressed in 1.6 is expected to have equal power components at all frequencies within the
readout bandwidth, so is call a “white” noise.
However, another definition of noise will be used in this work and is derived in an image processing
point of view. Indeed, the difference between a processed image compared to one reference image
can also be defined as “noise”. This noise is therefore independent of the physical alterations of the
signal and reflects only an error introduced by a specific post-processing process.
1.2.2
Relaxation phenomenon
When radiofrequency excitation stops, the magnetization vector tends to return to its original position along B0 field due to interactions between the spins and their surroundings. This phenomenon
is called spin-lattice relaxation and is governed by a specific relaxation time T1 . Another relaxation
effect is given by the interaction between the spins themselves that causes a dephasing resulting in a
reduction of transverse magnetization. This so-called spin-spin relaxation has a specifc T2 relaxation
time. These relaxation phenomenon are expressed by the Bloch equations:
Mx (t) =
My (t) =
Mz (t) =
e−t/T2 Mx (0) cos(ω0 t) + My (0) sin(ω0 t) ,
e−t/T2 My (0) cos(ω0 t) − Mx (0) sin(ω0 t) ,
Mz (0)e−t/T1 + M0 1 − e−t/T1 .
(1.8)
(1.9)
(1.10)
Figure 1.3 illustrates longitudinal (T1 ) and transversal (T2 ) relaxation for the myocardial tissue
and the blood at 3T (T1 = 1471 ms, T2 = 47 ms and T1 = 1932 ms, T2 = 275 ms respectively [12]).
1
1
Myocardium
Blood
0.8
0.8
Myocardium
Blood
0.6
Mz
Mxy
0.6
0.4
0.4
0.2
0.2
0
0
1000
2000
3000
t (ms)
4000
5000
0
0
50
100
150
t (ms)
200
250
300
Figure 1.3 — Illustration of longitudinal and transversal relaxation for myocardial tissue and blood at 3T
(T1 and T2 values were taken from [12]).
5
1.2. MRI basis
T1 and T2 parameters play an important role in contrast depending on the timing of the signal
acquisition. In real conditions the transverse relaxation is also altered by interactions with local
magnetic field inhomogeneities, this is called the T2∗ relaxation, with T2∗ < T2 . Figure 1.4 shows the
signal measured after a 90◦ pulse, after the application of the rf pulse, the magnetization is decaying
due to T2∗ relaxation. This signal is called free induction decay (FID). If we add after the 90◦ pulse
a 180◦ rf excitation, the spins are refocussed and produce a signal called the echo. This sequence
called “spin echo”, can be repeated over the time to produce an image. If we define TE the time
between the 90◦ excitation pulse and the maximum amplitude in the signal echo, and TR the time
separating two consecutive repetitions, the transverse magnetization (which represents the acquired
signal intensity) is given by:
Mxy (T E) = M0 1 − e−T R/T1 e−T E/T2 .
(1.11)
Figure 1.4 — Illustration of spin echo. (a) FID signal measured after the 90◦ pulse, spin refocalization after
the 180◦ pulse and echo formation, schematic representation of 2 spins with different rotating frequencies
in the transverse plane to illustrate the refocalizaiton process. (b) Repetition of the spin echo, definition of
TE and TR.
6
1.2.3
Spatial encoding of information
As seen above, ignoring relaxation effects, signal is related to the spin density image by the relation
1.4 where the precessing frequency ω depends on the static magnetic field. To spatially discriminate
the signal of protons, the position of each spin along one direction, for example z, can be encoded
with a spatially varying magnetic field along that direction. The proton’s precessing frequency now
depends on that gradient G:
ω(z, t) = ω0 + ωG (z, t),
(1.12)
ωG (z, t) = γzG(t).
(1.13)
Figure 1.5 illustrates this principle in the case of a gradient applied along the field direction z.
Gradients can then be applied in x and y directions to encode spatially the spins in the transverse
plane. If we define k(t) as:
Z t
~k(t) = γ−
~ ′ )dt′ ,
G(t
(1.14)
0
the signal is thus expressed as:
s(~k) =
Z
~
ρ(~r)e−i2πk·~r .
(1.15)
Figure 1.5 — Illustration of slice selection with application of a gradient. (a) Precession frequency of spins
in all the volume is the same, (b) after application of a gradient in z direction, each spin rotates at different
frequency along z.
Relation 1.15 describes the signal as the Fourier transform of the spin density ρ. The domain
of spatial frequency of the signal is referred to as the k-space. Figure 1.6 illustrates the relative
importance of low spatial frequencies and high spatial frequencies in the image composition. Most
of the energy of the image is contained in the center of k-space whereas the details of the image are
encoded into the high frequencies.
1.2. MRI basis
7
Usually, k-space is sampled line-by-line in order to be reconstructed with the Fast Fourier Transform (FFT) algorithm. Knowing the relative importance of low and high frequencies in the image
formation, other sampling strategies can be more advantageous like radial sampling or spiral sampling that spend more time sampling low frequencies than high ones. The straightforward reconstruction with FFT is therefore no longer possible since points are not placed on the Cartesian grid,
but has to be performed with other methods. An illustration can be found in section 4.1.2, p. 65.
Part of this thesis is devoted to the development and validation of such a trajectory, however, in
this introductory part, we will only focus on Cartesian sampling since it is the more widely used in
clinical routine.
Figure 1.6 — Illustration of relative importance of low and high frequencies in k-space sampling. Upper,
full k-space sampling and its corresponding image; middle, sampling of the center of k-space gives the
intensity information of the image; lower, sampling of high frequencies only gives the edges of the image and
not the main contrast.
8
1.3
1.3.1
Cardiac MRI
Sequences for function measurement
Signal acquisition performed line-by-line, takes a certain time to sample the entire k-space. The
acquisition duration is often not compatible with requirements of imaging moving objects. In cardiac
imaging, one of the strategies is to segment the k-space sampling and to trigger the acquisition
onto a physiological information like the R-wave of the ECG. Figure 1.7 represents the complete
acquisition of k-space segmented over several heart beats. With this technique it is possible to
acquire consecutive packets of data during each R-R cycle which give temporal information to the
resulting images, as it is illustrated in the lower image of figure 1.7. The reconstruction of all the
collected k-space informations give a movie of the beating heart, referred to as “cine” [13]. To avoid
the respiratory motion artifacts into the final images, we often perform this sequence under patient
breathold. Another technique if the patient is unable to retain his respiration during 10 to 20
seconds is to perform the acquisition with a respiratory navigator in order to reconstruct only data
that were acquired at the same phase of the respiratory cycle [14].
Figure 1.7 — Schematic representation of k-space sampling in cine sequence. (a) Acquisition of one phase
of the cardiac cycle; (b) the acquisition is performed for each cardiac phase and is repeated until the entire
k-space is sampled.
The cine sequence is the standard sequence for the evaluation of myocardial function [15, 16]
and is considered as diagnostic. Contraction deficit due to a myocardial infarction, for example, is
visible when the movie is played and measurements such as left ventricular cavity volume or wall
thickening between diastole and systole are performed on the images and are considered as diagnostically relevant by a consensus of scientists and healthcare professionals [17].
However, this sequence is limited in presence of arrhythmia in patients. In this case, one alternative is to perform real-time imaging [18]. This last option is however non-trivial and needs some
1.3. Cardiac MRI
9
acceleration methods as described in section 1.3.2 to maintain the acquisition time sufficiently short
to depict the cardiac motion, typically less than 200 ms in normal human heart.
1.3.2
Acquisition and reconstruction methods for rapid imaging
Parallel imaging
Phased-array coils are composed of several coil elements that together provide a signal comparable
to the one obtained with a single surface coil but extended to a larger FOV, see figure 1.8. They,
were first used to improve image SNR [19]. Indeed, combining the images obtained with each
separate coil results in an homogeneous image as if it was acquire with a larger coil but with a
noise that is reduced [8]. However, the potential of parallel imaging to accelerate acquisition was
quickly recognized when Sodickson et al. proposed the SMASH method [20], nearly followed by
Pruessmann et al. [21] with the SENSE algorithm. Then a large number of methods were proposed
but nowadays, the two most widely used parallel imaging techniques are still SENSE and GRAPPA
[22] (derived from SMASH).
Figure 1.8 — Schematic representation of phased array coil. Left, one coil element (in blue) covers a small
part of the subject, right, several elements composing a phased array coil cover a larger FOV.
Acquisition time depends linearly on the number of phase encoding lines. Acquiring a fraction
1/R of the complete k-space lines will reduce the acquisition time by a factor R. While GRAPPA
resolves the problem of missing data in k-space, SENSE solves the problem in the image domain.
We will not detail the algorithms here (for more information see [22] and [21]) but we note that
some features of parallel imaging methods are a reduced overall SNR and a non-uniform noise over
the image [8]. Parallel imaging is therefore most useful in applications where SNR of images is
important such as perfusion imaging or angiography [23].
Acceleration methods by sharing information along acquisition time
In addition to parallel imaging, other methods use the intrinsic spatio-temporal properties of the
object of interest to accelerate the acquisition. In cardiac imaging, the temporal variation of k-space
signal is more important at the center of k-space than at edges. Here the idea was to sample high
frequencies less often than low frequencies.
One of the first method exploiting that property was the keyhole method [24, 25]. As illustrated
in part a of figure 1.9, a reference scan is first acquired to have a complete spatial resolution of the
object and the subsequent acquisitions only contain a reduced number of k-space line (the keyhole
views) located around the k-space center. BRISK [26] is an extension of keyhole method that relies
on the same principle but that spreads the acquisition of the high resolution image over several
phases (see figure 1.9, b).
10
Figure 1.9 — Scheme of the Keyhole (a) and BRISK (b) methods. By combining informations acquired
over several phases the complete k-space can be recovered.
Acceleration method by filtering of x − f space
The UNFOLD method was proposed by Madore et al. [27] and uses some of the principles of parallel
imaging however without phased array coils. Indeed, like in parallel imaging, only a fraction 1/R of
k-space line is acquired but the recovery of missing information is done by using the temporal information instead of the signal provided by the different coil elements. Whereas UNFOLD is limited
to 2-fold acceleration (or 4-fold in particular conditions), the method presented by Tsao et al. [28],
referred to as k − t BLAST, offers larger acceleration factors. For example an 8 times acceleration
factor could be reached in cardiac application [29]. Figure 1.10 illustrates the underlying principle
of these methods.
If we consider a fully sampled k-space and observe one line profile over time (part a of figure
1.10), we notice that one part of the field of view is nearly stationary (chest wall, liver, ...) whereas
the other part moves periodically (heart). So, when we plot the Fourier transform of the profile
into the time direction we end up with an area with a relatively thin bandwidth (the no-moving
structures) and an area with a broad bandwidth (corresponding to myocardium).
Now, if k − t space is undersampled by acquiring for example only odd lines at odd time points
and even lines at even time points, we will end up with an aliasing artifact in the corresponding
images with structures lying on top of each other (part b of figure 1.10). Correspondingly, the x − f
space (x refers to a spatial dimension that can be either x or y axis) exhibits replications of the
main temporal spectra. However, the replications are not overlapping and are located such that a
simple filtering at edges of x − f space is sufficient to recover the main information of the data, and
thus to eliminate aliasing of the image. This corresponds to the UNFOLD method.
11
1.3. Cardiac MRI
Figure 1.10 — From left to right, scheme of the k − t space sampling (to save visibility only a selection of
line are plotted), one image of the time serie, white line profile in function of time denoted as y − t space,
result of the Fourier transform in temporal direction, y − f space. (a) represents the case of a full k-space
sampling, (b) represents the 2-fold undersampling with an alternation of line acquired in function of time,
(c) represents a 5-fold undersampled k-space, also with an alternation of line acquired in function of time.
When going further with the undersampling (part c of figure 1.10), the replications are now
overlapping in x − f space and a simple filtering is therefore not possible. However, with the help
of training data, one can recover the global shape of the main temporal spectra and extract it in
x − f space in order to remove the aliasing artifacts. The determination of this specific filter is
the particularity of the k − t BLAST method. Following equation 1.15, the signal can be expressed
slightly differently with a matrix formulation:
s = Eρx−f ,
(1.16)
where s is the acquired signal in k − t space, ρx−f is the image in x − f space and E is the
transformation between signal and image. The image ρx−f can be initialized with non-zero values,
it is therefore expressed as:
ρx−f = ρ̄x−f + W q,
(1.17)
where W is a weighting matrix and q is a solution of the following constrained minimization problem:
arg minks − EW qk22
(1.18)
12
k − t BLAST algorithm is then given by the following regularized minimization problem:
arg minks − E ρ̄x−f − EW qk22 + λkqk22
(1.19)
These two methods can be combined with parallel imaging to further increase the acceleration
factor, UNFOLD was coupled with SMASH in [30] and also with SENSE (referred to as TSENSE) in
[31] whereas k − t BLAST was enlarged to k − t SENSE to benefit from the signal given by multiple
coils [28].
1.3.3
Viability measurement
Even if cine imaging gives a good evaluation of myocardial anatomy and function, a complementary
tool is necessary to accurately evaluate the extent of myocardial diseases such as infarction [32].
Indeed, T1 of infarcted tissue was shown to vary slightly when compared to normal myocardium
[33], but the difference is difficult to highlight with classical sequences. Thus the use of contrast
agents was introduced in order to enhance the contrast between different areas, for example infarction
and viable myocardium.
Contrast agents
The most widely used contrast agents are derived from the Gd3+ ion. This paramagnetic ion reduces dramatically the T1 and T2 of its surroundings but is nevertheless highly toxic in its ionic form
[34]. All approved contrast agents are thus chelates of Gd3+ that have different pharmacological
properties. There are two broad categories of chelates, the macrocyclic molecules, where Gd3+ is
caged into the ligand, and the linear molecules. Two examples of contrast agents routinely used in
R cyclic molecule) and Gd-DTPA (Magnevist,
R linear molecule)
clinics are Gd-DOTA (Dotarem,
[35]. However, one severe adverse effect of Gd3+ chelate administration in patients is nephrogenic
systemic fibrosis (NFS). This potentially fatal complication is more likely to occur in patients with
a high degree renal impairment. It has been shown that the administration of some linear molecule
chelates induced NFS whereas it was not the case with cyclic molecule chelates [35].
The use of Gd3+ chelates for infarction visualization mainly relies on the fact that this extracellular contrast agent accumulates in the interstitial space that is enlarged in infarction but not
in viable tissue, it can also be trapped in scar tissue due to high concentration of collagen fibers
producing a signal enhancement due to the increased transit time in this area. Other types of Gd3+
chelates were specifically designed for other purposes, one example is intravascular agent that was
used to assess microvascular flow [36].
Even if Gd3+ chelates are the most routinely used in clinics, they have some limited specificity
that can be overcome with other types of paramagnetic ions. Indeed, intracellular contrast agents
have a great potential for molecular imaging. Mn2+ was quickly recognized as an efficient agent to
assess cellular viability in ischemia-induced injuries (in heart but also in brain [37]) and, after being
validated for hepatic dysfunction assessment [38], its chelate Mn-DPDP was even recognized as a
viability marker in patients with a myocardial infarction [39]. A part of this thesis was devoted to
the investigation of Mn2+ -enhanced infarction quantification.
Super-paramagnetic iron oxide particules (SPIO) have been used to label specific cells such as
macrophages in the context of tumor staging or lymph node detection [40]. The in vivo labeling
of macrophages with SPIO was even able to show their mobilization to myocardial infarction [41].
1.3. Cardiac MRI
13
Compared to Gd3+ and Mn2+ that modify the T1 relaxation of the surrounding tissue, here the
contrast mechanism is different since the iron acts on the T2∗ relaxation by creating local field
inhomogeneities.
T1 -weighted sequence
T1 contrast between viable myocardium and infarction is usually measured with sequences that are
very sensitive to T1 variations [16]. The most commonly used is the inversion-recovery sequence,
where the magnetization is first inverted with a 180◦ rf pulse. The time between the inversion and
the signal acquisition is called the inversion time T I and is usually set to the time corresponding to
a null signal in the viable myocardium, since this was shown to give the best contrast [42]. Figure
1.11 shows the signal recovery of infarction and viable myocardium signal intensity. The contrast
between these two areas depends mainly on the T I chosen, we observe that infarction first appears as
a hypointense signal compared to viable myocardium for small T I, whereas this contrast is inverted
for longer T I. A major improvement of this sequence is the used of phase information to recover
the sign of the longitudinal magnetization, this sequence is referred to as Phase-Sensitive Inversion
Recovery, PSIR [43]. With this sequence, infarcted myocardium always appear as an hyperintense
signal compared to normal myocardium and the contrast has the advantage of being less dependent
of the choice of T I. This is illustrated in the lower part of figure 1.11.
14
Figure 1.11 — Longitudinal magnetization relaxation in inversion recovery sequence for infarction (MI)
and viable myocardium (normal). (a) Acquired signal is proportional to the absolute value of longitudinal
magnetization, depending on the timing of the acquisition the contrast between MI and viable can be either
negative, null (around 220 ms) or positive. (b) With PSIR sequence the original sign of the magnetization
is recovered, the contrast is less dependent of the T I chosen and is exclusively positive.
1.4. Aim of the project
1.3.4
15
Translational research
Improvement of Human health is often done by translating the “bench” discoveries into the clinical
context, or ”bedside”. This “from bench-to-bedside” view of research, also referred to as “translational research” is clearly a two ways relation. Indeed, clinics can benefit from drug discoveries or
tool improvements and the new observations made in patients or existing clinical “gold standards”
then guide the fundamental research.
Translational research is the main motivation of developing small animal imaging onto clinical
MR scanners. In fact, tools developed on rodent experiments can be directly applied to human
studies since the same system is used. Similarly, recent developments in sequence design giving
access to advanced tools in clinics can directly be used in the context of fundamental research
without having to implement them on the experimental system. Another motivation is the limited
availability of dedicated scanners to most institutions, that makes small animal imaging on clinical
systems a more widely considered alternative [44].
1.4
Aim of the project
Nowadays, an important part of MRI research is devoted to the development of molecular imaging, nevertheless, complete cardiac assessment still consists of two main measurements. One is
function assessment to detect the contraction abnormalities, the other one is viability assessment,
giving insight into the precise extent and location of ischemic myocardial injuries. Several technical
challenges remain both in functional and in viability imaging. In a translational approach to the
problem, the aim of this work was to develop new techniques to provide an efficient characterization
of the myocardium using tools provided by MR physics and image processing, from mice to patients.
The manuscript is divided into two main parts. The first one presents techniques developed for
small animal viability and function imaging. The second part of is dedicated to spiral imaging and
its application to real-time in humans.
16
Part I
Small animal imaging (from
bench...)
17
2
Manganeseenhanced MRI
in mice
Part of this chapter has been published in: Delattre et al., “Myocardial infarction quantification with
Manganese-Enahnced MRI (MEMRI) in mice using a 3T clinical scanner” [45]
The two following chapters aim at presenting the challenges of cardiac imaging of mice on a
clinical 3T scanner. Function parameters measurements such as end-diastolic, end-systolic volumes
(EDV and ESV respectively) and ejection fraction (EF) as well as tissue characterization are needed
to evaluate infarction extension. The choice to perform small animal studies on a clinical scanner is
motivated by a translational research approach as it was presented in chapter 1 (p 15).
2.1
Manganese as a contrast agent
In cardiac magnetic resonance imaging (MRI), extracellular contrast agents such as Gadolinium
(Gd3+ ) chelates, are now routinely used in clinical practice, as well as in research protocols to assess myocardial perfusion or interstitial space remodeling [46–48]. Gd3+ is paramagnetic and thus
shortens the T1 of the surrounding environment. It therefore enhances the contrast between two
areas of interest that were initially difficult to discriminate with conventional sequences, as it is the
case for infarcted and viable myocardial tissues for example.
The main limitation of Gd3+ -based contrast agents is that the usually used chelates are nonspecific (see chapter 1 p 12). Even if they have proven their accuracy to depict the infarction volume,
the measurement of the hyperintensity related to the presence of Gd3+ stays an indirect method
of viability assessment and is restricted to models where the region of interest corresponds to an
extracellular space. This is not the case in stunning for example where cells are not contracting but
with a membrane still intact. This limitation is the main motivation for the use of more specific
contrast agents in cardiac MRI.
Intracellular MR contrast agents can provide additional information on the cellular ions exchanges. Manganese ion (Mn2+ ) was quickly recognized as an efficient MR contrast agent as it
19
20
Chapter 2. Manganese-enhanced MRI in mice
induces a strong T1 shortening effect [49] to surrounding tissues. It was then successfully applied
to activity detection in the brain and in the heart [37], and recently to pancreatic β-cells, where the
loss of function is involved in diabetes pathologies [50]. Even if all the transportation mechanisms
are not known [51, 52], it is admitted that Mn2+ enters cardiomyocytes mainly by L-type voltage
dependant Calcium (Ca2+ ) channels and stays into the cells for hours [53]. As an analog of the
Ca2+ ion, Mn2+ should therefore has the potential to assess Ca2+ homeostasis in-vivo, generating
an important interest for researchers [54]. Indeed, Ca2+ cycling is of vital importance to cardiac
cell function and plays an important role in ventricular dysfunction such as heart failure [55]. In
opposition to clinically used Gd3+ -based chelate (Gd-DOTA or Gd-DTPA [35]), T1 shortenning
induced signal is therefore depicting viable cells or indeed cells where Ca2+ influx is present.
The potential of M n2+ to depict cardiomyocyctes activity has been shown in presence of dobutamine and diltiazem (which are known to respectively increase and decrease Ca2+ influx into the
heart). T1-weighted images showed respectively an increased and a decreased Mn2+ -induced signal intensity (SI) with the addition of those drugs compared with Mn2+ only injection [56]. In the
context of cardiac pathologies, a reduced Mn2+ accumulation has also been observed in stunned cardiomyocytes [57] as well as in the zone adjacent to a myocardial infarct [54]. Manganese-Enhanced
MRI (MEMRI) has also been successfully used to assess myocardial infarction in various animal
models from pig to rat [53, 58–61]. In all these studies, it was shown that Mn2+ was retained into
viable cells allowing an accurate visualization of the infarction area. These results tend to support
the hypothesis that Mn2+ is a good marker of cell viability, however the role of perfusion and protein
binding into Mn2+ distribution in viable and infarcted compartments are not fully understood.
Only a few studies, however, investigated the possible use of MEMRI for myocardial infarction
assessment in mice [54, 62]. These studies used a model of permanent coronary occlusion where
infarct size determined by triphenyltetrazolium chloride (TTC) at 7 days was linearly correlated to
the infarct size measured from MEMRI [62]. However, a lower SI, suggesting a decreased Mn2+
accumulation was also observed in the peri-infarct area where ischemic tissue may also be present
[54]. The type of coronary occlusion, as well as the timing of examination after the induction of the
myocardial injury, may also impact MEMRI experiments. Gadolinium chelates (Gd-DTPA or GdDTPA-BMA) were largely used for assessment of myocardial infarction in mouse models [46, 63, 64].
However, it revealed some disadvantages over Mn2+ contrast enhancement that are presented below:
Timing: The time window during which assessment is possible is relatively short with Gd3+
(contrast agent is visible during approximately 1 hour [63]) compared with Mn2+ where ions
can stay in mitochondria for several hours [54]. Performing infarct quantification too early
with Gd3+ can lead to an overestimation of the infarct zone that restrains again this time
window between 20 and 60 min after injection [63–65]. From this point of view, Mn2+ allows
more flexibility.
Accuracy: Mn2+ makes the viable part of myocardium appear bright (with the usual sequences
used) which allows an easier segmentation of the myocardium and the infarct pattern than with
Gd3+ . In the latter case, parameters of the sequence are often chosen to null myocardium signal
prior to contrast agent addition in order to maximize the contrast between viable myocardium
and enhanced infarction [66]. As the Gd3+ induced SI decreases during the acquisition, care
must be taken to adapt the TI according to this decrease otherwise the size of the non-viable
zone will appear to decrease with time [53]. The accuracy of infarction delineation is thus
hardly dependent on the timing of imaging while it is less the case for Mn2+ .
2.2. Imaging myocardial viability in mice
21
Toxicity: An important drawback related to Mn2+ is its toxicity. It has been shown that Mn2+
can induce side effects from simple somnolence to tremor, convulsions or even cardiorespiratory
arrest have been observed in several animal models [37]. Moreover, the rate or the route of
injection play a major role in the potential toxicity of Mn2+ . In fact, the toxicity of Mn2+
is mainly due to the blockage of the normal Ca2+ influxes into cells, so rather than to be
related to the Mn2+ concentration itself the toxicity is more due to the equilibrium with the
extracellular Ca2+ concentration [67]. To avoid that, several solutions were found [53], one
was to chelate Mn2+ ions into Mn-DPDP (Dipiradamol diphosphate), this way Mn2+ ions
are released slowly avoiding the competition with Ca2+ ions leading to side effects. Another
solution was to inject Ca2+ ions at the same molarity as Mn2+ . Finally, injecting Mn2+ slowly
as an infusion instead of a bolus can limit the toxic effects. For mice the toxicity limit dose
was defined as 962 nmol/g for intraperitoneal (IP) route [37].
2.2
2.2.1
Imaging myocardial viability in mice
Material and methods
Experimental settings
Imaging was performed on a clinical 3T MR scanner (Magnetom TIM Trio, Siemens Medical Solutions, Erlangen Germany). MRI exams for small animals need specific instrumentations to monitor
the heart and respiratory rates as well as to gate the MR sequences onto the ECG R-wave. During
our experiments we used a dedicated monitoring and gating system (Model 1025, SA instruments,
Inc. NY, USA) as well as a dedicated 2 channels mouse receiver coil (Rapid biomedical GmbH,
Rimpar Germany). In practice, the anesthesia induction was done into the console room with isoflurane gas at 5 %, the anesthetized mouse was then placed into the coil bed on the MR table and
isoflurane was maintained between 1-2 % during the imaging session based on the respiratory rate
of the animal. Two electrodes were inserted subcutaneously under the front paws on each side of
the thoracic cage and connected to the ECG monitoring system, a small pressure pillow was used
to control the respiratory rate. Figure 2.1 illustrates the whlole experimental system.
Infarction model
Two main models are usually used to perform an infarction in the left ventricle, permanent coronary
occlusion and ischemia-reperfusion where the coronary artery flow is blocked and then reestblished.
We choose the ischemia-reperfusion model since it is closer to the clinical situation in which the
artery is ultimately reopened.
Animal surgery as well as ex-vivo analysis were performed by Dr Vincent Braunersreuther, from
the division of cardiology (Department of Medicine, University Hospital, Foundation for Medical
Researchers, Geneva, Switzerland). 15-20 week old C57BL/6J mice were anaesthetized with 4%
isoflurane and intubated. Mechanical ventilation was performed (150 µl at 120 breaths/min) using
a rodent respirator (model 683; Harvard Apparatus). Anaesthesia was maintained with 2% isoflurane
delivered in 100% O2 through the ventilator. A thoracotomy was performed and the pericardial sac
was then removed. An 8-0 prolene suture was passed under the left anterior descending (LAD)
coronary artery at the inferior edge of the left atrium and tied with a slipknot to produce occlusion.
A small piece of polyethylene tubing was used to secure the ligature without damaging the artery.
Ischemia was confirmed by the visualization of blanching myocardium, downstream of the ligation.
22
Figure 2.1 — Scheme of the experimental setup for MRI exams of mice.
23
After 60 minutes of ischemia, the LAD coronary artery occlusion was released and reperfusion
occurred. Reperfusion was confirmed by visible restoration of color to the ischemic tissue. The
chest was then closed and air was evacuated from the chest cavity. The ventilator was then removed
and normal respiration restored.
Ex-vivo analysis
Between 8 to 15 hours after MRI sessions, mice were re-anesthetized with 10 ml/kg of ketaminexylazine (12 mg/ml an 1.6 mg/ml, respectively) and the heart was rapidly excised. The heart
was then rinsed in NaCl 0.9%, frozen and manually sectioned into approximately 1 mm transverse
sections from apex to base (5-6 slices/heart). Heart slices were then incubated at 37◦ C with 1%
triphenyltetrazolium chloride (TTC) in phosphate buffer (pH 7.4) for 15 min, fixed in 10% formaldehyde solution and each side of the slices was photographed with a digital camera (Nikon Coolpix).
For collagen staining hearts isolated from animals were perfused with NaCl 0.9% to remove blood
and were frozen in OCT. They were then cut serially from the occlusion locus to the apex in 5 µm
cryosections. Eight to ten serial sections were stained with Masson-trichrome for studying collagen
in the injured myocardium.
MRI sequences
Two type of MR sequences were used for this study, the first one was a turboflash cine sequence
to assess myocardial function and the second one was a T1-weighted turboflash sequence to assess
tissue viability. These sequences were based on the clinical routine sequences used in patients. We
adapted the parameters to allow mouse imaging.
The cine sequence had the followings parameters: FOV 66 mm, in plane resolution 344 µm, slice
thickness 1 mm, typically 4 consecutive slices to cover the whole left ventricle, TR/TE 11/5 ms,
flip angle 30◦ , GRAPPA with acceleration factor 2, 3 averages, typical acquisition time per slice 3
min. The T1-weighted turboflash sequence used Phase Sensitive Inversion Recovery reconstruction
(PSIR) [43] with following parameters: FOV 80 mm, in plane resolution 156 µm, slice thickness 1
mm, typically 8 slices (50% slice overlap), TR/TE 438/7.54 ms, flip angle 45◦ , TI 380 ms, GRAPPA
with acceleration factor 2, 2 averages, typical acquisition time per slice 2 min 30. Fifty percent
slice overlap was chosen to diminish effects of partial volumes [68]. For this sequence a constant
TI was chosen to keep the same contrast properties in all images. Both sequences were respiratory
and ECG gated. Comparing with other previous studies on small animals conducted at high field,
gradient performances of our system allowed a similar spatial resolution (156 µm vs. 117 µm at
11.7T [63] or 100 µm at 9.4T [69]) leading to a precise infarction quantification. For comparison,
a Siemens 3T clinical scanner has a maximum slew-rate of 180 T/(m·s) and a maximum gradient
amplitude of 40 mT/m whereas a Brucker Biospin 9.4T scanner has a maximum slew-rate of 9090
T/(m·s) and a maximum gradient amplitude of 1 T/m which is respectively 51 and 25 times higher
than the clinical system.
Image analysis
All image processing was done with Osirix software (Open source http://www.osirix-viewer.com/).
For ejection fraction (EF) calculations, segmentation of the endocardial contour allowed evaluation
of the end-diastolic volume (EDV) and end-systolic volume (ESV) by summing the volumes of each
24
acquired slices based on the Simpson’s rule (see figure 2.2). ESV and EDV are calculated as follows:
EDV orESV =
X
nb of slices
endocardial area · (slice thickness + gap between slices).
EF is defined by:
EF =
EDV − ESV
.
EDV
(2.1)
(2.2)
Figure 2.2 — Scheme of EDV and ESV evaluation, following Simpson’s rule, an elliptic volume can be
evaluate as the sum of the area of finite circles.
Endocardial and epicardial contours were manually traced and excluded the papillary muscles
[70]. Wall thickening evaluation for regional function assessment was done using an in-house software
calculating the percentage of wall thickening of 100 rays covering the whole left ventricle. The
formulation is the following:
WT =
systolic radial length − diastolic radial length
.
diastolic radial length
(2.3)
Mean of results were then calculating for the different sectors covering the left ventricle. Sectors
were defined based on the American Heart Association (AHA) standardized guidelines for myocardial
segmentation [17]. Figure 2.3 shows a part of this segmentation. For infarction quantification,
segmentation of endocardial and epicardial contours was done again including this time the papillary
muscles.
Statistics
All presented values are mean ± standard deviation (SD), or if stated mean ± standard error of the
mean (SEM). Statistics were performed with PAWS Statistics 18 software. To compare two groups
of values student’s t-tests were used, while comparison of more than two groups was performed either
with analysis of variance (ANOVA) followed by Bonferroni post-hoc test or with a non-parametric
test (Friedman’s two ways ANOVA by ranks).
25
Figure 2.3 — Part of myocardial segmentation as defined by the American Heart Association [17]. Middle
slice of the left ventricle is divided into 6 sectors: A = Anterior, AL = Anterolateral, IL = Inferolateral, I
= Inferior, IS = Inferoseptal, AS = Anteroseptal.
2.2.2
Manganese optimal dose determination
The first step of the study was to determine the optimal Mn2+ dose to deliver, in order to have
the best enhancement of the myocardium with the sequence used while remaining under the toxic
level. This was an important step considering the toxicity of Mn2+ at high concentrations and delivery rate [37] and the variability of the different doses reported in literature for cardiac MEMRI. To
achieve this, normal mice underwent MEMRI exam with increasing Mn2+ doses. In practice, MnCl2
was diluted in NaCl 0.9% solution to obtain stock solution of 7.5 mM or 15 mM depending on the
experiment. An IP line was placed in the mouse before MRI exam in order to deliver the Mn2+
solution and a long line (4 m) was set between the exam room and the console, where the infusion
pump was placed. Due to the loss of flow along the long distance, the infusion pump was set to a
minimum of 2ml/h to ensure the delivery of the Mn2+ solution. Mice where divided in two groups,
one experienced low concentrations of Mn2+ (35-200 nmol/g) and the other high concentrations of
Mn2+ (250-480 nmol/g). The highest delivered dose was however largely under the toxicity limit
(962 nmol/g [37]). Table 2.1 summarizes the concentrations and settings used for these experiments.
All settings were calculated in order to inject a maximum volume of 1 ml of stock solution.
The heart rate reported in Table 2.1 was measured at least 15 min after the end of each Mn2+
injection. No arrhythmia or significant change in heart rate were encountered at high Mn2+ concentrations when comparing with the heart rate just before Mn2+ injection (p>0.2). However, for the
”low” concentrations as well as for the ”high” concentrations experiments, the successive injections
of Mn2+ led to a decrease of heart rate that became significant for the last doses when comparing
with the heart rate measured at the beginning of the experiment (p<0.03 and p<0.05 for ”low” and
”high” dose protocol respectively). Knowing that Mn2+ plasmatic half-life is approximately 3 min
[71], it suggests that the observed depression was more probably due to the cumulative effect of
the anesthesia than to a direct effect of Mn2+ injection. This can be explained by the fact that
anesthesia was driven in order to keep a stable respiratory rate and not necessarily a constant heart
rate. Moreover, the decreased heart rate observed at the end of the ”low” concentrations protocol
corresponding to 200 nmol/g of Mn2+ was not reproduced after the first injection of 250 nmol/g
Mn2+ in the ”high” concentrations protocol.
We measured signal enhancement in the septum as well as in the left ventricle free wall (around
the anterolateral area, where infarction has to occur in this model) to ensure that results were not
dependent on the localization in the myocardium. Figure 2.4 shows 2 examples of dose response
curves obtained for “low” and “high” Mn2+ concentrations.
26
Figure 2.4 — Examples of dose response curves for low (up) and high (down) Mn2+ concentrations. Signal
intensity of left ventricular blood pool, septum, free wall are reported. The green curve represent the step
of increasing Mn2+ concentrations.
27
Table 2.1 — Mn2+ injection parameters and heart rate (HR) measured at least 15 min after the end of
Mn2+ injection for “low” and “high” concentrations experiments and general information about Mn2+ solution
used in the measurement of dose versus signal enhancement curve. Symbols in brackets denote significant
difference in HR compared with beginning of the experiment, before any Mn2+ injection (§ p<0.01, non
stated values mean non-significant difference). BW stands for body weight.
Experimental
timing (min)
0
5
40
75
110
145
0
5
40
75
110
Low concentrations - Stock solution 7.5 mmol/l
Cumulative
Infusion time HR (bpm)
dose
(min)
(nmol/g)
BW
0
333±16
35
4.2
311±14
70
4.2
306±10
100
3.6
298±14
150
6
289±18 (§)
200
6
290±17 (§)
High concentrations - Stock solution 15 mmol/l
0
303±27
250
15
284±6
320
4.2
264±38
390
4.2
252±30
480
5.4
245±39 (§)
n
3
3
3
3
3
3
4
2
2
4
4
The time delay between two successive injections was at least 30 min for the SI to achieve a
steady-state. Figure 2.5 shows the synthesis of the dose experiments. The results indicated that
signal enhancement increases linearly with Mn2+ dose up to 200 nmol/g before reaching a plateau.
Linear regression results are y = 0.78x − 26.37 with R2 = 0.99 and p<0.001 for signal enhancement
measured in septum area and y = 0.81x − 21.77 with R2 = 0.99 and p<0.001 for measurement in
free wall. Slopes and intercepts are not significantly different between these two regions of interest
(p>0.05).
There are at least two hypothesis to explain the plateau observed above 200 nmole/g :
• It can be a saturation related problem that does not allow visualization of a further increase in
Mn2+ concentration in the myocardium after this level because of the limited dynamic of the
SI. Indeed, high shortening of T1 in tissue due to Mn2+ leads to a saturation of SI, dependent
on the inversion time chosen in the sequence (see figure 2.6).
• It can also be the result of a true physiological effect that could be either a limited Mn2+ accumulation in the cardiomyocytes, a limited entry of Mn2+ due to a saturation of binding sites,
or a limited relaxation rate change secondary to protein binding with Mn2+ ions. According
to Kang et al. [72], binding of Mn2+ ions to macromolecules leads to a more efficient dipolar
interaction with surrounding protons decreasing significantly proton T1.
Moreover, a recent study of Waghorn et al. [54], who mapped the T1 decrease in myocardium,
shows the same plateau in their measurements occurring above 197 nmol/g (which is comparable
to our results). They came to the conclusion that above this concentration, an increase in Mn2+
concentration in the myocardium did not lead to a decrease in T1 even if the absolute concentration
of Mn2+ in dry myocardium they measured increased. This thus tend to validate the hypothesis of
a limited change in relaxation rates. Our results could not discriminate between the two hypothesis
28
Figure 2.5 — Signal enhancement measured in septum and free wall area (see insert) for normal mice (SI
minus signal baseline measured before Mn2+ injection, S0) 35 min after Mn2+ injection versus Mn2+ dose.
1
0.8
0.6
0.4
Mz
0.2
0
T1 = 500 ms
T1 = 300 ms
T1 = 100 ms
T1 = 50 ms
−0.2
−0.4
−0.6
−0.8
−1
0
500
1000
1500
t(ms)
Figure 2.6 — Scheme of SI after an inversion pulse for different T1. With a chosen inversion time TI of
500 ms (dotted line), measured SI given by tissues with T1 less than 100 ms are identical.
29
but further experiments can be performed such as SI measurements with varying TI to ensure that
we do not saturate the SI, measurements of T1 on a solution of proteins with varying Mn2+ concentrations could give insights into the enhancement provided by this kind of binding.
Following these results we choose to use a Mn2+ dose of 200 nmol/g BW at a rate of 4 ml/h,
typical infusion duration was then 6 min for a 30 g BW mouse and the volume injected was 400 µl.
We used the stock solution of 15 mmol/l in order to minimize the injected volume as well as the
injection duration.
2.2.3
Myocardial function quantification
Function measurements were performed on 3 groups of mice. A group of normal mice (n=4), a
group of sham mice (n=4) and a group of mice with an infarction (n=6). The infarction model of
this last group was a 60 min ischemia (performed by ligation of the left anterior descending coronary
artery) followed by reperfusion (referred to as IR60). The sham group underwent the same surgery
as the IR60 group but with no ligation of the coronary artery. The MR exam was performed 24
hours after surgery. Cine slices covering the whole left ventricle were acquired in order to measure
global and regional function. For this study these measurement were performed before and after
Mn2+ injection in odrer to determine if the presence of the contrast agent had an influence on the
cine image quality and function measured. Examples of cine images for the left ventricle middle slice
of a control and an IR60 mouse are given in Figure 2.7. We observed a clear contraction deficit for
the IR60 mouse compared to the control mouse. In this particular example we also observe a larger
contraction of the septum in the IR60 mouse probably due to a compensation mechanism (however
this was not significantly relevant in the following measurements).
Figure 2.7 — Example of cine images (acquired before Mn2+ injection) of systolic (left) and diastolic
(middle) phases in middle slice of the heart, as well as T1-weighted PSIR image (right) for an IR60 mouse
(A) and a control mouse (B). In (A) the lateral part of the myocardium is not moving between systolic and
diastolic phases which is not the case in (B). The PSIR images illustrates the localization of the infarction
for the IR60 mouse
Global function results
Table 2.2 shows results of EDV, ESV, EF and heart rate for IR60, sham operated and control groups
for measurements done before and after Mn2+ injection. We obtained a significant decrease of EF
30
for the IR60 group compared to the sham (29%, p<0.01) and control (20%, p<0.05) groups before
Mn2+ injection. However, these differences were no longer significant after Mn2+ injection. We
could also note a significant decrease in EDV and ESV leading to an increase in EF for the control
group as well as a decrease in ESV for the IR60 group after Mn2+ injection. Heart rate was not
significantly different neither between the 3 groups (p>0.4) nor after Mn2+ injection (p>0.2).
Table 2.2 — End diastolic volume (EDV), end systolic volume (ESV), ejection fraction (EF) and heart
rate (HR) for IR60, sham and control groups, measured before and after Mn2+ injection. Values are mean
± SD. For the IR60 group, P indicates significant difference with the sham and control groups respectively
obtained with ANOVA followed by Bonferroni post-hoc test (* p<0.05, § p<0.01, † p<0.001). Symbols in
brackets indicate result of t-test for comparison of measurement done before and after Mn2+ in each group.
EDV (µl)
ESV (µl)
EF (µl)
HR (bpm)
before Mn
after Mn
before Mn
after Mn
before Mn
after Mn
before Mn
after Mn
control
sham
IR60
P, IR60
vs
sham
trol
40.8±6.1
30.3±4.1 (*)
16.6±2.0
8.7±1.6 (†)
0.59±0.03
0.71±0.05 (§)
348±29
317±33 (NS)
30.7±13.9
32.6±3.2 (NS)
10.2±5.4
8.4±3.8 (NS)
0.66±0.09
0.75±0.09 (NS)
291±53
283±37 (NS)
57.6±9.5
47.5±10.8 (NS)
30.9±5.3
20.7±5.1 (§)
0.47±0.06
0.57±0.07 (NS)
328±25
316±42 (NS)
§
NS
†
§
§
NS
NS
NS
NS
*
†
§
*
NS
NS
NS
P, IR60
vs con-
When comparing EDV and ESV results with another study for the same model [64] we have
higher results in volume estimation leading to decreased EF for the control group (59% vs 70%
[64]). This could be due to a deeper anesthesia of the animal. Indeed, it has been recently shown
that, in control mice, isoflurane anesthesia can reduce EF to 60% compared to 79% obtained with
a deep sedation only [73]. However, the hypothesis of corrupted results due to the lower spatial
resolution reached in our cine measurements (344 µm vs 100 µm [64]) would be rejected since we
would have underestimate volumes more probably leading to an overestimation of EF. Also, compared to baseline, before Mn2+ injection, we obtained a global increase in EF measurements after
Mn2+ injection that is significant for the control group and correlated with a decrease in EDV and
ESV. This is explained by the loss of contrast between blood and myocardium in presence of Mn2+ .
In fact, the determination of endocardial volume tends to be underestimated, more importantly
during the systole than the diastole (see figure 2.8). From the definition of EF, if EDV and ESV
are underestimated, with ESV in a larger manner, EF is therefore overestimated. As Mn2+ intake
is globally more significant in control mice myocardium than in other groups it can explain the
difference between measurements done before and after Mn2+ injection in this group.
As a consequence, estimation of EF should preferentially be done before Mn2+ injection. However, we must point out that observations from cine images still depict, or not, of a decreased
contraction after Mn2+ injection.
Quantification of regional function
Left ventricular regional function was assessed by comparing wall thickness between diastole and
systole for 6 sectors covering the whole left ventricle, leading to percentage wall thickening eval-
31
Figure 2.8 — Example of cine images before and after Mn2+ injection. After Mn2+ injection we observe a
diminished contrast between LV cavity and myocardium that make the contour definition more problematic,
notably in systole.
uation. Figure 2.9 shows the results obtained before and after Mn2+ injection for control, sham
operated and IR60 mice. Table 2.3 shows the mean wall thickening in the 6 different myocardial
sectors for the 3 groups.
Wall thickening was significantly reduced in the free wall (including anterior, anterolateral, inferolateral and inferior sectors) for infarcted mice compared with the control and sham groups both
before (p<0.001) and after (p<0.01) Mn2+ injection, while no significant difference was obtained
in the septum. No significant difference was measured, neither between control and sham groups
(p>0.4), nor between measurements done before and after Mn2+ injection (p>0.5). As an explanation, the underestimation of the endocardial contour affects this measurement less than for EDV
and ESV, where this error is multiplied by the number of slices. To check this hypothesis, we
measured the area of myocardium on the middle slice of the left ventricle, obtained by manually
tracing endocardial and epicardial contours before and after Mn2+ injection. Figure 2.10 shows the
results of these measurements. The area was not significantly different before and after injection for
the systolic and diastolic phases in the IR60, sham and control groups whereas we observed some
significant variations of endocardial area for control as well as for IR60 in systolic phase.
Contrary to EF, wall thickening measurement can be performed after Mn2+ injection, which
allows considerable acceleration of the imaging protocol by doing the injection 45 minutes before
the MRI exam.
32
Figure 2.9 — Wall thickening between diastolic and systolic phase for sectors defined in figure 2.3 before
and after Mn2+ injection, for control group (n=4) (A), sham operated group (n=4) (B), IR60 group (n=6)
(C), and comparison of wall thickening before Mn2+ injection for the 3 groups (D). Values are Mean±SD
(A=Anterior, AL=Anterolateral, IL=Inferolateral, I=Inferior, IS=Inferoseptal, AS=Anteroseptal). * indicates significant difference between groups (at least p<0.05).
33
Table 2.3 — Wall thickening between diastolic and systolic phase in the 6 left ventricle sectors (defined in
figure 2.3) for IR60, sham and control groups, before and after Mn2+ injection. Values are mean±SD. For
the IR60 group, P indicates significant difference with the sham and control groups respectively obtained
with post-hoc Bonferroni test (*p<0.05, §p<0.01, † p<0.001).
Before Mn2+
control
sham
IR60
anterior
anterolateral
inferolateral
inferior
inferoseptal
anteroseptal
After Mn2+
0.77±0.18
0.77±0.12
0.71±0.20
0.63±0.11
0.51±0.17
0.53±0.11
control
0.79±0.16
1.03±0.26
0.90±0.29
0.94±0.08
0.75±0.21
0.64±0.17
sham
0.16±0.08
0.11±0.07
0.15±0.10
0.31±0.27
0.69±0.30
0.65±0.10
IR60
anterior
anterolateral
inferolateral
inferior
inferoseptal
anteroseptal
0.82±0.08
0.74±0.11
0.62±0.20
0.73±0.20
0.70±0.13
0.71±0.05
0.78±0.11
0.78±0.35
0.83± 0.34
0.92±0.07
0.69±0.04
0.68±0.16
0.31±0.05
0.23±0.10
0.15±0.13
0.27±0.23
0.71±0.21
0.74±0.16
P, IR60 vs
sham
†
†
†
§
NS
NS
P, IR60 vs
sham
†
§
§
†
NS
NS
P, IR60 vs
control
†
†
§
NS
NS
NS
P, IR60 vs
control
†
§
*
§
NS
NS
Figure 2.10 — Left ventricular wall area (upper) and cavity area (lower) measured before and after Mn2+
injection for control, sham and IR60 groups. Symbols indicates significative differences (* p<0.05, § p<0.01).
34
2.2.4
Infarction quantification
During each experiment, PSIR images of the whole myocardium were acquired. A typical example
is shown in figure 2.11 (upper row).
Figure 2.11 — Selection of T1-weighted PSIR short axis images (upper row) and corresponding TTC
staining (bottom row) for an IR60 mouse (A) and a sham operated mouse (B). White colour present in TTC
slices for (A) indicating tissue necrosis correlates with hypointense signal in PSIR images whereas neither
white nor hypointense signal is present in (B).
We defined a contrast to noise ratio (CNR) measurement between signal measured in MI area
(or free wall) and septum area for control, sham operated and IR60 group was measured. Noise
was chosen to be the mean of the SD in both those areas. We chose to avoid using the background
noise since parallel imaging was performed leading to an inhomogeneous noise in different area of
the image [74]. Results are shown in fiugre 2.12. For the IR60 group CNR was significantly higher
(p<0.01) than sham operated, and control groups. Also, no significant difference was found between
the control and sham groups (p=1.000), and both values are not significantly different from zero
(p=0.7 and p=0.3 for control and sham respectively). The high variability obtained for CNR measurements can be explained by the method of noise calculation. The noise was estimated from the
standard deviation of the SI in the left ventricle wall. This part of the image is corrupted by a physiological noise due to flow artifacts added to the image noise that could be measured from another
area of the image. However, CNR was always sufficiently high to allow infarction segmentation in
the IR60 group. No significant difference of Mn2+ uptake between control and sham group was
obtained, indicating that our method could not detect any effect on Ca2+ homeostasis consecutively
to open chest surgical procedure.
