POLITECNICO DI MILANO Generation of a patient specific predictor

Transcription

POLITECNICO DI MILANO
Facoltá di ingegneria dei sistemi
Corso di Laurea Specialistica in Ingegneria Biomedica
Generation of a patient specific predictor for
osteoporotic risk of fracture of the femoral neck
Relatore:
Tomaso Villa Maria Tobia
Correlatore:
Marco Viceconti
Relazione della prova finale di:
Anna Caimi 770324
Gloria Casaroli 765186
Anno Accademico 2011/2012
Ringraziamenti
Desideriamo porgere il primo sentito ringraziamento al Prof. Marco Viceconti per
l’estrema disponibilitá e cortesia dimostrateci in questi mesi, per le grandi opportunitá
che ci ha offerto, per la realtá che ci ha fatto conoscere e nella quale ci ha introdotto;
un forte ringraziamento va al Prof. Tomaso Villa, che ci ha offerto la possibilitá di
compiere un’esperienza di tesi all’estero che è stata la migliore della nostra carriera,
sostenuto durante tutto il suo corso e ci ha sempre mostrato grande disponibilitá. Un
ringraziamento speciale va a Giovanna Farinella, che ci ha seguito passo a passo nello
svolgimento del nostro lavoro, fornendoci sempre grande aiuto e competenza, oltre che
aver stretto con noi un forte legame di amicizia.
Un grazie particolare va a tutti coloro che hanno condiviso con noi l’esperienza
a Sheffield rendendola davvero speciale, e in particolare alle nostre colleghe Sandra e
Francesca che sono state due perfette compagne d’avventura.
Grazie anche a tutti i nostri compagni di corso per gli anni di studio, per i progetti
di gruppo, per le risate e per tutti i momenti belli passati qui al Politecnico e durante
la vita normale di tutti i giorni.
Grazie anche a tutti gli amici di sempre ed alle persone noi care, per le serate, i
week-end, per averci parlato di tutto fuorché dello studio, per averci capito, distratto
e consolato con la loro simpatia e il loro affetto; in particolare grazie a Massimo e a
Riccardo, che ci sono stati accanto piú di ogni altro e ci hanno sostenuto nonostante
la grande distanza.
Un ringraziamento particolare va infine alle nostre famiglie, per averci incoraggiato
nelle scelte, sostenuto nei 5 anni, aver condiviso con noi i momenti di soddisfazione
e consolato in quelli piú difficili e per averci regalato la possibilitá di compiere questa
esperienza all’estero. Grazie, senza di voi, per noi, oggi sarebbe un giorno qualunque.
Anna e Gloria
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Contents
Contents
v
List of Figures
vii
List of Tables
xi
Abstract
xiii
Sommario
xix
1 Introduction
1
1.1
Osteoporosis: social and economic burden . . . . . . . . . . . . . . . . .
1
1.2
The Clinical State of the Art . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
The VPHOP project . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4
The aim of our work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2 Anatomy and Mechanics
13
2.1
Body’s reference system and planes . . . . . . . . . . . . . . . . . . . . .
13
2.2
Relative position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3
Musculoskeletal anatomy
. . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.3.1
Anatomy and physiology of bone . . . . . . . . . . . . . . . . . .
15
2.3.2
The pelvis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.3.3
The femur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.4
The hip joint and angle . . . . . . . . . . . . . . . . . . . . . . .
22
Bone tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.4.1
Composition of bone . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.4.2
Bone Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.4
v
vi
CONTENTS
2.5
2.6
2.4.3
Bone mechanobiology . . . . . . . . . . . . . . . . . . . . . . . .
29
2.4.4
Effects of underloading and overloading . . . . . . . . . . . . . .
31
Hip Biomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.5.1
Hip muscles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.5.2
Hip kinematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Definition and basic information . . . . . . . . . . . . . . . . . . . . . .
35
2.6.1
The importance of Gait analysis for this study . . . . . . . . . .
37
2.6.2
Effect of sub-optimal neuromotor control . . . . . . . . . . . . .
41
2.6.3
Hypothesis of our model . . . . . . . . . . . . . . . . . . . . . . .