Infarction quantification was performed with a threshold technique and compared to the quantification performed with ex vivo measurements. For ex vivo quantification, we used triphenyltetrazolium chloride (TTC) staining, which (as shown in figure 2.11) lets appear viable cells in red and
is considered here as a gold standard [75]. For the MR images of the IR60 group, left ventricle was
first segmented by manually tracing endocardial and epicardial contour. The threshold was defined
by the mean SI measured in the myocardium area presenting a significant contractile dysfunction
observed on cine images and a hypointense area on one representative slice (generally the middle
35
slice of the heart). The error in estimation of the infarct volume was evaluated by defining the
threshold obtained by repeated drawing of ROI in the same area. This deviation was 5 a.u. and was
the same for all mice in this group. Infarction volume was then evaluated taking initial threshold ±
5 a.u. Figure 2.13 illustrates an example of this segmentation method.
Figure 2.12 — Contrast to noise ratio for control, sham and IR60 groups, symbol indicates significant
difference from 0 (§, p<0.01).
Figure 2.13 — Segmentation method for infarction volume quantification. Upper row; example of slices
covering whole heart of an IR60 mouse with manually drawn endocardial and epicardial contours. Bottom
row; corresponding results of the threshold segmentation technique.
36
For the control and sham groups, the infarction volume was derived from the threshold of the
IR60 group. In fact, we observed that infarction area had a SI reduction of 60 a.u. compared to
septum area (mean taken for all IR60 mice, n=6), so the threshold was defined by:
threshold = SIm − 60 (a.u.),
(2.4)
where SIm was the mean SI between 2 ROI (one in septum and one in free wall). Error bars were
determined by evaluating infarction volume taking the deviation of 5 a.u. on the threshold as described for the IR60 group. Figure 2.14 shows the results of infarction quantification for all mice.
Infarction quantification performed with MEMRI is strongly correlated with the measurements derived from TTC staining. The maximal deviation obtained for the error bars was 9%.
The result of linear regression between MEMRI and TTC infarction measurements was y=0.94x
with R2 =0.91 and p<0.001. The Bland-Altman plot also shows a good agreement between the
two methods without significant bias. The bias between MEMRI and TTC is 1.7%. In both the
control and sham operated groups, the SI was homogenously enhanced over all the heart without
any large area of hypointense signal on the PSIR, as shown in figure 2.11 and confirmed by the
fact that CNR is not significantly different from zero for the control and sham groups respectively.
This was in agreement with the TTC staining that did not reveal myocardial infarct in both these
groups. These results are very similar to those obtained with Gd3+ chelates enhancement for the
same infarction model [64]. However, in both control and sham groups, infarction quantification led
to a non-zero value whereas TTC results did not indicate any necrosis. This was often caused by
darker pixels present at edges of the myocardium that were counted as part of an infarct. Although,
the erroneous pixels could be easily detected by visual inspection, we chose not to discard them but
rather to use a linear model with no intercept in order to decrease the contribution of these isolated
pixels. A possible refinement of the method would be to apply morphological operations (erosions
and dilatations) on the segmented images to eliminate contributions of isolated pixels at edges;
Figure 2.14 — Left; infarction volume quantification with MEMRI versus TTC staining for IR60 (n=6),
sham (n=4) and control (n=4) groups as a percentage of left ventricular wall volume. Error bars symbolize the volume variation obtained with the threshold segmentation technique. Right; Bland-Altman plot
for comparison of infarction quantification method between MEMRI and TTC staining. Bias (defined by
mean(MEMRI-TTC)) was 1.7% (solid line), dashed lines are 1.96*SD(MEMRI-TTC)=8.0% corresponding
to 95% confidence interval.
2.3. Manganese kinetics
37
An important result was that MEMRI did not overestimate the infarct size. In fact, no previous
study had described Mn2+ enhancement in acute phase of reperfused infarction, so it was not known
whether other mechanism such as stunning could affect Mn2+ uptake. Very little literature is found
on the assessment of stunning in mice, [76, 77] and no data is provided for our mouse model and
time point (24h after reperfusion). However, Krombach et al. [57] have previously shown that
Mn2+ can assess stunning in rats 30 minutes after repeated ischemia-reperfusion protocol. The
explanation of such mechanism remains controversial as Mn2+ was accumulating via Na+ /Ca2+
exchanger-mediated transport during hypoxia in an in vitro study on isolated perfused myocardium
[78]. Should Mn2+ uptake be reduced in stunned cardiomyocytes, it could induce an overestimation
of the infarction volume compared to TTC, therefore a positive bias. Such an effect was clearly not
present in our model. Finally, it was important to provide a running protocol to assess myocardial
infarction in the acute phase as it is the start point of all further longitudinal studies.
2.3
Manganese kinetics
In this part, we focus on the SI kinetics following perfusion of Mn2+ in representative areas of myocardium (infarction, remote and left ventricular (LV) blood pool). We measured those kinetics for
IR60 mice 24 hours but also 8 days after reperfusion. The main question was wether Mn2+ kinetics
could provide additional insights to discriminate between an acute and a chronic injury. indeed,
MRI remains currently limited since a similar extracellular Gd3+ -chelate contrast accumulation is
observed in both situations [79]. Additional information to date the infarct may be provided by
edema imaging using T2 MR sequences [80]. Progresses have been made in the MR sequence design
[81], but edema imaging using MRI remains challenging. Other authors observed different T1 between acute and chronic infarct after extracellular Gd3+ -chelate injection [33] but the robustness of
this technique has not been recognized. The specific detection of intracellular 23 Na could also give
insights into the date of infarction since concentration of 23 Na decreases with time after a reperfused infarction due to collagen deposition [82] but this technique needs specific hardware. In the
search of a better diagnostic tool to differentiate between acute and chronic myocardial infarct, the
most promising method today is based on the combination of both intravascular and extravascular
contrast agents. Intravascular contrast agents enhanced acute infarct but not chronic scar because
vascular integrity is destroyed in acutely infarcted myocardium but is intact in scar tissue [83].
The practical feasibility of this method that requires two contrast media injections remained to be
demonstrated in the clinical setting.
2.3.1
Animal groups
Mice were submitted to the same surgical protocol as previously to induce an infarction (60 min
ischemia followed by reperfusion). MRI was performed either 24 hours (n=10) and/or 8 days (n=12)
after the reperfusion (within this group of 12, 4 mice underwent MRI at both time points). Animals
were then sacrificed after 24 hours (n=6) or after the 8 days (n=12) MR examination to perform ex
vivo analysis. TTC staining was perfomed in all the mice sacrificed in the 24 hours group (n=6 /
10) and in 6 mice of the 8 days group (n = 6 /12). Collagen staining was performed in the remaining
mice of the 8 days group (n=6 /12).
2.3.2
Myocardial function quantification
Infarction was successfully induced in all the mice, as attested by a hypokinesia in the lateral wall
of the left ventricle on the cine sequences. Ejection fraction was measured for each animal before
38
the Mn2+ injection as it has been shown to be more accurate (see section 2.2.3, p 29). We observed
a decreased global function (ESV, EDV, EF) compared with normal mice (see values in table 2.4).
However, there was only a trend for the global function to worsen between acute (24 hours) and
chronic (8 days) time points, EF measured on the same animals (n=4) at the 2 time points were
not significantly different, (p=0.828). Compared to the values obtained by Yang et al. [64] 24 hours
after the reperfusion (for the same mouse model EDV = 33 µl, ESV = 22 µl and EF = 37 %) our
measured EDV, ESV and EF are in the same range considering the variability obtained in infarction
size. However, the results obtained for EDV and ESV are lower (and as a consequence the EF is
higher) than the values measured by Ross at al. [84] in a model of 2 hours ischemia followed by
reperfusion. They also have followed the EF several weeks after reperfusion and did not observed
any significant change of EF between measurements done 24 hours and 8 days after reperfusion
which correlates our observations.
Table 2.4 — EDV, ESV and EF measurements 24 hours and 8 days after reperfusion (mean ± SEM).
Symbols indicate significant difference compared with control values (* p<0.05, § p<0.01), P values indicate
the result of the t-test performed between both time points.
EDV (µl)
ESV (µl)
EF (%)
2.3.3
control
40.8±6.1
16.6±2.0
59.0± 3.2
24 hours
54.0±3.0
28.2±1.8*
47.1±3.1*
8 days
66.8±7.9
40.7±6.8
41.1±3.3§
p
0.144
0.085
0.828
Kinetic curves
To measure Mn2+ kinetics, 3 region of interest (ROI) were manually traced in the left ventricular blood pool (LV blood pool), the septum (referred to as remote) and the anterolateral sector
(referred to as infarct). This latter area was located in the hypointense signal area 45 min after
R (R2008b,
Mn2+ injection (see part 2.2.4, p 34). Mean kinetics were computed using MATLAB
The Mathworks, Inc, Natick, MA, USA). Signal intensity data were first resampled onto the same
time points using a linear interpolation between adjacent points, taking t = 0 as the time of the
beginning of Mn2+ injection. For each mouse, curves were normalized by the maximum SI measured
in the left ventricular blood pool. We chose this normalization since it is representative of the Mn2+
diffusion into the circulatory system which can be slightly different from one animal to another. The
normalized curves for the respective areas (remote, infarct and left ventricular blood pool) were then
averaged and mean kinetics were obtained for acute (24h) as well as for chronic (8 days) conditions.
The mean SI kinetics measured in remote, infarct and LV blood pool are described in Figure 2.15.
At both time points, the left ventricular blood pool SI rapidly enhanced during the Mn2+ infusion
(wash in phase, until 10-15 min after the beginning of the injection) followed by a slow decrease
during the wash-out. In contrast, the SI in the remote myocardium demonstrated a slower increase
during the wash in phase that is prolonged during the wash-out phase where a plateau is reached.
At 24 hours, the SI measured into infarct showed a rapid increase followed by a slow decrease during
the wash-out (similarly to LV blood pool kinetic), whereas at 8 days the SI increases slowly during
both phases before reaching a plateau. The signal intensities 55 min after the Mn2+ injection were
not significantly different between acute and chronic time points (p=0.475, p=0.580 and p=0.056
for remote, infarct and LV blood pool respectively). SI was lower in infarct than in remote, however
due to the dispersion of data this difference was statistically relevant only at 8 days (p=0.007).
39
The slopes describing the entry of Mn2+ into the 3 regions of interest are reported on Figure
2.16 (with detailed values in Table 2.5). Slopes were determined without user interaction on the
normalized kinetics by performing a linear regression on the points included in the interval t = 0
(arrival of the contrast media in the left cavity) and t = tmax corresponding to the peak of SI into
the left ventricular blood pool. At 24 hours, during the wash in, the SI increase was much faster
in the infarct area than the SI increase in the remote area (mean slopes are 0.110±0.013 a.u. and
0.053±0.009 a.u. for infarct and remote respectively, p=0.002) without any difference with the SI
of the LV blood pool (mean slope is 0.119±0.016 a.u., p =N.S.). However, at 8 days, the SI increase
of the infarct area was much slower during the wash in and similar to the SI increase in the remote
area (mean slopes are 0.037±0.008 a.u. and 0.052±0.011 a.u. for infarct and remote respectively,
p=0.307).
24 hours − n=10
1.6
1.4
8 days − n=12
remote
infarct
LV blood pool
1.4
1.2
1
SI normalized (−)
SI normalized (−)
1.2
0.8
0.6
0.4
1
0.8
0.6
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−10
remote
infarct
LV blood pool
1.6
0
10
20
30
40
50
60
−0.4
−10
0
10
20
t (min)
30
40
50
60
t (min)
Figure 2.15 — Time course of normalized signal enhancement after injection of Mn2+ in different areas
in the myocardium (remote, infarct and left ventricular blood pool), for mice with 60 min ischemia followed
by 24h (left) or 8 days (right) of reperfusion. Bold lines are mean normalized SI and thin lines are SEM.
Table 2.5 — Left: Manganese entry slopes and mean correlation coefficient for remote, infarct and LV
blood pool for acute and chronic time points. Right: P-values for non-parametric test (Friedman’s two ways
ANOVA by ranks) results and pairwise comparisons.
slopes
(a.u.)
remote
infarct
LV
blood
pool
24 hours
8 days
t-test
p
0.053±0.009
(R=0.745)
0.110±0.013
(R=0.880)
0.119±0.016
(R=0.904)
0.052±0.011
(R=0.801)
0.037±0.008
(R=0.668)
0.105±0.011
(R=0.941)
N.S.
Friedman’s
ANOVA
remote - infarct
infarct - LV
blood pool
p<0.001
N.S.
remote - LV
blood pool
24
hours
0.001
8
days
0.000
0.002
0.307
1.000
0.000
0.002
0.013
40
Figure 2.16 — Slopes of Mn2+ wash-in kinetics into septum, infarct an left ventricular blood pool for MR
exams taken 24 hours (n=10) and 8 days (n=12) after reperfusion (* p< 0.05, ** p<0.01, *** p< 0.001).
The early arrival of Mn2+ into infarction during the wash-in, observed 24h after surgery but not
at 8 days, lead to a positive contrast between the infarct and the remote myocardium. To illustrate
that, figure 2.17 shows an example of PSIR images taken at 4 time points over the Mn2+ kinetic,
for acute and chronic time point.
41
Figure 2.17 — Examples of PSIR images taken at representative time points during the MRI exam 24h
(upper) and 8 days (lower) after surgery: left to right, before Mn2+ injection, at the peak of enhancement
in the LV blood pool and the infarct area (10 min after Mn2+ injection), when SI of remote and infarction
are approximately the same (25 min post injection) and after wash-out of Mn2+ corresponding to late
enhancement image (55 min post injection). The window width and window level were set to enhance the
contrast between viable myocardium and infarction, this is why the change of SI in the LV cavity is not
visible.
2.3.4
Measurements of infarction extension
Figure 2.18 — Schematic representation of infarct extension measurement. The infarct extension length
is divided by the left ventricular length to obtain a percent of infarction extension.
For measurement of infarction extent in ex vivo pictures, the ex-vivo slice corresponding to the
slice selected for the MRI kinetic measurement was chosen for each animal (this slice was located
at the middle height of the left ventricle or slightly lower). Then MEMRI and ex vivo images
were treated with the same method. One line was traced all over the circumference of the left
ventricle in the middle of the wall giving a “left ventricular length” as it is illustrated in Figure 2.18.
Then a second line was traced on the same location but limited to the area where the infarction
was observed (hypointense signal on the MEMRI images, white area in the TTC images and blue
staining on the Masson-trichrome images) giving an “infarct extension length”. Infarct extension
42
was then calculated as follows:
Infarction extension =
infarct extension length
.
left ventricular length
(2.5)
Images were treated randomly in order to avoid bias in the measurements. Figure 2.19 shows
examples of ex vivo images taken 8 days after reperfusion.
Figure 2.19 — Comparison of MR images and ex-vivo colorations (TTC in a, Masson in b) for 4 examples
of images taken 8 days after surgery.
The myocardium infarction measured at late recirculation (55 min after Mn2+ injection and
referred as “wash-out”) appeared as a negative contrast by comparison to remote myocardium.
Measurements of infarction extension were compared to ex-vivo measurements (TTC for acute and
Masson for chronic infarctions). The detailed results of correlation between measurements are
presented in Table 2.6 and in figure 2.20. The infarct area derived from the hypointense SI during
the wash-out strongly correlated to the infarct extension measured by TTC either at 24 hours or 8
days (r> 0.9, p < 0.013). However, measurements of infarction extension using this positive contrast
(referred to as “wash-in”) showed that this zone is smaller than TTC staining area (p=0.002) and
than the hypointense area observed during wash-out (referred to as “wash-out”) (p=0.026). Wash-in
infarction extension was also poorly correlated with either the wash-out measurement or with the
TTC staining.
43
Table 2.6 — Linear regression between infarction extent measurements with MEMRI and ex-vivo methods
(TTC or Mason) for acute and chronic time points. No correlation is observed between the wash-in contrast
in MEMRI compared to TTC measurements, whereas all the other measurements are correlated. R is the
correlation coefficient and p is the result of the test in which the null hypothesis is a slope equal to zero.
24h
8d
MEMRI wash-in - wash-out
(n=6)
MEMRI wash-in - TTC (n=6)
MEMRI wash-out - TTC (n=6)
MEMRI wash-out - Masson
(n=5)
equation
y=-0.262x+57.8
R
0.228
p
0.663
y=0.052x+37.9
y=1.029x-13.3
y=1.128x-6.8
0.040
0.907
0.929
0.941
0.013
0.022
Figure 2.20 — Correlation between MEMRI and ex-vivo infarction extent measurements (TTC or Masson).
44
2.4
2.4.1
Discussion
Acute infarction
Saeed et al. [85] have shown that in a model of reperfused infarction in rat, the injection of MnCl2 24
hours after reperfusion gave rise to a different Mn2+ kinetic than the one that we observed. Indeed,
the maximum signal enhancement in the viable part of the myocardium as well as in the infarction
occurred at the same time point (5 min post injection). This rapid kinetic may be explained by the
fact that the contrast agent injection was done intravenously. However, Bremerich et al. [59] have
shown that, in the same model of reperfused infarction in rat, the intravenous injection of Mn-DPDP
lead to a rapid Mn2+ entry into MI before to be washed out and a slow entry of Mn2+ into viable
parts of the myocardium. This is very similar to our measurements but for the IP injection of Mn2+
instead of an IV injection of Mn-DPDP (note that for mice an IP injection is very similar to an IV
injection [86]).
This kinetic pattern shown in Bremerich et al. [59] demonstrates also the presence of a double
contrast between infarction and viable parts of the myocardium, however, this was not really emphasized in the article and no correlation was performed between this hyperintense area and TTC
measurements. Indeed, SI in infarction is first enhanced by accumulation of Mn2+ compared to
viable myocardium and this contrast is reversed for late measurements, when Mn2+ is washed out
from infarction. In acute time point Mn2+ kinetics in infarction has the same behavior than into LV
blood pool. Indeed, Mn2+ entry slopes in infarction and in LV blood pool are identical which tends
to support the hypothesis that entry of Mn2+ into infarction is not related to a cellular mechanism
but rather to a free diffusion mechanism. Actually, whereas extracellular space is reduced into viable
myocardium due to dense distribution of cardiomyocytes [87], it becomes larger in acute infarction
due to cell membrane rupture of necrotic cells as well as presence of massive interstitial edema [88].
The diffusion of Mn2+ into this new extracellular space is therefore easier and can explain this observation. Moreover, Mn2+ entry slope into remote area at both time points is significantly lower
than entry into blood pool, meaning that accumulation of Mn2+ into viable cells is a slow process in
good agreement with our previous hypothesis concerning the rapid entry of Mn2+ into the infarction.
Our results also showed that the normalized SI is more important in infarction than in blood pool.
This phenomenon was also observed with extracellular contrast agent [89] and may be explained by
the enhancement of paramagnetic ions relaxivity in presence of proteins released by necrotic cells [90].
This can also explain the apparently slower release of Mn2+ in infarction compartment compared to
blood pool. These findings are important since they show that enhancement of SI related to presence
of Mn2+ does not involve in this case an increase in cell activity or an intracellular Ca2+ increase.
Moreover, as shown in table 2.6 the area presenting an early arrival of Mn2+ is not correlated with
the TTC analysis, but is systematically smaller (infarct extent was 41.7±14.0 % and 72.6±10.7 %
for wash-in and TTC respectively, p=0.002). One hypothesis is that infarction area still contains
cells with intact membrane but not functional (e.g. stunned or apoptotic cells) in which Mn2+ can
not penetrate which thus reduces Mn2+ diffusion [63]. The presence of apoptotic cells was shown in
ischemia reperfusion injury [91] even if the maximum concentration of these cells seems to be earlier
than 24 hours after reperfusion [92].
2.4. Discussion
2.4.2
45
Chronic infarction
Mn2+ kinetics into infarction area measured 8 days after surgery is completely different than in
acute infarction: the wash-in of Mn2+ is slower and reduced compared with viable myocardium. It
can be explained by three main contributions:
• Firstly, the accumulation of collagen into infarction area as showed by Masson-trichrome measurements (see figure 2.19). Ojha et al. [63] showed that size of infarction is significantly
reduced between acute and chronic measurements due to the wound contracture and the proliferation of collagen due to scar maturation that is maximum 7 days after infarction. This
dense tissue therefore inhibits the diffusion of Mn2+ .
• Secondly, it has been shown that presence of inflammatory cells like macrophages is still
important 7 days after infarction [87], slow enhancement of SI in infarction can thus be due to
accumulation of Mn2+ into these cells, by a similar mechanism as into viable cardiomyocytes,
explaining the slow kinetic observed.
• Finally, as the scar tissue is relatively thin compared to acute infarction, partial volume effects
from surrounding viable tissue are not to be excluded.
2.4.3
Manganese as a marker of cell viability
The results we obtained showed that Mn2+ is able to accurately define infarction volume, for an
ischemia-reperfusion infarction model, 24 hours and 8 days after surgery. However, the Mn2+
kinetics, particularly in infarction compartment, brings to light that care must be taken with the
interpretation of Mn2+ contrast. Indeed, signal enhancement related to Mn2+ accumulation seems
to be related to the cell viability only at late recirculation time point (between 40 to 55 min after
Mn2+ injection), whereas transient enhancement do not describe intracellular phenomenons such as
Ca2+ changes. Following these observation, it seems that Mn2+ can be used as a cell viability only
for situations in which a steady state of Mn2+ enhancement has been reached.
46
Highly
time-resolved
functional
imaging
3
Part of this chapter was presented as a poster at ISMRM 16th annual congress [93].
In the previous chapter, we have set a running protocol to assess cardiac function and viability
on mice. The sequences used could depict efficiently the global as well as the regional function
parameters. However, compared to dedicated instrumentation, i.e. high field MR scanners with
higher gradient performances, we had to make a tradeoff on the spatial resolution to maintain a
sufficient temporal resolution. In this chapter we present an alternative sequence design to reduce
this tradeoff and finally achieve parameters that are better suited to cardiac mouse heart imaging.
3.1
Sequence presentation
Following Roussakis et al. [94], 11 phases per cardiac cycle are necessary to accurately evaluate
ejection fraction. Whereas it is easily achievable for human heart, it becomes challenging when
imaging rodents. Indeed, the dimensions of the heart as well as the heart rate of mice are about one
order of magnitude respectively smaller and higher than humans. Moreover, with clinical scanners,
the achievable time resolution is limited by gradient hardware. Currently, the available cine sequence
can provide a spatial resolution of 257 µm but with a limited time resolution of 13.5 ms. Nevertheless,
to accurately estimate cardiac function, spatial and temporal resolution we reach should be in the
order of 117 µm and 8.4 ms respectively [95]. Whereas the spatial resolution of 257 µm is sufficient
to depict accurately heart structures, the temporal resolution is too low. Indeed, if we consider
a mouse heart rate of 600 bpm (the physological heart rate of mouse varies between 468 and 615
bpm for anesthetized and conscious mice respectively [73]), the time resolution needed to acquire
11 phases per cardiac cycle is 9 ms. We thus need to accelerate our sequence to be able to assess
accurately cardiac function.
47
48
3.1.1
Chapter 3. Highly time-resolved functional imaging
Sequence parameters
The basic cine sequence was a segmented turboflash cine sequence with following parameters: FOV
111 mm, in plane resolution 257 µm, slice thickness 1 mm, TR/TE 13.5/6.2 ms, flip angle 30◦ ,
GRAPPA with acceleration factor 2, 2 averages, typical acquisition time per slice 3 min. Sequence
is respiratory and ECG gated. For each R-R cycle only one line of k-space is acquired in order
to minimize the acquisition time. With this configuration, time resolution is limited by gradient
hardware (maximum available amplitude and slew-rate) to 13.5 ms.
The proposed sequence (referred to as “interleaved cine”) is composed of the repetition of the
basic sequence where the second repetition is shifted of a time delay corresponding to the half of
the basic sequence’s TR (i.e. 6.8 ms). Figure 3.1 represents schematically the design of the basic
cine as well as the interleaved cine. Finally, the combination of the two repetitions gives a cine with
a time resolution of 6.8 ms instead of 13.5 ms. This scheme has the main advantage of keeping the
acquisition time constant between the two sequences.
Figure 3.1 — Schematic representation of sequence and reconstruction design. Basic cine (a) and interleaved cine (b), while the two repetitions of basic cine are averaged to form the final image serie, the second
repetition of the interleaved cine is shifted of 1/2 TR and the two combined repetitions form a serie with
enhanced time resolution.
3.1.2
Temporal regularization
The final combination of images in the interleaved cine was however more prone to noise compared
to the basic sequence since there was no more averaging. Moreover, since the two combined image
series are acquired successively in time, the flow artifacts are different between the two series and
introduce different ghosting artifacts into the two series. This results in a flickering artifact in the
final combination, which signal intensity variation is not in phase with the repetitions. To illustrate
the origin of this effect, figure 3.2 represents the first image of the cine series acquired in different
conditions. We observe that ghosting artifact propagates along the phase encoding direction, which
is an inherent property of artifacts due to flow or motion during acquisition. Moreover, the pattern
of this artifact is different if the acquisition is shifted of 1/2 TR but also if it is simply repeated
without any trigger delay addition.
49
3.1. Sequence presentation
Figure 3.2 — Illustration of the ghosting artifacts responsible for the final flickering artifact in the interleaved cine. Images are the first frame of cine series acquired at the same location, windowing was forced
on purpose in order to have a better visualization of the artifacts. (a) Cine acquired with no trigger delay,
(b) cine acquired after a trigger delay of 1/2 TR. (a) and (b) are the two cine composing the interleaved
cine, white arrows are pointing areas where signal intensity of ghosting artifact are different between the
two cine. (c) Repetition of (a), (d) same as (a) but with different phase encoding direction (“from left to
right” instead of “from anterior to posterior” for (a),(b)and (c)).
In order combine denoising and correction of the flickering artifact we took advantage of the
periodicity of the cardiac cycle and applied a Fourier filtering along the time direction for each
image pixels. This could be done efficiently by a soft threshold of Fourier coefficients. Indeed, the
algorithm can be written as the following minimization problem:
s = arg min
X
k
kfk − sk k22 + λ
X
k
kŝk1 ,
(3.1)
where k is the temporal indice, f is the image data, λ is the threshold used for the soft threshold,
with ŝ the Fourier transform in the temporal direction.
3.1.3
Validation experiments
The first step of our study was to validate our sequence, i.e. to check that the combination of the
two interleaved cine was able to retrieve the temporal information of the heart contraction usually
measured in a single cine and to estimate the efficiency of our denoising algorithm. Since the minimal TR was limited to 13.5 ms to achieve a spatial resolution of 257 µm, we used the basic cine
sequence with TR=13.5 ms and 2 averages as a reference and we set the TR of our interleaved cine
to 27 ms, in order to retrieve a final temporal resolution of 13.5 ms and compare it to the reference.
An example of the obtained results is illustrated in Figure 3.3. As shown on the temporal
profile of one pixel in the myocardium, the interleaved cine sequence is prone to a flickering artifact.
The images of difference between interleaved cine and reference cine showed some artifacts that are
present only one phase over two. However after the filtering provided by the temporal regularization
50
this artifact is greatly reduced. The observation of the images as well as their temporal profile showed
that the interleaved cine is able to retrieve the information present on the reference sequence in order
to describe the cardiac contraction even if the two repetitions were acquired successively.
Figure 3.3 — I: Four out of the 20 cardiac phases for the reference sequence (TR 13.5 ms, 2 averages,
images are taken on 2RR cycles) (a), the proposed sequence (TR 27 ms, no average, 2 repetitions) (b), and
the filtered proposed sequence (c). (d) and (e) are the differences between (b) and (c) respectively and the
reference sequence (a). White arrows point artifacts that are present only one phase over two, whereas there
are no more visible after temporal regularization (f). II: Time course of the white profile for (a), (b) and (c).
III: Time course of signal intensity pixel corresponding on white profile on II. Peak of signal enhancement
corresponds to a flow artifact.
Figure 3.4 — Effect of the threshold (th in a.u.) value on the data term kI − Iref k2 ,. In this example
th=500 a.u. leads to an important reduction of the flickering artifact.
The threshold λ chosen to reduce the flickering artifact is important since it acts directly on the
flickering artifact correction. Figure 3.4 illustrates the effect of the threshold on the data term which
was defined by kI −Iref k2 , with I the filtered interleaved cine and Iref the reference cine. Evaluation
51
3.1. Sequence presentation
of the flickering artifact reduction was done by computing the energy E of the temporal derivative
of the data term. Car must be taken by not choosing a too important value of the threshold since
it may smooth the data temporal profile and lead to information loss.
E=
X ∂
kI − Iref k2 .
∂t
t
(3.2)
In Figure 3.3 and in the following, the threshold λ=500 a.u. was chosen since it reduced significantly the value of E measured before and after denoising (p=0.037 when comparing log-transformed
values). Indeed, we measured a noise reduction of -6.0 ± 2.2 dB (n=8).
The sequence we propose achieves a temporal resolution comparable to those obtained on dedicated small animal scanners with a sufficient spatial resolution to depict the cardiac morphology (we
obtained TR=6.8 ms / in plane resolution = 257 µm compared to TR=7.5 ms / in plane resolution
=117 µm for a 11.7T scanner [63], or TR=7 ms / in plane resolution =100 µm for a 9.4T scanner
[69]).
The validation experiments showed that the principle of combining two acquisitions shifted of
one half TR can retrieve the temporal information of each cardiac phase. Following this principle
we could further enhance the time resolution by shifting the acquisition by smaller amount of time.
However, this method would work to a certain extent since the frequency band containing the main
information in k-space has a certain width. Therefore, the minimal time shift should be the time
needed to acquire this frequency band. Figure 3.5 shows an example of signal contained into the
k-space for one cardiac phase. We estimated that the readout time needed to acquire one line of
k-space is around 5 ms (time when ADC is on). Figure 3.6 shows signal acquisition as a function
of time, for basic and interleaved cine. With the proposed scheme we observe that the signals are
not overlapping and that we could further increase the time resolution to at least 5 ms, the length
of the ADC.
Figure 3.5 — Left, example of one image of interleaved cine, right, corresponding k-space where the
intensity scale varies from the min signal intensity to 5% of the max signal intensity for visualization
purpose.
52
Figure 3.6 — Signal time course of one representative k-space line for basic cine (upper) and for interleaved
cine (lower), blue and red curves correspond to the signal acquired during the first and the second repetition
respectively.
3.2
Mass and function measurements
The ability of interleaved cine to depict accurately cardiac contraction and morphology was validated
by measuring the mass of left and right ventricles in several mice as well as global function parameters
in normal mice and in mice with an infarction.
3.2.1
Animals
Two types of mice were used for this study, a group of male C57/BL6 wild type (WT) mice aged
between 8 and 12 weeks (n=7) as well as a group of female TG1306/R1 transgenic (TG) mice aged
of 24 weeks (n=23). This last group has a C57/BL6 background and is known to develop a left ventricle hypertrophy [96, 97]. For the function measurements, part of C57/BL6 WT mice underwent
a complete ligation of left descendant coronary artery to produce a myocardial infarction (MI) and
were scanned 24 hours after the surgery (n=3).
After the MRI session, mice belonging to the TG group as well as WT mice were sacrificed and
heart was removed. The left and right atria were excluded from the heart and the two ventricles
were weighted to provide a reference value. This operation was performed by the same operator (Dr
Corinne Pellieux) for all mice to avoid any bias introduced by the dissection method.
3.2.2
Image analysis
The combination of the two image series for interleaved cine as well as the temporal regularization
R
was performed with MATLAB(R2008b,
The Mathworks, Inc, Natick, MA, USA). Measurement
53
3.2. Mass and function measurements
of left and right ventricles mass as well as ejection fraction was done using Osirix software (open
source http://www.osirix-viewer.com/). For mass evaluation, epicardial and endocardial contours of
the left and right cavity were manually drawn on the the systolic phase of the short axis interleaved
cines. The mass was calculated as follows:
(LV + RV )mass = γ · Slice thickness
X
[epi area - (LV endo area + RV endo area)],
(3.3)
all slices
where γ is the specific gravity of the myocardium, γ=1.055 g/cm3 . For ejection fraction, endocardial
contours of left ventricle were manually drawn on all interleaved cine slices, excluding the papillary
muscles [70]. Ejection fraction was defined by (EDV-ESV)/EDV, where EDV is the end-diastolic
volume and ESV the end-systolic volume.
3.2.3
Results
After the validation experiment, the sequence was applied to the acquisition of short axis views
of mice heart with a high temporal resolution, i.e. TR=6.8 ms. The increase in time resolution
with the interleaved cine is depicted in Figure 3.7. To illustrate the improvement provided by the
temporal regularization, figure 3.8 shows the temporal profile of a pixel line before and after the
filtering.
Figure 3.7 — Images of one cardiac cycle for a mouse with an infarction (same mouse as in Figure 2).
Basic cine (a) and interleaved cine after filtering (b). Right, temporal evolution of the white profile.
54
Figure 3.8 — Left, image sample of the interleaved cine before temporal regularization; middle, temporal
profile of the white line in left, right, temporal profile after the regularization, the noise is reduced as well
as the flickering artifact.
We used the interleaved cine to measure the left and right ventricles mass of a group of control
mice (WT, n=4) and a group of mice that tend to develop a cardiac hypertrophy (TG, n=23).
The mass measured with interleaved cine was highly correlated with ex-vivo measurements as it is
shown on Figure 3.9. Linear regression results were y = 0.829x + 25.1, R2 =0.858 and we observed
no significant bias between methods (mean(MR, ex-vivo)=-2.62 mg). Moreover, the dispersion of
data obtained by Bland-Altmann plot was not so higher than the values reported by Yang et al.
[64] measured with a dedicated scanner (95% of data were located in the range of ± 27 mg and ±
15 mg, for our study and Yang et al. respectively).
Finally, we measured the ejection fraction of a group of control mice (WT, n=4) and a group
of mice with an infarction (MI, n=3). We measured a significant increase of ESV and EDV for MI
group (p<0.01 and p<0.05 respectively) as well as a significant decrease of EF compared to the
WT group (p<0.001) as reported in figure 3.10. The values we obtained for the WT group were in
agreement with values reported by Ross et al. [84] for the same mice breed and age (they obtained
48 µl, 22 µl and 50% for EDV, ESV and EF respectively).
3.2. Mass and function measurements
55
Figure 3.9 — Left, correlation of mass measurement between MRI and with ex-vivo methods, y = 0.829x+
25.1, R2 =0.858. Images show examples of short axis systolic phases taken at the middle of the heart for
a WT (left) and a TG mouse with hypertrophied myocardium (right), white bar represents 5 mm. Right,
Bland-Altman plot shows for comparison of MRI and ex-vivo methods. Solid line is mean(ex-vivo,MRI)=2.62, dashed lines are 1.96*SD(MRI-ex-vivo)=27.15 mg corresponding to 95% confidence interval.
Figure 3.10 — Ejection fraction for control mice, WT (n=4) and for mice with a myocardial infarction,
MI (n=3), values are mean ± SD, symbols represent significant difference (* p<0.05, § p<0.01, † p<0.001).
56
3.3
Going further with the image enhancement...
The temporal regularization proposed to reduce the flickering artifact as well as the noise in the
interleaved cine was shown to be relatively efficient. However, on several sets of data it failed to
enhance the image quality, even if the threshold for the temporal regularization λ was enhanced.
Figure 3.11 shows an example where the proposed regularization does not eliminate the flickering
artifact and where the temporal infomration of data are lost if the threshold is set too high.
Figure 3.11 — Example of temporal profile of interleaved cine for several threshold values used for the
temporal regularization. Whereas low values do not remove the flickering artifact, higher ones corrupt the
temporal definition of the data
3.3.1
Model presentation
During our experiments, we observed that the flickering artifact was mainly due to different flow
and motion artifacts between phases and combined cine series. This artifact was not necessarily in
phase with the repetitions. To refine the post-processing part in order to more robustly eliminate
this artifact, we propose the following model:
J(s, A, ω, φ) =
X
k
kfk − A sin (ωk + φ) − sk k22 +
λX
ksˆk k1 ,
2
k
(3.4)
3.3. Going further with the image enhancement...
57
where k is the temporal indice, f is the image data and we define fk′ = fk − A sin(ωk + φ). The
proposed algorithm to minimize J is the following:
1. Estimation of data variation, calculation of spatial mean signal of f → fm
2. Fit of ω and φ on fm
3. On each pixel of f, fit of A
4. Caluclate fk′ = f − A sin(ω · k + φ), the modified data
5. Soft threshold on Fouirer coefficients of s
To first test our algorithm we developed a numerical phantom composed of a vertical line moving
horizontally and periodically in the FOV. We added to the signal a component corresponding to
A0 sin(ω0 · k + φ0 ) as well as noise. Figure 3.12 shows this numerical phantom. The 2 first steps of
the algorithm consisted in calculating the mean signal intensity variation of the image in function of
time in order to estimate the parameters ω and φ by a least square fit of the data. Indeed, we made
the hypothesis that ω and φ are constant for all image pixels, whereas A is spatially dependent. The
next step is therefore to estimate parameter A knowing ω and φ. This is simply done by minimizing
J regarding to A (s is set to null value as an initial condition):
X
∂J
(A sin(ωk + φ) − fk ) · sin(ωk + φ) = 0,
=0→2
∂A
k
P
fk sin(ωk + φ)
A = Pk
.
2
k sin (ωk + φ)
(3.5)
Knowing then all the model parameters, the data s are calculated with the relation s = f −
A sin(ωk + φ). The flickering artifact is therefore removed and the last step of the algorithm,
consisting in the soft thresholding of Fourier coefficents of s further removes the noise present in
the images. In the ideal case of the numerical phantom the method works very well and can
retrieve efficiently the signal, as it is illustrated on figure 3.13. We observed that A is highly spatial
dependent.
Figure 3.12 — Illustration of the proposed algorithm to remove the flickering artifact on numerical phantom. Left, one image of the temporal serie, middle, temporal profile of one line of the image on left, right,
variation of the mean signal intensity along time fm and corresponding fit to evaluate ω and φ.
58
Figure 3.13 — Results on the numerical phantom, (b) original temporal profile, (c) temporal profile of s
obtained at the 4th step of the algorithm, (d) final result of s obtained after soft thresholding the Fourier
coefficients (λ = 0.5 a.u.). (a) shows the spatial variation of parameter A.
We then applied the same algorithm to real data. Figure 3.14 shows the results obtained for the
data previously presented in figure 3.11, where we could not get rid of the flickering artifact without
losing important information regarding myocardial contraction. The proposed algorithm was able
to remove efficiently the flickering artifact as well as to reduce the noise after the soft thresholding
step. Once again, the map of A shows the importance of fitting this parameter for each spatial
pixel since the variations are important between different areas of the image. We even observe sign
changes in some areas of the FOV indicating a phase inversion of the model.
3.3.2
Validation experiments
As for the first part of the study, we evaluated the denoising efficiency of our algorithm by computing
the data term between the interleaved cine and the reference sequence (see p 49). We already had
obtained an important reduction of the parameter E (the energy of the temporal derivative of the
data term). This reduction was even more important with the proposed denoising algorithm. Table
3.1 shows the results obtained when comparing E after the simple soft thresholding of Fourier
coefficient, referred to as “model 1” (see equation 3.1 p 49) and E after denoising with the full
algorithm, referred to as “model 2”. We obtained a significant better reduction of noise with the
complete algorithm than with only Fourier soft threshold (p=0.02).
59
3.3. Going further with the image enhancement...
Figure 3.14 — Results on real data, (c) original temporal profile, (d) temporal profile of s obtained at
the 4th step of the algorithm, (e) final result of s obtained after soft thresholding the Fourier coefficients
(λ = 500 a.u.). (a) shows one image example of the cine serie and (b) the spatial variation of parameter A.
Table 3.1 — Evaluation of the flickering artifact reduction with the two proposed denoising algroithms. p
values are the result of the t-test comparison with original value.
original
denoised with model 1
denoised with model 2
E
23.8 ± 1.0
23.2 ± 0.9 (p=0.23)
22.7 ± 0.7(p=0.037)
noise reduction vs original
−6.0 ± 2.2 dB
−10.4 ± 3.8 dB
60
3.3.3
Function and mass measurements
During our experiments, we observed that the efficiency of the cine denoising was very important
for contraction default visualization when the cine movies were played. However, the quality of the
denoising did not impact on the size of the ROI manually traced on the endocardial and epicardial
contours. Indeed, we tested this assumption by tracing repeatedly (3 times) the ROI on the image
denoised with model 1 and with model 2. We did that experiment on 3 different slices, on diastole
and systole (i.e. n=6). The endocardial cavity area was not significantly different for all cases, mean
p value obtained was 0.84±0.20. Following this result, we can consider that the mass and function
measured on images denoisd with model 1 are not significantly different from the ones that would
be evaluated on images denoised with model 2.
3.4
Discussion
The interleaved cine sequence provides an efficient tool to assess cardiac mass and function in mice.
This allows longitudinal studies to be performed on clinical scanners and thus opens the possibility
of new research protocols.
Moreover, the results showed that the proposed algorithm can correct efficiently the main artifacts provided by the combination of both cine series. However, an actual limitation is the choice of
the threshold value λ. Indeed, this value was empirically set to visually improve the image quality
without losing temporal definition of the myocardium contraction, but a refinement of the method
would be to determine objectively the optimal λ. To do this, the major difficulty is to define an
objective parameter to describe the image quality. Since there is no reference data, the use of a
cross validation method would be a solution. This work has to be performed in the future.
Finally, the proposed sequence design, even if limited to periodic motion assessment, can be
extended to other applications. Indeed, sequences in which TR is increased due to a long pulse
preparation, as it is the case for tagging, can benefit of several repetitions of the cine to improve the
time resolution. Tagging was shown to be very relevant for regional function assessment in small
animal imaging [98, 99] and can also be inserted in MEMRI protocols [100]. Cine interleaved would
therefore add a great benefit for tagging studies under stress conditions in rats or extending tagging
for mice studies.
Part II
Spiral imaging (...to bedside)
61
4
Spiral Sequence
Part of this chapter has been published as a review: Delattre et al., “Spiral demystified” [101].
4.1
4.1.1
What is spiral ?
Introduction
In many MRI applications it is crucial to reduce the acquisition time. One method to achieve
this can be the use of non-Cartesian k-space acquisition schemes, such as spiral trajectories [102].
Spiral sampling, including variable density spiral, has the advantage of the ability to cover k-space
in one single shot starting from the center of k-space. Moreover, spiral imaging is very flexible.
High temporal and spatial resolution, as required for specific applications like cardiac imaging and
functional MRI, can be obtained by tuning the number of interleaves and the variable density
parameter. Intrinsic properties of the spiral trajectory itself offer advantages that cannot be found
with other types of trajectories. The major ones are:
• an efficient use of the gradient performance of the system
• an effective k-space coverage since the corners are not acquired
• a large SNR provided by starting the acquisition at the center of k-space
In principle, spiral offers some inherent refocusing of motion and flow induced phase error, which
is not compensated by conventional sampling schemes [103]. Covering the center of k-space in each
interleave can be useful, as this information can be used for self-navigated sequences for example
[104]. Finally, very early acquisition of the k-space centre needed for ultra-short TE sequences (UTE)
can be fulfilled with spiral [105]. For these reasons, the main applications of spiral imaging lie in
dynamic MRI, such as cardiac imaging [18, 106, 107], coronary imaging [64, 108–110], functional
MRI [111–113] and also chemical shift imaging [114, 115].
63
64
Chapter 4. Spiral Sequence
Even though a broad range of applications for spiral imaging exists, we will focus in this chapter on examples from cardiac and head imaging to illustrate specific properties of spiral imaging.
Indeed, spiral sampling allows real-time cardiac MRI with a high in-plane resolution (1.5 mm2 )
[18]. 3D cine images acquired with variable density spirals show sharper images than comparable
Cartesian images with the same nominal spatial and temporal resolution (1.35 mm2 and 102 ms
respectively) [108]. Moreover, its use for 3D coronary angiography improved SNR and CNR by a
factor of 2.6 compared to current Cartesian approach [116] and reduced the acquisition time to a
single breath-hold with 1 mm isotropic spatial resolution [109]. Here, image quality was improved
by a fat suppression technique obtained with a spectral-spatial excitation pulse that further reduced off-resonance artifacts due to fat [117]. Finally, the usefulness of variable density spiral phase
contrast was shown with its application to real-time flow measurement at 3T [118]. The authors
showed the ability to monitor intracardiac, carotid and proximal flow in healthy volunteers with a
typical temporal resolution of 150 ms, a spatial resolution of 1.5 mm and no need for triggering or
breath-holding. These results are very promising for cardiac patients with dyspnea or arrhythmia.
Salerno et al. [119] showed another promising example of spiral application; myocardial perfusion
imaging. In conventional myocardial perfusion imaging the so named dark-rim artifact [120] is often
a problem. This artifact was minimized by the use of spiral acquisition schemes.
All of these examples show the potential of spiral imaging to improve clinical diagnostic imaging
by reducing the overall acquisition time without any penalty on the spatial and the temporal resolution and by reducing the negative effects of flow and motion on image quality. After describing
all the advantages of spiral imaging the remaining question is: Why is the spiral sequence not used
more often in clinical routine? A simple answer is that spiral imaging is more complex than Cartesian imaging. Practical difficulties make the implementation of spiral imaging quite challenging and
counterbalance the advantages of this method:
1. The design of gradient waveforms requires specific attention, because hardware is most often
optimized for linear waveforms. Indeed, calculation of the trajectory requires the non-trivial
resolution of differential equations and specific care must be taken to find a solution suitable
on the scanner.
2. The reconstruction of such images is no longer straightforward because points are not sampled
on a Cartesian grid. The simple use of Fast Fourier Transform (FFT) is therefore not possible,
and this also implies that parallel imaging algorithms such as SENSE or GRAPPA have to be
adapted to this non-Cartesian trajectory to benefit from the acceleration they can provide.
3. Spiral images are often prone to particular artifacts such as distortion and blurring that have
several physical origins, from gradient deviations to off-resonance effects due to B0 inhomogeneities and concomitant field, and that need to be measured in order to correct for them.
This last difficulty is probably the most limiting one in spiral imaging.
Due to the reasons listed above, spiral trajectories still lack popularity and seem to be reserved
for experts and a few specific applications. This chapter aims at reviwing the main challenges of
spiral imaging, showing the solutions that have been proposed to address these problems.
4.1.2
What is spiral ?
This preliminary section presents the theory necessary to understand the difficulties related to the
spiral trajectory. Image formation is only possible by encoding the spatial location of the spins
65
4.1. What is spiral ?
Figure 4.1 — Example of variable density spiral trajectory on a cartesian grid.
in the precessing frequency. This is done by application of varying fields, called gradients. The
location information is then contained in the phase of the rotating proton spin. Neglecting the
relaxation processes of the sample magnetization, the signal acquired, s, can be expressed by the
Fourier Transform of the proton density, ρ, (For simplicity, in the following the signal measured,
s, is implicitly considered proportional to the magnetization and the longitudinal magnetization at
equilibrium proportional to the spin density of the sample [7]):
s(~k) =
Z
~
ρ(~r)ei2πk·~r ,
(4.1)
where k is the k-space coordinate and is related to gradient fields:
~k(t) = γ−
Z
t
~ ′ )dt′ ,
G(t
(4.2)
0
where γ− = γ/(2π) and γ is the gyromagnetic ratio. The easiest way to reconstruct the image is
the use of the Fast Fourier Transform algorithm [121], which is computationally the most efficient
algorithm. Cartesian sampling is thus the most appropriate sampling scheme because points are
placed on a Cartesian grid. However, this trajectory has the disadvantage of being very slow, as
the coverage of k-space must be done line by line. On this first aspect, the spiral trajectory is more
interesting because it uses the gradient hardware very efficiently and can also cover k-space in a
single shot.
Another advantage is that the sampling density can be varied in the center of k-space, which can
be useful in several applications where more attention is given to low spatial frequencies. However,
reconstruction of the image is then not straightforward, as points are no longer placed onto the
Cartesian grid (see Figure 4.1).
Also, this kind of trajectory is more sensitive to field inhomogeneities because the readout is
usually longer than in Cartesian sampling. As a consequence, phase shifts from several origins can
accumulate during the relatively long readout time, resulting in image degradations.
66
4.1.3
Spiral trajectory
A major advantage of spiral is its ability to cover k-space in a single shot. This last property can
also be achieved with EPI readout but, as a Cartesian trajectory, it needs a fast change of gradient
intensity that produces important Eddy currents.
The general spiral trajectory can be written as:
k = λτ αD ejωτ ,
(4.3)
where k = k(t) is the complex location in k-space, λ = [0, 1] is a function of time t, αD is the variable
density parameter (αD = 1 corresponds to uniform density), ω = 2πn with n the number of turns
in the spiral, and λ = N/(2 ∗ F OV ) with N the matrix size.
The most common spiral scheme is Archimedeaan spiral, characterized by the fact that successive turnings of the spiral have a constant separation distance. This corresponds to the case
αD = 1 (uniform density spiral). Figure 4.2 shows an example of this trajectory. However, interest rapidly turned to variable density spirals (where successive turnings of the spiral are no longer
equidistant, αD 6= 1) instead of purely Archimedean because this enhances the flexibility of the
trajectory by sampling the center of k-space differently than the edges resulting in a reduction of
aliasing artifacts when undersampling the trajectory [122], as well as reducing motion artifacts [103].
While k(t) defines the spiral trajectory in k-space, the exact position of sampled data points
along that trajectory is defined by the choice of the function τ (t). For the special case that τ (t) = t,
the amount of time spent for each winding is constant, regardless of whether the acquired winding
is near the center or in the outer part of the spiral. In other words, the readout gradients reach their
maximum performance at the end of the acquisition. This acquisition scheme, initially proposed
by Ahn et al [102], is called the constant-angular-velocity spiral trajectory. The constant-angularvelocity spiral trajectory can easily be transformed into a so-called constant-linear-velocity spiral by
√
using τ (t) = t, in Eq. (4.3). It has been shown that the constant-linear-velocity spiral offers some
advantages in terms of SNR and gradient performance as compared to the constant-angular-velocity
spiral [123]. Although constant-linear-velocity spiral trajectories are more practical, the constantangular-velocity spiral trajectories have some interesting properties. In a constant-angular-velocity
Archimedean spiral, the number of sampling points per winding is constant. If this number is even,
all acquired data points are aligned along straight lines through the origin and are collinear with
the center point, as schematically shown in Figure 4.2.
With the desired k-space trajectory, the gradient waveforms G(t) and the slew-rate S(t) can be
defined by:
G(t) =
jωτ (t)
jωτ (t−∆t)
k̇(t)
τ̇ dk
λ τ (t)α
− τ (t − ∆t)α
De
De
=
=
,
γ
γ dτ
γ
∆t
S(t) = Ġ(t) =
τ̇ 2 d2 k
τ̈ dk
+
,
2
γ dτ
γ dτ
(4.4)
(4.5)
where G(t) = Gx (t) + iGy (t) is the gradient amplitude and S(t) = Sx (t) + iSy (t) is the gradient
slew-rate in both directions, ∆t is the time interval of the gradient waveform.
This formulation implies a sinusoidal waveform for Gx (t) and Gy (t). The major difficulty here is
to find an analytic equation for the gradient waveform G(t) by defining τ (t) in order to enable the
real-time calculation of the gradient waveform at the MR scanner. Even though sinusoidal gradient
67
Figure 4.2 — Constant-linear-velocity (left) versus constant-angular-velocity (right) spiral trajectory: data
points acquired or interpolated onto constant-angular-velocity spiral trajectory (right), indicated by gray
points, lie on straight lines through the origin and are collinear with the the center point (black point).
waveforms are smoother than the trapezoidal gradients used for Cartesian sampling, imperfections
in the realization of the trajectory are unavoidable. Inaccurate gradient fields generate an additional phase term, which accumulates during data acquisition and result in variations of the actual
trajectory from the calculated trajectory. This leads to image blurring, because the reconstruction
is performed with improper k-space position of the data points and thus introduces artifacts to the
whole image as it will be reviewed in details in the following sections.
4.1.4
Specific advantages of spiral trajectory
As mentioned in the introduction, spiral trajectory offers some inherent advantages over other types
of trajectory.
Insensitivity to flow and motion Due to its particular gradient waveforms, spiral trajectory is
relatively insensible to flow and motion artifacts. Indeed, considering the accumulated phase
from an isochromat located at position r at time t placed into a static field B0 and a gradient
field G(t) one obtains:
φ(r, t) = γB0 − γ
Z
t
G(t′ )r(t′ ).
(4.6)
0
The position r(t) can also be written with a Taylor series expansion:
φ(r, t) = γB0 − γ
Z
0
t
1 d2 r ′
dr
t
2
+
·
·
·
dt′ .
G(t′ ) r0 + ′ t′ +
dt
2 dt′2
(4.7)
Equation (4.7) can then be decomposed into the gradient moment expansion [8]:
φ(t) = γB0 + γM0 (t)r0 + γM1 (t)
where
dr
1
d2 r
+ γ M2 (t) 2 + · · · ,
dt
2
dt
(4.8)
68
Z
t
M0 =
G(t′ )dt′
0
Z t
M1 =
tG(t′ )dt′
0
Z t
M2 =
t2 G(t′ )dt′
, 0th order gradient moment,
, 1st order gradient moment,
(4.9)
, 2nd order gradient moment.
0
As a consequence, accumulated phase can be independent of position, speed and acceleration
if the gradient moments are null. In the case of spiral imaging, gradient moments are weak at
the center of k-space and increase slowly with time. Moreover, due to their sinusoidal forms,
gradients take periodically positive and negative comparable values that compensate phase
accumulation. Those reasons lead to a weak phase accumulation and give the spiral trajectory
a certain insensitivity to movement and flow artifacts. Furthermore, the symmetry of x and
y gradients do not lead to a phase accumulation in a preferred direction which would be the
case in EPI for example.
Robustness to aliasing Another advantage of spiral trajectory is its robustness to aliasing artifacts due to the possibility of oversampling the center of the k-space. Indeed, the image
spectrum is non-uniform in the k-space, with low spatial frequencies containing most of the
image energy. Undersampling uniformly the k-space will end to aliasing artifacts while sampling sufficiently the center of k-space by increasing the sampling density in this region will
drastically reduce them. Tsai et al. [122] demonstrated on a short axis cardiac image that
the severe aliasing artifacts produced by the chest wall with uniform density spiral scan were
suppressed with variable density spirals (scan parameters were: 17 interleaves, FOV=16cm,
in-plane resolution of 0.65mm, TE=15ms, readout time=16ms) [122]. Liao et al. [103] have
also shown that oversampling the center of k-space provides additional reduction of motion
artifacts. This is due to the fact that observed motion is a periodic phenomenon, of which the
frequency band is mainly contained in the low spatial frequencies (scan parameters to obtain a
cine with 16 frames were TR=50ms, FOV=30cm, matrix size of 185x185, acquisition time=40
s).
K-space center informations Finally, sampling the center of the k-space for each interleave gives
information about the position of the object and can be used as a navigator for abdominal
and cardiac applications [124]. Liu et al. [104] obtained highly improved image reconstruction
in the context of DWI by using a low resolution image given by the first variable density
interleave of the spiral trajectory to correct the phase of the high resolution image (obtained
with all the interleaves) (scan parameters were TE/TR=2.5/67ms, FOV=22cm, matrix size
of 256x256, 28 interleaves for one slice, acquisition time 8.1min for whole brain).
4.1.5
Eddy currents
Time varying gradient fields induce currents in the conducting elements composing the magnet and
the coils. These so-called Eddy currents create a magnetic field that opposes the change caused
by the original one (Lenz’s law), deteriorating the gradient waveform. In modern scanners, Eddy
currents are mainly corrected with actively shielded gradients but residual currents can still be
69
present. In a simple Eddy current model [8], the field generated by the Eddy current Ge (t) is given
by:
Ge (t) = −
dG
× e(t),
dt
(4.10)
where G is the applied gradient, × denotes the convolution and e(t) is the impulse response of the
system:
e(t) = H(t)
X
αn e−t/τn ,
(4.11)
n
where H(t) is the unit step function. Just a few terms in this summation are necessary to characterize
most of the Eddy current behavior. As it adds an unwanted magnetic field, the Eddy current effect
results in phase accumulation leading to image distortions. It is mainly responsible for the wellknown ghosting artifact in EPI imaging, while in the case of spiral trajectory it causes image blurring.
4.1.6
Sensibility to inhomogeneities
Spiral images are prone to blurring and distortions originating from several sources. Ignoring relaxation effects, Eq. (4.1) showed that the signal acquired from an object in a magnetic field is given
by:
ZZ
s(t) =
ρ(x, y)ei2π(kx x+ky y+φ(x,y,t)) dxdy,
(4.12)
where kx and ky are the k-space coordinates, ρ(x, y) the proton density of the object at (x, y) coordinates and φ(x, y, t) the arbitrary field inhomogeneities that is mainly composed of main field
inhomogeneity, gradient imperfections, residual Eddy currents, chemical shift between water and
other species or susceptibility differences between air and tissue.
This equation is general and applies to every sampling scheme. This means that even Cartesian
sampling is prone to field inhomogeneities. However, in Cartesian sampling, only one gradient is
varied at a time, which implies that dephasing affects only one direction. This results in a simple
shift of the object. It is more problematic with spiral because both in-plane gradients are varied
continuously at the same time, resulting in a shift of the object in all directions that causes image
blurring. The exact reconstruction of such a signal is given by:
ρ(x, y) =
Z
T
s(t)e−i2π(kx x+ky y) e−i2πφ(x,y,t) dt.
(4.13)
0
This is called the conjugate phase reconstruction, because before integration, the signal is multiplied by the conjugate of phase accrued due to field inhomogeneities. The inhomogeneity term
can often be written as a linear relation with time t, φ(x, y, t) = tφ(x, y) implying that it is more
significant when t is important. There are, though, two approaches to eliminate this effect:
• use interleaved spirals that have a short readout time and then limit phase accumulation
• correct for the inhomogeneities when reconstructing the image
Distortion and blurring induced by phase accumulation due to inhomogeneities are probably
the main reasons that explain the lack of success of spiral trajectories in clinical routine, however,
with the improvements of methods to measure and correct for these inhomogeneities, this situation
should not be definitive.
70
4.1.7
Concomitant fields
Another parameter that can alter the image quality is concomitant gradient fields. Indeed, Maxwell’s
equations imply that imaging gradients are accompanied by higher order, spatially varying fields
called concomitant fields. They can cause unwanted phase accumulation during readout resulting,
again, in image blurring in the special case of spiral, but once again, even though Cartesian sampling
is also affected by this additional dephasing, the effect on the image is simply less disturbing.
Considering identical x and y gradient coils with a relative orientation of 90◦ , the lowest order
concomitant field can be expressed as [125] :
Bc =
G2z
8B0
2
x +y
2
+
!
G2x + G2y
Gx Gz
Gy Gz
z2 −
xz −
yz,
2B0
2B0
2B0
(4.14)
where x,y,z are the laboratory directions, B0 the static field, Gx , Gy , Gz the gradients in the
laboratory system. Concomitant gradients cause phase accumulation during the readout gradient
that is expressed as:
fc (t) = γ−
Z
t
Bc dt.
(4.15)
∞
The knowledge of the analytical dependence of this effect with spatial coordinates is necessary
to correct for its contribution to image blurring.
4.2
Designing the trajectory
The first difficulty with spiral imaging is the design of the trajectory itself. Indeed, gradient solutions
must be found by solving the differential Eqs. (4.4) and (4.5) that are computationally intensive to
calculate even with the improvement of hardware capabilities. Closed-form equations are necessary
to be easily usable on a clinical scanner. To take maximum advantage of the gradient hardware
capabilities two regimes are defined. Indeed, near the center of k-space, the trajectory is only limited
by the gradient slew-rate since gradient amplitude is low. For this slew-rate limited regime, S(t) is
set to the maximum available slew-rate Sm . Then, when reaching the maximal gradient amplitude
one comes into the so called amplitude-limited regime where G(t) is equal to the maximum available
gradient amplitude Gm .
4.2.1
General solution
The spiral trajectory has to go as quickly as possible from center of k-space to the edges. For that,
the first part of the trajectory is only limited by the time needed to switch the gradients on, also
referred to as the gradient slew-rate. Then, the second limitation is the maximal amplitude of the
gradients. A simple analytical solution for constant density (αD = 1) was first given by Dyun et
al. [126] for the slew-rate limited case only and then extended by Glover [127] for the two regimes.
It was then generalized to variable density by Kim et al. [128]. They defined the function τ (t) as
follows:
 r
1/(αD /2+1)


Sm γ αD

+1 t

λω 2 2
τ (t) =
1/(αD +1)


γGm

(α
+
1)t

D
λω
slew-rate-limited regime
(4.16)
amplitude-limited regime
4.2. Designing the trajectory
71
Figure 4.3 — Examples of spiral trajectories calculated with Kim design [128] for 10 interleaved spirals,
FOV=100 mm, matrix size N=128, k-space values (left) and gradient waveforms (right) for constant density
spiral αD =1 (A), and variable densiy spiral αD =3 (B). Dotted lines represent the transition between slewrate limited regiem and amplitude-limited regime.
72
The trajectory starts in the slew-rate limited regime and switches to the amplitude limited regime
when t corresponds to G(t) = Gm , the maximum available gradient amplitude. This closed-form
solution has the advantage of being easily implemented on a clinical scanner. An example of the
trajectory obtained with this formulation, as well as the gradient waveform, is illustrated in Figure
4.3. However, when the number of interleaves is increased, this trajectory leads to large slew-rate
overflow for small k-space values (i.e. for t → 0) as it is illustrated by Figure 4. Depending on the
gradient performances, this can be a real problem with most clinical scanners because slew-rates are
limited and the execution of a trajectory with such overshoot is simply impossible.
For example, the spiral sequence we used in this work ∗ was implemented on a 3T Siemens clinical
scanner (Magnetom TIM Trio, Siemens Healthcare, Erlangen Germany) and used a maximum slewrate of Sm = 170 T/(m·s) and a maximum gradient amplitude of Gm = 26 mT/m. Simulations
of Figure 4.3 and Figure 4.4 were done for this system by taking a small security margin: Sm =
90/100·170 = 153 T/(m·s) and Gm = 90/100·26 = 23.4 mT/m.
Figure 4.4 — Upper; examples of τ (t) calculated with Kim [128] (bold plain line), Zhao [129] (plain line),
Glover [127] (dashed line) methods and zoom of the first 0.2 ms of τ (t) and sx (t). Lower; corresponding
slew rate sx (t), on the left y-axis scale was cut between ±200 T/m/s to have a better visualization; on
the right, zoom of the first 0.2 ms and full y-axis scale. Calculation proposed by Kim largely overshoot
maximum available slew rate in this case whereas it is not the case with the Zhao and Glover propositions.
(Parameters used for this simulation: FOV=10 cm, N=128, αD =1, 10 interleaves, Λ=6·10-3, L = 3.6·10−4 ).
∗ The source code of the sequence was provided by Gunnar Krüger, from Siemens Medical Solutions, Centre
d’Imagerie BioMedicale (CIBM), Lausanne
73
4.2. Designing the trajectory
4.2.2
Glover’s proposition to manage k-space center
As pointed out by Glover [127], an instability exists for small k-space values (i.e. k ≈ 0) because
the solution derived in the slew-rate limited regime is unbounded at the origin. He proposed an
alternative in the case of a uniform density spiral by setting the slew rate at the origin to be Sm/Λ,
instead of Sm, where Λ is tuned by the user. τ (t) is therefore defined by:
τ (t) =
1 2
2 βt
h
i1/3
Λ + 12 49 β 2 t4
with β =
Sm γ
.
λω 2
(4.17)
This solution ensures a smooth transition near the origin that avoids slew-rate overflow as illustrated in Figure 4.4. For this example Λ was chosen to be 6·10−3, as this corresponds to the minimum
value to avoid slew-rate overflow with the example of the Siemens 3T system characteristics.
4.2.3
Zhao’s adaptation to variable density spiral
Zhao et al. [129] proposed another solution to this problem adapted to the variable density case by
setting the slew rate to exponentially increase to its maximum value:
S(t) = Sm (1 − e−t/L )2 ,
(4.18)
where L is a parameter used to regularize the slew-rate at the origin. Then, they obtain:
τ (t) =
"r
Sm γ
λω 2
αD
+1
2
#1/(αD /2+1)
−t/L
t + Le
−L
.
(4.19)
The parameter L is chosen by setting S(t) = Sm /2 for the P-th data point:
L=−
P ∆t
√ ,
ln(1 − 1/ 2)
(4.20)
where ∆t is the time interval between 2 points of the trajectory. Figure 4.4 shows τ (t) and Sx (t)
obtained with these propositions by choosing P = 9. This value corresponds to the minimum value
to avoid slew-rate overshoot for the performance of the chosen scanner.
4.2.4
Comparison of the 3 propositions
As illustrated in Figure 4.4, Kim’s proposition [128] causes the slew-rate to overflow in the first
milliseconds of the trajectory which implies that the center of k-space is not correctly sampled. This
is a real problem because important information is contained in the center of k-space. Both Glover
[127] and Zhao [129] efficiently correct for this problem at a price of lengthening the trajectory
by 3.5% and 2.3% respectively in the particular example of Figure 4.4. In addition, by adapting
to variable density, Zhao [129] has the advantage of choosing an exponential variation of the time
parameter τ (t) instead of a time power, this results in a smaller spiral readout time than with
Glover’s proposition [127], which may be useful for some applications where acquisition time is
limited.
74
4.3
Coping with blurring in spiral images
Spiral images, unlike Cartesian images, are often subject to blurring and distortion. The origins of
such effects are well described and a lot of effort has been made to correct them. Here the main
solutions that have been proposed to correct contributions such as imperfect trajectory realization,
off-resonance artifacts and concomitant fields are described.
4.3.1
Measuring gradient deviations
Deviation from the targeted k-space trajectory due to hardware inadequacies or imperfect eddy
current correction can lead to image artifacts that are more disturbing in the case of spiral imaging
because phase accumulation in both gradient directions induces image blurring. A way to correct
for these deviations is first to measure them.
Trajectory measurement with a reference phantom
Mason et al. [130] proposed estimation of the actual k-space trajectory from the MR signal. The
method uses several calibration measurements on a small sphere reference phantom of tap water
placed at off-isocenter locations in the magnet bore (x0 , y0 ). The use of this “point phantom” allows
the measured signal to be considered as a simple combination of proton density in the phantom and
the dephasing term. Eq. (4.1) can thus be written as:
s(t) = ρ(x0 , y0 )ei2πφ(r) ,
(4.21)
where φ(t) = φ0 (t) + kx (t)x0 + ky (t)y0 . For each location (x0 , y0 ), the observed phase change ∆φn
between samples n and n − 1 is assumed to be due to a combination of a spatially invariant timedependant magnetic field, that induces a phase change ?0n, and the time-varying spatial gradients
in the Gx and Gy directions that induced incremental phase changes 2π∆kxn x0 and 2π∆kyn y0 . So
the total phase change is:
∆φn = φ0n + 2π(∆kxn x0 + ∆kyn y0 ).
(4.22)
From the several acquisition made at different location (x0 , y0 ) the data are fitted to determine
the function kx (t), ky (t) and φ(t) by a least squares algorithm. Some faster methods were also
proposed that do not involve the displacement of a reference phantom but use the signal from
the studied subject. Spin location is done by self-encoding gradients added just before readout
acquisition. Such gradients are calibrated gradients applied stepwise in the same direction as the
field gradient to be measured [131]. They then combine different acquisitions to obtain the phase
change from which the k-space trajectory can be deduced the same way as Mason et al. [130].
The method proposed by Alley et al. [132] uses the readout data acquired with normal gradient
waveforms and then with reversed waveforms on a 10 cm phantom filled with water. The signal
obtained after a Fourier transform in the phase encoding direction in one gradient direction can be
written as:
s± (x, t) = ρ(x, t)ei2πφ±(x,t) ,
(4.23)
where s+ (t) refers to the “normal” acquisition and s− (t) to the one acquired with the reversed
gradient waveform. The phase term can be separated in odd and even terms:
φ± (x, t) = θ(x, t) ± ψ(x, t) ± k(t)x.
(4.24)
4.3. Coping with blurring in spiral images
75
Subtraction of the two phases φ+ and φ− give access to the trajectory data by a least square
fit in the spatial direction. However, this method requires a high number of readout acquisitions to
characterize the whole k-space trajectory.
In-vivo trajectory measurement
The method proposed by Zhang et al. [133] is even faster as it uses only two slices positioned along
each gradient of interest. This method is based on the proposition of Duyn et al. [134] and has the
advantage of being used in vivo, thus avoiding fastidious preliminary calibration of the trajectory
with a phantom. The trajectory is obtained directly from the phase difference between the two
acquired signals. In fact, if one assumes an infinitely thin slice, the signal for a slice normal to the
x-axis located at x0 is:
ZZ
s(x0 , t) = ei2πφ(x0 ,t)
ρ(x0 , y, z)ei2πky (t)y dydz,
(4.25)
where the phase term is:
φ(x0 , t) = kx (t)x0 + ψ(t).
(4.26)
Then, the trajectory is obtained by subtracting the phase of signal from two close slices located
at x1 and x2 :
kx (t) =
φ(x2 , t) − φ(x1 , t)
.
x2 − x1
(4.27)
This method was compared with the small-phantom method from Mason et al. [130] and authors
have found a very good accordance between results. This method is thus the first that can be used
directly in a human subject. However, when high resolution is needed, some significant errors are
observed for high k-space values.
A correction recently proposed by Beaumont et al. [135] greatly improves this latter technique.
Indeed, from Eq. (4.13) they show that for a well-shimmed squared slice profile, the nonlinear spatial
variation of B0 field can be neglected so the term φ(x, y, t) becomes φ(t). Then, the signal is the
Fourier transform of the ?effective magnetization density? ρ(x, y)e−i2πφ(t) . If the slice is located in
the x plane at x = 0 and considering an homogenous sample, the effective magnetization density is
proportional to the slice profile, so the signal can be written as:
s(kx (t)) ∝
sin(πkx (t)∆s)
,
πkx (t)∆s
(4.28)
where ∆s is the slice thickness. This function has several zeros located at kx = n∆s−1 (n an integer
> 1). If kmax ∝ ∆s − 1, zero or very low signal can be encountered preventing the calculation of the
trajectory at these points. They solve this problem by adding another gradient that shifts k-space
points in order to avoid the nulling signal and allow recovery of the high k-space values.
Trajectory estimation by one-time system calibration
Recently, Tan et al. [136] described an efficient method to correct for the trajectory deviations. They
described the alteration of k-space trajectory by both anisotropic gradient delays in each physical
axis as well as Eddy currents. For trajectories like spiral, residual Eddy currents can cause severe
distortion in images. They proposed a model in which each contribution is separated and corrected
76
after applying system calibration. The model only depends on gradient system parameters in the
physical coordinates that can be found by measuring the real trajectory and comparing it with the
theoretical one.
Indeed, as it was explained by Aldefeld et al. [137], some timing delays exist in hardware between
the command and effective application of the gradient by gradient amplifiers. These delays can be
different in each physical axis so have to be characterized separately. The delayed gradients in the
logical coordinates are thus simply a rotation of those defined in the laboratory coordinates:
GdL


Gx (t − τx )


= RT Gd = RT  Gy (t − τy )  ,
Gz (t − τz )
(4.29)
where GdL is the delayed gradient in logical coordinates, RT the rotation matrix and τ the delays
in the 3 directions. Tan et al. [136] showed that Eddy currents induce a k-space trajectory which
can be simply modeled as the integration of the convolution of slew-rate of the desired gradient
waveform and the system impulse response. From Eq. (4.10) and considering the Taylor expansion
of the exponential contained in Eq. (4.11) they obtain:
ke (t) =
Z
0
t
S(t′ ) × H(t′ )dt′ ≈ A
Z
t
G(t′ )dt′ + B
0
Z
0
t
G(t′ )t′ dt′ − B
Z tZ
0
t
0
dG
τ dτ dt′ ,
dτ
(4.30)
P
P
where A = − n an and B =
n (an /bn ). This last form represents the scaling term for the
theoretical k-space trajectory in the first term and the two other terms are the system response
for gradient switching (for more details see [136]). The main advantage of this method is that the
measurement of system parameters τx,y,z , A and B needs to be done only once and can then be
used for any slice position. The authors compared the improvement of image quality with their
method to the simple anisotropic gradient compensation and noticed that eddy current model must
be added in order to correct efficiently for the residual k-space imperfections.
4.3.2
Correcting off-resonance effects
Off-resonance effects refer to signal contributions with resonance frequencies different to the central
water proton resonance frequency. These contributions mainly come from the chemical shift between water and other species, from susceptibility differences between different tissues or between
air and tissue and from main field inhomogeneities. Field inhomogeneities are especially important
for sequences where off-center slices are difficult to shim or in areas where susceptibility differences
and motion are important, for example in the thorax.
As the spiral sampling scheme usually has a longer readout time, it is more affected by off
resonance effects than the classical Cartesian scheme. These effects accumulate all along the readout
time and result in image blurring that can be important. One can easily see from Eq. (4.13) that
this exact conjugate phase reconstruction needs a lot of computation time because each pixel must
be reconstructed with its own off resonance frequency φ(x, y). Fortunately, faster alternatives to
this exact reconstruction have been developed and are described below.
77
Field map estimation
The correction for these inhomogeneities first needs knowledge of the field map to assess the term
φ(x, y) shown in Eq. (4.13). A rapid method proposed by Schneider [138] consists of the acquisition
of two datasets with different echo times. The image obtained for each acquisition is given by:
s1 (x, y) = ρ1 (x, y)ei2π(T E1 φ(x,y))
s2 (x, y) = ρ2 (x, y)ei2π(T E2 φ(x,y))
(4.31)
and
s∗1 s2 = ρ1 ρ2 ei2π(φ(x,y)(T E2 −T E1 )) ,
(4.32)
so the field inhomogeneity term is simply given by the phase of the two images:
φ(x, y) =
angle(s∗1 s2 )
angle(s∗1 s2 )
=
.
2π(T E2 − T E1 )
2π(∆t)
(4.33)
The phase has to be unwrapped or limited by choosing ∆t short enough.
Time-segmented reconstruction
In the method developed by Noll et al. [139], the time integral in Eq. (4.13) is broken into a finite
number of temporal boxes. In each of these temporal segments, the term eti φ(x,y) is assumed to
be constant and reconstruction is done for each segment. The final image is obtained by adding
together the integrals over all time segments:
ρ̄(x, y) =
T
X
s(ti )e−i2π(kx x+ky y+ti φ(x,y)) .
(4.34)
ti =0
Figure 4.5 — Synthetic scheme of the time-segmented reconstruction algorithm (adapted from [8]).
As shown in Figure 4.5, the computation time depends on the number of temporal segments
used for reconstruction and can thus be quite important.
78
Frequency-segmented reconstruction and multifrequency interpolation
A similar method was proposed by Noll et al. [139] and consists of segmenting the inhomogeneity
term, φ(x, y), into multiple constant frequencies, φn (x, y). For each of these frequencies, an image
was reconstructed and the final image was taken as a spatial combination of these different reconstructions based on spatial varying frequency. Figure 4.6 illustrates this algorithm. A refinement to
this method was developed by Man et al. [140] and consists of writing the inhomogeneity term as
a linear combination of constant frequency terms. This is the multifrequency interpolation being:
e−i2πtφ(x,y) =
X
cn [f (x, y)]e−i2πtφn (x,y) .
(4.35)
n
For each frequency φn (x, y), an inverse reconstruction is performed and the image obtained is :
ρ̄(x, y) =
Z
T
−i2πtφn (x,y)
s(t)e−i2π(kx x+ky y)e
dt
.
(4.36)
o
The resultant image is taken as a linear combination of those images:
ρ̄(x, y) =
X
cn [φ(x, y)]ρ¯n (x, y).
(4.37)
n
The coefficients cn are typically obtained from Eq. (4.35) with a least square algorithm. This
method is faster than the classical frequency segmented method because it allows reconstruction
of fewer frequencies, as unknown frequencies can be obtained as a linear combination of the two
nearest ones.
Figure 4.6 — Synthetic scheme of the frequency-segmented reconstruction algorithm (adapted from [8]).
Linear field map interpolation
The last two methods give good correction of field inhomogeneities, but suffer from time-consuming
algorithms, as several reconstructions must be done to obtain the final image. One fast and efficient
79
technique proposed by Irarrazabal et al. [141] is to consider only the first order variation of the
inhomogeneities, so the field map is fitted with linear terms using a least square algorithm :
φ̄(x, y, ) = φ0 + αx + βy.
The model of signal received then becomes:
ZZ
s̄(t) = ei2πtφ0
ρ(x, y)ei2π((kx +αt)x+(ky +βt)y) dxdy.
(4.38)
(4.39)
Knowing φ0 , α, β from the field map fit, the reconstruction of the data can be done by replacing
the trajectory points by the corrected ones: kx′ = kx + αt, ky′ = ky + βt and demodulating the signal
to frequency φ0 . Then, the signal is reconstructed in one single operation.
Off-resonance correction without field map acquisition
Another method proposed by Noll et al. [142] consists of correcting the blurring without knowledge
of a field map to help in the conjugate phase reconstruction process. They start from the idea that
it should be possible to minimize the blur of an image by reconstruction at various off resonance
frequencies and then choosing the least blurry pixels to form a composite image. To automate the
selection process one needs a quantification of the blurriness. For this, they propose that an image
reconstructed on resonance should be real, because all excited spins are in phase. In the presence of
field inhomogeneities this assumption is no longer true and the imaginary part of the reconstructed
image becomes important. The quantification of this imaginary part can thus be used as a criterion
for defining the extent of off-resonance:
ZZ
C[x, y, fi (x, y)] =
|Im{ρ̄[x, y, fi (x, y)]}|α dxdy,
(4.40)
where α is chosen empirically between 0.5 and 1 by the user. The minimization of this criterion
reduces the blurring during the reconstruction. This method works well when the range of offresonance frequencies is small, otherwise spurious minima can appear in the objective function,
increasing the risk of a wrong choice of frequency, which can cause artifacts in the final image. A
refinement of this method was proposed by Man et al. [140] and consists of, first, estimating a
coarse field map using relatively few demodulating frequencies to avoid spurious minima given by
the objective function. Then the minimization is repeated with a better estimation of field map (i.e.
using a higher number of frequencies) but constrained by the previous coarse estimation.
Recent improvements and propositions
Recently, Chen et al. [143] proposed a semi automatic method for off-resonance correction that
represents a significant improvement over previously described methods and makes reconstruction
more robust. They propose to first acquire a low-resolution field map and then perform a frequency
constrained off resonance reconstruction from the acquired map. The first strategy used to perform
the reconstruction is to incorporate a linear off-resonance correction term in the image as previously
done in [141] and to add a parameter used to search for non linear components of the off-resonance
frequency, here φn :
Z
ρ(x, y) = s(t)e−i2πφ0 e−i2π((kx x+αt)x+(ky y+βt)y) e−i2πtφn dt.
(4.41)
This modification can significantly improve the computational efficiency of the algorithm as it
searches for a range of non-linear terms, and not for the actual off resonance frequency directly.
80
The second strategy is to take into account regions where the field map varies non-linearly by
interpolating the field map with polynomial terms instead of only linear terms. Thus the model
becomes:
Z
ρ(x, y) = s(t)e−i2πt(φn +φp (x,y)) e−i2π(kx x+ky y) dt,
(4.42)
where φp (x, y) is the polynomial fit of field map and φn are constant offset frequencies. Reconstruction can then be done by multiple frequency interpolation [140]. Also recently, Barmet et al. [144]
proposed a conceptually different approach to assess field inhomogeneities. They proposed to simply
measure the magnetic field around the investigated object with an array of miniature field probes
that do not interfere with the main experiment. This has the main advantage of knowing the phase
accrued during the signal acquisition, i.e. in exactly the same conditions as the experiment, so it
does not require additional scan time for the acquisition of a field map. The efficiency of this method
was evaluated by Lechner at al. [145] in comparison with the so called “Duyn calibration Technique”
(DCT) which is the method of trajectory measurement proposed by Duyn et al. [134], Zhang et
al. [133] and improved by Beaumont et al. [135]. They found that both DCT and Magnetic Field
Monitoring (MFM) effectively detect k-space offsets and trajectory error propagation, and correct
for general error sources such as timing delays. Also, artifacts such as deformation and blurring
were dramatically reduced.
4.3.3
Managing with concomitant fields
As seen before, off-resonance effects can be assessed by a field map acquired with the method proposed by Schneider [138]. However, concomitant gradient effects are independent of acquisition time
(i.e. echo time TE) and can therefore not be assessed this way. Knowledge of the analytical dependence of this effect with spatial coordinates is necessary to correct for its contribution to image
blurring.
King et al. [146] have shown that the effect of concomitant gradients can be separated into 2
parts: the through plane effect and the in-plane effect which can be corrected with different methods.
The through-plane effect can be understood by considering a 2D axial scan (Gz =0). From Eq. (4.14):
BC =
G20
z2
2B0
with
G0 (t) =
q
G2x (t) + G2y (t).
(4.43)
This means that the concomitant field is 0 at isocenter and increases quadratically with off-center
distance z. It is however independent of position within any given axial plane. Therefore, with some
approximations, the phase shift due to this contribution can be seen as a time-dependant frequency
shift varying with plane position zc given by:
φc (zc , t) = γ−
zc2 2
G (t).
2B0 0
(4.44)
Then the signal received becomes:
s(t) = ei2πφc (z,t)
ZZ
ρ(x, y)ei2π(kx x+ky y) dxdy.
(4.45)
The correction for this effect can be done by demodulating the signal data over the time with
frequency φc (zc , t) before the reconstruction of the image. The in-plane effect of concomitant fields
can be understood by considering, this time, a 2D sagittal plane (i.e. Gx = 0). Again, from Eq.
81
(4.14) we have:
"
2 #
1 G2z x2
Gz y
Bc =
+
− Gy z
,
2B0
4
2
(4.46)
where the x2 term is a through-plane contribution, similar to the axial case, but its coefficient is 4
times smaller. The remaining terms depend on location within the slice and increase with off-centre
distance. King et al. [146] showed that the term Gy Gz gives a small contribution compared to the
others, so if the through-plane x2 term is removed by demodulating the signal like in the previous
section, and considering some approximations that can be done in the case of spiral scans, Eq. (4.46)
becomes a time independent frequency shift:
G2
φc (y, z) = γ− m
4B0
y2
2
+z ,
4
(4.47)
where Gm is the maximal amplitude of gradients.
Frequency-segmented deblurring can be applied to correct this offset by partitioning the range
of constant frequency offsets φc (y, z) into bins. The scan data are demodulated with the center
frequency of each bin and the resulting images are combined by pixel-dependant interpolation to
form the final deblurred image.
In the general case of an arbitrary plane, King et al. [146] proposed a formulation where phase
accumulation due to concomitant fields is described as a time-independent frequency offset :
φc (X, Y, Z) = γ−
G2m
(F1 X 2 + F2 Y 2 + F4 Y Z + F5 XZ + F6 XY ),
4B0
(4.48)
where X,Y ,Z are the read/phase/slice coordinates or “logical” coordinates and Fi are constants
depending only on the plane rotation matrix (for more details see Appendix of [146]).
Recent improvements
Recently, and just after the proposition of the semi-automatic off-resonance correction method [143]
(a fast alternative to conjugate phase reconstruction) Chen et al. [143] proposed the first fast phase
conjugate reconstruction correcting both off-resonance effects given by B0 field inhomogeneity and
concomitant gradient fields. The corrupted acquired signal is written as:
s(t) =
ZZ
ρ(x, y)ei2π(kx x+ky y) ei2πφ(x,y,t) dxdy,
(4.49)
where the phase accrued is composed of an off-resonance effect as described before φ(x, y) and a
frequency shift due to concomitant field φc (x, y) described by Eq. (4.48):
φ(x, y) = tφ(x, y) + tc φc (x, y).
(4.50)
In this method, the frequency off-resonance term is approximated by a Chebyshev polynomial
function of time t. This allows reconstruction of a set of images corrected for concomitant fields
and then application of the semi-automatic method for off-resonance correction. The proposed
algorithms are shown to be computationally efficient and the whole method seems well suited for
applications where the acquired field map is unreliable.
82
4.3.4
Summary
Table 4.1 compares the different methods proposed for correction of off resonance effects, such as B0
inhomogeneity and concomitant gradients. Methods presented to measure real k-space trajectories
can also be used in addition to these correction methods.
Table 4.1 — Comparison of proposed methods efficiency for deblurring images
Ref.
Corrects for
B0 inhomogeneity
B0 inhomogeneity
B0 inhomogeneity
Field map required ?
Yes, accurate
Yes, accurate
Yes, accurate
1
2
3
[139]
[147]
[141]
4
Speed
–
–
+
[140]
B0 inhomogeneity
No
—
5
[143]
Yes, low resolution
-
6
[148]
B0 inhomogeneity
and partially concomitant gradients
B0 inhomogeneity
and
concomitant
gradients
Yes, low resolution
–
Accuracy of correction
= to 2
= to 1
> 1, 2 and 4 but
worst in areas with
non linear inhomogeneities
= to 5 but still relatively prone to estimation errors
> 1, 2 and 3
> 1, 2, 3, 4 and 5,
great improvement
for scan planes far
from isocenter
1, Time-segmented reconstruction; 2, multifrequency interpolation; 3, linear field-map interpolation;
4, without field map estimation; 5, semi-automatic method; 6, reconstruction based on Chebyshev
approximation to correct for B0 field inhomogeneity and concomitant gradients.
In summary, when acquisition time is not constrained, the method using linear field-map interpolation [141] is the best choice, as it easily corrects for B0 field inhomogeneity with very low
computation time. On the other hand, for situations where acquisition time must stay short, the
field map acquired is often inaccurate and the semi-automatic method [143] should be used, as
it is more efficient in this situation. However, in cases when scan planes are placed far from the
isocenter of the magnet bore, the combined method with Chebyshev approximation [143] must be
chosen to correct also for concomitant gradient fields. In extreme situations where no field map can
be acquired because of time constraints, or because the quality obtained is very poor, a correction
method without field map acquisition can be used, but will suffer from an important computational
cost.
4.4
Spiral image reconstruction
Points of spiral trajectories are no longer on the Cartesian grid; therefore, the direct use of Fast
Fourier Transform (FFT) is not possible. Several propositions have been made to reconstruct images
from k-space data. One idea was to use the extension of the discrete Fourier transform, also called
phase conjugate reconstruction [149], but this method is very time consuming.
An alternative was then to extent the FFT to the case of non uniformly sampled data. For now,
the most commonly used method is the gridding algorithm [150, 151]. This basically consists of in-
4.4. Spiral image reconstruction
83
terpolation of k-space points on the Cartesian grid that then allows the application of an FFT, which
remains the most computationally efficient algorithm for reconstruction. However, interpolation in
k-space leads to errors that are spread over the whole image once reconstructed. This part must
therefore be done with caution and different propositions were made concerning the way to interpolate data, these methods all belong to the non-uniform FFT (NUFFT) family [152, 153]. Moreover,
variable density sampling encountered in spiral must be taken into account before interpolation onto
the Cartesian grid, this requires compensation for differently sampled areas by multiplying the data
by a density compensation function (DCF). This implies also postcompensation of data after the
gridding step.
To generate the gridded data points, the problem of data resampling can also be solved by
the BURS (block uniform resampling) algorithm [154], where a set of linear equation is given
an optimal solution using the pseudoinverse matrix computed with singular value decomposition
(SVD). Moreover, noise and artifact reduction can be obtain by using truncated SVD [155]. This
algorithm has the advantage to avoid the pre and post compensation steps of the gridding algorithm. Other alternatives to calculation of DCF were proposed either by iteratively reconstructing
data using matrices scaled larger than target matrix (INNG method) [156] or by using an iterative
deconvolution-interpolation algorithm (DING) [157]. Another proposition is to calculate a generalized FFT (GFFT) to reconstruct data which is mathematically the same algorithm than gridding
with a Gaussian kernel, however GFFT was shown to be more precise in the case of reconstruction
of small matrices [158]. We will here focus on the gridding algorithm because it is still the most
widely used and is the basis of a large number of other reconstruction alternatives.
Other reconstruction methods, that do not rely on the direct use of FFT, are called model based
reconstructions. This family of methods rely on the resolution of a variational problem in which
the image is the solution given by minimizing of a cost function. A review about these methods
can be found in Fessler et al. [159]. This cost function is usually composed of a data term that
corresponds to the least square cost function, to which is added a regularization that can have
several formulations depending on the constraints of the problem. One example of such method
can be found in Boubertakh et al. [160]. Even if these methods are already widely used in other
application domains (electrical and computer engineering), they were hardly adopted by the MR
community until recently probably because of the computational complexity they bring along. It is
however interesting to notice that despite the apparent difference between model-based and gridding
approach, Sedarat et al. [161] have shown that gridding was an approximation of the least square
solution.
4.4.1
Gridding algorithm
Density compensation function
As spiral sampling is not uniform over the k-space some compensation must be done in order to
avoid an overweighting of low spatial frequencies compared to high frequencies, which would result in signal intensity distortions. This can be done in several ways. For example, if the density
varies smoothly, Meyer et al. [162] showed good reconstructed image quality by using the analytical
formulation of the trajectory to compensate for the variable density. However, this method is no
longer reliable when the density varies sharply or in the case of less ideal spiral trajectories. Hoge
et al. [163] compared several analytical propositions with their method using the determinant of
the Jacobian matrix between Cartesian coordinates and the spiral sampling parameters of time and
84
interleave rotation angle used as a density compensation function. They could show the reliability
of their method even in the case of trapezoidal or distorted gradient waveforms.
However, when the real trajectory moves too far away from the theoretical one, another approach
independent of the sampling pattern should be used. This is based on the Voronoi diagram [164]
to calculate the area around each sampling point, figure 4.7 shows an example of this diagram on
a spiral trajectory. This area is then used to compensate for density variation, the bigger the area
around the sample (the size of the Voronoi cell), the smaller the density sampling. Rasche et al. [165]
have shown the power of this technique in the case of a distorted spiral trajectory, the advantage
being that this technique depends only on the sampling pattern and not on the acquisition order.
This technique nevertheless needs knowledge of the real trajectory (see section 4.3.1 p. 74) and may
be limited due to the high computational complexity of the Voronoi diagram.
0.5
0.03
0.4
0.02
0.3
0.2
0.01
ky (a.u.)
ky (a.u.)
0.1
0
0
−0.1
−0.01
−0.2
−0.3
−0.02
−0.4
−0.5
−0.5
0
kx (a.u.)
0.5
−0.03
−0.03
−0.02
−0.01
0
kx (a.u.)
0.01
0.02
0.03
Figure 4.7 — Example of Voronoi diagram for density compensation on a variable density spiral trajectory,
zoom on the center part of the diagram (right).
Interpolation onto the Cartesian grid
Once the samples have been corrected for the non-uniform density, they need to be interpolated onto
the Cartesian grid in order to perform the FFT algorithm for image reconstruction. O’Sullivan [151]
showed that the best way to interpolate samples was the use of an infinite sinc function. Samples in
the Fourier domain are convolved with a sinc function, resulting in a multiplication with the Fourier
transformation of a sinc function (a boxcar function) in the image domain. In practice, the use of an
infinite sinc function is not possible. In gridding, this function is replaced by compactly supported
kernels. However, the computational simplicity of the kernel must be balanced with the level of
artifacts in the resulting image, which is not an easy tradeoff. Jackson et al. [150] investigated
several kernels and showed that the Kaiser-Bessel kernel gave the best reconstructed image quality.
In this case however, the final image must be corrected by dividing by the Fourier transform of the
kernel to avoid distorted intensities due to the fact that it is no longer a rectangular function, this
is the so-called roll-off correction. Figure 4.8 illustrates the whole process in the case of 1D signal.
4.4. Spiral image reconstruction
85
Figure 4.8 — Illustration of the operation of the gridding method in the image domain. (a) Inverse
transform µ(x, y) (solid) of some measured signal s(u, v) and the convolution function inverse transform
c(x, y) (for example Kaiser-Bessel window). (b) After convolution in the Fourier domain, the result is equal
to µ(x, y) · c(x, y). (c) Sampling on to the Cartesian grid in the Fourier domain results in aliasing or folding
of the spectrum. (d) Convolution correction or roll-off correction by division with c(x, y). The dotted
component represents the error introduced by gridding. This figure is reproduced from [151].
86
4.4.2
Spiral imaging and parallel imaging
As mentioned earlier, spiral acquisition schemes are still not commonly used in the clinic, even
though spiral imaging has several advantages over standard Cartesian acquisitions. One main reason for this is that Cartesian acquisitions are routinely accelerated with parallel imaging, whereas
this is not trivial for spiral acquisitions. Without parallel imaging, the speed advantage of spiral
trajectories is compromised. However, recently introduced methods for non-Cartesian parallel imaging, in conjunction with improved computer performance, will enable the use of accelerated spiral
acquisition schemes for both clinical routine and research.
In parallel imaging, acceleration of the image acquisition is performed by reducing the sampling
density of the k-space data. The reduced sampling density of k-space is related to a reduced FOV.
If an object is larger than the reduced FOV, all parts of the object outside the reduced FOV will
be folded back into the reduced FOV. This effect, called aliasing or foldover artifact, is depicted in
Figure 4.9. The undersampling of Cartesian acquisitions along the phase encoding (PE) direction
leads to discrete aliasing artifacts along this direction. Compared to the discrete aliasing artifact
behavior of Cartesian acquisitions, aliasing in spiral imaging is different. In k-space, undersampling
of a spiral acquisition is affecting all directions. As a result, in the image domain, one image pixel
is folded with many other image pixels. This can be seen on the right-hand side of Figure 4.9.
Figure 4.9 — Simulated aliasing artifact behavior of Cartesian and spiral imaging: (a) In Cartesian
imaging, a factor of two undersampling along the phase encoding (PE) direction results in discrete aliasing.
In this aliased image, always one pixel is aliased onto another single pixel. (b) In comparison, undersampling
in spiral imaging, results in the situation that one image pixel is aliased with many other pixels. The
corresponding k-space trajectories with acquired (solid lines) and skipped (dashed lines) data points are
depicted at the bottom.
Today, there are two major parallel imaging methods routinely used, namely SENSE [21] and
GRAPPA [22]. While SENSE works completely in the image domain by unfolding aliased images,
GRAPPA works in k-space by reconstructing missing k-space data. For details about these methods
see [101].
Spline-based
image model for
spiral
reconstruction:
SPIRE
5
Part of this chapter was presented at ISMRM workshop on “Data sampling and Image reconstruction”
in Sedona, AZ, USA, January 2009 and as a poster at ISMRM 17th annual congress [166].
5.1
Model assumptions and justification
As discussed in the previous chapter, gridding algorithm is the most widely deployed method for
the reconstruction of non-Cartesian data because it has the main advantage of being very fast.
However, interpolation in k-space performed to replace the points onto the Cartesian grid introduces
unpredictable artifacts that are widespread over the image. This drawback becomes very important
when k-space is either undersampled or highly non-uniformly sampled. Figure 5.1 illustrates this
in the case of variable density spiral sampling. In our experiments, we want to take advantage of
variable density spiral properties, and will undersample the k-space to limit the acquisition time.
For this reason, we chose to adopt a model-based reconstruction.
We have already seen that the MRI signal s is related to the image f with the Fourier transform:
Z
s(kx , ky ) = f (x, y)ei2π(kx x+ky y) dxdy.
(5.1)
The image f is usually defined by its samples on a conventional M × M cartesian grid and a
sinc interpolation :
f (x, y) =
M−1
X
n1 ,n2 =0
cn1 ,n2 sinc(x − n1 )sinc(y − n2 ).
87
(5.2)
88
Chapter 5. Spline-based image model for spiral reconstruction: SPIRE
fully sampled
undersampled
variable density − undersampeld
Figure 5.1 — Illustration of weaknesses of gridding reconstruction in the case of spiral sampling. From
left to right, fully sampled k-space, undersampled k-space, undersampled k-space with a variable density
trajectory.
5.1.1
Spline-based image model
The sinc interpolation corresponds to the assumption that signal intensity pixels have a band-limited
frequency, which is in line with Shannon’s sampling theorem [167] that states:
Theorem 1 If a function f (t) contains no frequency compounds higher than W cps, it is completely
determined by giving its ordinates at a series of points spaced 1/2 W seconds apart.