43
3 Material and methods
45
3.1
Patients’ cohort and CT scanning . . . . . . . . . . . . . . . . . . . . . .
45
3.2
Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.2.1
Segmentation and morphing . . . . . . . . . . . . . . . . . . . . .
48
3.2.2
BoneMat software . . . . . . . . . . . . . . . . . . . . . . . . . .
53
3.2.3
Structure of msf file . . . . . . . . . . . . . . . . . . . . . . . . .
58
The mechanical load scenarios . . . . . . . . . . . . . . . . . . . . . . . .
64
3.3.1
Strength loading scenario . . . . . . . . . . . . . . . . . . . . . .
64
3.3.2
Femur-fall Charité database . . . . . . . . . . . . . . . . . . . . .
68
3.3
4 Results
71
5 Conclusion
87
6 Future development
91
Appendix
93
Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
The Mann-Whitney test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
ROC curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Logistic regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Bibliography
107
List of Figures
1.1
Image of the DXA of a femur. . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
FRAX interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
The graph shows fracture rates per 100 person-years by phenotypic and Tscore according to the National Osteoporosis Risk Assessment Study. Across
all phenotypic groups, low BMD is a consistent risk factor for fracture. . . .
6
1.4
Image representation of the VPHOP project. . . . . . . . . . . . . . . . . .
7
1.5
Biomed Town interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.6
Physiomspace interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.7
OpenClinica interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.8
VOP Hypermodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.1
Body planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2
Structure of long bone: external structure, epiphysis and diaphysis and a
section of bone with a view of internal structure and component. . . . . . .
16
2.3
Distribution of compact and cancellous bone in the upper part of the femur. 17
2.4
Structure of the cancellous and cortical bone: a representation of trabeculae
and of osteons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.5
Maximum strength with minimum weight. . . . . . . . . . . . . . . . . . . .
20
2.6
Structure of the pelvis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.7
A representation of forces that act on the pelvis from femur and from pelvis
to femur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.8
Anterior and posterior view of the femur. . . . . . . . . . . . . . . . . . . .
22
2.9
The coxofemoral joint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.10 CCD-angle in three configuration: coxa norma, coxa vara and coxa valga. .
25
vii
viii
List of Figures
2.11 Light micrograph of osteoclasts (arrows). Typical multinucleated osteoclast
nestled in its Howship’s lacuna. Bone (B), calcified cartilage (CC). Decalcified, methylene blue, and azure II stained section; original magnitude X
800. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12 Light micrograph of osteoblasts.
27
Spicule of calcified core line with os-
teoblasts (Ob) and thin osteoid (arrows). Osteoprogenitor cells (Opc) are
located between osteoblasts and blood vessel. Original magnification X600.
28
2.13 Diagrammatic representation of working hypothesis of bone resorption. A
typical resting bone surface is lined by a thin demineralized layer (OO), a
lamina lamitans (LL), and a flat bone-lining cells (BLC). . . . . . . . . . .
30
2.14 Hip joint lateral view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.15 Representation of the main hip muscles. . . . . . . . . . . . . . . . . . . . .
34
2.16 Principal movements’ angles . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.17 Representation of anatomical segments of human body and global and local
coordinate systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.18 Coordinate system for measured hip contact forces. The hip contact force
vector –F and its components –Fx , −Fy , −Fz acts from the pelvis to the
implant head and is measured in the femur coordinate system x, y, z. . . . .
38
2.19 Joint centres, reference points and coordinate system for gait analysis. . . .
39
2.20 Contact force F of typical patient NPA during nine activities. Contact force
F and its components –Fx , −Fy , −Fz . F and –Fz are nearly identical. . . .
40
2.21 Contact force vector F of typical patient NPA during nine activities. The
z-scales go up to 300% BW. Upper diagrams: Force vector F and direction
Ay of F in the frontal plane. Lower diagrams: Force vector F and direction
Az of F in the transverse plane. . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.22 Comparison of the predicted pattern of the hip load (solid black line) with
the variability of the hip load magnitude (grey band) measured on 4 subjects
through an hip prosthesis instrumented with a telemetric force sensor. . . .