Where cps are units of bandwidth W . The only functions that fulfill a limited bandwidth W and
which passes through given values at sampling points separated from 1/2 W seconds apart can be
reconstructed using a sinc function:
sinc(t) =
sin 2πW t
.
2πW t
(5.3)
Even if the sinc interpolation is largely used, it brings some inherent problems. The hypothesis of
finite bandwidth in k-space corresponds to a non-limited signal support in space, which is obviously
never the case. Sinc convolution then generates Gibbs oscillations in the image, also called “ringing
artifact”. Therefore, a more suitable (but non-bandlimited) interpolation between pixels can be
done with polynomial spline functions as it was shown to be an interesting alternative with many
advantages over the cardinal basis functions [168]. The image f will therefore be defined by:
f (x, y) =
M−1
X
n1 ,n2 =0
cn1 ,n2 β α1 (x − n1 )β α2 (y − n2 ),
(5.4)
where β is a B-spline function of degree α and cn1 ,n2 the corresponding spline coefficients.
Figure 5.2 shows B-spline function of different degree and figure 5.3 an example of polynomial spline
interpolation of a 1-D discrete signal.
B-spline functions are continuous symmetrical functions constructed from the (α + 1)-fold convolution of the β 0 rectangular pulse [168]:
β 0 (x) =


 1
1
2

 0
− 21 < x <
|x| = 12
otherwise
1
2
β n = β 0 ∗ β 0 ∗ β 0 ∗ ...β 0 .
|
{z
}
α+1 times
(5.5)
89
5.1. Model assumptions and justification
degree 0
degree 1
1
1
0.5
0.5
0
−3 −2
−1
0
1
2
0
−3 −2
3
−1
degree 3
1
0.5
0.5
−1
0
1
2
3
2
3
degree 5
1
0
−3 −2
0
1
2
0
−3 −2
3
−1
0
1
Figure 5.2 — B-spline functions of degree 0, 1, 3, 5
discrete signal
degree 0 interpolation
degree 1 interpolation
degree 3interpolation
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
0
0
2
4
6
8
0
0
2
4
6
8
0
0
2
4
6
8
0
0
2
Figure 5.3 — Spline interpolation of 1-D signal with spline function of degree 0, 1, 3.
4
6
8
90
The value of B-spline function of degree α can also be found by the analytical formulation [169]:
f (x) =
n+1
X
k=0
(−1)k
n
Γ(α + 2)
1
α+1
α+1
x+
− k sign x +
−k ,
Γ(α + 2 − k)Γ(α + 1) 2Γ(α + 1)
2
2
(5.6)
where Γ is the Gamma function, an extension of the factorial function to real and complex numbers,
if n is a positive integer Γ(n) = (n − 1)!.
B-splines have the advantage to be α times differentiable and to be easy and efficient to manipulate. For example their derivative reduces the spline degree:
1
1
dβ α (x)
= β α−1 (x + ) − β α−1 (x − ).
dx
2
2
(5.7)
Also, the Fourier transform of B-spline of degree α denoted β̂ α can be expressed as:
β̂ α (k) =
5.1.2
sin(k/2) α+1
k/2
.
(5.8)
Spline-based Image REconstruction: SPIRE
As stated before, we chose to adopt a model-based image reconstruction. The variational formulation
of the image reconstruction is the following:
J(f ) = ks − (Hf )k22 +µ R(f ) .
{z
}
| {z }
|
JLS
(5.9)
Jreg
The first term is a data term where s is the data, H the operator that transforms the image f
into the corresponding k-space, i.e. the Fourier transform following relation 5.1, and k · k2 denotes
the l2 -norm. As a reminder, the l2 -norm (also called Euclidian norm) of a vector x of n complex
coordinates is defined by:
v
uX
u n
kxk2 = t
|xj |2 .
(5.10)
j=1
The second term of 5.9 is the regularization that adds some a priori information to the model
or that constrains the solution to have certain desirable properties such as a small norm or smooth
transitions (in the case of an l2 -norm of a linear operator this is known as “Tikhonov regularization”
[170]). Since the problem is often ill-posed, i.e. there are less data samples than reconstructed
points, the regularization is important for the algorithm convergence.
The complex image solution f minimizes the functional J:
f = arg minks − (Hf )k22 + µR(f ).
5.2
(5.11)
Algorithm implementation
To implement this minimization we used a gradient descent method:
Ji+1 = Ji − λ
∂J
,
∂c
where c are the spline coefficients and are related to the image f by the relation 5.4.
(5.12)
91
5.2. Algorithm implementation
However, the gradient calculation needs the inversion of equation 5.9, which becomes complicated
for non-Cartesian trajectories [171] since it requires the inversion of a matrix of size M ×M ×N (with
M × M the image size and N the number of samples in k-space) that is usually out of reach. For
example, if we consider a spiral trajectory of 23780 points given by a typical MRI system (obtained
for 10 interleaved spirals with a FOV of 12.8 cm, an in-plane resolution of 1 mm and a bandwidth of
100 kHz on a Siemens 3T system) mapped on a 128x128 matrix, the storage of the matrix in float
precision would already require 1.5 GB.
To simplify the calculation, we generalize the method proposed by Wajer et al. [172] and Boubertakh et al. [160] to the spline-based image model formulation. This method computes the gradient
of J as a simple convolution between the current state of the image and a precomputed matrix.
Indeed, if we first concentrate on the data term of equation 5.9, we have:
JLS (f ) = ks − (Hf )k22 .
(5.13)
We can develop this in:
JLS (f ) =
L−1
X
l=0
|s − (Hf )|2 =
L−1
X
l=0
|sl |2 + |(Hf )l |2 − 2Re{sl (Hf )∗l },
(5.14)
where ∗ denotes the complex conjugate and l is the index of each sample along the k-space
trajectory. Including the image model described by equation 5.4 and the relation between signal
and image (equation 5.1), the second term of 5.14 become:
2
|(Hf )l |
=
=
=
=
1
N2
1
N2
1
N2
1
N2
N
−1
X
cn1 ,n2
Z
β α1 (x − n1 )β α2 (y − n2 )ei2π(kx n1 +ky n2 ) dxdy
cn1 ,n2
Z
β α1 (x′ )β α2 (y ′ )ei2π(kx (x +n1 )+ky (y +n2 )) dx′ dy ′
n1 ,n2 =0
N
−1
X
n1 ,n2 =0
N
−1
X
cn1 ,n2
n1 ,n2 =0
N
−1
X
"Z
l
l
β
α1
′
(x )β
α2
′
(y )e
′
l
l
i2π(kx
x+ky
y)
l
l
cn1 ,n2 β̂ α1 (kx )β̂ α2 (ky )ei2π(kx n1 +ky n2 ) ,
l
l
′
#
l
l
dx dy ei2π(kx n1 +ky n2 )
′
′
(5.15)
n1 ,n2 =0
where x′ = x − n1 and y ′ = y − n2 . Following the same development, the third term of 5.14
becomes:
(
)
N
−1
X
l
l
1
∗
∗
α1 ∗
α2 ∗
−i2π(kx
n1 +ky
n2 )
2Re{sl (Hf )l } = 2Re
sl
c
β̂ (kx )β̂ (ky )e
.
(5.16)
N n ,n =0 n1 ,n2
1
2
Then, the gradient of the functional J is given by the sum of the gradient of the 2 last terms
composing equation 5.14 since the first one is independent of the image model. Using the following
complex number properties for equation 5.15 and 5.16 respectively:
|z|2 = zz ∗,
2Re{z} = z + z ∗ .
(5.17)
92
For equation 5.15 we have:
∂|(Hf )l |2
∂
1
=
∂cm1 ,m2
∂cm1 ,m2 N 2
1
N2
N
−1
X
l
n1 ,n2 =0
N
−1
X
l
cn1 ,n2 β̂ α1 (kx )β̂ α2 (ky )ei2π(kx n1 +ky n2 ) ·
l
l
c∗n1 ,n2 β̂ α1 ∗ (kx )β̂ α2 ∗ (ky )e−i2π(kx n1 +ky n2 )
n1 ,n2 =0
!
.
(5.18)
After the derivation and some rearrangement, we finally get:
−1
i2 NX
l
l
∂|(Hf )l |2
2 h α1 ∗
α2 ∗
=
β̂ (kx )β̂ (ky )
cn1 ,n2 ei2π(kx (n1 −m1 )+ky (n2 −m2 )) .
∂cm1 ,m2
N
n ,n =0
1
(5.19)
2
Similarly for the equation 5.16 we have:
∂Re{sl (Hf )∗l }
∂
1
=
∂cm1 ,m2
∂cm1 ,m2 N
sl
N
−1
X
l
l
c∗n1 ,n2 β̂ α1 ∗ (kx )β̂ α2 ∗ (ky )e−i2π(kx n1 +ky n2 ) +
n1 ,n2 =0
s∗l
N
−1
X
l
l
c∗n1 ,n2 β̂ α1 (kx )β̂ α2 (ky )ei2π(kx n1 +ky n2 )
n1 ,n2 =0
!
.
(5.20)
After derivation and rearrangement we have:
l
l
2
= sl β̂ α1 (kx )β̂ α2 e−i2π(kx m1 +ky m2 ) .
∂cm1 ,m2
N
(5.21)
If we define 2 matrices G and D as follows:
Gm1 ,m2
=
L−1
i2
l
l
1 X h α1 ∗
α2 ∗
β̂
(k
)
β̂
(k
)
ei2π(kx m1 +ky m2 ) ,
x
y
2
N
(5.22)
l=0
Dm1 ,m2
=
L−1
l
l
1 X
sl β̂ α1 ∗ (kx )β̂ α2 ∗ (ky )e−i2π(kx m1 +ky m2 ) .
N
(5.23)
l=0
We can simply write the gradient as:
∂JLS (f )
= 2cm1 ,m2 ∗ Gm1 ,m2 − 2Dm1 ,m2 .
∂cm1 ,m2
(5.24)
Practically, the G matrix can be precomputed since it only depends on the k-space trajectory
whereas D matrix has to be calculated once for each data set. The convolution product is easily
performed in the Fourier domain by the product of the Fourier transforms of c and G. The computation becomes then largely simplified and is therefore closer to be implementable on a scanner
than in its initial form.
5.3
Regularization
The regularization is an important part of the algorithm since it describes the terms that are penalized during the image reconstruction. For example, a typical regularization is given by Tikhonov
93
5.3. Regularization
R(f ) = kAf k2 where A is a linear operator [170]. When A is the finite difference operator, it
penalizes the energy of the Laplacian of f and therefore assumes that the image contains smooth
transitions. If A is the identity, the regularization is also called pseudo-inverse and thus gives preference to images with smaller norms.
In our model, the regularization term is defined by:
Jreg = kDγ (f )k2 ,
(5.25)
where D is the differential operator of degree γ, with the condition γ ≤ α since B-spline are
α-times differentiable. The calculation of the gradient of Jreg was separated into two cases, γ = 0
and γ > 0. Whereas in the first case, images with a small norm are preferred, the second case
constraints the image to have smooth transitions.
1. Case γ = 0
Jreg
=
=
=
∂Jreg
∂c
=
kf k2
hc ∗ β α , c ∗ β α i
c, c ∗ β 2α+1
2c ∗ β 2α+1 ,
(5.26)
(5.27)
P
where hi denote the vectorial product, and where we expressed f (x) = k ck β α (x − k) with
the convolution product f = c ∗ β α . The transition in 5.26 was done by using the B-spline
Fourier transform (see equation 5.8).
2. Case γ > 0
Jreg
∂Jreg
∂c
= kDγ f k2
= c ∗ ∆γ ∗ β α−γ , c ∗ ∆γ ∗ β α−γ
= c, c ∗ ∆2 γ ∗ β 2α−2γ+1
= 2∆2γ ∗ c ∗ β 2α−2γ+1 ,
(5.28)
(5.29)
where ∆ is the discrete derivative, and where the transition in 5.28 is done with:
Dγ f = c ∗
dγ α
β = c ∗ ∆γ ∗ β α−γ .
dxγ
(5.30)
Table 5.1 sumarizes the results for different spline degree α and differential order γ:
Finally, the total gradient is expressed by the general formulation:
∂JLS (f )
= 2cm1 ,m2 ∗ Gm1 ,m2 − 2Dm1 ,m2 + 2µc ∗ ∆2γ ∗ β 2γ−2α+1 .
∂cm1 ,m2
5.3.1
(5.31)
Automatic parameter adjustment
The main difficulty in the reconstruction algorithm is to define the parameter µ that tunes the
weight to give to the a priori information. Indeed, if µ is set too low, not enough information is
added in order to make the algorithm converge whereas if it is set too high the algorithm would no
94
Table 5.1 — Gradient of the regularization term for different spline degree α and derivative order γ
B-spline
degree α
order
of
derivative γ
∂Jreg
∂c
0
0
2c
1
0
2β 3 ∗ c
1
3
0
1
2
2∆2 ∗ β 1 ∗ c
2β 7 ∗ c
2∆2 ∗ β 5 ∗ c
2∆6 ∗ β 1 ∗ c
more consider signal data to find the best solution. In order to set automatically this parameter we
implemented a cross validation method. For this, we used an extra set of data, sv referred to as the
“validation set” to determine if µ has to be increased or decreased. In our particular application, the
validation set was an extra spiral interleave placed at a different angle than the already acquired
samples.
After some iterations of the algorithm, we obtain a good estimation of the transform operator
H. We then calculate the data term when the validation set was reconstructed with this estimated
operator H and with variable µ, referred to as fv . The minimum data term value indicates which
µ is more suitable for the next iterations:

2

 ksv − Hfv (µ/2)k
2
min =
ksv − Hfv (µ)k

 ks − Hf (2µ)k2
v
v
underfitting µ → µ/2
µ stays equal
overfitting µ → 2µ
The re-evaluation of µ is done periodically. The stopping criteria of the algorithm is either the
stability of µ over a certain number of iterations or the reach of a given number of iterations. Figure
5.4 illustrates the automatic setting of µ.
Figure 5.4 — Scheme of the automatic setting of parameter µ for variable density spiral (vds) reconstruction example. Left, trajectory scheme, in blue the data and in red the validation set. Right, schematic
representation of the algorithm.
95
5.4. Evaluation on numercal Shepp-Logan phantom
5.4
5.4.1
Evaluation on numercal Shepp-Logan phantom
Methods
We have validated our reconstruction method on the Shepp-Logan phantom [173]. This numerical
phantom schematically represents an axial brain scan and has the advantage to provide signal
intensity for any arbitrary trajectory in k-space, see Figure 5.5.
Figure 5.5 — Shepp-Logan analytic phantom on a 256x256 matrix.
The trajectory used for the numerical experiments was a variable density spiral (VDS). We constructed 3 sets of data, the first one was a fully sampled k-space, the second one was an undersampled
k-space and the third one a highly undersampled k-space, see table 5.2. The undersampling was
defined with the formulation of the number of turns of the spiral (nt):
!1/αD !−1
nb of interleaves
nt = ceil 1 − 2
,
N · Nyquist
(5.32)
where here αD refers to the variable density parameter, N to the matrix size and “Nyquist” expresses
the undersampling factor. Nyquist = 1 is a fully sampled k-space whereas Nyquist < 1 corresponds
to radial undersampling of k-space while angular sampling stays constant.
Table 5.2 — Parameters used for the different experiment on Shepp-Logan numerical phantom.
k-space
Full
Undersampled
Highly undersampled
number of interleaves
26
23
21
αD
2
2
6
Nyquist
1
9/10
2/3
The number of interleaves varied between 21 and 26 in order to maintain the same TR between
experiments and to constrain the number of turns of the spiral to a minimum value of 4, thus the
trajectory keeps the spiral properties. Other general parameters were FOV = 40 mm, matrix size
N = 128 and TR = 11.5 ms. The high variable density parameter in the highly undersampled
experiment was chosen to preserve image quality. Figure 5.6 shows examples of the k-space spiral
trajectory used.
96
single interleave − α=2 − Nyquist=1
full k−space
0.5
0.5
0
0
−0.5
−0.5
0
0.5
−0.5
−0.5
single interleave − α=6 − Nyquist=2/3
0.5
0
0
0
0.5
0.5
full k−space
0.5
−0.5
−0.5
0
−0.5
−0.5
0
0.5
Figure 5.6 — Trajectories used in Shepp-Logan simulations for the fully sampled (upper) and the highly
undersampling (lower) cases. Left, detail of one interleave of the trajectory, right, full trajectory.
97
Gridding algorithm
In order to evaluate the performance of our reconstruction method we compared the results with
the widely used gridding method (see section 4.4, p. 82). We choose to use the Voronoı̈ diagram as
a density compensation function [165] and the interpolation onto the Cartesian grid was done with
a Kaiser-Bessel kernel (as it was shown to give good image quality [150]).
SNR quantification
To evaluate the quality of the reconstructed images, we calculated the SNR against the analytical
ground truth. Since we know the exact solution of the problem, we can evaluate the noise given
by the reconstruction algorithm and corresponding to thte difference with the reference image. The
SNR is given by:
P
|I(x) − Iref (x)|2
xP
SNR = −10 log10
,
(5.33)
2
x |Iref (x)|
where I is the considered image and Iref is the reference one. Before to calculate the SNR the
images need to be normalized since they are not in the same range of signal intensity. To perform
this normalization we calculated the mean signal intensity of two areas in the image corresponding
to the minimum (SImin ) and the maximum (SImax ) signal intensity in the reference image. Figure
5.7 shows the location of these areas. The normalization is then performed following the relation:
Inorm =
I − SImin
.
SImax − SImin
(5.34)
20
40
60
80
100
120
20
40
60
80
100
120
Figure 5.7 — Orange areas on the Shepp-Logan phantom represent the areas chosen for the images
normalization. The one located in black area corresponds to the minimum signal intensity and the one in
the white border of the object to the maximum signal intensity.
5.4.2
Results
Noiseless simulations
The SNR was calculated with the different calculated reconstructions, and detailed values are reported in table 5.3. We first observe that SNR was greatly improved with SPIRE method in almost
98
all tested reconstructions. For the full sampling and undersampling cases, cubic spline interpolation
combined with first order derivative in the regularization gave the best results while spline of degree
one was better in the highly undersampled case. Indeed, in this case, Shannon’s theorem is violated
since all the frequencies requested to reconstruct the image are not sampled (the minimum number
of frequencies to be sampled to uniquely define a signal is also referred to as Nyquist criteria), so
the use of sinc interpolation is no more valid. Considering that higher degree splines are closer to
the sinc interpolation, it can explain the fact that the spline of degree 1 gives better results than
the third one.
Table 5.3 — SNR in dB for reconstruction of the fully-sampled, undersampled and highly undersampled
Shepp-Logan phantom with gridding and SPIRE. Comparison of results with different spline degree α and
different order of the differential operator in the regularization γ. Higher SNR in each cases are in bold.
α
0
1
3
γ
0
0
1
0
1
2
gridding
Full
10.59
9.41
10.88
11.44
11.97
10.85
6.39
90%
10.05
9.34
9.56
10.79
10.88
10.02
5.85
67%
7.32
6.75
9.71
5.99
6.67
6.57
6.14
Figure 5.8 shows the results obtained with the gridding reconstruction and with the SPIRE
method with the parameters (α and γ) giving the best SNR.
In the spatial profiles of figure 5.8, we observe that gridding is very affected by Gibbs ringing
artifacts for the fully sampled case while this is not the case with SPIRE as it was expected with the
spline-based image model and the absence of the band-limited assumption of the signal. Moreover,
gridding reconstruction exhibit non-homogeneous signal intensity over the image, with a notable
enhancement of the signal intensity at the center of the image. SPIRE images are more homogeneous and the contrasts are better preserved. Finally, the artifacts in the undersampled and highly
undersampled cases are greatly reduced in SPIRE images. However, transitions appear smoother
than in gridding, giving visual impression of slightly lower spatial resolution.
99
gridding
α=3, γ=1
1
0.8
0.6
0.4
0.2
0
0
gridding
50
100
50
100
50
100
α=3, γ=1
1
0.8
0.6
0.4
0.2
0
0
gridding
α=1, γ=1
1
0.8
0.6
0.4
0.2
0
0
Figure 5.8 — Shepp-Logan reconstruction with gridding (left) and SPIRE (middle). Upper row, for fully
sampled k-space (Nyquist=1), middle row, for undersampled k-space (Nyquist=9/10), bottom row, for
highly undersampled k-space (Nyquist=2/3). Plot of profile of the middle horizontal line of the phantom
(orange line), for the gridding reconstruction (blue line) and for SPIRE (red line), compared to the reference
(black line).
100
Noisy simulations
Further experiments were performed on same sets of data but corrupted with additive Gaussian
noise. The noise was added to the k-space signal and SNR was 21 dB for fully sampled and undersampled sets, and 27 dB for highly undersampled set. Table 5.4 shows the SNR obtained for these
experiments and figure 5.9 the results of the reconstruction obtained with the best set of parameters
with the SPIRE method in the same three different sampling cases.
Similarly to the noiseless experiments, the first derivative order in the regularization associated
with the cubic spline interpolation gave the best results for the fully sampled and undersampled
cases while the spline of degree one interpolation was the best for the highly undersampled case.
Table 5.4 — SNR in (dB) for reconstruction of the fully-sampled, undersampled and highly undersampled
Shepp-Logan phantom with gridding, and SPIRE, and noise corruption of data. Comparison of results with
different spline degree α and different degree of the differential operator in the regularization γ. Higher SNR
in each cases are in bold.
α
0
1
3
γ
0
0
1
0
1
2
gridding
Full
6.99
5.88
6.79
7.88
9.59
8.67
4.59
90%
6.29
5.46
6.32
7.58
8.44
7.86
4.26
67%
7.95
7.26
8.47
6.32
6.95
6.96
3.05
The spatial profiles of figure 5.9 illustrate a great improvement of image quality with SPIRE
compared with gridding that is in relation with the nearly 2-fold increase in SNR measured. In
these experiments again, the undersampling artifacts visible in the gridding are totally removed
with SPIRE method.
101
gridding
α=3, γ=1
1
0.8
0.6
0.4
0.2
0
0
gridding
50
100
50
100
50
100
α=3, γ=1
1
0.8
0.6
0.4
0.2
0
0
gridding
α=1, γ=1
1
0.8
0.6
0.4
0.2
0
0
Figure 5.9 — Shepp-Logan reconstruction with gridding (left) and SPIRE (middle). K-space data are
corrupted with noise. Upper row, for fully sampled k-space (Nyquist=1), middle row, for undersampled
k-space (Nyquist=9/10), bottom row, for highly undersampled k-space (Nyquist=2/3). Plot of profile of
the middle horizontal line of the phantom (orange line), for the gridding reconstruction (blue line) and for
SPIRE (red line), compared to the reference (black line).
102
5.5
5.5.1
MRI experiments
Methods
Sequence implementation
A spiral sequence∗ was modified to provide variable density trajectories with the formulation proposed by Kim et al. [128]. Since this formulation gives gradient amplitude and slew rate overshoots
for the first points of the trajectory, we implemented the solution provided by Zhao et al. [129].
The sequence was then run on the Siemens 3T scanner. We acquired data of a phantom and of a
healthy volunteer heart.
The sequence parameters were the followings:
Phantom experiment FOV=240 mm, N=128, slice thickness 5 mm, flip angle 90◦ , TE/TR=1/100
ms (TR was set long for tests purpose but readout time was 23 ms), 10 interleaves, variable
densitiy parameter αD =3, Nyquist=1, spine coil used (reconstruction of a single channel centered on the field of view)
Volunteer heart experiment FOV=240 mm, N=128, slice thickness 5 mm, flip angle 30◦ , TE/TR=0.8/12
ms, 10 interleaves, variable densitiy parameter αD =3, Nyquist=1, body coil used (reconstruction of the 2 channels and combination with sum of square algorithm † )
Reconstruction
SPIRE method with automatic regularization parameter adjustment used a cubic spline interpolation and first order derivative in the regularization (α=3 and γ=1). The choice of these parameters
was given by the previous results on numerical simulations, indeed they were previously shown to
give the best reconstruction from an SNR point of view.
5.5.2
Results
Figure 5.10 shows the comparison between SPIRE and gridding methods for the phantom image
reconstruction while figure 5.11 shows an example of a healthy volunteer heart image. The typical
reconstruction time for a 128x128 matrix was 24 min, divided in the following tasks:
• 7 min for precomputation of matrix G (that needs to be done only once for each trajectory)
• 2 min for precomputation of matrix D
• 15 min for the iterative minimization of functional J
So further reconstructions with the same trajectory will take only 17 min. By comparison, gridding
reconstruction for a 128x128 matrix takes 5 sec. However, SPIRE calculation was not optimized for
the calculation time, and it is still possible to parallelize the algorithm.
In the phantom experiment, SPIRE resulted in an image that is more blurred than the gridding
reconstruction. However, the signal at the bottom of the image is better preserved and the artifacts
at the center of the image are greatly reduced. Once again, the SPIRE image is more homogeneous
∗ Source code provided by Gunnar Krueger, Siemens Medical Solutions, Centre d’Imagerie BioMédicale (CIBM),
Lausanne
† Sum of square combination is given by I = kI
2
2
ch1 k + kIch2 k , where I is the final image and Ich1 , Ich2 are the
images obtained with separate coils.
103
5.5. MRI experiments
SPIRE
gridding
Figure 5.10 — Comparison of SPIRE (left) and gridding (right) reconstruction of a phantom.
than gridding. In this particular example, however, it is not straightforward that SPIRE method
gives better results than gridding since the lower sharpness of edges is an important drawback for
clinicians compared to spiral artifacts that may be more easily recognizable.
In the volunteer heart example, the situation is more difficult since the area of interest is experiencing blood flow variations that bring along additional artifacts to the final image. Separate images
of the two channels of the body coil show reduced artifacts with SPIRE compared with gridding
and sum of square image gives more signal contrast.
We can notice the white border present in SPIRE reconstruction that is due to edge effects and
is present only when object of interest is near the border of the FOV.
104
Figure 5.11 — Comparison of SPIRE (upper) and gridding (lower) reconstruction of a volunteer heart. Images obtained with the two channels of the body
coil (left and middle) and sum of square reconstruction (right). Arrows indicate areas where noise is reduced in SPIRE compared with gridding and where
signal is enhanced is SOS images.
5.6. Discussion
5.6
105
Discussion
In terms of SNR obtained in the numerical simulations, SPIRE method was shown outperform gridding reconstruction. The automatic adjustment of the regularization parameter gave reproducible
and robust results. The innovative spline-based image model avoided the Gibbs ringing artifacts that
are often present after gridding reconstruction and allowed a better preservation of image as well as
an important reduction of undersampling artifacts. Similar observations were made in real data experiments, even if a drawback is a reduction of edge sharpness compared to gridding in certain cases.
We have to notice that we used the set of parameters (spline degree α and derivative order
γ) that gave the best results in the case of no data undersampling in the numerical simulations.
Whether this set of parameters are also the best for real data reconstructions was not investigated
and still need to be confirmed. Morevoer, the impact of the regularization parameter µ has to be
deeply explored since it may be responsible of the resulting blurriness of the images.
The presented results of MRI acquisitions may not be completely convincing, especially regarding
the results given by gridding reconstruction. However, SPIRE method has the important advantage
to propose a novel view of image reconstruction with a spline model. Moreover, the algorithm can
be easily adapted to benefit from existing regularization methods. While we only investigated the
effect of linear regularizers, the formulation can be extended to non-linear regularizations such as
total variation [174] or sparsity [175]. This last method, also known as “compressed sensing”, is
still a hot topic in MRI [176]. The main idea underlying this method is the ability to recover an
image with only a small number of samples by the mean of l1 -norm regularization in a sparsifying
transformation. Indeed, if, by a known transformation, the image can be defined by a small number
of samples and then be recovered without losing any determinant information, it opens the possibility to greatly accelerate the acquisition time. One well-known sparsifying transformation is the
wavelet transform [177] that is widely used nowadays in all computational domains (notably with
the JPEG2000 compression [178]).
Finally, the reconstruction time of SPIRE is not considered to be an important drawback since
the computation can be accelerated by the parallelization of the algorithm an its implementation on
new multiprocessor graphic cards (GPUs). Indeed, calculation of G and D matrices as well as FFT
operations needed in the convolution product can benefit from a parallelized calculation and thus
be accelerated proportionally to the number of available processors. For example, it has already
been shown that the reconstructions of real-time radial data could be accelerated by a factor 17
with its implementation on GPUs [179]. Acceleration can also be expected by using more complex
convex optimization algorithm such as conjugate gradient [180] that are known to converge faster
than gradient descent as it is actually done.
106
k − t SPIRE:
time extension
of spline-based
image model
applied to
real-time
6.1
6
Spatio-temporal model
Usually, one considers that a set of data is acquired at a single discrete time point. While this
erroneous assumption do not impact the image quality in a wide range of situations, it becomes
problematic when the object of interest is dynamically changing (e.g. moving) as fast as the acquisition time needed to sample the data set. Instead of modeling k-space data as being acquired
at discrete time points, we propose to consider the samples in the 3-D domain (kx , ky , t) as it is
illustrated in figure 6.1.
The image model is therefore the generalization of the spline interpolation of data in space and
also in time domain:
f (x, y, t) =
M−1
X
T
−1
X
n1 ,n2 =0 n3 =0
cn1 ,n2 ,n3 β α1 (x − n1 )β α2 (y − n2 )β α3 (t − n3 ),
(6.1)
where T is the number of time frames, n1 and n2 the spatial index (usually n1 = n2 ) and n3 the
temporal index. cn1 ,n2 ,n3 is the spline coefficient located at position n1 , n2 , n3 .
107
108
Chapter 6. k − t SPIRE: time extension of spline-based image model applied to
real-time
ky(a.u.)
0.5
0
−0.5
0.5
0
−0.5
0
0.005
0.01
kx(a.u.)
0.015
0.02
0.025
0.03
0.035
0.04
0.015
0.02
0.025
0.03
0.035
0.04
t (s)
ky(a.u.)
0.5
0
−0.5
0.5
0
−0.5
kx(a.u.)
0
0.005
0.01
t (s)
Figure 6.1 — k − t space representation of spiral trajectory with usual assumption of discrete time points
for data sampling (upper) and proposed consideration of dataset (lower).
109
6.2. Model fitting & implementation
6.2
Model fitting & implementation
We do a least-square regression as in the previous chapter:
JLS (f ) = ks − (Hf )k22 .
(6.2)
However, we have to calculate the gradient of the functional JLS for this new image formulation.
We start from the general development:
JLS (f ) =
L−1
X
|s − (Hf )|2 =
l=0
L−1
X
l=0
|sl |2 + |(Hf )l |2 − 2Re{sl (Hf )∗l },
(6.3)
where l is the sampling index, the space and time parameters are inherently included into this index.
Without repeating all the calculation steps we already developed in chapter 5 (see section 5.2, p
90), we end up with the following results for the second and third term of equation 6.3:
|(Hf )l |2 =
2Re{sl (Hf )∗l }
(
1
N2
1
= 2Re
sl
N n
X
l
n1 ,n2 ,n3
X
l
cn1 ,n2 ,n3 β̂ α1 (kxl )β̂ α2 (kyl )β α3 (t − n3 )ei2π(kx n1 +ky n2 ) , (6.4)
c∗n1 ,n2 ,n3 β̂ α1 ∗ (kxl )β̂ α2 ∗ (kyl )β α3 (t
1 ,n2 ,n3
− n3 )e
l
l
−i2π(kx
n1 +ky
n2 )
)
. (6.5)
The derivation thus gives:
l
l
∂|(Hf )l |2
2 h α1 ∗ l α2 ∗ l i2 X
β̂ (kx )β̂ (ky )
=
cn1 ,n2 ,n3 β α3 (t − n3 )ei2π(kx (n1 −m1 )+ky (n2 −m2 )) , (6.6)
∂cm1 ,m2 ,m3
N
n ,n ,n
1
2
3
l
l
2
= sl β̂ α1 (kxl )β̂ α2 β α3 (t − m3 )e−i2π(kx m1 +ky m2 ) . (6.7)
∂cm1 ,m2 ,m3
N
If we define matrices Gt and Dt :
Gtm1 ,m2 ,m3
=
L−1
l
l
1 X h α1 ∗ l α2 ∗ l i2 α3
β̂ (kx )β̂ (ky ) β (t − n3 )ei2π(kx m1 +ky m2 ) ,
2
N
(6.8)
l=0
t
Dm
1 ,m2 ,m3
=
L−1
l
l
1 X
sl β̂ α1 ∗ (kxl )β̂ α2 ∗ (kyl )β α3 (t − n3 )e−i2π(kx m1 +ky m2 ) .
N
(6.9)
l=0
We finally obtain:
∂JLS (f )
t
= 2cm1 ,m2 ,m3 ∗ Gtm1 ,m2 ,m3 − 2Dm
.
1 ,m2 ,m3
∂cm1 ,m2 ,m3
(6.10)
110
real-time
Spline interpolation in the time domain
Similarly to space domain, data can be seen as discrete samples measured from a continuum in the
time direction. The continuous space and time representation is created with a spline interpolation
that has two important parameters.
Scaling of the basis function For example if a complete k-space is defined by an ensemble of 10
spiral interleaves, the 0th degree spline basis function will have a scale of 10 interleaves and
each reconstructed time point will correspond to the time needed to acquire 10 interleaves.
However if a higher time resolution is needed, the scale of the basis function can be reduced
to 5 or even 3 interleaves.
Spline degree As we have seen in the previous chapter, the higher is the spline degree the larger
is the function, i.e. degree 0 is define over [0, 1], degree 1 over [0, 2], degree 3 over [0 4]
etc... If, as in the previous case the temporal definition is enhanced by reducing the scale of
the spline function, the degree 0 spline will not contain a full k-space. However, as the spline
degree increases more samples are taken into account into the reconstruction but are weighted
following the time.
Concretely, images are evaluated for each unit time point (one unit time correspond to the time
needed to acquire the data included in the scale of the basis spline function). However, since we
now have a continuum of data in time we can reconstruct image at arbitrary time points, for this
we resample data by taking a linear combination of the calculated data. Figure 6.2 represents this
operation for the reconstruction of a time point located between two unit time points.
111
6.2. Model fitting & implementation
0th degree spline − nearest neighbor
1
0.8
0.6
0.4
0.2
0
−2
0
2
4
6
8
10
6
8
10
6
8
10
6
8
10
st
1 degree spline
1
0.8
0.6
0.4
0.2
0
−2
0
2
4
rd
3 degree spline
1
0.8
0.6
0.4
0.2
0
−2
0
2
4
th
5 degree spline
1
0.8
0.6
0.4
0.2
0
−2
0
2
4
Figure 6.2 — Contribution of basis spline functions for the reconstruction of an image located at 3.2 (a.u.)
time point. Gray curves are the basis spline functions, color curves are the one involves into the image
reconstruction at that time point, circles indicate the weights of each spline function. The more the spline
degree is, the more basis function contribute to the reconstruction of one specific time point.
112
real-time
6.3
6.3.1
Evaluation on numerical phantom
Methods
Numerical phantom implementation
To simulate the cardiac contraction we created a numerical phantom based on Shepp-Logan [173].
This phantom was composed of two concentric circles representing endocardium and epicardium
and are contracting following a sinusoidal function to mimic the beating heart. The cardiac cycle
was set to 1 s. The sampling trajectory was a multi shot spiral, calculated with similar parameters
as a typical in-vivo trajectory (i.e. gradient amplitude and slew rate). The main parameters were
the followings:
• 10 interleaves
• matrix size N = 64
• FOV = 6.4 cm
• TR = 10.2 ms
• acquisition time for one frame = 102 ms
The reconstruction was then performed in order to achieve a time resolution of 30.6 ms (which
is the time needed to acquire 3 interleaves). we choose this time resolution since it is compatible
with cardiac stress-study requirements (following [94], 11 phases per heart cycle are needed to
correctly evaluate the global function; i.e., 30.6 ms corresponds to 178 bpm). The final image serie
contained 51 images. To evaluate the performance of our method we compared the reconstruction
obtained with different scales of basis spline function: 3 interleaves, 5 interleaves 7 interleaves and
10 interleaves and also with different spline degree (0, 1, 3, 5 referred to as sp0, sp1, sp3, sp5). The
time points on which the reconstruction was performed were kept identical for all reconstruction
method in order to allow comparison between them. Figure 6.3 shows the reference phantom in
systole and diastole and its temporal profile. We also varied the number of interleaves taken into
account for the reconstruction (3, 5, 7, 10 interleaves) to evaluate the effect of the scale of the basis
function.
Figure 6.3 — Numerical phantom used as a reference for k-t reconstruction, in diastole (left), in systole
(middle) and the temporal profile of the diagonal pixels of the image (right).
113
6.3. Evaluation on numerical phantom
Modification of the spiral trajectory
The classical spiral trajectory [127, 128] is based on a rotation of an angle 2π/Nb of interleaves.
This ensures an homogeneous coverage of k-space when all the interleaves are acquired. However, if
we consider only 3 consecutive interleaves, we end up with a very inhomogeneous k-space coverage
as it is illustrated on figure 6.4. One way to resolve this problem was first proposed in the context
of radial sampling by Winkelmann et al. [181] and then by Kim et al. for spiral sampling [182]. The
proposition is to rotate the spiral interleave by 222.4969◦ · (n − 1), where n is the actual interleave
indice, this angle is referred to as the golden-ratio angle. By using this angle we observe a more
homogeneous coverage of k-space.
k
y
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Figure 6.4 — Spiral trajectory, with constant rotation of 2π/Nb of interleaves between interleaves (upper)
and with golden ratio angle rotation between interleaves (lower). Left, 10 consecutive interleaves of the
trajectory, rigth, 4 sets of 3 consecutive interleaves that will be used for image reconstruction.
Reference method - reconstruction by gridding
As a reference method, we used the gridding reconstruction associated with the sliding window
method. The reconstruction time points were chosen the same as for the spline-based reconstructions to allow fair comparison between methods. The gridding algorithm was using the density
compensation function proposed by Johnson et al. [183] instead of the Voronoi diagram as it was
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done in the previous chapter, since the authors recently shown this method to be more efficient.
The sliding window technique was proposed by Riederee at al. [184] and allows the reconstruction
of images on chosen time points by using all informations needed to fill completely a k-space, illustrated in figure 6.5. In principle sliding window can enhance the time resolution however, the
intrinsic time resolution of one image is nevertheless constrained by the time needed to acquire the
entire k-space.
Figure 6.5 — Scheme of the classical reconstruction technique (upper) compared to the sliding window
technique (lower). In the classical method, the reconstructed time resolution depends on the time needed
to acquire the entire k-space(in this example composed of 10 spiral interleaves) while in the sliding window
technique the reconstructed time points can be arbitrary chosen, in this case, one interleave is involved into
the reconstruction of several frames.
6.3.2
Results
SNR evaluation
Figure 6.6 represents the diagonal profile of the images as a function of time for gridding and k − t
SPIRE. To evaluate the reconstruction quality we calculated the SNR against the analytical ground
truth with the relation 5.33 (see p. 97). In figure 6.7, we show the results obtained.
115
sp0
sp1
sp3
sp5
time (a.u.)
time (a.u.)
time (a.u.)
time (a.u.)
time (a.u.)
10 interleaves
7 interleaves
5 interleaves
3 interleaves
gridding
Figure 6.6 — Temporal profile of pixels on diagonal for all methods (gridding and spline-based with
different degree, sp0, sp1, sp3, sp5) and the different number of interleaves taken as basis scale (3, 5, 7, 10).
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Figure 6.7 — SNR for the different reconstruction methods (gridding, sp0, sp1, sp3, sp5) as a function of
the scale of the basis function (3,5,7,and 10 interleaves respectively).
The temporal profiles as well as the SNR measurements illustrate clearly that increasing the
spline degree or enhancing the scale of the basis function (i.e. the number of interleaves) both enhance the SNR. Indeed, we note few improvement of SNR with sp5 compared to sp3, in fact cubic
B-spline is known to already perform very well in many interpolation applications [185]. However,
this observation is true up to a certain limit. Indeed, SNR measured for sp3 and sp5 decreases
when increasing the number of interleaves due to the oversampling of data over the time direction.
This fact is correlated with the observation of blurred edges because the data considered in the
reconstruction cover a time interval during which the motion of the object become important. This
observation is also true for sp1 but the decrease of SNR occurs later, i.e. for 10 interleaves. Gridding
and sp0 exhibit a constant increase in SNR when increasing the number of interleaves.
Another important result is that the gridding method exhibit a reduced SNR compared with
k − t SPIRE, all basis function scales taken together. In particular, sp0 on 10 interleaves has a
better SNR than gridding associated with sliding window. As a recall, sp0 is the nearest neighbor
reconstruction (or sliding window), thus no time weighting is introduced in this model. That means
that the model-based reconstruction, with only a spatial spline image model already gives better
results than the gridding reference method. These results are comparable with the one observed in
the previous chapter with the SPIRE method.
Evaluation of the edge preservation
SNR is an important global measure for the evaluation of an image reconstruction technique, but
another critical factor is the conservation of edge definition, which depends on two factors:
The sampling over k-space: Blurring can be introduced due to data undersampling.
The sampling over the time: If the motion of the object during the time needed to sample the
k-space is non negligible, it can introduce blurring. This will be referred to as blurring due to
an oversampling in time.
117
To evaluate the edge preservation we define a metric called “edge SNR”. This factor is calculated
by angularly averaging the edge profile for each normalized image in the time serie and by calculating
the gradient in y direction∗ . The normalization is an important step to allow comparison between
the different reconstructions. The detailed algorithm is the following:
1. Image serie normalization between 0 and 1
2. Angular averaging of the image profile
3. Repeat step 2 for each time frame
4. Perform the gradient on vertical direction
5. Select only the area corresponding to the endocardial transition (this choice is arbitrary since
endocardial and epicardial transition visually give the same gradient, so globally the same
information about edge preservation). Select means put all other pixels of the image equal to
zero
6. Select only phases corresponding to motion i.e., the columns 6 to 15, since they correspond to
images where edge definition is more critical
7. Perform SNR on the resulting image with relation 5.33, referred to as “edge SNR”
Figure 6.8 illustrates some of these steps and figure 6.9 shows the gradient obtained for the
different reconstructions performed.
Figure 6.8 — Representative steps performed to determine the edge SNR. The example is given on the
reference images. The angularly averaged profile (marked with the blue line) is computed on the normalized
image (a). The result in (b) is plotted as a function of time (c). The endocardial contour on the gradient
image is selected (d) and the SNR is performed on the signal corresponding to the highest motion (blue
square corresponds to phases 6 to 15).
∗ Angular
averaging makes sense for a numerical phantom.
real-time
gridding
sp0
sp1
sp3
sp5
time (a.u.)
time (a.u.)
time (a.u.)
time (a.u.)
time (a.u.)
10 interleaves
7 interleaves
5 interleaves
3 interleaves
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Figure 6.9 — Gradient of the angularly averaged profile as a function of time for all tested reconstructions.
Only the endocardial border is represented. The window width and window level were kept identical between
all images. These images will be used to calculate edge SNR.
119
Figure 6.10 — Edge SNR for the different reconstructions (gridding, sp0, sp1, sp3, sp5) as a function of
the scale of the basis function (3, 5, 7 and 10 interleaves respectively).
Gridding and sp0 both show an increase of the edge SNR when interleave number is increased,
corresponding to better k-space sampling. This observation is also true for sp1 and sp3 up to 5 interleaves after which the edge SNR falls down in this case due to an oversampling of data over time.
Similarly, for sp5 the edge SNR decreases while the number of interleaves increases. These results are
correlated with the observation of blurry gradient profiles on figure 6.9. The results suggest that the
sp1 on 5 interleaves is the method giving the best edge SNR, closely followed by sp0 on 10 interleaves.
Moreover, edge SNR associated to gridding with sliding window (10 interleaves) is lower than at
least one of the proposed k − t SPIRE independently of the interleave number. This is an important result signifying that k − t SPIRE is able to better preserve edge definition in an image serie
even with a significant undersampling of data ( 3 or 5 interleaves) than the classical reference method.
Finally, we plotted the edge SNR versus the global image SNR in figure 6.11 to observe the
tradeoff between both parameters knowing that the ideal case would be a high SNR with a preserved
edge SNR. In order to increase the time resolution of an image serie, the choice of sp5 on 3 interleaves
gives the largest SNR with moreover a better edge SNR than the one obtained with the reference
reconstruction (gridding + sliding window). Other reconstructions give also very good tradeoff
between SNR and edge SNR like sp3 on 3i or sp5 on 5i. However, in some situations which would
require a high edge SNR tolerating a reduction of global SNR, the choice should therefore be either
sp0 on 10i or sp1 on 5i or 7i. In the case of sp0 on10i, care must be taken regarding the blurring
that can be introduced by the too important object motion during the same acquisition that may
not be resolved by the sampling of the full k-space in certain situations.
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Figure 6.11 — Edge SNR versus SNR for the different reconstructions with 3, 5, 7 and 10 interleaves.
Points of each curve respectively represents the gridding, sp0, sp1, sp3, sp5 reconstructions.
6.4. k − t SPIRE for real-time cardiac imaging
6.4
6.4.1
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k − t SPIRE for real-time cardiac imaging
Methods
In vivo real-time images were acquired in a healthy volunteer, the sequence was adapted to use the
golden ratio angle rotation between interleaves (see section 6.3.1, p.113) and the parameters were
the followings:
• 10 interleaves
• variable density αD = 3
• matrix size N= 64
• FOV = 128 mm
• TE/TR = 0.8 / 12 ms
• slice thickness = 5 mm
• fa = 30◦
• matrix body coil (2 elements)
• number of slices = 2
The two slices were located one near the apex and one near the base and are acquired simultaneously. The time needed to acquire the two full k-space is then 10x2x12 ms = 240 ms. The
reconstruction time points were chosen to give a final time resolution of 60 ms, corresponding to
the time needed to acquire 3 interleaves. As in the numerical experiment, reconstructions were
performed with a basis function scale of 3, 5, 7 and 10 interleaves and the spline degree chosen were
0, 1, 3 and 5.
6.4.2
Results
Figure 6.12 shows the reconstructions obtained for the apex and the basis slices with the reference
method and with k − t SPIRE for sp5 on 3 interleaves. These images are compared to the cine
sequence that was acquired as the anatomical reference. Figure 6.13 shows the resulting images for
the slice located in the apex.
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Figure 6.12 — Images of apex and basis slices reconstructed with gridding and k −t SPIRE compared with
cine sequence in diastole. Gridding images (middle column) were reconstructed on 10 interleaves whereas
k − t SPIRE was performed with sp5 on 3 interleaves in this example. White line in gridding shows the
pixels where the temporal profiles will be done.
Figure 6.14 shows the temporal profile of pixels under the white line in figure 6.13. Visual
inspection reveals similar results as the ones obtained with the numerical phantom, i.e. reconstructions performed with a low number of interleaves need a high degree spline to have a correct image
definition, whereas with higher number of interleaves too high degree spline (sp3 or sp5) introduce
a temporal blurring. Figure 6.15 shows a selection of performed reconstructions. The reference
sequence (gridding on 10 interleaves) is compared to k − t SPIRE images, apparently exhibiting a
reasonable image definition without too important temporal blurring.
sp0
sp1
sp3
sp5
10 interleaves
7 interleaves
5 interleaves
3 interleaves
Gridding
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Figure 6.13 — Gridding and k − t SPIRE images for apex slice, for different scale of the basis function (3,
5, 7 and 10 interleaves).k − t SPRE was performed with sp0, sp1, sp3 and sp5.
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The myocardium contraction occurring at systole (black arrow in figure 6.15) is better defined in
sp3 and sp5 on 3 interleaves than in gridding on 10 interleaves. The other reconstructions, exhibit
an enlarged transition like in gridding, signifying that they are affected by temporal blurring as well.
To illustrate this observation, figure 6.16 shows the systolic contraction with gridding and with sp5
on 3i. We observe that the left ventricular cavity has not the same size and shape between both
reconstruction methods. Since the k − t SPIRE is performed with a basis function of 3 interleaves it
is more likely that this method is able to better retrieve the real contraction informations compared
with gridding that is performed on 10 interleaves.
sp0
sp1
sp3
sp5
time (a.u.)
time (a.u.)
time (a.u.)
time (a.u.)
time (a.u.)
10 interleaves
7 interleaves
5 interleaves
3 interleaves
Gridding
Figure 6.14 — Temporal profiles of apex slice for gridding and k − t SPIRE with different scales of the basis function.
125
Figure 6.15 — Selection of temporal profiles of apex slices for reconstructions visually giving a good image
definition without an important temporal blurring.
Figure 6.16 — Reconstructed frames around the systole for, upper, the reference sequence (gridding on
10 interleaves) and, lower, k − t SPIRE sp5 on 3 interleaves. The left ventricular cavity has not the same
size and shape when comparing the 2 reconstruction methods.
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real-time
Figure 6.17 shows the gradient in vertical direction performed onto the temporal profiles of
gridding and k − t SPIRE. Figure shows the same informations obtained with CINE images. We
observe that this sequence, which is very well resolved in time, gives thinner gradients at edges.
Black arrows show the area where gridding has larger gradients compared to k − t SPIRE where
gradients are thinner, meaning sharper edge transition since gradient widening is another indicator
of temporal blurring. Figure 6.18 shows the same result for slice located at basis. Gradient is thinner
in epicardial transition in k − t SPIRE than in gridding and is better visible at endocardium.
Figure 6.17 — Comparison of temporal profiles and gradient of CINE, gridding (on 10 interleaves) and
k − t SPIRE (sp5 on 3 interleaves) for apex slice. Black arrows indicate location where gradient is thinner
in k − t SPIRE than in gridding, indicating less blurring by time oversampling.
127
Figure 6.18 — Comparison of temporal profiles and gradient of CINE, gridding (on 10 interleaves) and
k − t SPIRE (sp5 on 3 interleaves) for basis slice. Black arrows indicate location where gradient is thinner
(at epicardium), and also where it is better visible (at endocardium) in k − t SPIRE than in gridding.
6.4.3
Reconstruction artifacts
Point spread function of k − t SPIRE
We can define the point spread function (PSF) as the resulting image when k-space data contains
only values equal to one. It is the equivalent to observing the effect of the sampling on the reconstructed data for a given method of reconstruction. Under ideal conditions, when the full k-space is
sampled, the PSF would be a Dirac Delta function. The visualization of the PSF is a good indicator
of the reconstruction method and the potential image quality.
Figure 6.19 shows the PSF obtained with the different reconstructions methods. We observe
that undersampling k-space introduces important signal from side lobes. This signal is decreasing
when the interleaves number is increasing (or approaching Nyquist criterion) or when spline degree
is increased.
The temporal profiles of the PSF show the presence of a repeated pattern of side lobes in undersampled trajectories (3 and 5 interleaves) that is due to the different k-space trajectories between
each reconstructed phase. This artifact was also visible in the temporal profile of reconstructed real
data (see figure 6.14) and is mainly due to the undersampling. Once again, this artifact is drastically
reduced by enhancing the spline degree in k − t SPIRE reconstruction.
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Finally, we observe that gridding on 10 interleaves contains a lot of signal in the side lobes,
corresponding to the sinc interpolation of this algorithm, as attended spline interpolation largely
reduces these side lobes.
sp0
sp1
sp3
sp5
10 interleaves
7 interleaves
5 interleaves
3 interleaves
Gridding
Figure 6.19 — PSF for gridding and k − t SPIRE for different number of interleaves (3, 5, 7, and 10) and
different spline degree (0, 1, 3 and 5). Images are scaled between 0 and 10% of the peak value.
sp0
sp1
sp3
sp5
10 interleaves
7 interleaves
5 interleaves
3 interleaves
Gridding
129
Figure 6.20 — Temporal profile of the central line of the PSF for gridding and k − t SPIRE, for different
number of interleaves (3, 5, 7, and 10) and different spline degree (0, 1, 3 and 5). Images are scaled between
0 and 10% of the peak value.
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White border artifact
In figures 6.12 and 6.13, we observe that the k − t SPIRE images exhibit a white bordering effect,
reminding the vignetting of old photographs. This artifact can easily be removed either by cropping
the image or by reconstructing it into a larger support (i.e. N=128 instead of 64) and cropping to
recover the original size. We did not use this last option since the increase of matrix size implies
also the increasing of reconstruction time.
Since this artifact was not present with the reconstructions of the numerical phantom which
objects are entirely contained into the field of view, we suppose that this signal mainly comes from
structures in contact with edges of the image enhanced by the multiple convolutions present into
the algorithm.
Grid-like artifact
We observed in figure 6.13 that the k − t SPIRE images present a shadow representing a grid.
This artifact is more pronounced for highly undersampled data and low spline degree. To try to
understand the origin of such an artifact we used the numerical phantom and we performed several
reconstructions. We first change the matrix size for the reconstruction and we observe (see figure
6.21) that the size of the grid depends on this parameter. When the support is larger, the grid is
also larger, and if it is high enough it can be completely removed, as it is the case in this example
with N=256. We also reconstructed the same data but with a variable density trajectory instead of
uniformly sampled data. In this case, the grid artifact is no more present.
Figure 6.21 — From left to right, k − t SPIRE images of the sp0 on 3 interleaves reconstruction of the
numerical phantom for a uniform density spiral sampling on N=64, 128, 256 and with a variable density
spiral (αD = 3) for N=64. Bottom line represents zooms of the central part of the reconstructions.
Similarly, when we reconstruct our real data on a larger support (N=128), we observe the
disappearance of the grid-like artifact as it is illustrated in figure 6.22 on an image where this artifact
is important. This would be a very simple way to resolve this artifact as well as the white border
6.5. Comparison with existing reconstruction methods
131
artifact, however, to achieve a comparable image convergence we had to perform 1000 iterations
instead of 200 for the smaller matrix (N=64) which is more time consuming. As an example the
complete calculation of the image serie (G and D matrices calculations + iterative algorithm) with
N=64 took 1 hour 40 min and 8 hours 18 min with N= 128. The detailed calculation time is given
in table 6.1.
Table 6.1 — Detailed calculation times for sp5 on 3 interleaves on matrix size N=64 and N=128. Reconstruction of 45 frames for one of the two channel coils.
G
D
iterative algorithm
N=64
75’
20’
5’
N=128
4h36’
1h08’
2h34’
Figure 6.22 — From left to right, k − t SPIRE images of the sp5 on 3 interleaves reconstruction of the apex
slice on N=64, 128 and the zoom of the central part of N=128. Images show the reconstruction performed
for only one of the two channel coils.
6.5
Comparison with existing reconstruction methods
Among the large variety of methods proposed to accelerate data acquisitions, we have seen (section
1.3.2), that methods such as UNFOLD [27] or k − t BLAST [28] use information contained in the
temporal spectrum of the data to remove aliasing from images. Moreover, k−t BLAST was extended
to non-Cartesian sampling by Hansen et al. [186] and another method to resolve aliasing by using
temporal frequency information specifically applied to undersampled spiral k-space trajectories was
proposed by Shin et al. [187].
These methods share the property that aliasing due to undersampling can be resolved under
certain conditions by unfolding the aliased spectra, which implies decoupling of the a previous reconstruction of images and restoring the resolution in x − f space. In k − t SPIRE, we use the
temporal information of samples directly in the reconstruction process and thus resolve the aliasing
problem in one single operation. Moreover, the image reconstruction proposed with the previous
methods is done with the gridding algorithm. In the context of spiral sampling, this implies the
assumption that a set of spiral interleaves is acquired at one single time-point, which is not true.
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Figure 6.23 shows the result of the reconstruction obtained if we considered in k −t SPIRE that a set
of 3 interleaves was acquired at a single time point, the reconstruction shown is performed with sp5
on 3 interleaves. We observe that we loose the temporal accuracy previously demonstrated, together
with the enhancement of artifacts. This demonstrates the weakness of this too simple assumption.
To our knowledge, no reconstruction method has yet considered explicitly the time sampling
information. In Cartesian sampling, a group of line in k-space or at least one line is always considered
to be acquired at a single time-point. For non-Cartesian sampling, the same approach is used
considering either that a group of rays or a set of spiral interleaves are acquired at a single time-point.
This is also related to the fact that gridding reconstruction is very often used (alone or associated
with regularization methods or iterative algorithms to resolve the undersampling artifacts). Gridding
is inherently supposing that all data samples are acquired at the same temporal time point since
the samples are placed in a plane, or a grid located at one time of interest. Even though the latter
simplification on time sampling assumption is not a problem for a large variety of applications,
we have seen that introducing the real temporal information about each data sample improves the
image accuracy in the case of highly accelerated acquisitions.
Figure 6.23 — Comparison of temporal profiles for apex slice of gridding (on 10 interleaves), k − t SPIRE
(sp5 on 3 interleaves) and k −t SPIRE with the assumption that sets of 3 interleaves are acquired at the same
time point. Black arrows indicate location of the degradation of temporal resolution as well as enhancement
of artifacts.
Finally, although not our main objective, we have demonstrated the ability of our method to
produce real-time images with a time resolution of 60 ms for a simultaneous acquisition of 2 slices,
meaning only one slice can be acquired with 30 ms time resolution. This is comparable with recent
studies that reported a temporal resolution of 20 ms with spiral [188] and radial [179] sampling
schemes. In this last study, parallel imaging was introduced to further accelerate the acquisition.
We did not yet considered the improvement that can be given by combining parallel imaging with
our method, but we can expect further acceleration. Still, this point has to be further explored in
future studies.
Furthermore, we have to recall that in k − t SPIRE no assumption is made about the spatiotemporal distribution of the data and no a priori information is added into the model, this technique
can thus be integrated with a lot of other existing algorithms, and applications that benefit from
accelerated acquisitions. Most notably, additional regularization terms could be considered, such as
6.5. Comparison with existing reconstruction methods
133
sparsity in the wavelet domain and compressed sensing [175, 176]. Moreover, since the least-square
model fitting already works very well, some more acceleration or further undersampling may be
excepted by the addition of a regularization term.
Finally, applications of k − t SPIRE may also have a real interest in situations where quantitative
assessment of contrast is important and varies importantly in a short time window, such as cardiac
perfusion [189] or hyperpolarized studies [190, 191].
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real-time
7
Conclusions and
perspectives
The aim of this thesis was to adopt a translational approach for the development of new techniques
to provide an efficient characterization of the myocardium by means of viability and function assessment. At the “bench” level, this was performed by developing MEMRI and a highly time-resolved
cine sequence. Concerning the “bedside”, investigation of spiral imaging combined with a new image
reconstruction method appeared to be well-suited to real-time application in humans.
7.1
MEMRI and highly time-resolved cine
Manganese is known to be an efficient marker of viability in various animal models. In this work,
the ability to use MEMRI in a mouse model on a clinical scanner was first demonstrated, and the
accuracy of infarction quantification with this technique was also confirmed in an ischemia reperfusion model that has not previously been explored. Moreover, Manganese kinetics showed that acute
and chronic infarction expressed different accumulation behavior of this contrast agent, indeed we
observed a fast entry of Mn2+ into acute infarction whereas scar tissue experienced a slow accumulation of Mn2+ .
On one hand MEMRI offered an efficient tool for the differentiation of acute and chronic infarction but in the other hand it opened up several questions concerning the accumulation mechanism of
this ion into cells. This mechanism is altered by several factors such as perfusion, diffusion, extracellular space and specific channel activity. The answers to these questions are not straightforward and
need deeper studies. The study of kinetics in other infarction models such as permanent ligation,
where perfusion is altered in a different manner, and at longer time points where inflammatory cells
are no longer present at the infarction site, can give some clues to better understand the underlying
mechanisms. Other specific contrast agents can be tested in combination with Mn2+ such as extracellular or intravascular Gd3+ chelates to correlate their respective distributions in the different
compartments to our observations. In parallel, histological studies of the cells at the infarct site as
well as immunohistochemistry experiments will help the characterization of the tissue composition
and organization.
135
136
Chapter 7. Conclusions and perspectives
Small animal studies presented also a technical challenge concerning cine imaging. The combination of high spatial and temporal resolution requirements needs specific alternatives to overcome
the gradient limitations imposed by clinical scanners. A hardware solution could be to increase
gradient strength by using specific gradient inserts. Otherwise, we have to act on the software part
of the system, i.e. data acquisition. The proposed cine interleaved sequence allowed enhancement
of the temporal resolution up to 6.5 ms and was validated with mass and global function measurements. The next step is to apply this sequence to cardiac stress studies and ideally to compare the
global function results with a reference method that could be performed on a dedicated scanner.
In parallel, the potential benefit interleaved cine can provide to sequences where TR is lengthened
by preparations or complex combinations, such as tagging, has to be explored. Indeed, our group
has an important experience with tagging in rats and cine interleaved could allow tagging to be
performed in mice and/or in stress studies.
7.2
Spiral imaging
In this work, we have proposed an innovative reconstruction method applied to the spiral sampling
scheme, but that can also be used with arbitrary trajectories. We introduced a spline-based image
model, that is in itself already widely recognized but that was not yet been applied to MR image
reconstruction. This model has several conceptual advantages over the classical sinc interpolation.
The evaluation of Spline-based image reconstruction (SPIRE) gave robust and significantly improved
SNR results compared to the gridding reconstruction, but nevertheless introduced some blurring artifacts that may be a limitation for clinical applications. However, the potential of SPIRE was really
revealed with its extension to the time domain. The important improvement given by k − t SPIRE
compared with gridding and sliding window reconstruction on numerical simulations was confirmed
with real-time cardiac data reconstruction. The introduction of the temporal information related
to each data sample is the main improvement given to the algorithm and allows efficient aliasing
artifact resolution.
Obviously, these first experiments have to be confirmed, in particular in the context of cardiac
stress studies where high temporal resolution real-time imaging is the most challenging. Quantitative evaluation of the benefit provided by k − t SPIRE should be compared to more classical
methods (gridding and sliding window, highly accelerated imaging and parallel imaging) in in-vivo
conditions. Moreover, the coupling of this method with parallel imaging techniques such as SENSE
should be performed either to further accelerate, or to enhance the image spatial resolution.The
addition of a regularization method has also to be investigated, in particular in the current context
of continuously growing interest on compressed sensing.
As a further perspective, spiral imaging and k − t SPIRE should be applied to small animal
functional imaging. We started to investigate this application but due to technical limitations this
could not be achieved during the time of this thesis. However, the requirements of rodent imaging
could be efficiently fulfilled with this strategy in order to improve the quantitative evaluation of
function. Finally, other applications can benefit from acceleration potential of k − t SPIRE, in
particular in which quantification of signal intensity is important, such as perfusion imaging or
hyperpolarized studies.
Bibliography
[1] “World Health Organization. Cardiovascular diseases (CVDs)”,
source
http://www.who.int/mediacentre/factsheets/fs317/en/, vol. Fact sheet NÂ◦ 317, 2009.
:
[2] S Capewell, E Ford, J Croft, J Critchley, K Greenlundc, and D Labarthe, “Cardiovascular risk
factor trends and potential for reducing coronary heart disease mortality in the united states
of america”, Bulletin of the World Health Organization, vol. 88, pp. 120–130, Jan 2010.
[3] Valentin Fuster and Torsten P Vahl, “The role of noninvasive imaging in promoting cardiovascular health”, J Nucl Cardiol, vol. 17, no. 5, pp. 781–790, Jan 2010.
[4] Edward T D Hoey, “Cardiovascular mri in patients with ischaemic heart disease: from science
to practice?”, Postgrad Med J, vol. 86, no. 1019, pp. 513–4, Sep 2010.
[5] American College of Cardiology Foundation Task Force on Expert Consensus Documents,
W Gregory Hundley, David A Bluemke, J Paul Finn, Scott D Flamm, Mark A Fogel,
Matthias G Friedrich, Vincent B Ho, Michael Jerosch-Herold, Christopher M Kramer, Warren J Manning, Manesh Patel, Gerald M Pohost, Arthur E Stillman, Richard D White, and
Pamela K Woodard, “Accf/acr/aha/nasci/scmr 2010 expert consensus document on cardiovascular magnetic resonance: a report of the american college of cardiology foundation task
force on expert consensus documents”, J Am Coll Cardiol, vol. 55, no. 23, pp. 2614–62, Jun
2010.
[6] C. P. Slichter, Principles of Magnetic Resonance, Springer-Verlag Berlin Heidelberg New
York, 1978.
[7] E. M. Haacke, R. Brown, M. R. Thompson, and R. Venkatesan, Magentic resonance imaging
: physical priniples and sequence design, John Wiley & Sons, 1999.
[8] M. A. Bernstein, K. King, and X. J. Zhou, Handbook of MRI Pulse Sequences, Elsevier
Academic Press, 2004.
[9] II Rabi, JR Zacharias, S Millman, and P Kusch, “A new method of measuring nuclear magnetic
moment”, Phys Rev, vol. 53, no. 4, pp. 318–318, Jan 1938.
[10] P. C. Lauterbur, “Image formation by induced local interactions - examples employing nuclear
magnetic resonance”, Nature, vol. 242, no. 5394, pp. 190–191, Jan 1973.
[11] P. Mansfield and P. K. Grannell, “Nmr diffraction in solids”, J Phys C Solid State, vol. 6, no.
22, pp. L422–L426, Jan 1973.
137
138
Bibliography
[12] G. J. Stanisz, E. E. Odrobina, J. Pun, M. Escaravage, S. J. Graham, M. J. Bronskill, and
R. M. Henkelman, “T1, T2 relaxation and magnetization transfer in tissue at 3T”, Magn
Reson Med, vol. 54, no. 3, pp. 507–12, 2005.
[13] D. J. Tyler, L. E. Hudsmith, S. E. Petersen, J. M. Francis, P. Weale, S. Neubauer, K. Clarke,
and M. D. Robson, “Cardiac cine MR-imaging at 3T: FLASH vs SSFP”, J Cardiovasc Magn
Reson, vol. 8, no. 5, pp. 709–15, 2006.
[14] Dana C Peters, Reza Nezafat, Holger Eggers, Christian Stehning, and Warren J Manning,
“2d free-breathing dual navigator-gated cardiac function validated against the 2d breath-hold
acquisition”, J Magn Reson Imaging, vol. 28, no. 3, pp. 773–7, Sep 2008.
[15] Scott Beach and Mushabbar A Syed, “Current and upcoming roles of ct and mri in clinical
cardiac imagery”, Curr Cardiol Rep, vol. 9, no. 5, pp. 420–7, Sep 2007.
[16] O. Wieben, C. Francois, and S. B. Reeder, “Cardiac MRI of ischemic heart disease at 3T:
Potential and challenges”, Eur J Radiol, vol. 65, no. 1, pp. 15–28, 2008.
[17] M. D. Cerqueira, N. J. Weissman, V. Dilsizian, A. K. Jacobs, S. Kaul, W. K. Laskey, D. J.
Pennell, J. A. Rumberger, T. Ryan, and M. S. Verani, “Standardized myocardial segmentation and nomenclature for tomographic imaging of the heart: a statement for healthcare
professionals from the Cardiac Imaging Committee of the Council on Clinical Cardiology of
the American Heart Association”, Circulation, vol. 105, no. 4, pp. 539–42, 2002.
[18] K. S. Nayak, C. H. Cunningham, J. M. Santos, and J. M. Pauly, “Real-time cardiac MRI at
3 tesla”, Magn Reson Med, vol. 51, no. 4, pp. 655–60, 2004.
[19] P B Roemer, W A Edelstein, C E Hayes, S P Souza, and O M Mueller, “The nmr phased
array”, Magn Reson Med, vol. 16, no. 2, pp. 192–225, Nov 1990.
[20] D K Sodickson and W J Manning, “Simultaneous acquisition of spatial harmonics (smash):
fast imaging with radiofrequency coil arrays”, Magn Reson Med, vol. 38, no. 4, pp. 591–603,
Oct 1997.
[21] K. P. Pruessmann, M. Weiger, M. B. Scheidegger, and P. Boesiger, “SENSE: sensitivity
encoding for fast MRI”, Magn Reson Med, vol. 42, no. 5, pp. 952–62, 1999.
[22] M. A. Griswold, P. M. Jakob, R. M. Heidemann, M. Nittka, V. Jellus, J. Wang, B. Kiefer,
and A. Haase, “Generalized autocalibrating partially parallel acquisitions (GRAPPA)”, Magn
[23] M Weiger, K P Pruessmann, A Kassner, G Roditi, T Lawton, A Reid, and P Boesiger,
“Contrast-enhanced 3d mra using sense”, J Magn Reson Imaging, vol. 12, no. 5, pp. 671–7,
Nov 2000.
[24] R A Jones, O Haraldseth, T B Müller, P A Rinck, and A N Oksendal, “K-space substitution:
a novel dynamic imaging technique”, Magn Reson Med, vol. 29, no. 6, pp. 830–4, Jun 1993.
[25] J J van Vaals, M E Brummer, W T Dixon, H H Tuithof, H Engels, R C Nelson, B M Gerety,
J L Chezmar, and J A den Boer, “”keyhole” method for accelerating imaging of contrast agent
uptake”, J Magn Reson Imaging, vol. 3, no. 4, pp. 671–5, Jan 1993.
Bibliography
139
[26] M Doyle, E G Walsh, G G Blackwell, and G M Pohost, “Block regional interpolation scheme
for k-space (brisk): a rapid cardiac imaging technique”, Magn Reson Med, vol. 33, no. 2, pp.
163–70, Feb 1995.
[27] B Madore, G H Glover, and N J Pelc, “Unaliasing by fourier-encoding the overlaps using the
temporal dimension (unfold), applied to cardiac imaging and fmri”, Magn Reson Med, vol. 42,
no. 5, pp. 813–28, Nov 1999.
[28] J. Tsao, P. Boesiger, and K. P. Pruessmann, “k-t BLAST and k-t SENSE: dynamic MRI with
high frame rate exploiting spatiotemporal correlations”, Magn Reson Med, vol. 50, no. 5, pp.
1031–42, 2003.
[29] Michael S Hansen, Sebastian Kozerke, Klaas P Pruessmann, Peter Boesiger, Erik M Pedersen,
and Jeffrey Tsao, “On the influence of training data quality in k-t blast reconstruction”, Magn
Reson Med, vol. 52, no. 5, pp. 1175–83, Nov 2004.
[30] Jeffrey Tsao, “On the unfold method”, Magn Reson Med, vol. 47, no. 1, pp. 202–7, Jan 2002.
[31] P Kellman, F H Epstein, and E R McVeigh, “Adaptive sensitivity encoding incorporating
temporal filtering (tsense)”, Magn Reson Med, vol. 45, no. 5, pp. 846–52, May 2001.
[32] Han W Kim, Afshin Farzaneh-Far, and Raymond J Kim, “Cardiovascular magnetic resonance
in patients with myocardial infarction: current and emerging applications”, Dec 29 2009.
[33] Daniel R Messroghli, Kevin Walters, Sven Plein, Patrick Sparrow, Matthias G Friedrich,
John P Ridgway, and Mohan U Sivananthan, “Myocardial T1 mapping: application to patients
with acute and chronic myocardial infarction”, Jul 1 2007.
[34] Marie-France Bellin and Aart J Van Der Molen, “Extracellular gadolinium-based contrast
media: an overview”, Eur J Radiol, vol. 66, no. 2, pp. 160–7, May 2008.
[35] Kendra M Hasebroock and Natalie J Serkova, “Toxicity of mri and ct contrast agents”, Expert
Opin Drug Metab Toxicol, vol. 5, no. 4, pp. 403–16, Apr 2009.
[36] J Schwitter, M Saeed, M F Wendland, N Derugin, E Canet, R C Brasch, and C B Higgins,
“Influence of severity of myocardial injury on distribution of macromolecules: extravascular
versus intravascular gadolinium-based magnetic resonance contrast agents”, J Am Coll Cardiol,
vol. 30, no. 4, pp. 1086–94, Oct 1997.
[37] A. C. Silva, J. H. Lee, I. Aoki, and A. P. Koretsky, “Manganese-enhanced magnetic resonance
imaging (MEMRI): methodological and practical considerations”, NMR Biomed, vol. 17, no.
8, pp. 532–43, 2004.
[38] M E Bernardino, S W Young, J K Lee, and J C Weinreb, “Hepatic mr imaging with mn-dpdp:
safety, image quality, and sensitivity”, Radiology, vol. 183, no. 1, pp. 53–8, Apr 1992.
[39] A. Skjold, B. H. Amundsen, R. Wiseth, A. Stoylen, O. Haraldseth, H. B. Larsson, and P. Jynge,
“Manganese dipyridoxyl-diphosphate (MnDPDP) as a viability marker in patients with myocardial infarction”, J Magn Reson Imaging, vol. 26, no. 3, pp. 720–7, 2007.
[40] Jeff W M Bulte and Dara L Kraitchman, “Iron oxide mr contrast agents for molecular and
cellular imaging”, NMR in biomedicine, vol. 17, no. 7, pp. 484–99, Nov 2004.
140
Bibliography
[41] Karin Montet-Abou, Jean-Luc Daire, Jean-Noël Hyacinthe, Manuel Jorge-Costa, Kerstin Grosdemange, François Mach, Alke Petri-Fink, Heinrich Hofmann, Denis R Morel, Jean-Paul Vallée, and Xavier Montet, “In vivo labelling of resting monocytes in the reticuloendothelial
system with fluorescent iron oxide nanoparticles prior to injury reveals that they are mobilized to infarcted myocardium”, Eur Heart J, vol. 31, no. 11, pp. 1410–20, Jun 2010.
[42] O. P. Simonetti, R. J. Kim, D. S. Fieno, H. B. Hillenbrand, E. Wu, J. M. Bundy, J. P. Finn,
and R. M. Judd, “An improved MR imaging technique for the visualization of myocardial
infarction”, Radiology, vol. 218, no. 1, pp. 215–23, 2001.
[43] P. Kellman, A. E. Arai, E. R. McVeigh, and A. H. Aletras, “Phase-sensitive inversion recovery
for detecting myocardial infarction using gadolinium-delayed hyperenhancement”, Magn Reson
Med, vol. 47, no. 2, pp. 372–83, 2002.
[44] W. D. Gilson and D. L. Kraitchman, “Cardiac magnetic resonance imaging in small rodents
using clinical 1.5 T and 3.0 T scanners”, Methods, vol. 43, no. 1, pp. 35–45, 2007.
[45] Bénédicte M A Delattre, Vincent Braunersreuther, Jean-Noël Hyacinthe, Lindsey A Crowe,
François Mach, and Jean-Paul Vallée, “Myocardial infarction quantification with manganeseenhanced mri (memri) in mice using a 3t clinical scanner”, NMR in biomedicine, vol. 23, no.
5, pp. 503–13, Jun 2010.
[46] F. H. Epstein, “MR in mouse models of cardiac disease”, NMR Biomed, vol. 20, no. 3, pp.
238–55, 2007.
[47] A. Jacquier, C. B. Higgins, and M. Saeed, “MR imaging in assessing cardiovascular interventions and myocardial injury”, Contrast Media Mol Imaging, vol. 2, no. 1, pp. 1–15, 2007.
[48] H. Sakuma, “Magnetic resonance imaging for ischemic heart disease”, J Magn Reson Imaging,
vol. 26, no. 1, pp. 3–13, 2007.
[49] M. H. Mendoncadias, E. Gaggelli, and P. C. Lauterbur, “Paramagnetic Contrast Agents in
Nuclear Magnetic-Resonance Medical Imaging”, Seminars in Nuclear Medicine, vol. 13, no. 4,
pp. 364–376, 1983.
[50] Patrick F Antkowiak, Sarah A Tersey, Jeffrey D Carter, Moriel H Vandsburger, Jerry L Nadler,
Frederick H Epstein, and Raghavendra G Mirmira, “Noninvasive assessment of pancreatic betacell function in vivo with manganese-enhanced magnetic resonance imaging”, Am J Physiol
Endocrinol Metab, vol. 296, no. 3, pp. E573–8, Mar 2009.
[51] V. C. Culotta, M. Yang, and M. D. Hall, “Manganese transport and trafficking: lessons learned
from Saccharomyces cerevisiae”, Eukaryot Cell, vol. 4, no. 7, pp. 1159–65, 2005.
[52] S. Himeno, T. Yanagiya, and H. Fujishiro, “The role of zinc transporters in cadmium and
manganese transport in mammalian cells”, Biochimie, vol. 91, no. 10, pp. 1218–22, 2009.
[53] M. F. Wendland, “Applications of manganese-enhanced magnetic resonance imaging (MEMRI)
to imaging of the heart”, NMR Biomed, vol. 17, no. 8, pp. 581–94, 2004.
[54] B. Waghorn, T. Edwards, Y. Yang, K. H. Chuang, N. Yanasak, and T. C. Hu, “Monitoring
dynamic alterations in calcium homeostasis by T (1)-weighted and T (1)-mapping cardiac
manganese-enhanced MRI in a murine myocardial infarction model”, NMR Biomed, vol. 21,
no. 10, pp. 1102–11, 2008.
Bibliography
141
[55] V. Lukyanenko, A. Chikando, and W. J. Lederer, “Mitochondria in cardiomyocyte Ca2+
signaling”, Int J Biochem Cell Biol, vol. 41, no. 10, pp. 1957–71, 2009.
[56] T. C. Hu, R. G. Pautler, G. A. MacGowan, and A. P. Koretsky, “Manganese-enhanced MRI of
mouse heart during changes in inotropy”, Magn Reson Med, vol. 46, no. 5, pp. 884–90, 2001.
[57] G. A. Krombach, M. Saeed, C. B. Higgins, V. Novikov, and M. F. Wendland, “Contrastenhanced MR delineation of stunned myocardium with administration of MnCl(2) in rats”,
Radiology, vol. 230, no. 1, pp. 183–90, 2004.
[58] Y. Yang, M. L. Gruwel, J. Sun, P. Gervai, X. Yang, and V. V. Kupriyanov, “Manganeseenhanced MRI of acute cardiac ischemia and chronic infarction in pig hearts: kinetic analysis
of enhancement development”, NMR Biomed, vol. 22, no. 2, pp. 165–73, 2009.
[59] J. Bremerich, M. Saeed, H. Arheden, C. B. Higgins, and M. F. Wendland, “Normal and
infarcted myocardium: differentiation with cellular uptake of manganese at MR imaging in a
rat model”, Radiology, vol. 216, no. 2, pp. 524–30, 2000.
[60] T. C. Hu, T. F. Christian, A. H. Aletras, J. L. Taylor, A. P. Koretsky, and A. E. Arai,
“Manganese enhanced magnetic resonance imaging of normal and ischemic canine heart”, Magn
[61] S. Flacke, J. S. Allen, J. M. Chia, J. H. Wible, M. P. Periasamy, M. D. Adams, I. K. Adzamli,
and C. H. Lorenz, “Characterization of viable and nonviable myocardium at MR imaging: comparison of gadolinium-based extracellular and blood pool contrast materials versus manganesebased contrast materials in a rat myocardial infarction model”, Radiology, vol. 226, no. 3, pp.
731–8, 2003.
[62] T. C. Hu, W. Bao, S. C. Lenhard, T. R. Schaeffer, T. L. Yue, R. N. Willette, and B. M. Jucker,
“Simultaneous assessment of left-ventricular infarction size, function and tissue viability in a
murine model of myocardial infarction by cardiac manganese-enhanced magnetic resonance
imaging (MEMRI)”, NMR Biomed, vol. 17, no. 8, pp. 620–6, 2004.
[63] N. Ojha, S. Roy, J. Radtke, O. Simonetti, S. Gnyawali, J. L. Zweier, P. Kuppusamy, and
C. K. Sen, “Characterization of the structural and functional changes in the myocardium
following focal ischemia-reperfusion injury”, Am J Physiol Heart Circ Physiol, vol. 294, no. 6,
pp. H2435–43, 2008.
[64] P. C. Yang, P. Nguyen, A. Shimakawa, J. Brittain, J. Pauly, D. Nishimura, B. Hu, and
M. McConnell, “Spiral magnetic resonance coronary angiography–direct comparison of 1.5
Tesla vs. 3 Tesla”, J Cardiovasc Magn Reson, vol. 6, no. 4, pp. 877–84, 2004.
[65] J. N. Oshinski, Z. Yang, J. R. Jones, J. F. Mata, and B. A. French, “Imaging time after GdDTPA injection is critical in using delayed enhancement to determine infarct size accurately
with magnetic resonance imaging”, Circulation, vol. 104, no. 23, pp. 2838–42, 2001.
[66] Ankur Gupta, Vivian S Lee, Yiu-Cho Chung, James S Babb, and Orlando P Simonetti, “Myocardial infarction: optimization of inversion times at delayed contrast-enhanced mr imaging”,
Radiology, vol. 233, no. 3, pp. 921–6, Dec 2004.
[67] M. Bruvold, W. Nordhoy, H. W. Anthonsen, H. Brurok, and P. Jynge, “Manganese-calcium
interactions with contrast media for cardiac magnetic resonance imaging: a study of manganese
chloride supplemented with calcium gluconate in isolated Guinea pig hearts”, Invest Radiol,
vol. 40, no. 3, pp. 117–25, 2005.
142
Bibliography
[68] J. L. Daire, J. N. Hyacinthe, I. Tatar, K. Montet-Abou, M. K. Ivancevic, K. Masterson,
M. Jorge-Costa, D. R. Morel, and J. P. Vallee, “In vivo myocardial infarct area at risk
assessment in the rat using manganese enhanced magnetic resonance imaging (MEMRI) at
1.5T”, Magn Reson Med, vol. 59, no. 6, pp. 1422–30, 2008.
[69] E. M. Winter, R. W. Grauss, D. E. Atsma, B. Hogers, R. E. Poelmann, R. J. van der Geest,
C. Tschope, M. J. Schalij, A. C. Gittenberger-de Groot, and P. Steendijk, “Left ventricular
function in the post-infarct failing mouse heart by magnetic resonance imaging and conductance catheter: a comparative analysis”, Acta Physiol (Oxf ), vol. 194, no. 2, pp. 111–22,
2008.
[70] R J van der Geest, V G Buller, E Jansen, H J Lamb, L H Baur, E E van der Wall, A de Roos,
and J H Reiber, “Comparison between manual and semiautomated analysis of left ventricular
volume parameters from short-axis mr images”, J Comput Assist Tomogr, vol. 21, no. 5, pp.
756–65, Jan 1997.
[71] J. P. Mahoney and W. J. Small, “Studies on manganese. 3. The biological half-life of radiomanganese in man and factors which affect this half-life”, J Clin Invest, vol. 47, no. 3, pp.
643–53, 1968.
[72] Y. S. Kang and J. C. Gore, “Studies of tissue NMR relaxation enhancement by manganese.
Dose and time dependences”, Invest Radiol, vol. 19, no. 5, pp. 399–407, 1984.
[73] C. J. Berry, D. R. Thedens, K. Light-McGroary, J. D. Miller, W. Kutschke, K. A. Zimmerman,
and R. M. Weiss, “Effects of deep sedation or general anesthesia on cardiac function in mice
undergoing cardiovascular magnetic resonance”, J Cardiovasc Magn Reson, vol. 11, no. 1, pp.
16, 2009.
[74] O. Dietrich, J. G. Raya, S. B. Reeder, M. F. Reiser, and S. O. Schoenberg, “Measurement
of signal-to-noise ratios in MR images: influence of multichannel coils, parallel imaging, and
reconstruction filters”, J Magn Reson Imaging, vol. 26, no. 2, pp. 375–85, 2007.
[75] M. T. Vivaldi, R. A. Kloner, and F. J. Schoen, “Triphenyltetrazolium staining of irreversible
ischemic injury following coronary artery occlusion in rats”, Am J Pathol, vol. 121, no. 3, pp.
522–30, 1985.
[76] R. Bolli and E. Marban, “Molecular and cellular mechanisms of myocardial stunning”, Physiol
Rev, vol. 79, no. 2, pp. 609–34, 1999.
[77] T. G. Hampton, I. Amende, K. E. Travers, and J. P. Morgan, “Intracellular calcium dynamics
in mouse model of myocardial stunning”, Am J Physiol, vol. 274, no. 5 Pt 2, pp. H1821–7,
1998.
[78] D. C. Medina, D. M. Kirkland, M. F. Tavazoie, Jr. Springer, C. S., and S. E. Anderson,
“Na+/Ca2+-exchanger-mediated Mn2+-enhanced (1)H2O MRI in hypoxic, perfused rat myocardium”, Contrast Media Mol Imaging, vol. 2, no. 5, pp. 248–57, 2007.
[79] Raymond J Kim, Timothy S E Albert, James H Wible, Michael D Elliott, John C Allen, Jennifer C Lee, Michele Parker, Alicia Napoli, Robert M Judd, and Gadoversetamide Myocardial
Infarction Imaging Investigators, “Performance of delayed-enhancement magnetic resonance
imaging with gadoversetamide contrast for the detection and assessment of myocardial infarction: an international, multicenter, double-blinded, randomized trial”, Circulation, vol. 117,
no. 5, pp. 629–37, Feb 2008.
Bibliography
143
[80] Hassan Abdel-Aty, Anja Zagrosek, Jeanette Schulz-Menger, Andrew J Taylor, Daniel Messroghli, Andreas Kumar, Michael Gross, Rainer Dietz, and Matthias G Friedrich, “Delayed
enhancement and t2-weighted cardiovascular magnetic resonance imaging differentiate acute
from chronic myocardial infarction”, Circulation, vol. 109, no. 20, pp. 2411–6, May 2004.
[81] P. Kellman, “Erratum to Kellman P, McVeigh ER. Image reconstruction in SNR units: a
general method for SNR measurement. Magn Reson Med. 2005;54:1439â1447.”, Magnetic
Resonance in Medicine, vol. 58, pp. 211–212, 2007.
[82] Hanns B Hillenbrand, Lewis C Becker, Reza Kharrazian, Kai Hu, Carlos E Rochitte, Raymond J Kim, Enn L Chen, Georg Ertl, Ralph H Hruban, and JoÃ£o A C Lima, “23Na MRI
combined with contrast-enhanced 1H MRI provides in vivo characterization of infarct healing”,
Apr 1 2005.
[83] Maythem Saeed, Oliver Weber, Randall Lee, Loi Do, Alastair Martin, David Saloner, Philip
Ursell, Philippe Robert, Claire Corot, and Charles B Higgins, “Discrimination of myocardial
acute and chronic (scar) infarctions on delayed contrast enhanced magnetic resonance imaging
with intravascular magnetic resonance contrast media”, Nov 21 2006.
[84] A. J. Ross, Z. Yang, S. S. Berr, W. D. Gilson, W. C. Petersen, J. N. Oshinski, and B. A.
French, “Serial MRI evaluation of cardiac structure and function in mice after reperfused
myocardial infarction”, Magn Reson Med, vol. 47, no. 6, pp. 1158–68, 2002.
[85] M. Saeed, C. B. Higgins, J. F. Geschwind, and M. F. Wendland, “T1-relaxation kinetics
of extracellular, intracellular and intravascular MR contrast agents in normal and acutely
reperfused infarcted myocardium using echo-planar MR imaging”, Eur Radiol, vol. 10, no. 2,
pp. 310–8, 2000.
[86] M. F. Flessner, “The transport barrier in intraperitoneal therapy”, Am J Physiol Renal
Physiol, vol. 288, no. 3, pp. F433–42, 2005.
[87] F. Yang, Y. H. Liu, X. P. Yang, J. Xu, A. Kapke, and O. A. Carretero, “Myocardial infarction
and cardiac remodelling in mice”, Exp Physiol, vol. 87, no. 5, pp. 547–55, 2002.
[88] D. Garcia-Dorado and J. Oliveras, “Myocardial oedema: a preventable cause of reperfusion
injury?”, Cardiovasc Res, vol. 27, no. 9, pp. 1555–63, 1993.
[89] M. F. Wendland, M. Saeed, K. Lauerma, N. Derugin, J. Mintorovitch, F. M. Cavagna, and
C. B. Higgins, “Alterations in T1 of normal and reperfused infarcted myocardium after GdBOPTA versus GD-DTPA on inversion recovery EPI”, Magn Reson Med, vol. 37, no. 3, pp.
448–56, 1997.
[90] P. Marzola and S. Cannistraro, “Influence of paramagnetic ions bound to human serum
albumin on water 1HNMR relaxation times”, Physiol Chem Phys Med NMR, vol. 18, no.
4, pp. 263–73, 1986.
[91] Z. Q. Zhao, D. A. Velez, N. P. Wang, K. O. Hewan-Lowe, M. Nakamura, R. A. Guyton,
and J. Vinten-Johansen, “Progressively developed myocardial apoptotic cell death during late
phase of reperfusion”, Apoptosis, vol. 6, no. 4, pp. 279–90, 2001.
[92] R. S. Whelan, V. Kaplinskiy, and R. N. Kitsis, “Cell death in the pathogenesis of heart disease:
mechanisms and significance”, Annu Rev Physiol, vol. 72, pp. 19–44.
144
Bibliography
[93] B. Delattre, V. Braunersreuther, J-N Hyacinthe, J. P. Vallee, and Van De Ville D., “Fast
time-resolved cine sequence using temporal regularization for small animal cardiac imaging on
a clinical 3t scanner”, In Proceedings of the 18th Annual Meeting of the ISMRM, Stockholm,
Sweden, 2010.
[94] A. Roussakis, P. Baras, I. Seimenis, J. Andreou, and P. G. Danias, “Relationship of number
of phases per cardiac cycle and accuracy of measurement of left ventricular volumes, ejection
fraction, and mass”, J Cardiovasc Magn Reson, vol. 6, no. 4, pp. 837–44, 2004.
[95] F Wiesmann, J Ruff, S Engelhardt, L Hein, C Dienesch, A Leupold, R Illinger, A Frydrychowicz, K H Hiller, E Rommel, A Haase, M J Lohse, and S Neubauer, “Dobutamine-stress magnetic
resonance microimaging in mice : acute changes of cardiac geometry and function in normal
and failing murine hearts”, Circ Res, vol. 88, no. 6, pp. 563–9, Mar 2001.
[96] A. A. Domenighetti, Q. Wang, M. Egger, S. M. Richards, T. Pedrazzini, and L. M. Delbridge,
“Angiotensin II-mediated phenotypic cardiomyocyte remodeling leads to age-dependent cardiac dysfunction and failure”, Hypertension, vol. 46, no. 2, pp. 426–32, 2005.
[97] C. Pellieux, E. Aasum, T. S. Larsen, C. Montessuit, I. Papageorgiou, T. Pedrazzini, and
R. Lerch, “Overexpression of angiotensinogen in the myocardium induces downregulation of
the fatty acid oxidation pathway”, J Mol Cell Cardiol, vol. 41, no. 3, pp. 459–66, 2006.
[98] Marko K Ivancevic, Jean-Luc Daire, Jean-Noel Hyacinthe, Gerard Crelier, Sebastian Kozerke,
Karin Montet-Abou, Idil Gunes-Tatar, Denis R Morel, and Jean-Paul Vallée, “High-resolution
complementary spatial modulation of magnetization (cspamm) rat heart tagging on a 1.5 tesla
clinical magnetic resonance system: a preliminary feasibility study”, Invest Radiol, vol. 42,
no. 3, pp. 204–10, Mar 2007.
[99] Jean-Luc Daire, Jean-Pascal Jacob, Jean-Noel Hyacinthe, Pierre Croisille, Karin MontetAbou, Sophie Richter, Diomidis Botsikas, Matthieu Lepetit-Coiffé, Denis Morel, and JeanPaul Vallée, “Cine and tagged cardiovascular magnetic resonance imaging in normal rat at 1.5
t: a rest and stress study”, J Cardiovasc Magn Reson, vol. 10, pp. 48, Jan 2008.
[100] J. N. Hyacinthe, M. K. Ivancevic, J. L. Daire, and J. P. Vallee, “Feasibility of complementary
spatial modulation of magnetization tagging in the rat heart after manganese injection”, NMR
Biomed, vol. 21, no. 1, pp. 15–21, 2008.
[101] Bénédicte M A Delattre, Robin M Heidemann, Lindsey A Crowe, Jean-Paul Vallée, and JeanNoël Hyacinthe, “Spiral demystified”, Magn Reson Imaging, vol. 28, no. 6, pp. 862–81, Jul
2010.
[102] C. B. Ahn, J. H. Kim, and Z. H. Cho, “High-speed spiral-scan echo planar NMR imaging-I”,
IEEE Trans Med Imaging, vol. 5, no. 1, pp. 2–7, 1986.
[103] J. R. Liao, J. M. Pauly, T. J. Brosnan, and N. J. Pelc, “Reduction of motion artifacts in cine
MRI using variable-density spiral trajectories”, Magn Reson Med, vol. 37, no. 4, pp. 569–75,
1997.
[104] C. Liu, R. Bammer, D. H. Kim, and M. E. Moseley, “Self-navigated interleaved spiral
(SNAILS): application to high-resolution diffusion tensor imaging”, Magn Reson Med, vol.
52, no. 6, pp. 1388–96, 2004.
Bibliography
145
[105] Y. Qian and F. E. Boada, “Acquisition-weighted stack of spirals for fast high-resolution threedimensional ultra-short echo time MR imaging”, Magn Reson Med, vol. 60, no. 1, pp. 135–45,
2008.
[106] G. Narayan, K. Nayak, J. Pauly, and B. Hu, “Single-breathhold, four-dimensional, quantitative
assessment of LV and RV function using triggered, real-time, steady-state free precession MRI
in heart failure patients”, J Magn Reson Imaging, vol. 22, no. 1, pp. 59–66, 2005.
[107] S. Ryf, K. V. Kissinger, M. A. Spiegel, P. Bornert, W. J. Manning, P. Boesiger, and M. Stuber,
“Spiral MR myocardial tagging”, Magn Reson Med, vol. 51, no. 2, pp. 237–42, 2004.
[108] B. Kressler, P. Spincemaille, T. D. Nguyen, L. Cheng, Z. Xi Hai, M. R. Prince, and Y. Wang,
“Three-dimensional cine imaging using variable-density spiral trajectories and SSFP with application to coronary artery angiography”, Magn Reson Med, vol. 58, no. 3, pp. 535–43, 2007.
[109] A. V. dos Santos, S. Matias, P. Saraiva, and A. Goulao, “MR imaging features of brain stem
hypoplasia in familial horizontal gaze palsy and scoliosis”, AJNR Am J Neuroradiol, vol. 27,
no. 6, pp. 1382–3, 2006.
[110] A. M. Taylor, J. Keegan, P. Jhooti, P. D. Gatehouse, D. N. Firmin, and D. J. Pennell, “A comparison between segmented k-space FLASH and interleaved spiral MR coronary angiography
sequences”, J Magn Reson Imaging, vol. 11, no. 4, pp. 394–400, 2000.
[111] C. C. Chen, C. W. Tyler, and H. A. Baseler, “Statistical properties of BOLD magnetic
resonance activity in the human brain”, Neuroimage, vol. 20, no. 2, pp. 1096–109, 2003.
[112] M. Klarhofer, M. Barth, and E. Moser, “Comparison of multi-echo spiral and echo planar
imaging in functional MRI”, Magn Reson Imaging, vol. 20, no. 4, pp. 359–64, 2002.
[113] T. K. Truong and A. W. Song, “Single-shot dual-z-shimmed sensitivity-encoded spiral-in/out
imaging for functional MRI with reduced susceptibility artifacts”, Magn Reson Med, vol. 59,
no. 1, pp. 221–7, 2008.
[114] E. Adalsteinsson, J. Star-Lack, C. H. Meyer, and D. M. Spielman, “Reduced spatial side lobes
in chemical-shift imaging”, Magn Reson Med, vol. 42, no. 2, pp. 314–23, 1999.
[115] Y. S. Levin, D. Mayer, Y. F. Yen, R. E. Hurd, and D. M. Spielman, “Optimization of fast
spiral chemical shift imaging using least squares reconstruction: application for hyperpolarized
(13)C metabolic imaging”, Magn Reson Med, vol. 58, no. 2, pp. 245–52, 2007.
[116] P. Bornert, M. Stuber, R. M. Botnar, K. V. Kissinger, P. Koken, E. Spuentrup, and W. J.
Manning, “Direct comparison of 3D spiral vs. Cartesian gradient-echo coronary magnetic
resonance angiography”, Magn Reson Med, vol. 46, no. 4, pp. 789–94, 2001.
[117] C. H. Meyer, J. M. Pauly, A. Macovski, and D. G. Nishimura, “Simultaneous spatial and
spectral selective excitation”, Magn Reson Med, vol. 15, no. 2, pp. 287–304, 1990.
[118] C. Y. Liu, P. Varadarajan, G. M. Pohost, and K. S. Nayak, “Real-time color-flow MRI at 3 T
using variable-density spiral phase contrast”, Magn Reson Imaging, vol. 26, no. 5, pp. 661–6,
2008.
[119] M. Salerno, C.T. Sica, C. M. Kramer, and C. H. Meyer, “Myocardial perfusion imaging using
spiral trajectories”, ISMRM Data Sampling and Reconstruction Workshop, 2009.