42
3.1
Phantom and patient set-up during CT-scan. . . . . . . . . . . . . . . . . .
47
3.2
Points of morphing:head, greater trochanter, under below greater trochanter,
and lesser trochanter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
53
Points of morphing:the four points on the basis of the femur: anterior,
posterior, medial and lateral. . . . . . . . . . . . . . . . . . . . . . . . . . .
54
List of Figures
3.4
ix
Steps of morphing algorithm: (a) original template mesh, (b) original STL
mesh, (c) result from morphing the template mesh on the STL using RBF
method, (d) results after projection (c) on (b), (e) result from the Laplacian
smoothing, (f) final result represented in a high quality mesh of the STL
geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
Representation of the regression line between experimental and predicted
strain [STM+ 07]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
55
58
BoneMat of the femur, with the range of Young’s modulus between 18000MPa
and the maximum value obtained from the software (19832 MPa). . . . . .
59
3.7
final structure of VME data tree (final structure of msf file). . . . . . . . . .
60
3.8
TF_IF_CH reference system. . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.9
CHA reference system; blue line is z-axis, red line is x-axis and green line
is y-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10 Disposition of the Ansys key-points on CHA reference system.
62
. . . . . . .
62
3.11 Frontal plane and transversal plane. . . . . . . . . . . . . . . . . . . . . . .
63
3.12 Femur’s local coordinate system and keypoints. . . . . . . . . . . . . . . . .
66
3.13 Representetion of the nominal direction of load. . . . . . . . . . . . . . . . .
67
3.14 Representetion of all direction of load. . . . . . . . . . . . . . . . . . . . . .
67
4.1
Visualization of the distribution of principal deformation on the femur (anterior view). The highest deformation is located on the top of the neck. . .
72
4.2
Visualization of the maximum strain point in posterior view of the femur. .
73
4.3
Box plot of BMD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.4
Box plot of FRAX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4.5
Box plot of Strength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.6
Box plot of WorkFlow 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.7
ROC curve of BMD. AUC is of 0.73. . . . . . . . . . . . . . . . . . . . . . .
79
4.8
ROC curve of FRAX. AUC is of 0.64. . . . . . . . . . . . . . . . . . . . . .
79
4.9
ROC curve of strength. AUC is of 0.72. . . . . . . . . . . . . . . . . . . . .
80
4.10 ROC curve of WF2. AUC is of 0.75. . . . . . . . . . . . . . . . . . . . . . .
80
4.11 ROC curve of strength, WF2, N_BMD, TH_BMD and FRAX. AUC is of
0.84. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
4.12 ROC curve of strength and FRAX. AUC is of 0.74. . . . . . . . . . . . . . .
83
4.13 ROC curve of WF2 and FRAX. AUC is of 0.76. . . . . . . . . . . . . . . . .
83
x
List of Figures
4.14 ROC curve of strength, WF2, and FRAX. AUC is of 0.80. . . . . . . . . . .
84
4.15 ROC curve of strength and WF2. AUC is of 0.80. . . . . . . . . . . . . . .
84
4.16 ROC curve of strength, WF2, N_BMD and TH_BMD. AUC is of 0.83. . .
85
1
Representation of a box plot. The dots indicate the outliers. . . . . . . . . .
96
2
Representation of Mann-Whitney method. In (a) the different treatment
cause different effects, while in (b) they don’t produce any differences. . . .
97
3
Gaussian distribution of two population completely separated.
. . . . . . . 101
4
Gaussian distribution of two population with overlapping area. . . . . . . . 101
5
Contingency table.TP represents the true positive results, TN represents
the true negative results, FP represents the false positive results and FN
represents the false negative results. . . . . . . . . . . . . . . . . . . . . . . 102
6
Representation of the ROC curve as a function of specificity (Sp) and sensitivity (Se). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7
Representation of the AUC, the area under ROC curve. . . . . . . . . . . . 104
8
Logistic function with β0 = −1 and β1 = 2. . . . . . . . . . . . . . . . . . . 105
List of Tables
2.1
Peak loads of single and average patients, cycle times and body weight for
average patient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.1
CT scan parameters and setting. . . . . . . . . . . . . . . . . . . . . . . . .
46
4.1
Descriptive Statistic of control group. . . . . . . . . . . . . . . . . . . . . .
73
4.2
Descriptive Statistic of fracture group. . . . . . . . . . . . . . . . . . . . . .
74
4.3
Results of Mann-Whitney test. This test show which are the significant
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
Descriptive Statistic: the mean value and the standard deviation for Age,
Weight, Height, Femoral neck BMD and total BMD. . . . . . . . . . . . . .
4.5
81
In this table are reported all the variables in the equation obtained with the
logistic regression of WF2 and FRAX. . . . . . . . . . . . . . . . . . . . . .
4.9
81
logistic regression of strength and FRAX. . . . . . . . . . . . . . . . . . . .
4.8
76
logistic regression of strength, WF2, N_BMD, TH_BMD and FRAX. . . .
4.7
76
Descriptive Statistic: the mean value and the standard deviation for risk of
fracture of WF2, strength and risk of fracture of FRAX. . . . . . . . . . . .
4.6
75
81
logistic regression of strength, WF2 and FRAX. . . . . . . . . . . . . . . . .
81
4.10 In this table are reported all the variables in the equation obtained with the
logistic regression of strength and WF2. . . . . . . . . . . . . . . . . . . . .
82
4.11 In this table are reported all the variables in the equation obtained with the
logistic regression of strength, WF2, TH_BMD and N_BMD. . . . . . . . .
xi
82
Abstract
Osteoporosis, which literally means porous bone, is a disease in which the density
and quality of bone are reduced. As bones become more porous and fragile, the risk
of fracture is greatly increased. The loss of bone occurs silently and progressively.
Often there are no symptoms until the first fracture occurs. Around the world, 1 in
3 women and 1 in 5 men are at risk of osteoporotic fracture. In fact, an osteoporotic
fracture is estimated to occur every 3 seconds. The most common fractures associated
with osteoporosis occur at the hip, spine and wrist. The likelihood of these fractures
occurring, particularly at the hip and spine, increases with age in both women and
men.
Due to its prevalence worldwide, osteoporosis is considered as a serious public health
concern. Currently it is estimated that over 200, 000, 000 people worldwide suffer from
this disease. Approximately 30% of all postmenopausal women have osteoporosis in the
United States and in Europe. At least 40% of these women and 15 − 30% of men will
sustain one or more fragility fractures in their remaining lifetime. Aging of populations
worldwide will be responsible for a major increase of the incidence of osteoporosis in
postmenopausal women.
Of particular concern are vertebral (spinal) and hip fractures. Vertebral fractures
can result in serious consequences, including loss of height, intense back pain and
deformity. Hip fracture often requires surgery and may result in loss of independence
or death. It has been shown that an initial fracture is a major risk factor for a new
fracture. An increased risk of 86% for any fracture has been demonstrated in people
that have already sustained a fracture. Likewise, patients with a history of vertebral
fracture have a 2.3-fold increased risk of future hip fracture and a 1.4-fold increased in
risk of distal forearm fracture.
The social burden of this disease is very high: approximately four million osteoxiii
xiv
ABSTRACT
porotic bone fractures cost the European health system more than 30 billion Euro per
year. This figure could double by 2050.
Current fracture prediction is based on historical, fracture-patient data sets to identify key factors that contribute to the increased probability of an osteoporotic fracture.
This approach oversimplifies the mechanisms leading to an osteoporotic fracture and
fails to take into account numerous hierarchical factors which are unique to the individual. These factors span cell-level to body-level functions, i.e.:
• body: musculoskeletal anatomy and neuromotor control define the daily loading
spectrum, including para-physiological overloading events;
• organ: fracture events occur at organ level and are influenced by the elasticity
and geometry of bone;
• tissue: bone elasticity and geometry are determined by tissue morphology;
• cell: cell activity changes tissue morphology and composition over time;
• constituents: constituents of the extracellular matrix are the prime determinants
of tissue strength.