146
Bibliography
[120] E. V. Di Bella, D. L. Parker, and A. J. Sinusas, “On the dark rim artifact in dynamic contrastenhanced MRI myocardial perfusion studies”, Magn Reson Med, vol. 54, no. 5, pp. 1295–9,
2005.
[121] J. W. Cooley and J. W. Tukey, “An Algorithm for Machine Calculation of Complex Fourier
Series”, Mathematics of Computation, vol. 19, no. 90, pp. 297–&, 1965.
[122] C. M. Tsai, L. C. Man, and D. G. Nishimura, “Off-centered spiral trajectories”, Magn Reson
Med, vol. 43, no. 3, pp. 446–51, 2000.
[123] A. Macovski and C. H. Meyer, “A novel fast-scanning system”, In: Works in progress of the
5th Annual Meeting of the SMRM, Montreal, Canada, pp. 156–157, 1986.
[124] A. C. Brau and J. H. Brittain, “Generalized self-navigated motion detection technique: Preliminary investigation in abdominal imaging”, Magn Reson Med, vol. 55, no. 2, pp. 263–70,
2006.
[125] M. A. Bernstein, X. J. Zhou, J. A. Polzin, K. F. King, A. Ganin, N. J. Pelc, and G. H. Glover,
“Concomitant gradient terms in phase contrast MR: analysis and correction”, Magn Reson
Med, vol. 39, no. 2, pp. 300–8, 1998.
[126] J. H. Duyn and Y. Yang, “Fast spiral magnetic resonance imaging with trapezoidal gradients”,
J Magn Reson, vol. 128, no. 2, pp. 130–4, 1997.
[127] G. H. Glover, “Simple analytic spiral K-space algorithm”, Magn Reson Med, vol. 42, no. 2,
pp. 412–5, 1999.
[128] D. H. Kim, E. Adalsteinsson, and D. M. Spielman, “Simple analytic variable density spiral
design”, Magn Reson Med, vol. 50, no. 1, pp. 214–9, 2003.
[129] T. Zhao, Y. Qian, Y.-K. Hue, T.S. Ibrahim, and F. Boada, “An improved analytical solution
for vairable density spiral design”, Proceedings of International Society of Magnetic Resonance
in Medicine, vol. 16, pp. 1342, 2008.
[130] G. F. Mason, T. Harshbarger, H. P. Hetherington, Y. Zhang, G. M. Pohost, and D. B. Twieg,
“A method to measure arbitrary k-space trajectories for rapid MR imaging”, Magn Reson
Med, vol. 38, no. 3, pp. 492–6, 1997.
[131] T. Onodera, S. Matsui, K. Sekihara, and H. Kohno, “A Method of Measuring Field-Gradient
Modulation Shapes - Application to High-Speed Nmr Spectrscopic Imaging”, Journal of
Physics E-Scientific Instruments, vol. 20, no. 4, pp. 416–419, 1987.
[132] M. T. Alley, G. H. Glover, and N. J. Pelc, “Gradient characterization using a Fourier-transform
technique”, Magn Reson Med, vol. 39, no. 4, pp. 581–7, 1998.
[133] Y. Zhang, H. P. Hetherington, E. M. Stokely, G. F. Mason, and D. B. Twieg, “A novel k-space
trajectory measurement technique”, Magn Reson Med, vol. 39, no. 6, pp. 999–1004, 1998.
[134] J. H. Duyn, Y. Yang, J. A. Frank, and J. W. van der Veen, “Simple correction method for
k-space trajectory deviations in MRI”, J Magn Reson, vol. 132, no. 1, pp. 150–3, 1998.
[135] M. Beaumont, L. Lamalle, C. Segebarth, and E. L. Barbier, “Improved k-space trajectory
measurement with signal shifting”, Magn Reson Med, vol. 58, no. 1, pp. 200–5, 2007.
Bibliography
147
[136] H. Tan and C. H. Meyer, “Estimation of k-space trajectories in spiral MRI”, Magn Reson
Med, vol. 61, no. 6, pp. 1396–404, 2009.
[137] B. Aldefeld and P. Bornert, “Effects of gradient anisotropy in MRI”, Magn Reson Med, vol.
39, no. 4, pp. 606–14, 1998.
[138] E. Schneider and G. Glover, “Rapid in vivo proton shimming”, Magn Reson Med, vol. 18, no.
2, pp. 335–47, 1991.
[139] D. C. Noll, “”Reconstruction techniques for magnetic resonance imaging” PhD. Thesis, Stanford University”, 1991.
[140] L. C. Man, J. M. Pauly, and A. Macovski, “Improved automatic off-resonance correction
without a field map in spiral imaging”, Magn Reson Med, vol. 37, no. 6, pp. 906–13, 1997.
[141] P. Irarrazabal, C. H. Meyer, D. G. Nishimura, and A. Macovski, “Inhomogeneity correction
using an estimated linear field map”, Magn Reson Med, vol. 35, no. 2, pp. 278–82, 1996.
[142] D. C. Noll, J. M. Pauly, C. H. Meyer, D. G. Nishimura, and A. Macovski, “Deblurring for
non-2D Fourier transform magnetic resonance imaging”, Magn Reson Med, vol. 25, no. 2, pp.
319–33, 1992.
[143] W. Chen and C. H. Meyer, “Semiautomatic off-resonance correction in spiral imaging”, Magn
[144] C. Barmet, N. D. Zanche, and K. P. Pruessmann, “Spatiotemporal magnetic field monitoring
for MR”, Magn Reson Med, vol. 60, no. 1, pp. 187–97, 2008.
[145] S. M. Lechner, P. T. Sipila, F. Wiesinger, A. B. Kerr, and M. W. Vogel, “Spiral imaging
artifact reduction: a comparison of two k-trajectory measurement methods”, J Magn Reson
Imaging, vol. 29, no. 6, pp. 1485–92, 2009.
[146] K. F. King, A. Ganin, X. J. Zhou, and M. A. Bernstein, “Concomitant gradient field effects
in spiral scans”, Magn Reson Med, vol. 41, no. 1, pp. 103–12, 1999.
[147] L. C. Man, J. M. Pauly, and A. Macovski, “Multifrequency interpolation for fast off-resonance
correction”, Magn Reson Med, vol. 37, no. 5, pp. 785–92, 1997.
[148] W. Chen, C. T. Sica, and C. H. Meyer, “Fast conjugate phase image reconstruction based on
a Chebyshev approximation to correct for B0 field inhomogeneity and concomitant gradients”,
Magn Reson Med, vol. 60, no. 5, pp. 1104–11, 2008.
[149] A. Maeda, K. Sano, and T. Yokoyama, “Reconstruction by weighted correlation for MRI with
time-varying gradients”, IEEE Trans Med Imaging, vol. 7, no. 1, pp. 26–31, 1988.
[150] J. I. Jackson, C. H. Meyer, D. G. Nishimura, and A. Macovski, “Selection of a convolution
function for Fourier inversion using gridding [computerised tomography application]”, IEEE
Trans Med Imaging, vol. 10, no. 3, pp. 473–8, 1991.
[151] J. D. O’Sullivan, “A fast sinc function gridding algorithm for fourier inversion in computer
tomography”, IEEE Trans Med Imaging, vol. 4, no. 4, pp. 200–7, 1985.
[152] JA Fessler and BP Sutton, “Nonuniform fast fourier transforms using min-max interpolation”,
Ieee T Signal Proces, vol. 51, no. 2, pp. 560–574, Jan 2003.
148
Bibliography
[153] J. A. Fessler, “On NUFFT-based gridding for non-Cartesian MRI”, J Magn Reson, vol. 188,
no. 2, pp. 191–5, 2007.
[154] D. Rosenfeld, “An optimal and efficient new gridding algorithm using singular value decomposition”, Magn Reson Med, vol. 40, no. 1, pp. 14–23, 1998.
[155] H. Moriguchi and J. L. Duerk, “Modified block uniform resampling (BURS) algorithm using truncated singular value decomposition: fast accurate gridding with noise and artifact
reduction”, Magn Reson Med, vol. 46, no. 6, pp. 1189–201, 2001.
[156] H. Moriguchi and J. L. Duerk, “Iterative Next-Neighbor Regridding (INNG): improved reconstruction from nonuniformly sampled k-space data using rescaled matrices”, Magn Reson
Med, vol. 51, no. 2, pp. 343–52, 2004.
[157] R. E. Gabr, P. Aksit, P. A. Bottomley, A. B. Youssef, and Y. M. Kadah, “Deconvolutioninterpolation gridding (DING): accurate reconstruction for arbitrary k-space trajectories”,
[158] G. E. Sarty, R. Bennett, and R. W. Cox, “Direct reconstruction of non-Cartesian k-space data
using a nonuniform fast Fourier transform”, Magn Reson Med, vol. 45, no. 5, pp. 908–15, 2001.
[159] Jeffrey A Fessler, “Model-based image reconstruction for mri”, IEEE Signal Process Mag, vol.
27, no. 4, pp. 81–89, Jul 2010.
[160] R. Boubertakh, J. F. Giovannelli, A. De Cesare, and A. Herment, “Non-quadratic convex
regularized reconstruction of MR images from spiral acquisitions”, Signal Processing, vol. 86,
no. 9, pp. 2479–2494, 2006.
[161] H Sedarat and D G Nishimura, “On the optimality of the gridding reconstruction algorithm”,
IEEE Trans Med Imaging, vol. 19, no. 4, pp. 306–17, Apr 2000.
[162] C. H. Meyer, B. S. Hu, D. G. Nishimura, and A. Macovski, “Fast spiral coronary artery
imaging”, Magn Reson Med, vol. 28, no. 2, pp. 202–13, 1992.
[163] R. D. Hoge, R. K. Kwan, and G. B. Pike, “Density compensation functions for spiral MRI”,
[164] F. Aurenhammer, “Voronoi Diagrams - a Survey of a Fundamental Geometric Data Structure”,
Computing Surveys, vol. 23, no. 3, pp. 345–405, 1991.
[165] V. Rasche, R. Proksa, R. Sinkus, P. Bornert, and H. Eggers, “Resampling of data between
arbitrary grids using convolution interpolation”, IEEE Trans Med Imaging, vol. 18, no. 5, pp.
385–92, 1999.
[166] B. Delattre, J. N. Hyacinthe, J. P. Vallee, and D. Van de Ville, “Spline-based variational
reconstruction of variable density spiral k-space data with automatic parameter adjustment”,
In: Proceedings of the of the 17th Annual Meeting of the ISMRM, Hawai’i, USA, p. 2066,
2009.
[167] CE SHANNON, “Communication in the presence of noise”, P Ire, vol. 37, no. 1, pp. 10–21,
Jan 1949.
[168] M. Unser, “Splines - A perfect fit for signal and image processing”, Ieee Signal Processing
Magazine, vol. 16, no. 6, pp. 22–38, 1999.
Bibliography
149
[169] M Unser and T Blu, “Fractional splines and wavelets”, Siam Rev, vol. 42, no. 1, pp. 43–67,
Jan 2000.
[170] AN Tikhonov, “Regularization of incorrectly posed problems”, Dokl Akad Nauk Sssr+, vol.
153, no. 1, pp. 49–&, Jan 1963.
[171] R Van de Walle, H H Barrett, K J Myers, M I Altbach, B Desplanques, A F Gmitro, J Cornelis,
and I Lemahieu, “Reconstruction of mr images from data acquired on a general nonregular
grid by pseudoinverse calculation”, IEEE Trans Med Imaging, vol. 19, no. 12, pp. 1160–7, Dec
2000.
[172] F. Wajer and K. P. Pruessmann, “Major speedup of reconstruction for sensitivity encoding
with arbitrary trajectories”, Proceedings of International Society of Magnetic Resonance in
Medicine, vol. 9, pp. 767, 2001.
[173] LA SHEPP and BF LOGAN, “Fourier reconstruction of a head section”, Ieee T Nucl Sci, vol.
NS21, no. 3, pp. 21–43, Jan 1974.
[174] LI RUDIN, S OSHER, and E FATEMI, “Nonlinear total variation based noise removal algorithms”, Physica D, vol. 60, no. 1-4, pp. 259–268, Jan 1992.
[175] M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing
for rapid MR imaging”, Magn Reson Med, vol. 58, no. 6, pp. 1182–95, 2007.
[176] DL Donoho, “Compressed sensing”, Ieee T Inform Theory, vol. 52, no. 4, pp. 1289–1306, Jan
2006.
[177] M Unser and T Blu, “Wavelet theory demystified”, Ieee T Signal Proces, vol. 51, no. 2, pp.
470–483, Jan 2003.
[178] DS Taubman and MW Marcellin, “Jpeg2000: Standard for interactive imaging”, P Ieee, vol.
90, no. 8, pp. 1336–1357, Jan 2002.
[179] Martin Uecker, Shuo Zhang, and Jens Frahm, “Nonlinear inverse reconstruction for real-time
mri of the human heart using undersampled radial flash”, Magn Reson Med, vol. 63, no. 6,
pp. 1456–62, Jun 2010.
[180] M. R. Hestenes and E. Stiefel, “Methods of Conjugate Gradients for Solving Linear Systems”,
Journal of Research of the National Bureau of Standards, vol. 49, no. 6, pp. 409–436, 1952.
[181] Stefanie Winkelmann, Tobias Schaeffter, Thomas Koehler, Holger Eggers, and Olaf Doessel,
“An optimal radial profile order based on the golden ratio for time-resolved mri”, IEEE Trans
Med Imaging, vol. 26, no. 1, pp. 68–76, Jan 2007.
[182] Yoon-Chul Kim, Shrikanth S Narayanan, and Krishna S Nayak, “Flexible retrospective selection of temporal resolution in real-time speech mri using a golden-ratio spiral view order”,
Magn Reson Med, Dec 2010.
[183] Kenneth O Johnson and James G Pipe, “Convolution kernel design and efficient algorithm for
sampling density correction”, Magn Reson Med, vol. 61, no. 2, pp. 439–47, Feb 2009.
[184] S J Riederer, T Tasciyan, F Farzaneh, J N Lee, R C Wright, and R J Herfkens, “Mr fluoroscopy:
technical feasibility”, Magn Reson Med, vol. 8, no. 1, pp. 1–15, Sep 1988.
150
Bibliography
[185] P. Thévenaz, T. Blu, and M. Unser, Image Interpolation and Resampling, Academic Press,
San Diego CA, USA, 2000.
[186] Michael S Hansen, Christof Baltes, Jeffrey Tsao, Sebastian Kozerke, Klaas P Pruessmann,
and Holger Eggers, “k-t blast reconstruction from non-cartesian k-t space sampling”, Magn
Reson Med, vol. 55, no. 1, pp. 85–91, Jan 2006.
[187] Taehoon Shin, Jon-Fredrik Nielsen, and Krishna S Nayak, “Accelerating dynamic spiral mri
by algebraic reconstruction from undersampled k–t space”, IEEE Trans Med Imaging, vol. 26,
no. 7, pp. 917–24, Jul 2007.
[188] Uygar Sümbül, Juan M Santos, and John M Pauly, “Improved time series reconstruction for
dynamic magnetic resonance imaging”, IEEE Trans Med Imaging, vol. 28, no. 7, pp. 1093–104,
Jul 2009.
[189] Michael Salerno, Christopher T Sica, Christopher M Kramer, and Craig H Meyer, “Optimization of spiral-based pulse sequences for first-pass myocardial perfusion imaging”, Magn.
Reson. Med., pp. n/a–n/a, Jan 2011.
[190] Dirk Mayer, Yi-Fen Yen, Atsushi Takahashi, Sonal Josan, James Tropp, B K Rutt, Ralph E
Hurd, Daniel M Spielman, and Adolf Pfefferbaum, “Dynamic and high-resolution metabolic
imaging of hyperpolarized [1-(13)c]-pyruvate in the rat brain using a high-performance gradient
insert”, Magn Reson Med, Nov 2010.
[191] Angus Z Lau, Albert P Chen, Nilesh R Ghugre, Venkat Ramanan, Wilfred W Lam, Kim A
Connelly, Graham A Wright, and Charles H Cunningham, “Rapid multislice imaging of hyperpolarized 13c pyruvate and bicarbonate in the heart”, Magn Reson Med, vol. 64, no. 5, pp.
1323–31, Nov 2010.
Appendix
Publication 1 Bénédicte M. A. Delattre, Vincent Braunersreuther, Jean-Noël Hyacinthe, Lindsey
A. Crowe, François Mach, and Jean-Paul Vallée, “Myocardial infarction quantification with
Manganese-Enhanced MRI (MEMRI) in mice using a 3t clinical scanner”, NMR in biomedicine,
vol. 23, no. 5, pp. 503-13, June 2010.
Publication 2 Bénédicte M. A. Delattre, Robin M. Heidemann, Lindsey A. Crowe, Jean-Paul
Vallée, and Jean-Noël Hyacinthe, “Spiral demystified”, Magnetic Resonance Imaging, vol. 28,
no. 6, pp. 862-81, July 2010.
151
Research Article
Received: 2 July 2009,
Revised: 13 November 2009,
Accepted: 14 November 2009,
Published online in Wiley InterScience: 2010
(www.interscience.wiley.com) DOI:10.1002/nbm.1489
Myocardial infarction quantification with
Manganese-Enhanced MRI (MEMRI) in mice
using a 3T clinical scanner
Bénédicte M. A. Delattrea *, Vincent Braunersreutherb, Jean-Noël Hyacinthea,
Lindsey A. Crowea, François Machb and Jean-Paul Valléea
Manganese (Mn2R) was recognized early as an efficient intracellular MR contrast agent to assess cardiomyocyte
viability. It had previously been used for the assessment of myocardial infarction in various animal models from pig to
mouse. However, whether Manganese-Enhanced MRI (MEMRI) is also able to assess infarction in the acute phase of a
coronary occlusion reperfusion model in mice has not yet been demonstrated. This model is of particular interest as it
is closer to the situation encountered in the clinical setting. This study aimed to measure infarction volume taking TTC
staining as a gold standard, as well as global and regional function before and after Mn2R injection using a clinical 3T
scanner. The first step of this study was to perform a dose-response curve in order to optimize the injection protocol.
Infarction volume measured with MEMRI was strongly correlated to TTC staining. Ejection fraction (EF) and percent
wall thickening measurements allowed evaluation of global and regional function. While EF must be measured
before Mn2R injection to avoid bias introduced by the reduction of contrast in cine images, percent wall thickening
can be measured either before or after Mn2R injection and depicts accurately infarct related contraction deficit. This
study is the first step for further longitudinal studies of cardiac disease in mice on a clinical 3T scanner, a widely
available platform. Copyright ! 2010 John Wiley & Sons, Ltd.
Keywords: manganese; mouse; cardiac MRI; occlusion reperfusion; myocardial infarction; 3T
INTRODUCTION
In cardiac magnetic resonance imaging (MRI), extracellular
contrast agents such as various gadolinium (Gd3þ) chelates,
are now routinely used in clinical practice, as well as in research
protocols to assess myocardial perfusion or interstitial space
remodeling (1–3). Intracellular MR contrast agents can provide
additional information on the cellular metabolism. Manganese
ion (Mn2þ) was quickly recognized as an efficient intracellular MR
contrast agent as it enters excitable cells via L-Type voltage
dependant channels to accumulate in mitochondria and also
induces a strong T1 shortening effect (4). As an analog of the
Calcium ion (Ca2þ), Mn2þ can assess Ca2þ homeostasis in-vivo
generating an important interest for researchers (5). In fact, Ca2þ
cycling is of vital importance to cardiac cell function and plays an
important role in ventricular dysfunction such as heart failure (6).
Many conditions can influence the Mn2þ uptake by cardiomyocytes. The presence of dobutamine (which is known to
increase Ca2þ influx into the heart) during Mn2þ infusion
increases signal enhancement in T1-weighted images whereas
the calcium channel blocker diltiazem reduces it (7). A
reduced Mn2þ accumulation has also been observed in stunned
cardiomyocytes (8) as well as in the zone adjacent to a myocardial
infarct (5). Manganese-Enhanced MRI (MEMRI) has also been used
to assess myocardial infarction in various animal models from pig
to rat (9–12). Only a few studies, however, investigated the
possible use of MEMRI for myocardial infarction assessment in
mice (5,13). All of these studies used a model of permanent
coronary occlusion. In this mouse model, infarct size determined
by TTC at 7 days was linearly correlated to the infarct size
measured from MEMRI (13) but a lower signal intensity,
suggesting a decreased Mn2þ accumulation was also observed
in the peri-infarct area where ischemic tissue may also be present
(5). The type of coronary occlusion model, as well as the timing of
examination after the induction of the myocardial injury, may
also impact MEMRI experiments. By comparison to permanent
coronary occlusion, the occlusion reperfusion model is closer to
the clinical situation in which the occluded coronary artery is
ultimately reperfused. The occlusion reperfusion model also
induces a reduced myocardial remodeling and infarct size
expansion in comparison to permanent occlusion (14) which can
* Correspondence to: B. M. A. Delattre, Department of Radiology – CIBM, Geneva
University Hospital, Rue Gabrielle-Perret-Gentil, 4, 1211 Geneva 14, Switzerland.
E-mail: [email protected]
a B. M. A. Delattre, J.-N. Hyacinthe, L. A. Crowe, J.-P. Vallée
Faculty of Medicine, University of Geneva, Geneva, Switzerland
b V. Braunersreuther, F. Mach
Division of Cardiology, Department of Medicine, University Hospital,
Foundation for Medical Researchers, Geneva, Switzerland
Contract/grant sponsor: Swiss National Science Foundation; contract/grant
number: PPOOB3-116901.
Contract/grant sponsor: The Center for Biomedical Imaging (CIBM), Lausanne
and Geneva, Switzerland.
Abbreviations used: BW, body weight; CNR, contrast to noise ratio; EDV,
end-diastolic volume; EF, ejection fraction; ESV, end-systolic volume; IP, intraperitoneal; IV, intravenous; MEMRI, manganese-enhanced MRI; MI, myocardial
infarct; PSIR, phase-sensitive inversion recovery; ROI, region of interest; SI,
signal intensity; TTC, triphenyltetrazolium chloride.
1
NMR Biomed. (2010)
Copyright ! 2010 John Wiley & Sons, Ltd.
B. M. A. DELATTRE ET AL.
affect Mn2þ uptake. Massive proliferation of fibroblasts and
collagen deposition begins 7 days after the induction of
myocardial infarct (15) whereas acute conditions are encountered 1 day after infarction, where inflammation and stunning
prevails. Whether MEMRI is also able to assess myocardial infarct
or a larger area including peri-infarct stunned myocardium in the
acute phase in a mouse model of coronary occlusion reperfusion
is largely unknown. Therefore, the purpose of this study was to
investigate the use of MEMRI in the assessment of acute
myocardial infarct in a clinically relevant coronary occlusion
reperfusion model in mice.
The secondary aim of this study was the set up of a MEMRI
protocol to assess subendocardial myocardial infarction on a 3T
clinical MR scanner. The infarction model was chosen to be a
60 min coronary occlusion followed by 24 h of reperfusion.
As Mn2þ toxicity is known to be a critical point (16), a dose
response study was performed in order to maximize the
enhancement provided for infarction quantification by injecting
an optimal dose of Mn2þ without suffering from side effects [as
arrhythmia, somnolence with general depressed activity, ataxia
and respiratory stimulation (16)]. Cardiac function measurements
were also performed systematically before and after Mn2þ
injection in order to determine if the presence of Mn2þ alters the
results. Finally this study aims to provide an efficient tool for
further research in myocardial disease in mice as the MR system
used in this experiment is a clinical scanner much more available
than dedicated small animal MR systems.
METHODS
Animal preparation
15–20 week old C57BL/6J mice were anaesthetized with 4%
isoflurane and intubated. Mechanical ventilation was performed
(150 ml at 120 breaths/min) using a rodent respirator (model 683;
Harvard Apparatus). Anaesthesia was maintained with 2%
isoflurane delivered in 100% O2 through the ventilator. A
thoracotomy was performed and the pericardial sac was then
removed. An 8–0 prolene suture was passed under the left
anterior descending (LAD) coronary artery at the inferior edge of
the left atrium and tied with a slipknot to produce occlusion. A
small piece of polyethylene tubing was used to secure the
ligature without damaging the artery. Ischemia was confirmed by
the visualization of blanching myocardium, downstream of the
ligation. After 60 min of ischemia, the LAD coronary artery
occlusion was released and reperfusion occurred. Reperfusion
was confirmed by visible restoration of color to the ischemic
tissue. The chest was then closed and air was evacuated from the
chest cavity. The ventilator was then removed and normal
respiration restored. This group is named IR60 in the following
(n ¼ 6). Sham operated animals were subjected to the same
protocol without LAD coronary occlusion (n ¼ 4). Animals from
the control group did not experience any surgery before MRI
exam (n ¼ 4). Twenty-four hours after surgical procedure, animals
were submitted to MRI analysis and then sacrificed to perform
histological staining as described in ‘ex-vivo evaluation of infarct
size’ section.
Manganese injection protocol
MnCl2 was diluted in NaCl 0.9% solution to obtain stock solution
of 7.5 mM or 15 mM depending on the experiment. An
intraperitoneal (IP) line was placed in the mouse before MRI
exam in order to deliver the Mn2þ solution. To optimize the
injection protocol we performed dose versus signal enhancement curves in control mice for several values of Mn2þ
concentrations. Table 1 summarizes the concentrations and
settings used for these injections. All settings were calculated in
order to inject a maximum volume of 1 ml of stock solution. We
measured signal enhancement in the septum as well as in the left
ventricle free wall (around the anterolateral area, where infarction
usually occurs with this model) to ensure that results were not
dependent on the localization in the myocardium. Following the
dose response results obtained, Mn2þ concentration delivered in
the infarction quantification protocol was chosen to be 200 nmol/
g body weight (BW) at a rate of 4 ml/h, typical infusion duration
was 6 min for a 30 g BW mouse and the volume injected was
400 ml.
Table 1. Mn2þ injection parameters and heart rate (HR) measured at least 15 min after the end of Mn2þ injection for ‘low’ and ‘high’
concentrations experiments and general information about Mn2þ solution used in the measurement of dose vs signal enhancement
curve. Symbols in brackets denote significant difference in HR compared with beginning of the experiment, before any Mn2þ
injection (xp < 0.01, non stated values mean non-significant difference)
experimental
timing (min)
0
5
40
75
110
145
Low concentrations
High concentrations
Stock solution 7.5 mmol/l
Stock solution 15 mmol/l
cumulative
dose (nmol/g)
infusion
time (min)
0
35
70
100
150
200
4.2
4.2
3.6
6
6
HR (bpm)
n
experimental
timing (min)
333 # 16
3
0
311 # 14
3
5
306 # 10
3
40
298 # 14
3
75
289 # 18 (x)
3
110
290 # 17 (x)
3
infusion rate 2 ml/h
cumulative
dose (nmol/g)
infusion
time (min)
0
250
320
390
480
15
4.2
4.2
5.4
HR (bpm)
n
303 # 27
284 # 6
264 # 38
252 # 30
245 # 39 (x)
4
2
2
4
4
2
www.interscience.wiley.com/journal/nbm
NMR Biomed. (2010)
INFARCTION QUANTIFICATION WITH MEMRI IN MICE USING A 3T SCANNER
MR imaging
During the MRI exam mice were anesthetized with isoflurane
1–2%. Imaging was performed on a clinical 3T MR scanner
(Magnetom TIM Trio, Siemens Medical Solutions, Erlangen
Germany) with a dedicated 2-channel mouse receiver coil (Rapid
biomedical GmbH, Rimpar Germany) 24 h after reperfusion. A
turboflash cine sequence assessed myocardial function before
and after Mn2þ injection: FOV 66 mm, in plane resolution 344 mm,
slice thickness 1 mm, typically 4 consecutive slices to cover the
whole left ventricle (no slice overlap), TR/TE 11/5 ms, flip angle
308, GRAPPA with acceleration factor 2, 3 averages, typical
acquisition time per slice 3 min. A T1-weighted turboflash
sequence using Phase Sensitive Inversion Recovery reconstruction (PSIR) (17) assessed myocardial viability: FOV 80 mm, in plane
resolution 156 mm, slice thickness 1 mm, typically 8 slices (50%
slice overlap), TR/TE 438/7.54 ms, flip angle 458, TI 380 ms,
GRAPPA with acceleration factor 2, 2 averages, typical acquisition
time per slice 2 min 30. Fifty percent slice overlap was chosen to
diminish effects of partial volumes (18). For this sequence a
constant TI was chosen to keep the same contrast properties in all
images. Both sequences were respiratory and ECG gated.
Ex-vivo evaluation of infarct size
After MR imaging, the mice were re-anaesthetized with 10 ml/kg
of ketamine-xylazine (12 mg/ml and 1.6 mg/ml, respectively) and
the animals were sacrificed by a rapid excision of the heart. The
heart was then rinsed in NaCl 0.9%, frozen and manually
sectioned into 1–2 mm transverse sections from apex to base
(5–6 slices/heart). These slices were incubated at 378C with 1%
triphenyltetrazolium chloride (TTC) in phosphate buffer (pH 7.4)
for 15 min, fixed in 10% formaldehyde solution and each side of
the slices was photographed with a digital camera (Nikon
Coolpix). The different zones (left ventricular wall myocardium
and infarction) were determined using a computerized planimetric technique (MetaMorph v6.0, Universal Imaging Corporation). Animals from the control and sham groups were also
submitted to histological staining after the MRI exam.
Figure 1. Myocardial segmentation as defined by the American Heart
Association (17). Middle slice of the left ventricle is divided into six
sectors: A ¼ anterior, AL ¼ anterolateral, IL ¼ inferolateral, I ¼ inferior,
IS ¼ inferoseptal, AS ¼ anteroseptal.
myocardium area presenting a significant contractile dysfunction
observed on cine images and a hypointense area on one
representative slice (generally the middle slice of the heart). The
error in estimation of the infarct volume was evaluated by
defining the threshold obtained by repeated drawing of ROI in
the same area. This deviation was 5 a.u. and was the same for all
mice in this group. Infarction volume was then evaluated taking
initial threshold # 5 a.u. For the control and sham groups, the
infarction volume was derived from the threshold of the IR60
group. In fact, we observed that infarction area had a SI reduction
of 60 a.u. compared to septum area (mean taken for all IR60 mice,
n ¼ 6), so the threshold was defined as SIm-60 where SIm was the
mean SI between 2 ROI (one in septum and one in free wall). Error
bars were determined by evaluating infarction volume taking the
deviation of 5 a.u. on the threshold as described for the IR60
group.
Statistics
All values are quoted as mean # standard deviation (SD).
Statistics were performed with SPSS Statistics 17.0 software. To
compare results of the three groups (control, sham operated and
IR60) analysis of variance (ANOVA) was performed, followed by a
Bonferroni post-hoc test. To compare results obtained before and
after Mn2þ injection, a student’s t-test was performed. The
significance was set to p < 0.05.
Image analysis
Ejection fraction (EF) calculation and infarction quantification
were performed with Osirix software (Open source http://
www.osirix-viewer.com/). For EF calculations, segmentation of
the endocardial contour allowed evaluation of the end-diastolic
volume (EDV) and end-systolic volume (ESV), with EF defined as
(EDV-ESV)/EDV. Wall thickening evaluation for regional function
assessment was done using in-house software allowing
calculation of percentage wall thickening in sectors covering
the whole left ventricle. Sectors were defined by the American
Heart Association (AHA) standardized guidelines for myocardial
segmentation (19). Figure 1 shows this segmentation. For
the Mn2þ dose study and infarction quantification, segmentation
was done by manually drawing a region of interest (ROI) in
septum area, myocardial infarction (MI) or free wall. The mean ROI
size was 1.22 # 0.51 mm2. The free wall ROI was chosen to be in
the same myocardial area for the control and sham-operated
groups as for the IR60 group. Infarction quantification was
performed with a threshold technique and compared to the
quantification performed with TTC staining, which is considered
as a gold standard (20). For the IR60 group, the threshold was
defined as the mean signal intensity (SI) measured in the
RESULTS
Manganese dose determination
The first step of our study was to determine the lowest Mn2þ dose
to deliver in order to have the best enhancement of the
myocardium with our PSIR sequence while remaining under
the toxic level. This was important considering the toxicity
of Mn2þ at high concentrations and delivery rate (16) and the
variability of the different doses reported in literature for cardiac
MEMRI. We observed a maximal enhancement of the myocardium 45 min after Mn2þ injection. The results shown in
Figure 2 indicate that signal enhancement increases linearly
with Mn2þ dose up to 200 nmol/g before reaching a plateau.
Linear regression results are y ¼ 0.78x–26.37 with R2 ¼ 0.99 and
p < 0.001 for signal enhancement measured in septum area and
y ¼ 0.81x–21.77 with R2 ¼ 0.99 and p < 0.001 for measurement in
free wall. Slopes and intercepts are not significantly different
between these two regions of interest ( p > 0.05). No arrhythmia
or significant change in heart rate were encountered at
high Mn2þ concentrations when comparing with the heart
rate just before Mn2þ injection ( p > 0.2). However, for the
3
NMR Biomed. (2010)
in order to minimize the injected volume as well as the injection
duration which was thus 400 ml and 6 min respectively for a 30 g
mouse).
Infarct quantification
Figure 2. Signal enhancement measured in septum and free wall area
(see insert) for control mice (signal intensity, SI, minus signal baseline
measured before Mn2þ injection, S0) 45 min after Mn2þ injection
versus Mn2þ dose.
‘low’ concentration as well as for the ‘high’ concentration
experiments, the successive injections of Mn2þ led to a decrease
of heart rate that became significant for the last doses when
compared with the heart rate measured at the beginning of the
experiment ( p < 0.03 and p < 0.05 for ‘low’ and ‘high’ dose
protocol respectively). We have to note that even for the highest
dose (480 nmol/g) we were well below the acute toxicity limit that
is 962 nmol/g for IP route (16). For the following experiments, i.e.
infarction quantification, we choose an MnCl2 IP injection of
200 nmol/g BW at a rate of 4 ml/h from a stock solution of
15 mmol/l (the stock solution of higher concentration was chosen
During each experiment, PSIR images of the whole myocardium
were acquired. A typical example is shown in Figure 3 (upper
row). Infarction area is represented as hypointense signal
corresponding to a reduced uptake of Mn2þ compared to viable
myocardium where Mn2þ accumulates. Contrast to noise ratios
(CNR) between signal measured in MI area (or free wall) and
septum area for the control, sham operated and IR60 groups were
measured. Noise was chosen to be the mean of the SD in both
those areas. For the IR60 group CNR was 4.00 # 1.59 which is
significantly higher ( p < 0.01) than sham operated, 0.66 # 1.04
and control groups, $0.13 # 0.60. Also, no significant difference
was found between the control and sham groups ( p ¼ 1.000), and
both values are not significantly different from zero ( p ¼ 0.7 and
p ¼ 0.3 for control and sham respectively).
Figure 3 shows an example of the comparison between MEMRI
and TTC staining. An ROI was drawn in the infarction area on
MEMRI images as described in the Methods section, and the mean
SI of this area defined the threshold used for infarction quantification. Figure 4 illustrates an example of this segmentation
method. Infarction quantification performed with MEMRI is
strongly correlated with the measurements derived from TTC
staining, as shown by Figure 5. The error bars aim to indicate the
maximal deviation that could be made by applying the
segmentation method. The maximal deviation obtained was
9%. The result of linear regression was y ¼ 0.94x with R2 ¼ 0.91
and p < 0.001. The Bland-Altman plot also shows a good
agreement between the two methods without significant bias.
The mean of the differences between MEMRI and TTC is 1.7%. In
both the control and sham operated groups, the signal intensity
was homogenously enhanced over all the heart without any large
Figure 3. Selection of T1-weighted PSIR short axis images (upper row) and corresponding TTC staining (bottom row) for an IR60 mouse (A) and a sham
operated mouse (B). White colour present in TTC slices for (A) indicating tissue necrosis correlates with hypointense signal in PSIR images whereas neither
white nor hypointense signal is present in (B).
4
NMR Biomed. (2010)
Figure 4. Segmentation method for infarction volume quantification. Upper row; example of slices covering whole heart of an IR60 mouse with
manually drawn endocardial and epicardial contours. Bottom row; corresponding results of the threshold segmentation technique.
area of hypointense signal on the PSIR, as shown in Figure 3 and
confirmed by the fact that CNR is not significantly different from
zero for the control and sham groups respectively. This was in
agreement with the TTC staining that did not reveal myocardial
infarct in both these groups.
Function assessment
Cine slices covering the whole left ventricle were acquired in
order to measure global and regional function before and
after Mn2þ injection. Examples of cine images for the left ventricle
middle slice of a control mouse and an IR60 mouse are given in
Figure 6. We observed a clear contraction decrease for the IR60
mouse compared to the control mouse. Global function was
determined by measurement of the ejection fraction (EF). Table 2
shows results of EDV, ESV, EF and heart rate for IR60, sham
operated and control groups for measurements done before and
after Mn2þ injection. We obtained a significant decrease of EF for
the IR60 group compared to the sham (29%, p < 0.01) and control
(20%, p < 0.05) groups before Mn2þ injection. However, these
differences were no longer significant after Mn2þ injection. We
could also note a significant decrease in EDV and ESV, leading to
an increase in EF for the control group as well as a decrease in ESV
for the IR60 group after Mn2þ injection. Heart rate was not
significantly different neither between the three groups ( p > 0.4)
nor after Mn2þ injection ( p > 0.2).
NMR Biomed. (2010)
5
Figure 5. Left; infarction volume quantification with MEMRI versus TTC staining for IR60 (n ¼ 6), sham (n ¼ 4) and control (n ¼ 4) groups as a percentage
of left ventricular wall volume. Error bars symbolize the volume variation obtained with the threshold segmentation technique. Right; Bland–Altman plot
for comparison of infarction quantification method between MEMRI and TTC staining. Solid line is mean(MEMRI-TTC) ¼ 1.7%, dashed lines are
1.96%SD(MEMRI-TTC) ¼ 8.0% corresponding to 95% Confidence Interval.
Figure 6. Example of cine images of systolic (left) and diastolic (middle) phases in middle slice of heart, as well as T1-weighted PSIR image (right) for an
IR60 mouse (A) and a control mouse (B). In (A) the lateral part of the myocardium is not moving between systolic and diastolic phases which is not the case
in (B).
Left ventricular regional function was assessed by comparing
wall thickness between diastole and systole for 6 sectors
covering the whole left ventricle, leading to percentage wall
thickening evaluation. Figure 7 shows the results obtained before
and after Mn2þ injection for control, sham operated and IR60
mice. Table 3 shows the mean wall thickening in the six different
myocardial sectors for the three groups. Wall thickening was
significantly reduced in every free wall area (including anterior,
anterolateral, inferolateral and inferior sectors) for infarcted mice
compared with the control and sham groups ( p < 0.001
before Mn2þ injection and p < 0.01 after), while no significant
difference was obtained in the septum sectors. As an example,
before Mn2þ injection, wall thickening is 0.77 # 0.12 in the
anterolateral sector for the control group and is reduced to
0.11 # 0.07 for the IR60 group, which constitutes a decrease of
86%. Also, no significant difference was measured, neither
between control and sham groups ( p > 0.4), nor between
measurements done before and after Mn2þ injection ( p > 0.5).
Figure 8 shows the correlation between wall thickening and
signal intensity for each sector of the left ventricle. To avoid any
bias between mice, signal intensity was normalized to signal
intensity of the anteroseptal sector. For the IR60 group we
obtained a strong correlation between regional function and
signal intensity, i.e. presence of Mn2þ, in different sectors
(y ¼ 22.3x–21.7, R2 ¼ 0.824, p ¼ 0.01). However, no correlation
was found for the control (R2 ¼ 0.003, p ¼ 0.9) and sham groups
(R2 ¼ 0.290, p ¼ 0.3).
DISCUSSION
Summary of results
As the main result of the study, a strong relationship, close to
unity, was observed for the infarction volume measurement by
MEMRI and TTC staining in the mouse model of acute myocardial
infarct induced by an occlusion reperfusion. Global function
could be assessed, as well as regional contraction deficit relative
to infarction induction. As an additional result, we validated the
use of a clinical 3T MR system for the study of myocardial infarct
in mice by a careful optimization of the protocols including the
dose and timing of the delivered Mn2þ, the cine and T1 weighted
MR sequences.
Table 2. End diastolic volume (EDV), end systolic volume (ESV), ejection fraction (EF) and heart rate (HR) for IR60, sham and control
groups, measured before and after Mn2þ injection. Values are mean # SD. For the IR60 group, P indicates significant difference with
the sham and control groups respectively obtained with post-hoc Bonferroni test (%p < 0.05; xp < 0.01; yp < 0.001). Symbols in
brackets indicate result of t-test for comparison of measurement done before and after Mn2þ in each group
P
EDV (ml)
ESV (ml)
EF
HR (bpm)
before Mn
after Mn
before Mn
after Mn
before Mn
after Mn
before Mn
after Mn
IR60
sham
control
IR60 vs sham
IR60 vs control
57.6 # 9.5
47.5 # 10.8 (NS)
30.9 # 5.3
20.7 # 5.1 (x)
0.47 # 0.06
0.57 # 0.07 (NS)
328 # 25
316 # 42 (NS)
30.7 # 13.9
32.6 # 3.2 (NS)
10.2 # 5.4
8.4 # 3.8 (NS)
0.66 # 0.09
0.75 # 0.09 (NS)
291 # 53
283 # 37 (NS)
40.8 # 6.1
30.3 # 4.1 (%)
16.6 # 2.0
8.7 # 1.6 (y)
0.59 # 0.03
0.71 # 0.05 (x)
348 # 29
317 # 33 (NS)
x
NS
y
x
x
NS
NS
NS
NS
%
y
x
%
NS
NS
NS
6
NMR Biomed. (2010)
Figure 7. Wall thickening between diastolic and systolic phase for sectors defined in Figure 1 before and after Mn2þ injection, for control group (n ¼ 4)
(A), sham operated group (n ¼ 4) (B), IR60 group (n ¼ 6) (C), and comparison of wall thickening before Mn2þ injection for the 3 groups (D). Values are
Mean # SD (A ¼ anterior, AL ¼ anterolateral, IL ¼ inferolateral, I ¼ inferior, IS ¼ inferoseptal, AS ¼ anteroseptal). % indicates significant difference between
groups (at least p < 0.05).
Table 3. Wall thickening between diastolic and systolic phase in the 6 left ventricle sectors (defined in Fig. 1) for IR60, sham and
control groups, before and after Mn2þ injection. Values are mean # SD. For the IR60 group, P indicates significant difference with the
sham and control groups respectively obtained with post-hoc Bonferroni test (%p < 0.05; xp < 0.01; yp < 0.001)
P
Before Mn2þ
IR60
sham
control
IR60 vs sham
IR60 vs control
anterior
anterolateral
inferolateral
inferior
inferoseptal
anteroseptal
0.16 # 0.08
0.11 # 0.07
0.15 # 0.10
0.31 # 0.27
0.69 # 0.30
0.65 # 0.10
0.79 # 0.16
1.03 # 0.26
0.90 # 0.29
0.94 # 0.08
0.75 # 0.21
0.64 # 0.17
0.77 # 0.18
0.77 # 0.12
0.71 # 0.20
0.63 # 0.11
0.51 # 0.17
0.53 # 0.11
y
y
y
x
NS
NS
y
y
x
NS
NS
NS
P
After Mn2þ
IR60
sham
control
IR60 vs sham
IR60 vs control
anterior
anterolateral
inferolateral
inferior
inferoseptal
anteroseptal
0.31 # 0.05
0.23 # 0.10
0.15 # 0.13
0.27 # 0.23
0.71 # 0.21
0.74 # 0.16
0.78 # 0.11
0.78 # 0.35
0.83 # 0.34
0.92 # 0.07
0.69 # 0.04
0.68 # 0.16
0.82 # 0.08
0.74 # 0.11
0.62 # 0.20
0.73 # 0.20
0.70 # 0.13
0.71 # 0.05
y
x
x
y
NS
NS
y
x
%
x
NS
NS
7
NMR Biomed. (2010)
Figure 8. Wall thickening versus signal intensity for the 6 sectors covering the left ventricle (A ¼ Anterior, AL ¼ Anterolateral, IL ¼ Inferolateral,
I ¼ Inferior, IS ¼ Inferoseptal, AS ¼ Anteroseptal). For each mouse, signal
intensity of each sector was normalized to signal intensity of AS sector.
Values are mean # SD for IR60 (n ¼ 6), sham (n ¼ 4) and control group
(n ¼ 4). Linear fit and R2 coefficient relates to the correlation obtained for
IR60 group values.
Manganese dose determination
8
The Mn2þ dose-response study was necessary for two reasons.
First, all previous studies on cardiac MEMRI in mice were
done on a dedicated small animal system, either with a 9.4T
(13) or a 7T magnet (5,7) which are known to give a better
sensitivity and have higher gradient performances than
clinical systems. We choose to work on a 3T magnet to take
advantage of the implementation of clinical sequences and
features like PSIR and GRAPPA. We reached a spatial and
temporal resolution comparable to those used in other
studies with the performance of this system. Secondly, we
used a T1-weighted PSIR sequence to assess signal enhancement that is different from sequences used in those previous
studies (FLASH or T1-mapping sequence). The inversion
recovery (IR) sequence is the best choice for the assessment
of myocardial injury in patients because of its high T1
sensitivity (21). However, image quality can be altered by an
inappropriate choice of inversion time (TI). Phase sensitive
reconstruction from the PSIR sequence prevents this problem
and gives more consistent image quality than an IR sequence
(17). A better consistency in data is expected with a
T1-mapping sequence, however this kind of measurement
is for the moment very long [around 45 min (5)].
During our experiments, we observed a maximal enhancement
of viable myocardium 45 min after Mn2þ injection, therefore
images for infarction quantification were taken after such a
delay. This delay is in agreement with other previous studies
(7,13) where injection was made intravenously (IV). This delay is
mainly explained by the time necessary for Mn2þ ions to be
accumulated in mitochondria after entering the cardiomyocytes
via L-type calcium channels, as IP injection is comparable to IV
in mice (22).
During the dose response experiments, heart rate measured
at least 15 min after the end of each Mn2þ injection did
not decrease significantly compared to before each injection.
However, the heart rate decreased progressively during
the experiment to finally become significantly different from
the one measured at the beginning of the experiment, before
the first injection. Knowing that Mn2þ plasma half-life is
approximately 3 min (23), it suggests that the observed
depression was more probably due to the cumulative effect
of the anesthesia than to a direct effect of Mn2þ injection.
This can be explained by the fact that anesthesia was driven
in order to keep a stable respiratory rate and not necessarily
a constant heart rate. Moreover, the decreased heart rate
observed at the end of the ‘low’ concentrations protocol
corresponding to 200 nmol/g of Mn2þ was not reproduced
after the first injection of 250 nmol/g Mn2þ in the ‘‘high’’
concentrations protocol. The results in Table 2 confirm this
hypothesis as no significant change in heart rate was
measured after Mn2þ injection in each group (the dose used
for infarction quantification was 200 nmol/g).
In Figure 2, a plateau of signal enhancement is reached
for Mn2þ concentrations higher than 200 nmol/g. This plateau
indicates either a saturation-related problem that does not allow
visualization of a further increase in Mn2þ concentration in the
myocardium after this level because of the limited dynamic of the
SI or is a result of a true physiological effect. Indeed, significant
shortening of T1 in tissue due to Mn2þ leads to a saturation of SI,
dependent on the inversion time chosen in the sequence. The
physiological effect could be due to either a limited Mn2þ
accumulation in the cardiomyocytes or a limited relaxation rate
change secondary to protein binding with Mn2þ ions. According
to Kang et al. (24), binding of Mn2þ ions to macromolecules leads
to a more efficient dipolar interaction with surrounding protons
significantly decreasing proton T1. Our experiments could not
answer this question as further investigations were beyond the
scope of this study. However, a recent study of Waghorn et al. (5),
who mapped the T1 decrease in myocardium, shows the same
plateau in their measurements occurring above 197 nmol/g
(which is comparable with our results). They came to the
conclusion that above this concentration, an increase in Mn2þ
concentration in the myocardium did not lead to a decrease in T1
even if the absolute concentration of Mn2þ in dry myocardium
increased. This would tend to validate the hypothesis of a limited
change in relaxation rates.
Infarction quantification
Gadolinium chelates are largely used for assessment of
myocardial infarction in mouse models (1,25,26). However, even
if this method has largely proven its efficiency, it has some
disadvantages over Mn2þ late enhancement. First, the time
window during which assessment is possible is relatively short
with Gd3þ [around 1 h (25)] compared with Mn2þ where ions can
stay in mitochondria for several hours (5). Also, performing infarct
quantification too early with Gd3þ can lead to an overestimation
of the infarct zone that restrains again this time window between
20 and 60 min after injection (25–27). From this point of
view, Mn2þ allows more flexibility. Thirdly Mn2þ makes the viable
part of myocardium appear bright which allows an easier
segmentation of the myocardium and the infarct pattern than
with Gd3þ, where parameters of the sequence are often chosen
to null myocardium signal prior to contrast agent addition.