By now, the clinical instruments for osteoporosis diagnosis are DXA and FRAX. DXA
is a low radiation X-ray capable of detecting quite small percentages of bone loss. It
is used to measure spine and hip bone mineral density (BMD), and can also measure
BMD of the whole skeleton. FRAX is a scientifically validated risk assessment tool,
endorsed by the World Health Organization (WHO) and now integrated into an increasing number of national osteoporosis guidelines around the world. It is considered
a major milestone in helping health professionals to improve identification of patients
at high risk of fracture. The web-based FRAX calculator assesses the ten-year risk of
osteoporosis fracture based on individual risk factors, with or without BMD values.
Both these tools are not patient-specific and in most studies is reported a predictive
accuracy in the range of 65 − 75%. Accuracy could be dramatically improved if a more
deterministic approach was used which accounts for those factors and their variation
between individuals.
Subject-specific finite element (FE) models of bones generated from the patient’s
CT data have been proposed to improve the fracture risk prediction, as they take into
xv
account the structural determinants of bone strength and the variety of external loads
acting on bones.
The aim of our work is to create a predictor of the fracture risk that improves
the actual clinical methods based on DXA imaging and the FRAX epidemiological
risk predictor. Such predictor should be patient-specific, based on the mechanical
properties of the patient’s femur. In order to assess the predictive accuracy of the FEbased predictor we developed, we used data of a retrospective clinical study involving a
cohort of 92 patients, 48 of them with a femoral neck fracture and 44 control patients,
who have been computed tomography-scanned (CT-scanned). For each of them, we
extracted the geometry of their femurs; then we meshed the specific geometry applying
the morphing method described in Grassi et al. [GHS+ 11] and assigned the material
properties using the Bonemat_V3 software [TSH+ 07]; in this way we built the patientspecific models. We calculated the risk of fracture as a function of bone strength, under
two different loading scenarios: the first scenario represents the physiological case of
the gait, in which some forces are applied on the head of the femur along different
directions. The second scenario, proposed in the VPHOP project, models side fall using
pre-computed loads using whole body dynamics model and a probabilistic formulation
of the force damping due to the soft tissues. For the first scenario we defined some
loads and boundary conditions in ANSYS to calculate the fracture load, while for
the second scenario we uploaded our models on an informatics platform realized for
VPHOP project. Finally we calculated the fracture risk composing the results obtained
by the different scenarios with the ones obtained by DXA and by FRAX using MannWhitney test, logistic regression, and the ROC curve. We used SPSS software to
calculate Mann-Whitney test and logistic regression and MATLAB to calculate ROC
curve.
Our results demonstrate that patient-specific predictors are better than clinical instruments: side fall scenario has a 10 years predictivity of 75%, while the physiological
scenario has a predictivity of 72%. For our cohort of patients DXA has a predictivity
of 73% in contrast with literature, while FRAX has one of 64%. Combining together
the proposed scenarios with clinical instrument DXA we obtained a predictor with an
accuracy of 83%, adding FRAX we obtained a predictor with an accuracy of 84%.
Combining the mechanical predictors without clinical instruments we obtained an accuracy of 80%, that is better than both FRAX and DXA. Thus, we can retain to have
xvi
ABSTRACT
obtained a predictor that is better than the ones used by clinicians.
The scientific importance of this work is the possibility to calculate with good accuracy the fracture risk of the femoral neck in osteoporotic patients. If this tool will
be used by clinicians, there would be a high saving of health and socio-economic burden, in addition to the improvement of the life’s quality of a lot of patients, because
the femoral neck fracture is one of the most frequent causes of death in these people,
especially for the elderly. The strengths of this work are the possibility to create a
patient-specific mechanical model through the pre-processing phase and then to evaluate the fracture risk; segmentation permits to build the femur, morphing allows to
generate a mesh quickly and to compare different geometries, and BoneMat assigns
materials in function of patient’s CT-images.
The limits of the work are high user-dependency of the pre-processing phase and
the high temporal and computational costs, especially for side fall scenario.
This thesis has some possible future developments, like the possibility to make the
pre-processing phase less user-dependent and to create a side fall scenario implemented
in ANSYS on the local disk: this would permit a saving in temporal and computational
costs.
A brief description of the content of each chapter is shown below:
• Chapter 1: this introductive chapter contains a simple description of osteoporosis, its aetiology and the relative social and economic burden followed by an
exposition of the clinical state of the art in diagnosing the pathology and the
presentation of the VPHOP project.