Comparing with other previous studies conducted at high field,
gradient performances of our system allowed a similar spatial
resolution [156 mm vs 117 mm at 11.7T (25) or 100 mm at 9.4T (28)]
leading to a precise infarction quantification. Indeed, we
obtained an excellent correlation with TTC measurement
NMR Biomed. (2010)
(y ¼ 0.94x, R2 ¼ 0.91), with no bias between methods confirmed
by Bland–Altman plot. These results are very similar to those
obtained with Gd3þ enhancement for the same infarction model
(26). This was reinforced by CNR results that showed a strong
contrast between viable and infarct area in the IR60 group but, as
expected, not in the control and sham operated groups. The high
variability obtained for CNR measurements can be explained by
the method of noise calculation. The noise was estimated from
the standard deviation of the SI in the left ventricle wall. It is
therefore a physiological noise much higher than image noise
that could be measured from an empty area of the images.
However, CNR was always sufficiently high to allow infarction
segmentation in the IR60 group.
These results did not bring to light any significant difference
of Mn2þ uptake between control and sham group, indicating that
the open chest surgical procedure do not affect Ca2þ homeostasis
sufficiently to be detected by our method. However, in both
control and sham groups, infarction quantification led to a
non-zero value whereas TTC results did not indicate any necrosis.
This was often caused by darker pixels present at edges of the
myocardium that were counted as part of an infarct. Although,
the erroneous pixels could be easily detected by visual inspection,
we chose not to discard them but rather to use a linear model
with no intercept in order to decrease the contribution of these
isolated pixels. As a work in progress, a possible refinement of the
method would be to apply morphological operations (erosions
and dilatations) on the segmented images to eliminate
contributions of isolated pixels at edges.
An important result was that MEMRI did not overestimate
the infarct size. In fact, no previous study has described Mn2þ
enhancement in acute phase of reperfused infarction, so it was
not known whether other mechanism such as stunning could
affect Mn2þ uptake. Very little literature is found on the
assessment of stunning in mice (29,30) and no data is provided
for our mouse model and time point (24 h after injury). However,
Krombach et al. (8) have previously shown that Mn2þ can assess
stunning in rats 30 min after repeated ischemia-reperfusion
protocol. The explanation of such mechanism remains controversial as Mn2þ was accumulating via Naþ/Ca2þ exchangermediated transport during hypoxia in an in vitro study of isolated
perfused myocardium (31). Should Mn2þ uptake be reduced in
stunned cardiomyocytes, it could induce an overestimation of the
infarction volume compared to TTC. Such an effect was clearly
not present in our model. Finally, it was important to provide a
running protocol to assess myocardial infarction in the acute
phase as it is the start point of all further longitudinal studies.
Function assessment
As shown in Table 2, we obtained a significant decrease of EF
for the IR60 group compared with the control and sham operated
groups (20% p < 0.05 and 29% p < 0.01 respectively). However,
when comparing EDV and ESV with another study for the same
model (26) we have higher results in volume estimation leading
to decreased EF for the control group [59% vs 70% (26)]. This
could be due to the lower spatial resolution reached in our cine
measurements [344 mm vs 100 mm (26)], but also to a deeper
anesthesia of the animal. Indeed, it has been recently shown that,
in control mice, isoflurane anesthesia can reduce EF to 60%
relative to 79% obtained with a deep sedation only (32).
Also, compared to baseline, before Mn2þ injection, we
obtained a global increase in EF measurements after Mn2þ
injection that is significant for the control group and correlated
with a decrease in EDV and ESV. This is explained by the loss of
contrast between blood and myocardium in presence of Mn2þ. In
fact, the determination of endocardial contours tends to be
underestimated. From the definition of EF, if EDV and ESV
are underestimated EF is therefore overestimated. As Mn2þ
intake is globally more significant in control mice myocardium
than in other groups it can explain the difference between
measurements done before and after Mn2þ injection in this
group. As a consequence, estimation of EF should preferentially
be done before Mn2þ injection. However, we must point out
that observations from cine images still show the presence, or
not, of a decreased contraction after Mn2þ injection. Some
flow artifacts are still present in cine images due to longer TE,
especially at phases when a significant change in flow
occurs, however those artifacts are far from the systolic and
diastolic phases so they do not alter function quantification.
Reduction of TE is limited by gradient performance, so a possible
way to assess this problem would be to use a sequence that
allows compensation of gradient moments such as spiral
acquisitions (33).
Regional function showed a significant decrease in wall
thickening from the anterior to the inferior part of the
myocardium for the IR60 group compared to the control and
sham groups. From Table 3 no significant difference was
observed in wall thickening measurement between control
and sham group. Moreover, globally no significant difference
is obtained in measurements done before and after Mn2þ. This
is explained by the fact that the underestimation of the
endocardial contour affects this measurement less than for
EDV and ESV, where this error is multiplied by the number of
slices. To check this hypothesis, we measured the area of
myocardium obtained by manually tracing endocardial and
epicardial contours before and after Mn2þ injection. The area was
not significantly different before and after injection for the
systolic and diastolic phases in the IR60, sham and control groups
(results not shown). This measurement can therefore be
done after Mn2þ injection, which allows considerable acceleration of the imaging protocol by doing the injection 45 min
before the MRI exam. Also, Figure 8 showed a strong
correlation between wall thickening and normalized SI for
each sector of the left ventricle showing that Mn2þ is mostly
present in functional parts of the myocardium supporting
the hypothesis that Mn2þ effectively depicts extracellular
Ca2þ uptake through L-type channels. No correlation was
found, either for sham, or for control groups due to the lack
of dispersion of wall thickening and normalized signal intensity
values. However, even if these results depicted the contraction
deficit relative to infarction, an important improvement in
function evaluation would be to perform assessment of
intramyocardial strains, either by displacement-encoded imaging
with stimulated echos (DENSE) (34) or with myocardial tagging, as
was previously reported in rat myocardium (35) and also in
mice (36,37). Moreover, it has been shown that myocardial
tagging can also provide accurate EF measurement (38), so
both regional and global function can be acquired in one
single acquisition.
Further applications
This study shows the feasibility of acute infarction assessment
and quantification at 3T. This opens various possibilities for
9
NMR Biomed. (2010)
applications such as longitudinal studies of infarction evolution
with different ischemic models, risk zone assessment as
previously done in other animal models (18,39), and investigation of drug effect on infarction size. Indeed, in rats,
injecting Mn2þ at the beginning of occlusion instead of after
reperfusion led to an accumulation of ions in the well perfused
myocardium, showing the area at risk as a hypointense signal in
MEMRI (18). The ratio of infarct area over the risk zone could be
accurately measured in mice with this method, thus decreasing
the variability related to variation of the occlusion site and or
coronary anatomy.
CONCLUSIONS
Infarction assessment in a mouse model of reperfused
myocardial infarction in the acute phase of the injury with
MEMRI has not previously been reported. This study has shown
assessment and quantification of even non-transmural infarcts
with an excellent correlation to standard TTC staining results.
Ejection fraction and percentage wall thickening measurements allowed evaluation of global and regional function.
While EF must be measured before Mn2þ injection to avoid bias
introduced by the reduction of contrast in cine images, percent
wall thickening can be measured either before or after Mn2þ
injection and accurately depicts infarct-related contraction
deficit. Finally, this MEMRI protocol allows longitudinal study of
cardiac disease in the mouse on a clinical 3T scanner, a widely
available platform.
Acknowledgements
This work was partially supported by the Swiss National Science
Foundation (grant PPOOB3-116901) and the Center for Biomedical Imaging (CIBM), Lausanne and Geneva, Switzerland.
REFERENCES
1. Epstein FH. MR in mouse models of cardiac disease. NMR Biomed.
2007; 20(3): 238–255.
2. Jacquier A, Higgins CB, Saeed M. MR imaging in assessing cardiovascular interventions and myocardial injury. Contrast Media Mol.
Imaging 2007; 2(1): 1–15.
3. Sakuma H. Magnetic resonance imaging for ischemic heart disease.
J. Magn. Reson. Imaging. 2007; 26(1): 3–13.
4. Mendoncadias MH, Gaggelli E, Lauterbur PC. Paramagnetic Contrast
Agents in Nuclear Magnetic-Resonance Medical Imaging. Sem. Nucl.
Med. 1983; 13(4): 364–376.
5. Waghorn B, Edwards T, Yang Y, Chuang KH, Yanasak N, Hu TC.
Monitoring dynamic alterations in calcium homeostasis by T
(1)-weighted and T (1)-mapping cardiac manganese-enhanced MRI
in a murine myocardial infarction model. NMR Biomed. 2008.
6. Lukyanenko V, Chikando A, Lederer WJ. Mitochondria in cardiomyocyte Ca2þ signaling. Int. J. Biochem. Cell. Biol. 2009; 41(10):
1957–1971.
7. Hu TC, Pautler RG, MacGowan GA, Koretsky AP. Manganese-enhanced
MRI of mouse heart during changes in inotropy. Magn. Reson. Med.
2001; 46(5): 884–890.
8. Krombach GA, Saeed M, Higgins CB, Novikov V, Wendland MF.
Contrast-enhanced MR delineation of stunned myocardium with
administration of MnCl(2) in rats. Radiology. 2004; 230(1): 183–190.
9. Wendland MF. Applications of manganese-enhanced magnetic resonance imaging (MEMRI) to imaging of the heart. NMR Biomed. 2004;
17(8): 581–594.
10. Yang Y, Gruwel ML, Sun J, Gervai P, Yang X, Kupriyanov VV. Manganese-enhanced MRI of acute cardiac ischemia and chronic infarction
in pig hearts: kinetic analysis of enhancement development. NMR
Biomed. 2009; 22(2): 165–173.
11. Bremerich J, Saeed M, Arheden H, Higgins CB, Wendland MF. Normal
and infarcted myocardium: differentiation with cellular uptake of
manganese at MR imaging in a rat model. Radiology 2000; 216(2):
524–530.
12. Bruvold M, Nordhoy W, Anthonsen HW, Brurok H, Jynge P. Manganese-calcium interactions with contrast media for cardiac magnetic
resonance imaging: a study of manganese chloride supplemented
with calcium gluconate in isolated Guinea pig hearts. Invest. Radiol.
2005; 40(3): 117–125.
13. Hu TC, Bao W, Lenhard SC, Schaeffer TR, Yue TL, Willette RN, Jucker BM.
Simultaneous assessment of left-ventricular infarction size, function
and tissue viability in a murine model of myocardial infarction by
cardiac manganese-enhanced magnetic resonance imaging (MEMRI).
NMR Biomed. 2004; 17(8): 620–626.
14. Michael LH, Ballantyne CM, Zachariah JP, Gould KE, Pocius JS, Taffet
GE, Hartley CJ, Pham TT, Daniel SL, Funk E, Entman ML. Myocardial
infarction and remodeling in mice: effect of reperfusion. Am. J.
Physiol. 1999; 277(2 Pt 2): H660–668.
15. Yang F, Liu YH, Yang XP, Xu J, Kapke A, Carretero OA. Myocardial
infarction and cardiac remodelling in mice. Exp. Physiol. 2002; 87(5):
547–555.
16. Silva AC, Lee JH, Aoki I, Koretsky AP. Manganese-enhanced magnetic
resonance imaging (MEMRI): methodological and practical considerations. NMR Biomed. 2004; 17(8): 532–543.
17. Kellman P, Arai AE, McVeigh ER, Aletras AH. Phase-sensitive inversion
recovery for detecting myocardial infarction using gadoliniumdelayed hyperenhancement. Magn. Reson. Med. 2002; 47(2):
372–383.
18. Daire JL, Hyacinthe JN, Tatar I, Montet-Abou K, Ivancevic MK, Masterson K, Jorge-Costa M, Morel DR, Vallee JP. In vivo myocardial infarct
area at risk assessment in the rat using manganese enhanced
magnetic resonance imaging (MEMRI) at 1.5T. Magn. Reson. Med.
2008; 59(6): 1422–1430.
19. Cerqueira MD, Weissman NJ, Dilsizian V, Jacobs AK, Kaul S, Laskey WK,
Pennell DJ, Rumberger JA, Ryan T, Verani MS. Standardized
myocardial segmentation and nomenclature for tomographic imaging of the heart: a statement for healthcare professionals from
the Cardiac Imaging Committee of the Council on Clinical Cardiology of the American Heart Association. Circulation 2002; 105(4):
539–542.
20. Vivaldi MT, Kloner RA, Schoen FJ. Triphenyltetrazolium staining of
irreversible ischemic injury following coronary artery occlusion in rats.
Am. J. Pathol. 1985; 121(3): 522–530.
21. Simonetti OP, Kim RJ, Fieno DS, Hillenbrand HB, Wu E, Bundy JM, Finn
JP, Judd RM. An improved MR imaging technique for the visualization
of myocardial infarction. Radiology 2001; 218(1): 215–223.
22. Bohl S, Lygate CA, Barnes H, Medway D, Stork LA, Schulz-Menger J,
Neubauer S, Schneider JE. Advanced methods for quantification of
infarct size in mice using three-dimensional high-field late gadolinium enhancement MRI. Am. J. Physiol. Heart Circ. Physiol. 2009;
296(4): H1200–1208.
23. Mahoney JP, Small WJ. Studies on manganese. 3. The biological
half-life of radiomanganese in man and factors which affect this
half-life. J. Clin. Invest. 1968; 47(3): 643–653.
24. Kang YS, Gore JC. Studies of tissue NMR relaxation enhancement by
manganese. Dose and time dependences. Invest. Radiol. 1984; 19(5):
399–407.
25. Ojha N, Roy S, Radtke J, Simonetti O, Gnyawali S, Zweier JL,
Kuppusamy P, Sen CK. Characterization of the structural and functional changes in the myocardium following focal ischemia-reperfusion injury. Am. J. Physiol. Heart Circ. Physiol. 2008;
294(6): H2435–2443.
26. Yang Z, Berr SS, Gilson WD, Toufektsian MC, French BA. Simultaneous
evaluation of infarct size and cardiac function in intact mice by
contrast-enhanced cardiac magnetic resonance imaging reveals contractile dysfunction in noninfarcted regions early after myocardial
infarction. Circulation 2004; 109(9): 1161–1167.
27. Oshinski JN, Yang Z, Jones JR, Mata JF, French BA. Imaging time
after Gd-DTPA injection is critical in using delayed enhancement to
determine infarct size accurately with magnetic resonance imaging.
Circulation 2001; 104(23): 2838–2842.
10
NMR Biomed. (2010)
28. Winter EM, Grauss RW, Atsma DE, Hogers B, Poelmann RE, van der
Geest RJ, Tschope C, Schalij MJ, Gittenberger-de Groot AC, Steendijk P.
Left ventricular function in the post-infarct failing mouse heart by
magnetic resonance imaging and conductance catheter: a comparative analysis. Acta Physiol. (Oxf ). 2008; 194(2): 111–122.
29. Bolli R, Marban E. Molecular and cellular mechanisms of myocardial
stunning. Physiol. Rev. 1999; 79(2): 609–634.
30. Hampton TG, Amende I, Travers KE, Morgan JP. Intracellular calcium
dynamics in mouse model of myocardial stunning. Am. J. Physiol.
1998; 274(5 Pt 2): H1821–1827.
31. Medina DC, Kirkland DM, Tavazoie MF, Springer CS Jr, Anderson SE.
Naþ/Ca2þ-exchanger-mediated Mn2þ-enhanced (1)H2O MRI in
hypoxic, perfused rat myocardium. Contrast Media Mol. Imaging
2007; 2(5): 248–257.
32. Berry CJ, Thedens DR, Light-McGroary K, Miller JD, Kutschke W,
Zimmerman KA, Weiss RM. Effects of deep sedation or general
anesthesia on cardiac function in mice undergoing cardiovascular
magnetic resonance. J. Cardiovasc. Magn. Reson. 2009; 11(1): 16.
33. Nayak KS, Hargreaves BA, Hu BS, Nishimura DG, Pauly JM, Meyer CH.
Spiral balanced steady-state free precession cardiac imaging. Magn.
Reson. Med. 2005; 53(6): 1468–1473.
34. Gilson WD, Yang Z, French BA, Epstein FH. Measurement of myocardial mechanics in mice before and after infarction using multislice
displacement-encoded MRI with 3D motion encoding. Am. J. Physiol.
Heart. Circ. Physiol. 2005; 288(3): H1491–1497.
35. Hyacinthe JN, Ivancevic MK, Daire JL, Vallee JP. Feasibility of complementary spatial modulation of magnetization tagging in the rat
heart after manganese injection. NMR Biomed. 2008; 21(1): 15–21.
36. Epstein FH, Yang Z, Gilson WD, Berr SS, Kramer CM, French BA.
MR tagging early after myocardial infarction in mice demonstrates
contractile dysfunction in adjacent and remote regions. Magn. Reson.
Med. 2002; 48(2): 399–403.
37. Zhou R, Pickup S, Glickson JD, Scott CH, Ferrari VA. Assessment of
global and regional myocardial function in the mouse using cine and
tagged MRI. Magn. Reson. Med. 2003; 49(4): 760–764.
38. Dornier C, Somsen GA, Ivancevic MK, Osman NF, Didier D, Righetti A,
Vallee JP. Comparison between tagged MRI and standard cine MRI for
evaluation of left ventricular ejection fraction. Eur. Radiol. 2004; 14(8):
1348–1352.
39. Natanzon A, Aletras AH, Hsu LY, Arai AE. Determining canine myocardial area at risk with manganese-enhanced MR imaging. Radiology
2005; 236(3): 859–866.
11
NMR Biomed. (2010)
Available online at www.sciencedirect.com
Magnetic Resonance Imaging 28 (2010) 862 – 881
Spiral demystified
Bénédicte M.A. Delattre a,⁎, Robin M. Heidemann b , Lindsey A. Crowe a ,
Jean-Paul Vallée a , Jean-Noël Hyacinthe a
a
Radiology Clinic, Geneva University Hospital and Faculty of Medicine, University of Geneva, 1211 Geneva 14, Switzerland
b
Department of Neurophysics, Max Planck Institute for Human Cognitive and Brain Sciences, 04103 Leipzig, Germany
Received 2 December 2009; revised 24 February 2010; accepted 5 March 2010
Abstract
Spiral acquisition schemes offer unique advantages such as flow compensation, efficient k-space sampling and robustness against motion
that make this option a viable choice among other non-Cartesian sampling schemes. For this reason, the main applications of spiral imaging
lie in dynamic magnetic resonance imaging such as cardiac imaging and functional brain imaging. However, these advantages are
counterbalanced by practical difficulties that render spiral imaging quite challenging. Firstly, the design of gradient waveforms and its
hardware requires specific attention. Secondly, the reconstruction of such data is no longer straightforward because k-space samples are no
longer aligned on a Cartesian grid. Thirdly, to take advantage of parallel imaging techniques, the common generalized autocalibrating
partially parallel acquisitions (GRAPPA) or sensitivity encoding (SENSE) algorithms need to be extended. Finally, and most notably, spiral
images are prone to particular artifacts such as blurring due to gradient deviations and off-resonance effects caused by B0 inhomogeneity and
concomitant gradient fields. In this article, various difficulties that spiral imaging brings along, and the solutions, which have been developed
and proposed in literature, will be reviewed in detail.
© 2010 Elsevier Inc. All rights reserved.
Keywords: Spiral; Non-cartesian; Gradient; Blurring; Off-resonance; Parallel imaging
1. Introduction
In many magnetic resonance imaging (MRI) applications,
it is crucial to reduce the acquisition time. One method to
achieve this can be the use of non-Cartesian k-space
acquisition schemes, such as spiral trajectories [1]. Spiral
sampling, including variable density spiral, has the advantage of the ability to cover k-space in one single shot starting
from the center of k-space. Moreover, spiral imaging is very
flexible. High temporal and spatial resolution, as required for
specific applications like cardiac imaging and functional
MRI, can be obtained by tuning the number of interleaves
and the variable density parameter. Intrinsic properties of
the spiral trajectory itself offer advantages that cannot be
found with other types of trajectories. The major ones are an
⁎ Corresponding author. Clinic of Radiology – CIBM, Geneva
University Hospital, 1211 Geneva 14, Switzerland. Tel.: +41 22 37 25 212;
fax: +41 22 37 27 072.
E-mail addresses: [email protected], [email protected]
(B.M.A. Delattre).
0730-725X/$ – see front matter © 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.mri.2010.03.036
efficient use of the gradient performance of the system, an
effective k-space coverage since the corners are not acquired,
as well as a large signal-to-noise ratio (SNR) provided
by starting the acquisition at the center of k-space. In principle, spiral offers some inherent refocusing of motion and
flow-induced phase error, which is not compensated by
conventional sampling schemes [2]. Covering the center of
k-space in each interleave can be useful, as this information
can be used for self-navigated sequences for example [3].
Finally, very early acquisition of the k-space centre needed
for ultra-short TE sequences can be fulfilled with spiral [4].
For these reasons, the main applications of spiral imaging lie
in dynamic MRI, such as cardiac imaging [5–7], coronary
imaging [8–11], functional MRI [12–14] and also chemical
shift imaging [15,16]. Even though a broad range of applications for spiral imaging exists, this review article is focused
on examples from cardiac and head imaging to illustrate
specific properties of spiral imaging. Indeed, spiral sampling
allows real-time cardiac MRI with a high in-plane resolution
(1.5 mm2) [6]. 3D Cine images acquired with variable density
spirals show sharper images than comparable Cartesian
B.M.A. Delattre et al. / Magnetic Resonance Imaging 28 (2010) 862–881
images with the same nominal spatial and temporal resolution
(1.35 mm2 and 102 ms, respectively) [8]. Moreover, its use
for 3D coronary angiography improved SNR and contrast-tonoise ratio (CNR) by a factor of 2.6 compared to current
Cartesian approach [17] and reduced the acquisition time to a
single breath-hold with 1-mm isotropic spatial resolution [9].
Here, image quality was improved by a fat suppression
technique obtained with a spectral-spatial excitation pulse
that further reduced off-resonance artifacts due to fat [18].
Finally, the usefulness of variable density spiral phase
contrast was shown with its application to real-time flow
measurement at 3T [19]. The authors showed the ability to
monitor intracardiac, carotid and proximal flow in healthy
volunteers with a typical temporal resolution of 150 ms, a
spatial resolution of 1.5 mm and no need for triggering or
breath-holding. These results are very promising for cardiac
patients with dyspnea or arrhythmia. Salerno et al. [20]
showed another promising example of spiral application;
myocardial perfusion imaging. In conventional myocardial
perfusion imaging, the so-named dark-rim artifact [21] is
often a problem. This artifact was minimized by the use of
spiral acquisition schemes.
All of these examples show the potential of spiral imaging
to improve clinical diagnostic imaging by reducing the
overall acquisition time without any penalty on the spatial
and the temporal resolution and by reducing the negative
effects of flow and motion on image quality. After describing
all the advantages of spiral imaging, the remaining question
is, why is the spiral sequence not used more often in clinical
routine? A simple answer is that spiral imaging is more
complex than Cartesian imaging. Practical difficulties make
the implementation of spiral imaging quite challenging and
counterbalance the advantages of this method. Firstly, the
design of gradient waveforms requires specific attention
because hardware is most often optimized for linear
waveforms. Indeed, calculation of the trajectory requires
the nontrivial resolution of differential equations, and
specific care must be taken to find a solution suitable on
the scanner. Secondly, the reconstruction of such images is
no longer straightforward because points are not sampled on
a Cartesian grid. The simple use of Fast Fourier Transform
(FFT) is therefore not possible, and this also implies that
parallel imaging algorithms such as sensitivity encoding
(SENSE) or generalized autocalibrating partially parallel
acquisitions (GRAPPA) have to be adapted to this nonCartesian trajectory to benefit from the acceleration they can
provide. Thirdly, and probably the most limiting factor, is
that spiral images are often prone to particular artifacts such
as distortion and blurring that have several physical origins,
from gradient deviations to off-resonance effects due to B0
inhomogeneities and concomitant field, and that need to be
measured in order to correct for them.
Due to the reasons listed above, spiral trajectories still
lack popularity and seem to be reserved for experts and a few
specific applications. This review article aims to guide the
reader through the main challenges of spiral imaging,
863
showing the solutions that have been proposed to address
these problems.
2. What is spiral?
This preliminary section presents the theory necessary to
understand the difficulties related to the spiral trajectory.
Image formation is only possible by encoding the spatial
location of the spins in the precessing frequency. This is
done by application of varying fields called gradients. The
location information is then contained in the phase of the
rotating proton spin. Neglecting the relaxation processes of
the sample magnetization, the signal acquired, s, can be
expressed as the Fourier Transform of the proton density, ρ
(for simplicity, in the following, the signal measured, s, is
implicitly considered proportional to the magnetization and
the longitudinal magnetization at equilibrium proportional to
the spin density of the sample [22]):
YY
Y R Y
r
ð1Þ
sðkÞ = qðrÞe − i2p k r d Y
where k is the k-space location and is related to gradient
fields:
Zt
Y
Y
ð2Þ
kðtÞ = Pc Gðt VÞdt V
0
where γ %=γ/(2π)
and γ is the gyromagnetic ratio. The
̵
easiest way to reconstruct the image is the use of the FFT
algorithm [23], which is computationally the most efficient
algorithm. Cartesian sampling is thus the most appropriate
sampling scheme because points are placed on a Cartesian
grid. However, this trajectory has the disadvantage of being
very slow, as the coverage of k-space must be done line by
line. On this first aspect, the spiral trajectory is more
interesting because it uses the gradient hardware very
efficiently and can also cover k-space in a single shot.
Another advantage is that the sampling density can be varied
in the center of k-space, which can be useful in several
applications where more attention is given to low spatial
Fig. 1. Example of variable density spiral trajectory on a Cartesian grid.
864
frequencies. However, reconstruction of the image is then
not straightforward, as points are no longer placed onto the
Cartesian grid (see Fig. 1). Also, this kind of trajectory is
more sensitive to field inhomogeneities because the readout is
usually longer than in Cartesian sampling. As a consequence,
phase shifts from several origins can accumulate during the
relatively long readout time, resulting in image degradations.
2.1. Spiral trajectory
The spiral trajectory has the advantage of the possibility
to cover k-space in a single shot. This last property can also
be achieved with echoplanar imaging (EPI) readout, but as a
Cartesian trajectory, it needs a fast change of gradient
intensity that produces important Eddy currents. The general
spiral trajectory can be written as:
k = ksa ejxs
ð3Þ
where k=k(t) is the complex location in k-space, τ=[0,1] is a
function of time t, α is the variable density parameter (α=1
corresponds to uniform density), ω=2πn with n the number
of turns in the spiral and λ=N/(2*FOV) with N the matrix
size. The most common spiral scheme is Archimedeaan
spiral, characterized by the fact that successive turnings of the
spiral have a constant separation distance. This corresponds
to the case α=1 (uniform density spiral). Fig. 2 shows an
example of this trajectory. However, interest rapidly turned to
variable density spirals (where successive turnings of the
spiral are no longer equidistant, α≠1) instead of purely
Archimedean because this enhances the flexibility of the
trajectory by sampling the center of k-space differently than
the edges resulting in a reduction of aliasing artifacts when
undersampling the trajectory [24], as well as reducing motion
artifacts [2]. While k(t) defines the spiral trajectory in kspace, the exact position of sampled data points along that
trajectory is defined by the choice of the function τ(t). For the
special case that τ(t)=t, the amount of time spent for each
winding is constant, regardless of whether the acquired
winding is near the center or in the outer part of the spiral. In
other words, the readout gradients reach their maximum
performance at the end of the acquisition. This acquisition
scheme, initially proposed by Ahn et al. [1], is called the
constant-angular-velocity spiral trajectory. The constantangular-velocity spiral trajectory can easily be transformed
into a so-called constant-linear-velocity spiral by using
τ(t)=√t, in Eq. (3). It has been shown that the constantlinear-velocity spiral offers some advantages in terms of
SNR and gradient performance as compared to the
constant-angular-velocity spiral [25]. Although constantlinear-velocity spiral trajectories are more practical, the
constant-angular-velocity spiral trajectories have some
interesting properties. In a constant-angular-velocity Archimedean spiral, the number of sampling points per winding
is constant. If this number is even, all acquired data points
are aligned along straight lines through the origin and are
collinear with the center point, as schematically shown in
Fig. 2.
With the desired k-space trajectory, the gradient waveforms G(t) and the slew-rate S(t) can be defined as:
:
:
k sðtÞa ejxsðtÞ − sðt − DtÞa e jxsðt − DtÞ
kðtÞ s dk
ð4Þ
G ðt Þ =
=
=
c ds
Dt
c
c
:
:
s2 d 2 k s̈ dk
ð5Þ
+
S ðt Þ = Gðt Þ =
c ds2 c ds
where G(t)=Gx(t)+iGy(t) is the gradient amplitude and
S(t)=Sx(t)+iSy(t) is the gradient slew-rate in both directions,
Δt is the time interval of the gradient waveform. This
formulation implies a sinusoidal waveform for Gx(t) and
Gy(t). The major difficulty here is to find an analytic
equation for the gradient waveform G(t) by defining τ(t) in
order to enable the real-time calculation of the gradient
waveform at the magnetic resonance (MR) scanner. Even
though sinusoidal gradient waveforms are smoother than the
trapezoidal gradients used for Cartesian sampling, imperfections in the realization of the trajectory are unavoidable.
Inaccurate gradient fields generate an additional phase term,
which accumulates during data acquisition and result in
variations of the actual trajectory from the calculated trajectory. This leads to image blurring, because the reconstruction is performed with improper k-space position of the
data points and thus introduces artifacts to the whole image.
2.2. Specific advantages of spiral trajectory
Fig. 2. Constant-linear-velocity (left) versus constant-angular-velocity
(right) spiral trajectory: Data points acquired or interpolated onto a
constant-angular-velocity spiral trajectory (right), indicated by gray points,
lie on straight lines through the origin and are collinear with the center point
(black point).
As mentioned in the introduction, spiral trajectory offers
some inherent advantages over other types of trajectory. For
example, it is relatively insensitive to flow and motion
artifacts. Indeed, considering the accumulated phase from an
isochromat located at position r at time t placed into a static
field B0 and a gradient field G(t), one obtains:
Zt
/ðr; tÞ = cB0 ðtÞ − c
Gðt VÞrðt VÞdt V
0
ð6Þ
The position r(t) can also be written with a Taylor
series expansion:
Zt
dr
1 d2 r 2
t
V
+
:::
dt V
t V+
/ðr; t Þ =cB0 ðt Þ +c Gðt VÞ r0 +
dt V
2 dt V2
0
ð7Þ
Eq. (7) can then be decomposed into the gradient moment
expansion [26]:
/ðr; t Þ = cB0 ðt Þ + cM0 ðt Þr0 + cM1 ðt Þ
dr
1
d2 r
+ c M2 ðt Þ 2 + :::
dt
2
dt
ð8Þ
Zt
Gðt VÞdt V; 0 order gradient moment
th
0
Zt
tGðt VÞdt V; 1st order gradient moment
M1 =
ð9Þ
0
Zt
t 2 Gðt VÞdt V; 2nd order gradient moment
M2 =
were TR=50 ms, FOV=300 mm, matrix size of 185×185,
acquisition time=40 s).
Finally, sampling the center of k-space for each interleave
gives information about the position of the object and can be
used as a navigator for abdominal and cardiac applications
[27]. Liu et al. [3] obtained highly improved image
reconstruction in the context of DWI by using a low
resolution image given by the first variable density interleave
of the spiral trajectory to correct the phase of the high
resolution image (obtained with all the interleaves) (scan
parameters were TR/TE=67/2.5 ms, FOV=220 mm, matrix
size of 256×256, 28 interleaves for one slice, acquisition
time 8.1 min for whole brain).
2.3. Eddy currents
Where
M0 =
865
0
As a consequence, accumulated phase can be independent
of position, speed and acceleration if the gradient moments
are null. In the case of spiral imaging, gradient moments are
weak at the center of k-space and increase slowly with time.
Moreover, due to their sinusoidal forms, gradients take
periodically positive and negative comparable values that
compensate phase accumulation. Those reasons lead to a
weak phase accumulation and give the spiral trajectory a
certain insensitivity to movement and flow artifacts.
Furthermore, the symmetry of x and y gradients do not
lead to a phase accumulation in a preferred direction which
would be the case in EPI for example.
Another advantage of spiral trajectory is its robustness to
aliasing artifacts due to the possibility of oversampling the
center of k-space. Indeed, the image spectrum is sparse in
k-space, with low spatial frequencies containing most of
the image energy. Undersampling uniformly k-space will
end to aliasing artifacts while sampling sufficiently the
center of k-space by increasing the sampling density in this
region will drastically reduce them. Tsai et al. [24]
demonstrated on a short-axis cardiac image that the severe
aliasing artifacts produced by the chest wall with uniform
density spiral scan were suppressed with variable density
spirals (scan parameters were: 17 interleaves, field of view
(FOV)=160 mm, in-plane resolution of 0.65 mm, TE=15 ms,
readout time=16 ms) [24]. Liao et al. [2] have also shown
that oversampling the center of k-space provides additional
reduction of motion artifacts. This is due to the fact that
observed motion is a periodic phenomenon, of which the
frequency band is mainly contained in the low spatial
frequencies (scan parameters to obtain a cine with 16 frames
Time varying gradient fields induce currents in the
conducting elements composing the magnet and the coils.
These so-called Eddy currents create a magnetic field that
opposes the change caused by the original one (Lenz's law),
deteriorating the gradient waveform. In modern scanners,
Eddy currents are mainly corrected with actively shielded
gradients but residual currents can still be present. In a
simple Eddy current model [26], the field generated by the
Eddy current Ge(t) is given by:
Ge ðt Þ = −
dG
× eð t Þ
dt
ð10Þ
where G is the applied gradient, × denotes the convolution
and e(t) is the impulse response of the system:
X
eðtÞ = HðtÞ
an e − t = s n
ð11Þ
n
where H(t) is the unit step function. Just a few terms in this
summation are necessary to characterize most of the Eddy
current behavior. As it adds an unwanted magnetic field, the
Eddy current effect results in phase accumulation leading to
image distortions. It is mainly responsible for the wellknown ghosting artifact in EPI, while in the case of spiral
trajectory, it causes image blurring.
2.4. Sensibility to inhomogeneities
Spiral images are prone to blurring and distortions
originating from several sources. Ignoring relaxation effects,
Eq. (1) showed that the signal acquired from an object in a
magnetic field is given by:
RR
ð12Þ
sðtÞ =
qðx; yÞe − i2p½kx x + ky y + /ðx;y;tÞ dxdy
where kx and ky are the k-space coordinates; ρ(x,y), the
proton density of the object at (x,y) coordinates and ϕ(x,y,t),
the arbitrary field inhomogeneities that is mainly composed
of main field inhomogeneity, gradient imperfections,
residual Eddy currents, chemical shift between water and
other species or susceptibility differences between air and
tissue. This equation is general and applies to every sampling
866
scheme. This means that even Cartesian sampling is prone to
field inhomogeneities. However, in Cartesian sampling, only
one gradient is varied at a time, which implies that dephasing
affects only one direction. This results in a simple shift of
the object. It is more problematic with spiral because both
in-plane gradients are varied continuously at the same time,
resulting in a shift of the object in all directions that causes
image blurring. The exact reconstruction of such a signal is
given by:
Z T
sðtÞei2pðkx x + ky yÞ ei2p /ðx;y;tÞ dt
ð13Þ
qðx; yÞ =
0
This is called the conjugate phase reconstruction, because
before integration, the signal is multiplied by the conjugate
of phase accrued due to field inhomogeneities. The
inhomogeneity term can often be written as a linear relation
with time t, ϕ(x,y,t)=tϕ(x,y) implying that it is more
significant when t is important. There are, though, two
approaches to eliminate this effect: the first one is to use
interleaved spirals that have a short readout time t and then
limit phase accumulation; the second is to correct for the
inhomogeneities when reconstructing the image. Distortion
and blurring induced by phase accumulation due to
inhomogeneities are probably the main reasons that explain
the lack of success of spiral trajectories in clinical routine;
however, with the improvements of methods to measure and
correct for these inhomogeneities, this situation should not
be definitive.
2.5. Concomitant fields
Another parameter that can alter the image quality is
concomitant gradient fields. Indeed, Maxwell's equations
imply that imaging gradients are accompanied by higher
order, spatially varying fields called concomitant fields.
They can cause unwanted phase accumulation during
readout resulting, again, in image blurring in the special
case of spiral, but once again, even though Cartesian
sampling is also affected by this additional dephasing,
the effect on the image is simply less disturbing. Considering identical x and y gradient coils with a relative
orientation of 90°, the lowest order concomitant field can
be expressed as [28]:
!
2
Gy Gz
Gz 2 2 G2x + G2y 2
Gx Gz
x +y +
xz −
yz
z −
Bc =
8B0
2B0
2B0
2B0
ð14Þ
where x, y, z are the laboratory directions, B0 the static
field, Gx, Gy, Gz the gradients in the laboratory system.
Concomitant gradients cause phase accumulation during
the readout gradient that is expressed by:
Zt
fc ðtÞ = Pc
Bc dt
−l
ð15Þ
The knowledge of the analytical dependence of this effect
with spatial coordinates is necessary to correct for its
contribution to image blurring.
3. Designing the trajectory
The first difficulty with spiral imaging is the design of
the trajectory itself. Indeed, gradient solutions must be
found by solving the differential Eqs. (4) and (5) that are
computationally intensive to calculate even with the improvement of hardware capabilities. Closed-form equations
are necessary to be easily usable on a clinical scanner. To
take maximum advantage of the gradient hardware
capabilities, two regimes are defined. Indeed, near the
center of k-space, the trajectory is only limited by the
gradient slew-rate since gradient amplitude is low. For this
slew-rate-limited regime, S(t) is set to the maximum
available slew-rate Sm. Then, when reaching the maximal
gradient amplitude, one comes into the so called amplitudelimited regime where G(t) is equal to the maximum available
gradient amplitude Gm.
3.1. General solution
A simple analytical solution for constant density (α=1)
was first given by Dyun et al. [29] for the slew-rate limited
case only and then extended by Glover [30] for the two
regimes. It was then generalized to variable density by Kim
et al. [31]. They defined the function τ(t) as follows:
8 "rffiffiffiffiffiffiffiffiffi
#
a 1=ða=2+1Þ
>
>
S
c
m
>
>
+1 t
slew−rate−limited regime
<
kx2 2
sð t Þ = 1 = ða + 1Þ
>
>
cGm
>
>
ða + 1Þt
amplitude−limited regime
:
kx
ð16Þ
The trajectory starts in the slew-rate-limited regime and
switches to the amplitude-limited regime when t corresponds
to G(t)=Gm, the maximum available gradient amplitude.
This closed-form solution has the advantage of being easily
implemented on a clinical scanner. An example of the
trajectory obtained with this formulation, as well as the
gradient waveform, is illustrated in Fig. 3. However, when
the number of interleaves is increased, this trajectory leads
to large slew-rate overflow for small k-space values (i.e.,
for t→0) as illustrated in Fig. 4. Depending on the gradient
performances, this can be a real problem with most clinical
scanners because slew-rates are limited and the execution of
a trajectory with such overshoot is simply impossible. For
example, a spiral sequence was implemented on 3T Siemens
clinical scanner (Magnetom TIM Trio, Siemens Healthcare,
Erlangen Germany) using a maximum slew-rate of Sm=170
T/(m·s) and a maximum gradient amplitude of Gm=26 mT/m
have been used. Simulations of Figs. 3 and 4 were done
867
Fig. 3. Examples of spiral trajectories calculated with the Kim design [31] for 10 interleaved spirals, FOV=100 mm, matrix size N=128. k-Space values (left)
and gradient waveforms (right) for constant density spiral α=1 (a), and variable density spiral α=3 (b). Dotted lines represent the transition between slew-rate
limited regime and amplitude-limited regime.
with this system by taking a small security margin: Sm=90/
100·170=153 T/(m·s) and Gm=90/100·26=23.4 mT/m.
the minimum value to avoid slew-rate overflow with the
example of the Siemens 3T system characteristics.
3.2. Glover's proposition to manage k-space center
3.3. Zhao's adaptation to variable density spiral
As pointed out by Glover [30], an instability exists for
small k-space values (i.e., k≈0) because the solution derived
in the slew-rate limited regime is unbounded at the origin.
He proposed an alternative in the case of a uniform density
spiral by setting the slew rate at the origin to be Sm/Λ,
instead of Sm, where Λ is tuned by the user. τ(t) is therefore
defined by
Zhao et al. [32] proposed another solution to this problem
adapted to the variable density case by setting the slew rate to
exponentially increase to its maximum value:
SðtÞ = Sm ð1− e − t = L Þ2
ð18Þ
ð17Þ
where L is a parameter used to regularize the slew-rate at the
origin. Then, they obtain:
"rffiffiffiffiffiffiffiffiffi
#
1 = ða = 2 + 1Þ
Sm c a
−t=L
+ 1 t + Le
−L
ð19Þ
sð t Þ =
kx2 2
This solution ensures a smooth transition near the origin
that avoids slew-rate overflow as illustrated in Fig. 4. For this
example, Λ was chosen to be 6·10−3, as this corresponds to
The parameter L is chosen by setting S(t)=Sm/2 for the
P-th data point:
PDt
pffiffiffiffiffi
ð20Þ
L= −
lnð1 − 1 = 2Þ
1 2
bt
Sm c
2
sð t Þ =
1 = 3 with b =
kx2
1 4 24
b t
L+
2 9
868
Fig. 4. Upper; examples of τ(t) calculated with Kim [31] (bold plain line), Zhao [32] (plain line), Glover [30] (dashed line) methods and zoom of the first 0.2 ms
of τ(t) and sx(t). Lower; corresponding slew rate sx(t), on the left y-axis scale was cut between±200 T/m/s to have a better visualization; on the right, zoom of the
first 0.2 ms and full y-axis scale. Calculation proposed by Kim largely overshoot maximum available slew rate in this case whereas it is not the case with the Zhao
and Glover propositions (parameters used for this simulation: FOV=100 mm, N=128, α=1, 10 interleaves, Λ=6·10−3, L=3.6·10−4).
where Δt is the time interval between 2 points of the
trajectory. Fig. 4 shows τ(t) and Sx(t) obtained with these
propositions by choosing P=9. This value corresponds to the
minimum value to avoid slew-rate overshoot for the
performance of the chosen scanner.
3.4. Comparison of the three propositions
As illustrated in Fig. 4, the proposition of Kim et al. [31]
causes the slew-rate to overflow in the first milliseconds of
the trajectory which implies that the center of k-space is not
correctly sampled. This is a real problem because important
information is contained in the center of k-space. Both
Glover [30] and Zhao [32] efficiently correct for this
problem at a price of lengthening the trajectory by 3.5%
and 2.3% respectively in the particular example of Fig. 4. In
addition, by adapting to variable density, Zhao [32] has the
advantage of choosing an exponential variation of the time
parameter τ(t) instead of a time power, this results in a
smaller spiral readout time than with Glover's proposition
[30], which may be useful for some applications.
4. Coping with blurring in spiral images
Spiral images, unlike Cartesian images, are often subject
to blurring and distortion. The origins of such effects are well
described and a lot of effort has been made to correct them.
Here the main solutions that have been proposed to correct
contributions such as imperfect trajectory realization, offresonance artifacts and concomitant fields are described.
4.1. Measuring gradient deviations
Deviation from the targeted k-space trajectory due to
hardware inadequacies or imperfect eddy current correction
can lead to image artifacts that are more disturbing in the
case of spiral imaging because phase accumulation in both
gradient directions induces image blurring. A way to correct
for these deviations is first to measure them.
4.1.1. Trajectory measurement with a reference phantom
Mason et al. [33] proposed estimation of the actual
k-space trajectory from the MR signal. The method uses
several calibration measurements on a small sphere reference
phantom of tap water placed at off-isocenter locations in the
magnet bore (x0, y0). The use of this “point phantom” allows
the measured signal to be considered as a simple combination of proton density in the phantom and the dephasing
term. Eq. (1) can thus be written as:
sðtÞ = qðx0 ; y0 Þe − i2p/ðtÞ
ð21Þ
where ϕ(t)=ϕ0(t)+kx(t)x0+ky(t)y0. For each location (x0,
y0), the observed phase change Δϕn between samples n
and n-1 is assumed to be due to a combination of a
spatially invariant time-dependant magnetic field, that
induces a phase change ϕ0n, and the time-varying spatial
gradients in the Gx and Gy directions that induced
incremental phase changes 2πΔkxnx0 and 2πΔkyny0. So
the total phase change is:
D/n = /0n + 2pðDkxn x0 + Dkyn y0 Þ
ð22Þ
From the several acquisitions made at different locations
(x0, y0) the data are fitted to determine the function kx(t),
ky(t) and ϕ(t) by a least squares algorithm.
Some faster methods were also proposed that do not
involve the displacement of a reference phantom but use
the signal from the studied subject. Spin location is done
by self-encoding gradients added just before readout
acquisition. Such gradients are calibrated gradients
applied stepwise in the same direction as the field
gradient to be measured [34]. They then combine different
acquisitions to obtain the phase change from which the
k-space trajectory can be deduced the same way as
Mason et al. [33].
The method proposed by Alley et al. [35] uses the
readout data acquired with normal gradient waveforms and
then with reversed waveforms on a 10 cm phantom filled
with water. The signal obtained after a Fourier transform in
the phase encoding direction in one gradient direction can
be written as:
sF ðx; tÞ = qðx; tÞe − i2p/F ðx;tÞ
ð23Þ
where s+(t) refers to the “normal” acquisition and s-(t) to the
one acquired with the reversed gradient waveform. The
phase term can be separated in odd and even terms:
/F ðx; tÞ = hðx; tÞ F wðx; tÞ F kðtÞx
ð24Þ
Subtraction of the two phases ϕ+ and ϕ− give access to
the trajectory data by a least square fit in the spatial direction.
However, this method requires a high number of readout
acquisitions to characterize the whole k-space trajectory.
4.1.2. In-vivo trajectory measurement
The method proposed by Zhang et al. [36] is even faster
as it uses only two slices positioned along each gradient
of interest. This method is based on the proposition of
Duyn et al. [37] and has the advantage of being used in
vivo, thus avoiding fastidious preliminary calibration of
the trajectory with a phantom. The trajectory is obtained
directly from the phase difference between the two
acquired signals. In fact, if one assumes an infinitely thin
slice, the signal for a slice normal to the x-axis located at
x0 is:
RR
ð25Þ
sðx0 ; tÞ = e − i2p/ðx0 ;tÞ qðx0 ; y; zÞe − i2pky ðtÞy dydz
where the phase term is:
/ðx0 ; tÞ = kx ðtÞx0 + wðtÞ
ð26Þ
Then, the trajectory is obtained by subtracting the phase
of signal from two close slices located at x1 and x2:
kx ð t Þ =
/ðx2 ; tÞ − /ðx1 ; tÞ
x2 − x1
ð27Þ
This method was compared with the small-phantom
method from Mason et al. [33], and authors have found a
869
very good accordance between results. This method is thus
the first that can be used directly in a human subject.
However, when high resolution is needed, some significant
errors are observed for high k-space values.
A correction recently proposed by Beaumont et al. [38]
greatly improves this latter technique. Indeed, from Eq.
(13), they show that for a well-shimmed squared slice
profile, the nonlinear spatial variation of B0 field can
be neglected so the term ϕ(x,y,t) becomes ϕ(t). Then, the
signal is the Fourier transform of the “effective magnetization density” ρ(x,y)e−i2πϕ(t). If the slice is located in the
x plane at x=0 and considering an homogenous sample, the
effective magnetization density is proportional to the slice
profile, so the signal can be written as:
sðkx ðt ÞÞ~
sinðp kx ðtÞDsÞ
p kx ðtÞDs
ð28Þ
where Δs is the slice thickness. This function has several
zeros located at kx=nΔs−1 (n an integer N1). If kmax≥Δs−1,
zero or very low signal can be encountered preventing the
calculation of the trajectory at these points. They solve this
problem by adding another gradient that shifts k-space
points in order to avoid the nulling signal and allow
recovery of the high k-space values.
4.1.3. Trajectory estimation by one-time system calibration
Recently, Tan et al. [39] described an efficient method
to correct for the trajectory deviations. They described the
alteration of k-space trajectory by both anisotropic gradient
delays in each physical axis as well as Eddy currents. For
trajectories like spiral, residual Eddy currents can cause
severe distortion in images. They proposed a model in
which each contribution is separated and corrected after
applying system calibration. The model only depends on
gradient system parameters in the physical coordinates that
can be found by measuring the real trajectory and
comparing it with the theoretical one.
Indeed, as explained by Aldefeld et al. [40], some timing
delays exist in hardware between the command and effective
application of the gradient by gradient amplifiers. These
delays can be different in each physical axis so have to be
characterized separately. The delayed gradients in the logical
coordinates are thus simply a rotation of those defined in the
laboratory coordinates:
#
"
Gx ðt − sx Þ
ð29Þ
GdL = RT Gd = RT Gy ðt − sy Þ
Gz ðt − sz Þ
where GdL is the delayed gradient in logical coordinates; RT, the rotation matrix and τ, the delays in the
three directions.
Tan et al. [39] showed that Eddy currents induce a
k-space trajectory which can be simply modeled as the
integration of the convolution of slew-rate of the desired
gradient waveform and the system impulse response. From
870
Eq. (10), and considering the Taylor expansion of the
exponential contained in Eq. (11) they obtain:
Zt
ke ðtÞ =
Sðt VÞ × Hðt VÞdt V
0
Zt
Zt Z t V
Zt
ðdG = dsÞsdsdt V
cA Gðt VÞdt V + B Gðt VÞt Vdt V− B
0
0
0
the classical Cartesian scheme. These effects accumulate all
along the readout time and result in image blurring that can
be important. One can easily see from Eq. (13) that this exact
conjugate phase reconstruction needs a lot of computation
time because each pixel must be reconstructed with its own
off resonance frequency ϕ(x,y). Fortunately, faster alternatives to this exact reconstruction have been developed and
are described below.
0
ð30Þ
where A=−Σnan and B=Σn(an/bn). This last form represents the scaling term for the theoretical k-space trajectory
in the first term and the two other terms are the system
response for gradient switching (for more details see [39]).
The main advantage of this method is that the measurement of system parameters τx,y,z, A and B needs to be
done only once and can then be used for any slice
position. The authors compared the improvement of image
quality with their method to the simple anisotropic gradient
compensation and noticed that eddy current model must
be added in order to correct efficiently for the residual
k-space imperfections.
4.2.1. Field map estimation
The correction for these inhomogeneities first needs
knowledge of the field map to assess the term ϕ(x,y) shown
in Eq. (13). A rapid method proposed by Schneider [41]
consists of the acquisition of two datasets with different
echo times. The image obtained for each acquisition is
given by:
4.2. Correcting off-resonance effects
so the field inhomogeneity term is simply given by the
phase of the two images:
Off-resonance effects refer to signal contributions with
resonance frequencies different to the central water proton
resonance frequency. These contributions mainly come from
the chemical shift between water and other species, from
susceptibility differences between different tissues or
between air and tissue and from main field inhomogeneities.
Field inhomogeneities are especially important for sequences
where off-center slices are difficult to shim or in areas where
susceptibility differences and motion are important, for
example, in the thorax.
As the spiral sampling scheme usually has a longer
readout time, it is more affected by off resonance effects than
s1 ðx; yÞ = q1 ðx; yÞe − i2pðTE1 /ðx;yÞÞ
s2 ðx; yÞ = q2 ðx; yÞe − i2pðTE2 /ðx;yÞÞ
ð31Þ
and:
s41 s2 = q1 q2 e − i2pð/ðx;yÞðTE2 − TE1 ÞÞ
/ð x; yÞ =
angleðs41 s2 Þ
angleðs41 s2 Þ
=
2pðTE2 − TE1 Þ
2p Dt
ð32Þ
ð33Þ
The phase has to be unwrapped or limited by choosing Δt
short enough.
4.2.2. Time-segmented reconstruction
In the method developed by Noll et al. [42], the time
integral in Eq. (13) is broken into a finite number of temporal
boxes. In each of these temporal segments, the term etiϕ(x,y)
is assumed to be constant and reconstruction is done for each
Fig. 5. Synthetic scheme of the time-segmented reconstruction algorithm (adapted from [26]).
segment. The final image is obtained by adding together the
integrals over all time segments:
qðx; yÞ =
T
X
sðti Þei2pðkx x
+ ky y + ti /ðx;yÞÞ
ð34Þ
ti = 0
As shown in Fig. 5, the computation time depends on the
number of temporal segments used for reconstruction and
can thus be quite important.
4.2.3. Frequency-segmented reconstruction and
multifrequency interpolation
A similar method was proposed by Noll et al. [43] and
consists of segmenting the inhomogeneity term, ϕ(x,y), into
multiple constant frequencies, ϕi(x,y). For each of these
frequencies, an image was reconstructed and the final image
was taken as a spatial combination of these different reconstructions based on spatial varying frequency. Fig. 6 illustrates this algorithm. A refinement to this method was
developed by Man et al. [44] and consists of writing the inhomogeneity term as a linear combination of constant frequency
terms. This is the multifrequency interpolation being:
X
ci ½f ðx; yÞ ei2p t/i ðx;yÞ
ð35Þ
ei2p t/ðx;yÞ =
i
For each frequency ϕi(x,y), an inverse reconstruction is
performed and the image obtained is:
Z T
qi ðx; yÞ =
sðtÞei2pðkx x + ky yÞ ei2p t/i ðx;yÞ dt
ð36Þ
0
The resultant image is taken as a linear combination of
those images.
X
qðx; yÞ =
ci ½/ðx; yÞqi ðx; yÞ
ð37Þ
i
The coefficients ci are typically obtained from Eq. (35)
with a least square algorithm. This method is faster than the
871
classical frequency segmented method because it allows
reconstruction of fewer frequencies, as unknown frequencies can be obtained as a linear combination of the two
nearest ones.
4.2.4. Linear field map interpolation
The last two methods give good correction of field
inhomogeneities but suffer from time-consuming algorithms,
as several reconstructions must be done to obtain the final
image. One fast and efficient technique proposed by
Irarrazabal et al. [45] is to consider only the first order
variation of the inhomogeneities, so the field map is fitted
with linear terms using a least square algorithm:
/ðx; yÞ = /0 + a x + b y
ð38Þ
The model of signal received then becomes:
RR
sðtÞ = e − i2p t /0 qðx; yÞe − i2pððkx + a tÞx + ðky + b tÞyÞ dxdy ð39Þ
Knowing ϕ 0 , α, β from the field map fit, the
reconstruction of the data can be done by replacing the
trajectory points by the corrected ones: kx′=kx+αt, ky′=ky+βt
and demodulating the signal to frequency ϕ0. Then, the
signal is reconstructed in one single operation.
4.2.5. Off-resonance correction without field
map acquisition
Another method proposed by Noll et al. [46] consists of
correcting the blurring without knowledge of a field map
to help in the conjugate phase reconstruction process. They
start from the idea that it should be possible to minimize
the blur of an image by reconstruction at various off
resonance frequencies and then choosing the least blurry
pixels to form a composite image. To automate the
selection process, one needs a quantification of the
blurriness. For this, they propose that an image reconstructed on resonance should be real, because all excited
Fig. 6. Synthetic scheme of frequency-segmented reconstruction algorithm (adapted from [26]).
872
spins are in phase. In the presence of field inhomogeneities,
this assumption is no longer true and the imaginary part of
the reconstructed image becomes important. The quantification of this imaginary part can thus be used as a criterion
for defining the extent of off-resonance:
C½x; y; fi ðx; yÞ =
RR
jImfq½x; y; fi ðx; yÞg j a dxdy
ð40Þ
where α is chosen empirically between 0.5 and 1 by the
user. The minimization of this criterion reduces the blurring
during the reconstruction. This method works well when
the range of off-resonance frequencies is small, otherwise
spurious minima can appear in the objective function,
increasing the risk of a wrong choice of frequency, which
can cause artifacts in the final image.
A refinement of this method was proposed by Man et al.
[47] and consists of, first, estimating a coarse field map using
relatively few demodulating frequencies to avoid spurious
minima given by the objective function. Then the minimization is repeated with a better estimation of field map (i.e.,
using a higher number of frequencies) but constrained by the
previous coarse estimation.
4.2.6. Recent improvements and propositions
Recently, Chen et al. [48] proposed a semi automatic
method for off-resonance correction that represents a
significant improvement over previously described methods
and makes reconstruction more robust. They propose to first
acquire a low-resolution field map and then perform a
frequency constrained off resonance reconstruction from the
acquired map. The first strategy used to perform the
reconstruction is to incorporate a linear off-resonance
correction term in the image as previously done in [45]
and to add a parameter used to search for non linear
components of the off-resonance frequency, here ϕi:
R
qðx; yÞ = sðtÞe − i2pt/0 e − i2pððkx + a tÞx + ðky + b tÞyÞ e − i2pt/i dt
ð41Þ
This modification can significantly improve the computational efficiency of the algorithm as it searches for a range
of non-linear terms, and not for the actual off resonance
frequency directly.
The second strategy is to take into account regions where
the field map varies nonlinearly by interpolating the field
map with polynomial terms instead of only linear terms.
Thus, the model becomes:
R
qðx; yÞ = sðtÞe − i2ptð/i
+ /p ðx;yÞÞ − i2pðkx x + ky yÞ
e
dt
ð42Þ
where ϕp(x,y) is the polynomial fit of field map and ϕi are
constant offset frequencies. Reconstruction can then be done
by multiple frequency interpolation [44].
Also recently, Barmet et al. [49] proposed a conceptually
different approach to assess field inhomogeneities. They
proposed to simply measure the magnetic field around the
investigated object with an array of miniature field probes
that do not interfere with the main experiment. This has the
main advantage of knowing the phase accrued during the
signal acquisition, i.e., in exactly the same conditions as
the experiment, so it does not require additional scan time
for the acquisition of a field map. The efficiency of this
method was evaluated by Lechner at al. [50] in comparison
with the so called “Duyn calibration Technique” (DCT)
which is the method of trajectory measurement proposed by
Duyn et al.[37] and Zhang et al. [36] and improved by
Beaumont et al. [38]. They found that both DCT and
magnetic field monitoring effectively detect k-space offsets
and trajectory error propagation, and correct for general error
sources such as timing delays. Also, artifacts such as
deformation and blurring were dramatically reduced.
4.3. Managing with concomitant fields
As seen before, off-resonance effects can be assessed
by a field map acquired with the method proposed by
Schneider [41]. However, concomitant gradient effects
are independent of acquisition time (i.e., echo time TE)
and can therefore not be assessed this way. Knowledge
of the analytical dependence of this effect with spatial
coordinates is necessary to correct for its contribution to
image blurring.
King et al. [51] have shown that the effect of concomitant
gradients can be separated into 2 parts: the through-plane
effect and the in-plane effect which can be corrected with
different methods. The through-plane effect can be understood by considering a 2D axial scan (Gz=0). From Eq. (14):
Bc =
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G20 2
z with G0 ðt Þ = G2x ðtÞ + G2y ðtÞ
2B0
ð43Þ
This means that the concomitant field is 0 at isocenter and
increases quadratically with off-center distance z. It is,
however, independent of position within any given axial
plane. Therefore, with some approximations, the phase shift
due to this contribution can be seen as a time-dependant
frequency shift varying with plane position zc given by:
/c ðzc ; t Þ = Pc
z2c 2
G ðt Þ
2B0 0
ð44Þ
Then the signal received becomes:
sðtÞ = ei2p/c ðzc ;tÞ
RR
qðx; yÞei2pðkx x +
ky yÞ
dxdy
ð45Þ
The correction for this effect can be done by demodulating the signal data over the time with frequency ϕc(zc,t)
before the reconstruction of the image.
The in-plane effect of concomitant fields can be
understood by considering, this time, a 2D sagittal plane
(i.e. Gx=0). Again, from Eq. (14):
"
2 #
1 G2z x2
Gz y
+
− Gy z
Bc =
ð46Þ
4
2B0
2
The x2 term is a through-plane contribution, similar to the
axial case, but its coefficient is 4 times smaller. The
remaining terms depend on location within the slice and
increase with off-centre distance. King et al. [51] showed
that the term GyGz gives a small contribution compared to
the others, so if the through-plane x2 term is removed by
demodulating the signal like in the previous section, and
considering some approximations that can be done in the
case of spiral scans, Eq. (46) becomes a time independent
frequency shift:
G 2 y2
+ z2
ð47Þ
/c ð y; zÞ = Pc m
4B0 4
where Gm is the maximal amplitude of gradients.
Frequency-segmented deblurring can be applied to
correct this offset by partitioning the range of constant
frequency offsets ϕc(y,z) into bins. The scan data are
demodulated with the center frequency of each bin and the
resulting images are combined by pixel-dependant interpolation to form the final deblurred image.
In the general case of an arbitrary plane, King et al. [51]
proposed a formulation where phase accumulation due to
concomitant fields is described as a time-independent
frequency offset:
G2
/c ð X ; Y ; Z Þ = Pc m
4B0
F1 X 2 + F2 Y 2 + F4 YZ + F5 XZ + F6 XY
ð48Þ
4.3.1. Recent improvements
Recently, and just after the proposition of the semiautomatic off-resonance correction method [48] (a fast
alternative to conjugate phase reconstruction), Chen et al.
[52] proposed the first fast phase conjugate reconstruction
correcting both off-resonance effects given by B0 field
inhomogeneity and concomitant gradient fields. The corrupted acquired signal is written as:
RR
qðx; yÞe − i2pðkx x
+ ky yÞ − i2p /ðx;y;tÞ
e
dxdy
ð49Þ
where the phase accrued is composed of an off-resonance
effect as described before ϕ(x,y) and a frequency shift due to
concomitant field ϕc(x,y) described by Eq. (48):
/ðx; yÞ = t /ðx; yÞ + tc /c ðx; yÞ
Table 1
Comparison of proposed methods efficiency for deblurring images
Ref.
Corrects for
Field map
required?
Speed
Accuracy of
correction
1
2
3
[60]
[62]
[63]
B0 inhomogeneity
B0 inhomogeneity
B0 inhomogeneity
Yes, accurate
Yes, accurate
Yes, accurate
−−
−−
+
4
[65]
B0 inhomogeneity
No
−−−
5
[66]
Yes, low
resolution
−
6
[70]
B0 inhomogeneity
and partially
concomitant
gradients
B0 inhomogeneity
and concomitant
gradients
= to 2
= to 1
N 1, 2 and 4 but
worst in areas
with non
linear inhom.
= to 5 but still
relatively prone
to estimation
errors
N 1, 2 and 3
Yes, low
resolution
−−
N 1, 2, 3, 4 and 5,
great improvement
for scan planes far
from isocenter
1, Time-segmented reconstruction; 2, multifrequency interpolation; 3, linear
field-map interpolation; 4, without field map estimation; 5, semi-automatic
method; 6, reconstruction based on Chebyshev approximation to correct for
B0 field inhomogeneity and concomitant gradients.
posed algorithms are shown to be computationally efficient
and the whole method seems well suited for applications
where the acquired field map is unreliable.
4.4. Summary
where X, Y, and Z are the read/phase/slice coordinates or
“logical” coordinates and Fi are constants depending only
on the plane rotation matrix (for more details see Appendix
of [51]).
sðtÞ =
873
ð50Þ
In this method, the frequency off-resonance term is
approximated by a Chebyshev polynomial function of time t.
This allows reconstruction of a set of images corrected for
concomitant fields and then application of the semiautomatic method for off-resonance correction. The pro-
Table 1 compares the different methods proposed for
correction of off resonance effects, such as B0 inhomogeneity and concomitant gradients. Methods presented to
measure real k-space trajectories can also be used in addition
to these correction methods.
In summary, when acquisition time is not constrained, the
method using linear field-map interpolation [45] is the best
choice, as it easily corrects for B0 field inhomogeneity with
very low computation time. On the other hand, for situations
where acquisition time must stay short, the field map
acquired is often inaccurate and the semi-automatic method
[48] should be used, as it is more efficient in this situation.
However, in cases when scan planes are placed far from the
isocenter of the magnet bore, the combined method with
Chebyshev approximation [52] must be chosen to correct
also for concomitant gradient fields. In extreme situations
where no field map can be acquired because of time
constraints, or because the quality obtained is very poor, a
correction method without field map acquisition can be used,
but will suffer from an important computational cost.
5. Spiral image reconstruction
Points of spiral trajectories are no longer on the Cartesian
grid; therefore, the direct use of FFT is not possible. Several
propositions have been made to reconstruct images from
874
k-space data. One idea was to use the extension of the
discrete Fourier transform, also called phase conjugate
reconstruction [53], but this method is very time consuming.
Other methods are based on regularized model based
reconstructions [54,55]. For now, the most commonly used
method is the regridding algorithm [56,57]. This basically
consists of interpolation of k-space points on the Cartesian
grid that then allows the application of an FFT, which
remains the most computationally efficient algorithm for
reconstruction. However, interpolation in k-space leads to
errors that are spread over the whole image once reconstructed. This part must therefore be done with caution.
Moreover, variable density sampling encountered in spiral
must be taken into account before interpolation onto the
Cartesian grid, this requires compensation for differently
sampled areas by multiplying the data by a density
compensation function (DCF). This implies also postcompensation of data after the gridding step. To generate the
gridded data points, the problem of data resampling can
also be solved by the BURS (block uniform resampling)
algorithm [58], where a set of linear equation is given an
optimal solution using the pseudoinverse matrix computed
with singular value decomposition (SVD). Moreover, noise
and artifact reduction can be obtained by using truncated
SVD [59]. This algorithm has the advantage to avoid the
pre and post compensation steps of the gridding algorithm.
Other alternatives to calculation of DCF were proposed
either by iteratively reconstructing data using matrices
scaled larger than target matrix [60] or by using an iterative
deconvolution-interpolation algorithm [61]. Another proposition is to calculate a generalized FFT (GFFT) to
reconstruct data which is mathematically the same algorithm than regridding with a Gaussian kernel; however,
GFFT was shown to be more precise in the case of
reconstruction of small matrices [62]. This review focuses
on the regridding algorithm because it is still the most
widely used and is the basis of a large number of other
reconstruction alternatives.
5.1. Density compensation function
As spiral sampling is not uniform over k-space some
compensation must be done in order to avoid an overweighting of low spatial frequencies compared to high
frequencies, which would result in signal intensity distortions. This can be done in several ways. For example, if the
density varies smoothly, Meyer et al. [63] showed good
reconstructed image quality by using the analytical formulation of the trajectory to compensate for the variable density.
However, this method is no longer reliable when the density
varies sharply or in the case of less ideal spiral trajectories.
Hoge et al. [64] compared several analytical propositions
with their method using the determinant of the Jacobian
matrix between Cartesian coordinates and the spiral
sampling parameters of time and interleave rotation angle
used as a density compensation function. They could show
the reliability of their method even in the case of trapezoidal
or distorted gradient waveforms.
However, when the real trajectory moves too far away
from the theoretical one, another approach independent of
the sampling pattern should be used. This is based on the
Voronoi diagram [65] to calculate the area around each
sampling point. This area is then used to compensate for
density variation, the bigger the area around the sample (the
size of the Voronoi cell), the smaller the density sampling.
Rasche et al. [66] have shown the power of this technique in
the case of a distorted spiral trajectory, the advantage being
that this technique depends only on the sampling pattern and
not on the acquisition order. This technique nevertheless
needs knowledge of the real trajectory (see section
“Measuring gradient deviation” in “Coping with blurring
in spiral images”) and may be limited due to the high
computational complexity of the Voronoi diagram.
5.2. Interpolation onto the Cartesian grid
Once the samples have been corrected for the non-uniform
density, they need to be interpolated onto the Cartesian
grid in order to perform the FFT algorithm for image
reconstruction. O'Sullivan [57] showed that the best way to
interpolate samples was the use of an infinite sinc function.
Samples in the Fourier domain are convolved with a sinc
function, resulting in a multiplication with the Fourier
transformation of a sinc function (a boxcar function) in the
image domain. In practice, the use of an infinite sinc
function is not possible. In regridding, this function is
replaced by compactly supported kernels. However, the
computational simplicity of the kernel must be balanced
with the level of artifacts in the resulting image, which is not
an easy tradeoff. Jackson et al. [56] investigated several
kernels and showed that the Kaiser-Bessel kernel gave the
best reconstructed image quality. In this case, however, the
final image must be corrected by dividing by the Fourier
transform of the kernel to avoid distorted intensities due to
the fact that it is no longer a rectangular function; this is the
so-called roll-off correction.
5.3. Spiral imaging and parallel imaging
As mentioned earlier, spiral acquisition schemes are still
not commonly used in the clinic, even though spiral
imaging has several advantages over standard Cartesian
acquisitions. One main reason for this is that Cartesian
acquisitions are routinely accelerated with parallel imaging,
whereas this is not trivial for spiral acquisitions. Without
parallel imaging, the speed advantage of spiral trajectories
is compromised. However, recently introduced methods for
non-Cartesian parallel imaging, in conjunction with improved computer performance, will enable the use of
accelerated spiral acquisition schemes for both clinical
routine and research.
In parallel imaging, acceleration of the image acquisition
is performed by reducing the sampling density of the k-space
875
each other. Compared to the discrete aliasing artifact
behavior of Cartesian acquisitions, aliasing in spiral imaging
is different. In k-space, undersampling of a spiral acquisition
is affecting all directions. As a result, in the image domain,
one image pixel is folded with many other image pixels. This
can be seen on the right-hand side of Fig. 7.
Today, there are two major parallel imaging methods
routinely used, namely SENSE [67] and GRAPPA [68].
While SENSE works completely in the image domain by
unfolding aliased images, GRAPPA works in k-space by
reconstructing missing k-space data. In the following, the
basic principles behind the SENSE method are described.
In principle, an MR image is the result of the spin density
variation of the underlying object multiplied by the coil
sensitivity of the receiver coil used for data acquisition.
This is depicted on the left-hand side of Fig. 8 for a
single coil with a certain two dimensional coil sensitivity
[C1(x,y)].
Iðx; yÞ = C1 ðx; yÞ qðx; yÞ
Fig. 7. Simulated aliasing artifact behavior of Cartesian and spiral imaging:
(a) In Cartesian imaging, a factor of two undersampling along the PE
direction results in discrete aliasing. In this aliased image, always one pixel is
aliased onto another single pixel. (b) In comparison, undersampling in spiral
imaging results in the situation that one image pixel is aliased with many
other pixels. The corresponding k-space trajectories with acquired (solid
lines) and skipped (dashed lines) data points are depicted at the bottom.
data. The reduced sampling density of k-space is related to
a reduced FOV. If an object is larger than the reduced
FOV, all parts of the object outside the reduced FOV will be
folded back into the reduced FOV. This effect, called aliasing or foldover artifact, is depicted in Fig. 7. The undersampling of Cartesian acquisitions along the phase-encoding
(PE) direction leads to discrete aliasing artifacts along
this direction. In the simulation shown on the left side of
Fig. 7, a factor of two undersampling along phase encoding
results into aliasing of exactly two image pixels on top of
ð51Þ
Now, let us assume that the FOV is reduced by a factor of
two, which is shown on the right hand side of Fig. 8. In this
aliased image, a single image pixel (I1) is the superposition
of two image pixels from different locations (ρ1 and ρ2) with
different coil sensitivities (C11 and C12). This can be written
as a linear equation:
I1 = C11 q1 + C12 q2
ð52Þ
To unfold the aliased image pixel, one has to separate I1
into the two different spin density contributions ρ1 and ρ2.
Even if the coil sensitivity is derived at both positions
beforehand, it is not possible to solve this problem, because
there is only one linear equation for two unknowns. If such
an aliased image is acquired with two independent receiver
coils, then two linear equations can be set up for each aliased
image pixel. In this case, there are two linear equations for
two unknowns. This means it is possible to find an exact
solution to the problem. In comparison, aliasing in spiral
imaging means that a single image pixel is aliased with many
other image pixels. In other words, a very large system of
Fig. 8. An MR image is a result of the coil sensitivity (C1) multiplied by the proton density of the object (ρ). One image pixel (I1) in an aliased image, here the
reduction factor R is two, is always the superposition of two image pixels from two different positions.
876
Fig. 9. Schematic depiction of the Conjugate Gradient SENSE reconstruction. Symbols are: C*N, conjugate of the sensitivity map of the Nth coil, IFFT, inverse
fast Fourier transform, GRID, gridding algorithm, DEGRID, degridding algorithm, CN, Sensitivity map of the Nth coil.
linear equations has to be solved for each folded image pixel.
Therefore, this kind of direct solution of the problem is
impractical due to computational burden. To address this
problem, an iterative method, the Conjugate-Gradient (CG)
method [69], was proposed for non-Cartesian SENSE
reconstructions [70,71].
A schematic depiction of the Conjugate-Gradient
SENSE algorithm is shown in Fig. 9 (adapted from [71]).
The iterative process starts with the undersampled,
originally acquired, single-coil spiral k-space data. In a
first step, the spiral data are density corrected, gridded and
Fourier-transformed. Then the single coil images are
multiplied by the corresponding complex conjugate coil
sensitivities and combined using a complex sum. This
initial image is the input for the CG algorithm as the first
guess for the actual image. The CG algorithm provides
iteratively refined guesses by minimizing the residual
between those guesses and the initial image. During each
iteration, the current guess is multiplied with the individual
coil sensitivities, which results in a separation into single
coil images. The single coil images are Fourier-transformed
and the resulting k-space data are degridded to obtain the
updated spiral k-space data. The subsequent procedure of
gridding, Fourier transform, conjugate sensitivity multiplication and summation is the same as for the originally
acquired data. A regularization term may be factored into
the iteration. A stopping criterion for the iteration has to be
defined, which is difficult, because stopping the process too
early will yield aliasing artifacts, while stopping too late
will introduce noise to the reconstruction.
Even though the CG SENSE method significantly
reduces the computational burden, and allowed Weiger
et al. [72] to demonstrate the benefits of accelerated spiral
acquisitions, the major hindrance of this method was still
the reconstruction time. Parallel image reconstruction times
for 2D spiral acquisitions have been reported to require
several hours and sometimes even days to reconstruct all
images from a typical functional MRI (fMRI) investigation
[72–74]. Whereas these previous investigations have
performed the reconstruction using coil sensitivity information from the image domain, recent approaches for parallel
imaging of non-Cartesian k-space sampling by means of the
GRAPPA algorithm, work completely in k-space and
represent a promising approach to overcome computational
limits [75–77]. While Heberlein et al. [76] described a
direct application of the radial GRAPPA approach to spiral
imaging by Griswold et al. [75], Heidemann et al. [77]
developed a more sophisticated algorithm. Both methods
enable accelerated single-shot spiral acquisitions, without
the need of a fully sampled spiral reference data set.
However, Heidemann et al. [77] included an interpolation
and reordering of the k-space data to generate radial
symmetry in the data, which simplifies the reconstruction
process significantly.
It has been shown earlier that a constant-angularvelocity trajectory has a radial symmetry (see Fig. 2).
Another important property of such a spiral is that the
distances in k-space between sampled data points along
such a line through the origin are constant, except for
those points directly adjacent to the central k-space point.
The symmetry of the constant-angular-velocity spiral can
be used to simplify the parallel image reconstruction. For a
direct spiral GRAPPA reconstruction, this symmetry
enables the use of a conventional Cartesian GRAPPA
reconstruction. The whole process is depicted in Fig. 10.
The spiral data are acquired along a constant-angularvelocity trajectory to benefit from the advantages of this
trajectory. The data are then interpolated onto a constant-
877
Fig. 10. A schematic depiction of the direct spiral GRAPPA reconstruction. The original spiral data are acquired onto a constant-linear-velocity trajectory.
Interpolation of the data onto a constant-angular-velocity trajectory introduces a radial symmetry to the k-space data. The pseudo-projections (highlighted by
gray lines) are reordered into a new hybrid space. The next step is to divide the hybrid space into segments. For each segment, a Cartesian GRAPPA
reconstruction is performed. The regenerated full hybrid space is reordered back into k-space.
angular-velocity trajectory. Since the only difference
between both trajectories is the position of the sampled
data points along the spiral trajectory, along the time axis,
a simple one-dimensional interpolation along the time axis
is sufficient to transform a constant-linear-velocity into a
constant-angular-velocity trajectory and vice versa. After
the interpolation, the k-space data are reordered into a new
hybrid space with coordinates projection angle and
winding number of the spiral. Compared to a radial hybrid
space with coordinates projection angle and radius, data
points in the spiral hybrid space are not aligned on an
equidistant grid. This is because the distance of a sampled
spiral data point to the origin (the radius in the radial case)
is not constant. To address this, the spiral hybrid space is
divided into segments. This segmentation of the hybrid
space corresponds to a division of the spiral data set into
sectors. Each segment is treated as a Cartesian k-space,
and a Cartesian GRAPPA reconstruction with an adapted
2D reconstruction kernel [78] is applied to each segment.
After regenerating a full hybrid space with the GRAPPA
reconstruction, the data are reordered back into k-space
and can then passed to a gridding procedure. Since no
iterations are necessary, this direct parallel imaging
reconstruction is very fast. The reconstruction time is
comparable to the gridding process. The in vivo examples
shown in Fig. 11 demonstrate that with spiral GRAPPA it
is possible to acquire high resolution T2* weighted singleshot acquisitions, which are suitable for fMRI. The
following parameters have been used: TR=2500 ms,
TE=30 ms, flip angle=75°, slice thickness 3.5 mm, FOV
220 mm, image matrix=256×256. Typically for fMRI
experiments, the repetition time is relatively long, about
2000–3000 ms. In the current study, the acquisition of
a spiral image with eight interleaves (see left-hand side
of Fig. 11 at the bottom) would take 20 s. In general, long
acquisition times make use of multi-shot approaches
inappropriate for fMRI experiments, where a high temporal resolution is crucial. Compared to the multi-shot
acquisitions (see left-hand side of Fig. 11), the undersampled single-shot acquisitions show severe aliasing
artifacts (see middle column of Fig. 11). Those aliasing
artifacts are completely removed by the spiral GRAPPA
reconstruction (see right-hand side of Fig. 11). These
images, obtained in less than 30 ms, with a high
acceleration factor of eight, show no aliasing artifacts,
but a reduced SNR compared to the multi-shot acquisitions. For fMRI and diffusion-weighted imaging, singleshot acquisitions are well established as the method of
choice to address problems related to physiological noise.
In Fig. 12 a comparison between a conventional singleshot spiral and an accelerated spiral is shown. The readout
duration of the conventional single-shot spiral acquisition is 146 ms. The spiral GRAPPA acquisition enables
reduction of the readout duration down to 18 ms. Due
to the shortened readout duration of accelerated spirals,
artifacts related to off-resonance effects and image blurring
878
Fig. 11. High resolution (0.86×0.86×3.5 mm3) T*2 weighted conventional multi-shot spiral versus single-shot spiral GRAPPA examples at 3T, acquired with a 32
channel prototype head array: Full acquisition reference images with two, four and eight interleaves (left column from top to bottom). Only a single interleaf of
each spiral data set is used, resulting in two-, four- and eight-times undersampled images (center column from top to bottom). Spiral GRAPPA reconstructions
with acceleration factors of two, four and eight (right column from top to bottom). This figure is reprinted from Ref. [77].
due to T2* relaxation are significantly reduced. The use of
parallel imaging enables acquisition of single-shot spiral
images with the off-resonance behavior of a multishot
approach. Through the development of the previously
described methods, the reconstruction times for accelerated
spiral acquisitions using parallel imaging are now
approaching those of Cartesian images. Due to synergy
effects, higher acceleration factors can be achieved with
non-Cartesian parallel imaging than possible with their
Cartesian counterparts. These properties and the advances
of spiral trajectories will pave the way for an implemen-
tation of such methods on MR scanners and, thus, also
offer the possibility for their use in clinical routine.
6. Conclusion
In conclusion, the spiral acquisition scheme provides
intrinsic advantages that cannot be obtained with other kinds
of trajectory, but however brings along some challenges in
gradient design, parallel imaging, gradient deviations and
off-resonance artifacts. These difficulties are well described
in the literature and have been addressed with various
Fig. 12. High-resolution (0.86×0.86×3.5 mm3) T*2 weighted single-shot
spiral acquisitions at 3T, acquired with a 32 channel prototype head array.
Especially in basal regions of the brain, the conventional single-shot spiral
acquisition suffers from severe off-resonance artifacts (left side). The singleshot spiral GRAPPA acquisition with acceleration factor of eight shows a
significantly improved off-resonance behavior (right side).
propositions adapted to different situations. A lot of effort
was concentrated on measuring the exact trajectory and
finding efficient and robust algorithms to correct for all
phase error accumulation and new approaches to reconstruct
well deblurred images were proposed even recently. Also,
the adaptation of parallel imaging techniques to spiral
trajectory significantly improved scan time and stays a hot
topic of research. Finally, combination of these techniques
contributes to reducing the gap between the spiral sequence
and more conventional techniques in order to make spiral
even more commonly used in clinical routines.
Acknowledgments
The authors would like to thank Rolf Grütter and Gunnar
Krüger for helpful discussions. This work was partially
supported by the Swiss Science Foundation (grant PPOO33116901) and the Centre d'Imagerie BioMédicale (CIBM) of
the UNIL, UNIGE, HUG, CHUV, EPFL.
References
[1] Ahn CB, Kim JH, Cho ZH. High-speed spiral-scan echo planar NMR
imaging-I. IEEE Trans Med Imaging 1986;5(1):2–7.
[2] Liao JR, Pauly JM, Brosnan TJ, Pelc NJ. Reduction of motion artifacts
in cine MRI using variable-density spiral trajectories. Magn Reson
Med 1997;37(4):569–75.
[3] Liu C, Bammer R, Kim DH, Moseley ME. Self-navigated interleaved
spiral (SNAILS): application to high-resolution diffusion tensor
imaging. Magn Reson Med 2004;52(6):1388–96.
[4] Qian Y, Boada FE. Acquisition-weighted stack of spirals for fast highresolution three-dimensional ultra-short echo time MR imaging. Magn
Reson Med 2008;60(1):135–45.
[5] Narayan G, Nayak K, Pauly J, Hu B. Single-breathhold, fourdimensional, quantitative assessment of LV and RV function using
triggered, real-time, steady-state free precession MRI in heart failure
patients. J Magn Reson Imaging 2005;22(1):59–66.
879
[6] Nayak KS, Cunningham CH, Santos JM, Pauly JM. Real-time cardiac
MRI at 3 Tesla. Magn Reson Med 2004;51(4):655–60.
[7] Ryf S, Kissinger KV, Spiegel MA, Bornert P, Manning WJ, Boesiger
P, et al. Spiral MR myocardial tagging. Magn Reson Med 2004;51(2):
237–42.
[8] Kressler B, Spincemaille P, Nguyen TD, Cheng L, Xi Hai Z, Prince
MR, et al. Three-dimensional cine imaging using variable-density
spiral trajectories and SSFP with application to coronary artery
angiography. Magn Reson Med 2007;58(3):535–43.
[9] Santos JM, Cunningham CH, Lustig M, Hargreaves BA, Hu BS,
Nishimura DG, et al. Single breath-hold whole-heart MRA using
variable-density spirals at 3T. Magn Reson Med 2006;55(2):371–9.
[10] Taylor AM, Keegan J, Jhooti P, Gatehouse PD, Firmin DN, Pennell
DJ. A comparison between segmented k-space FLASH and interleaved
spiral MR coronary angiography sequences. J Magn Reson Imaging
2000;11(4):394–400.
[11] Yang PC, Nguyen P, Shimakawa A, Brittain J, Pauly J, Nishimura D,
et al. Spiral magnetic resonance coronary angiography–direct comparison of 1.5 Tesla vs. 3 Tesla. J Cardiovasc Magn Reson 2004;6(4):877–84.
[12] Chen CC, Tyler CW, Baseler HA. Statistical properties of BOLD
magnetic resonance activity in the human brain. Neuroimage 2003;20
(2):1096–109.
[13] Klarhofer M, Barth M, Moser E. Comparison of multi-echo spiral and
echo planar imaging in functional MRI. Magn Reson Imaging 2002;20
(4):359–64.
[14] Truong TK, Song AW. Single-shot dual-z-shimmed sensitivityencoded spiral-in/out imaging for functional MRI with reduced
susceptibility artifacts. Magn Reson Med 2008;59(1):221–7.
[15] Adalsteinsson E, Star-Lack J, Meyer CH, Spielman DM. Reduced
spatial side lobes in chemical-shift imaging. Magn Reson Med 1999;42
(2):314–23.
[16] Levin YS, Mayer D, Yen YF, Hurd RE, Spielman DM. Optimization
of fast spiral chemical shift imaging using least squares reconstruction:
application for hyperpolarized (13)C metabolic imaging. Magn Reson
Med 2007;58(2):245–52.
[17] Bornert P, Stuber M, Botnar RM, Kissinger KV, Koken P, Spuentrup
E, et al. Direct comparison of 3D spiral vs. Cartesian gradient-echo
coronary magnetic resonance angiography. Magn Reson Med 2001;46
(4):789–94.
[18] Meyer CH, Pauly JM, Macovski A, Nishimura DG. Simultaneous
spatial and spectral selective excitation. Magn Reson Med 1990;15(2):
287–304.
[19] Liu CY, Varadarajan P, Pohost GM, Nayak KS. Real-time color-flow
MRI at 3 T using variable-density spiral phase contrast. Magn Reson
Imaging 2008;26(5):661–6.
[20] Salerno M, Sica CT, Kramer CM, Meyer CH. Myocardial perfusion
imaging using spiral trajectories. In: Proceedings of the ISMRM Data
Sampling and Reconstruction Workshop, Sedona AZ, USA; 2009.
[21] Di Bella EV, Parker DL, Sinusas AJ. On the dark rim artifact in
dynamic contrast-enhanced MRI myocardial perfusion studies. Magn
Reson Med 2005;54(5):1295–9.
[22] Haacke EM, Brown R, Thompson MR, Venkatesan R. Magnetic
resonance imaging: physical priniples and sequence design. New York:
John Wiley & Sons; 1999.
[23] Cooley JW, Tukey JW. An algorithm for machine calculation of
complex Fourier series. Math Comput 1965;19(90):297–&.
[24] Tsai CM, Nishimura DG. Reduced aliasing artifacts using variabledensity k-space sampling trajectories. Magn Reson Med 2000;43(3):
452–8.
[25] Macovski A, Meyer CH. A novel fast-scanning system. Works in
progress of the 5th Annual Meeting of the SMRM; 1986. p. 156–7.
Montreal, Canada.
[26] Bernstein MA, King K, Zhou XJ. Handbook of MRI pulse sequences.
San Diego, CA: Elsevier Academic Press; 2004.
[27] Brau AC, Brittain JH. Generalized self-navigated motion detection
technique: Preliminary investigation in abdominal imaging. Magn
Reson Med 2006;55(2):263–70.
880
[28] Bernstein MA, Zhou XJ, Polzin JA, King KF, Ganin A, Pelc NJ, et al.
Concomitant gradient terms in phase contrast MR: analysis and
correction. Magn Reson Med 1998;39(2):300–8.
[29] Duyn JH, Yang Y. Fast spiral magnetic resonance imaging with
trapezoidal gradients. J Magn Reson 1997;128(2):130–4.
[30] Glover GH. Simple analytic spiral k-space algorithm. Magn Reson
Med 1999;42(2):412–5.
[31] Kim DH, Adalsteinsson E, Spielman DM. Simple analytic variable
density spiral design. Magn Reson Med 2003;50(1):214–9.
[32] Zhao T, Qian Y, Hue Y-K, Ibrahim TS, Boada F. An improved
analytical solution for variable density spiral design. Proc Int Soc
Magn Reson Med 2008;16:1342.
[33] Mason GF, Harshbarger T, Hetherington HP, Zhang Y, Pohost
GM, Twieg DB. A method to measure arbitrary k-space
trajectories for rapid MR imaging. Magn Reson Med 1997;38(3):
492–6.
[34] Onodera T, Matsui S, Sekihara K, Kohno H. A method of measuring
field-gradient modulation shapes — application to high-speed NMR
spectroscopic imaging. J Phys E 1987;20(4):416–9.
[35] Alley MT, Glover GH, Pelc NJ. Gradient characterization using
a Fourier-transform technique. Magn Reson Med 1998;39(4):
581–7.
[36] Zhang Y, Hetherington HP, Stokely EM, Mason GF, Twieg DB. A
novel k-space trajectory measurement technique. Magn Reson Med
1998;39(6):999–1004.
[37] Duyn JH, Yang Y, Frank JA, van der Veen JW. Simple correction
method for k-space trajectory deviations in MRI. J Magn Reson
1998;132(1):150–3.
[38] Beaumont M, Lamalle L, Segebarth C, Barbier EL. Improved k-space
trajectory measurement with signal shifting. Magn Reson Med
2007;58(1):200–5.
[39] Tan H, Meyer CH. Estimation of k-space trajectories in spiral MRI.
Magn Reson Med 2009;61(6):1396–404.
[40] Aldefeld B, Bornert P. Effects of gradient anisotropy in MRI. Magn
Reson Med 1998;39(4):606–14.
[41] Schneider E, Glover G. Rapid in vivo proton shimming. Magn Reson
Med 1991;18(2):335–47.
[42] Noll DC, Meyer CH, Pauly JM, Nishimura DG, Macovski A. A
homogeneity correction method for magnetic resonance imaging
with time-varying gradients. IEEE Trans Med Imaging 1991;10(4):
629–37.
[43] Noll DC. “Reconstruction techniques for magnetic resonance imaging”
PhD. Thesis, Stanford University. 1991.
[44] Man LC, Pauly JM, Macovski A. Multifrequency interpolation for fast
off-resonance correction. Magn Reson Med 1997;37(5):785–92.
[45] Irarrazabal P, Meyer CH, Nishimura DG, Macovski A. Inhomogeneity
correction using an estimated linear field map. Magn Reson Med
1996;35(2):278–82.
[46] Noll DC, Pauly JM, Meyer CH, Nishimura DG, Macovski A.
Deblurring for non-2D Fourier transform magnetic resonance imaging.
Magn Reson Med 1992;25(2):319–33.
[47] Man LC, Pauly JM, Macovski A. Improved automatic off-resonance
correction without a field map in spiral imaging. Magn Reson Med
1997;37(6):906–13.
[48] Chen W, Meyer CH. Semiautomatic off-resonance correction in spiral
imaging. Magn Reson Med 2008;59(5):1212–9.
[49] Barmet C, Zanche ND, Pruessmann KP. Spatiotemporal magnetic field
monitoring for MR. Magn Reson Med 2008;60(1):187–97.
[50] Lechner SM, Sipila PT, Wiesinger F, Kerr AB, Vogel MW. Spiral
imaging artifact reduction: a comparison of two k-trajectory measurement methods. J Magn Reson Imaging 2009;29(6):1485–92.
[51] King KF, Ganin A, Zhou XJ, Bernstein MA. Concomitant gradient
field effects in spiral scans. Magn Reson Med 1999;41(1):103–12.
[52] Chen W, Sica CT, Meyer CH. Fast conjugate phase image
reconstruction based on a Chebyshev approximation to correct for
B0 field inhomogeneity and concomitant gradients. Magn Reson Med
2008;60(5):1104–11.
[53] Maeda A, Sano K, Yokoyama T. Reconstruction by weighted
correlation for MRI with time-varying gradients. IEEE Trans Med
Imaging 1988;7(1):26–31.
[54] Boubertakh R, Giovannelli JF, De Cesare A, Herment A. Nonquadratic convex regularized reconstruction of MR images from spiral
acquisitions. Signal Process 2006;86(9):2479–94.
[55] Delattre B, Hyacinthe JN, Vallee JP, Van de Ville D. Spline-based
variational reconstruction of variable density spiral k-space data with
automatic parameter adjustment. Proceedings of the of the 17th Annual
Meeting of the ISMRM, Hawai'i, USA; 2009. p. 2066.
[56] Jackson JI, Meyer CH, Nishimura DG, Macovski A. Selection of a
convolution function for Fourier inversion using gridding [computerised tomography application]. IEEE Trans Med Imaging 1991;10
(3):473–8.
[57] O'Sullivan JD. A fast sinc function gridding algorithm for Fourier
inversion in computer tomography. IEEE Trans Med Imaging 1985;4
(4):200–7.
[58] Rosenfeld D. An optimal and efficient new gridding algorithm
using singular value decomposition. Magn Reson Med 1998;40(1):
14–23.
[59] Moriguchi H, Duerk JL. Modified block uniform resampling (BURS)
algorithm using truncated singular value decomposition: fast accurate
gridding with noise and artifact reduction. Magn Reson Med 2001;46
(6):1189–201.
[60] Moriguchi H, Duerk JL. Iterative next-neighbor regridding
(INNG): improved reconstruction from nonuniformly sampled kspace data using rescaled matrices. Magn Reson Med 2004;51
(2):343–52.
[61] Gabr RE, Aksit P, Bottomley PA, Youssef AB, Kadah YM.
Deconvolution-interpolation gridding (DING): accurate reconstruction
for arbitrary k-space trajectories. Magn Reson Med 2006;56(6):
1182–91.
[62] Sarty GE, Bennett R, Cox RW. Direct reconstruction of non-Cartesian
k-space data using a nonuniform fast Fourier transform. Magn Reson
Med 2001;45(5):908–15.
[63] Meyer CH, Hu BS, Nishimura DG, Macovski A. Fast spiral coronary
artery imaging. Magn Reson Med 1992;28(2):202–13.
[64] Hoge RD, Kwan RK, Pike GB. Density compensation functions for
spiral MRI. Magn Reson Med 1997;38(1):117–28.
[65] Aurenhammer F. Voronoi diagrams — a survey of a
fundamental geometric data structure. Comput Surv 1991;23(3):
345–405.
[66] Rasche V, Proksa R, Sinkus R, Bornert P, Eggers H. Resampling of
data between arbitrary grids using convolution interpolation. IEEE
Trans Med Imaging 1999;18(5):385–92.
[67] Pruessmann KP, Weiger M, Scheidegger MB, Boesiger P. SENSE:
sensitivity encoding for fast MRI. Magn Reson Med 1999;42(5):
952–62.
[68] Griswold MA, Jakob PM, Heidemann RM, Nittka M, Jellus V, Wang
J, et al. Generalized autocalibrating partially parallel acquisitions
(GRAPPA). Magn Reson Med 2002;47(6):1202–10.
[69] Hestenes MR, Stiefel E. Methods of conjugate gradients for solving
linear systems. J Res Natl Bur Stand 1952;49(6):409–36.
[70] Kannengiesser SAR, Brenner AR, Noll TG. Accelerated image
reconstruction for sensitivity encoded imaging with arbitrary k-space
trajectories. Proceedings of the 8th Annual meeting of ISMRM,
Denver, CO; 2000. p. 155.
[71] Pruessmann KP, Weiger M, Bornert P, Boesiger P. Advances in
sensitivity encoding with arbitrary k-space trajectories. Magn Reson
Med 2001;46(4):638–51.
[72] Weiger M, Pruessmann KP, Osterbauer R, Bornert P, Boesiger P,
Jezzard P. Sensitivity-encoded single-shot spiral imaging for reduced
susceptibility artifacts in BOLD fMRI. Magn Reson Med 2002;48(5):
860–6.
[73] Stenger VA, Sutton BP, Boada F, Fessler JA, Noll DC. Reversed spiral
SENSE for fMRI. Proceedings of the 12th Scientific Meeting ISMRM,
Kyoto, Japan; 2004. p. 1024.
[74] Yeh EN, McKenzie CA, Ohliger MA, Sodickson DK. Parallel
magnetic resonance imaging with adaptive radius in k-space
(PARS): constrained image reconstruction using k-space locality in
radiofrequency coil encoded data. Magn Reson Med 2005;53(6):
1383–92.
[75] Griswold MA, Heidemann RM, Jakob PM. Direct parallel imaging
reconstruction of radially sampled data using GRAPPA with relative
shifts. Proceedings of the 11th Annual Meeting of the ISMRM,
Toronto, Canada; 2003. p. 2349.
881
[76] Heberlein K, Hu X. Auto-calibrated parallel spiral imaging. Magn
Reson Med 2006;55(3):619–25.
[77] Heidemann RM, Griswold MA, Seiberlich N, Kruger G, Kannengiesser SA, Kiefer B, et al. Direct parallel image reconstructions
for spiral trajectories using GRAPPA. Magn Reson Med 2006;56
(2):317–26.
[78] Griswold MA. Advanced k-space techniques. Proceedings of the 2nd
International Workshop on Parallel Imaging, Zuerich, Switzerland;
2004. p. 16–8.

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