• Chapter 2: this chapter contains some definitions of the utilized medical terminology, including a part on muscle-skeletal anatomy and a part on the mechanobiology of bone tissue. Then a description of the hip biomechanics and kinematics,
an introductive explanation of gait analysis and of the forces that act on the hip
during daily activities are reported. Finally a short paragraph about the effect
of sub-optimal neuromotor control and some considerations on our femur model
are presented.
• Chapter 3: in this chapter we present the cohort of patients involved in this
study and the explanation of all the phases of the work, with a final description
of the loading scenarios we applied.
xvii
• Chapter 4: this chapter contains the descriptive statistic of the cohort of patients, the results of the different loading scenarios and a statistical analysis of
them.
• Chapter 5: discussion.
• Chapter 6: future development.
Sommario
L’osteoporosi è una malattia metabolica che provoca una riduzione della densità delle
ossa (BMD) e un aumento della loro fragilità; con l’aumento della porosità si ha un aumento del rischio di frattura. Tale processo avviene progressivamente e senza mostrare
sintomi nel paziente.
Circa 1 donna su 3 e un uomo su 5 sono a rischio di frattura osteoporotica: si stima
che ogni 3 secondi nel mondo avvenga una frattura di questo tipo. Le zone più soggette
a frattura sono l’anca, le vertebre e i polsi, e la probabilità che si presenti questo tipo
di fratture aumenta con l’età. L’osteoporosi colpisce circa 200.000.000 di persone in
tutto il mondo: circa il 30% di donne in menopausa in Europa e negli Stati Uniti soffre
di questa malattia e si stima che il 40% delle donne e il 15 − 30% degli uomini subirà
almeno una frattura dovuta alla fragilità ossea nella loro vita.
L’osteoporosi resta ad oggi una patologia poco diagnosticata con conseguenze piuttosto gravi nella vita dell’individuo, in particolare a seguito di fratture che possono
coinvolgere la spina dorsale e il femore. Tra i due tipi di frattura, quello femorale è
sicuramente il più grave, anche se non è il più frequente, poiché porta ad un drastico
peggioramento dello stile di vita del paziente e ad un aumento del rischio di mortalità
nel primo anno a seguito della frattura. È stato dimostrato che per i pazienti che hanno
già subito una frattura legata alla patologia, il rischio che essa si ripresenti è superiore
all’86%; inoltre i pazienti che hanno nella loro storia clinica delle fratture vertebrali
hanno un rischio di frattura 2, 3 volte maggiore all’anca e 1, 4 volte alle zone distali.
I costi socio sanitari dell’osteoporosi sono molto elevati: in Europa ogni anno si
spendono circa e30.000.000.000; questa cifra è destinata a raddoppiare entro il 2050.
Le cure preventive per l’osteoporosi sono poco diffuse e gli strumenti diagnostici
attualmente utilizzati in clinica sono la DXA e la FRAX. La DXA è uno strumento
a bassa radiazione di raggi X in grado di rilevare lievi diminuzioni di BMD. Viene
xix
xx
SOMMARIO
utilizzata per misurare la BMD delle vertebre e dell’anca ma può essere utilizzata
per misurare anche quella relativa all’intero scheletro. La FRAX è uno strumento
elettronico che calcola il rischio di frattura a 10 anni approvato dall’Organizzazione
Mondiale della Sanità (WHO) e ormai incluso in molte linee guida in tutto il mondo.
È disponibile gratuitamente sul web e viene utilizzato dai clinici come aiuto nel definire
quali pazienti sono maggiormente soggetti a rischio di frattura e a delineare eventuali
linee terapeutiche. Entrambi questi strumenti hanno una predittività compresa tra
55 − 65%. Questi strumenti però sono basati sulla storia clinica del paziente senza
considerare una serie di fattori specifici a diversi livelli come:
• il corpo: l’anatomia muscoloscheletrica e il controllo neuromuscolare definisco lo
scenario di carico quotidiano, includendo sovraccarichi parafisiologici;
• l’organo: le fratture che avvengono sull’organo sono influenzate dalla geometria
e dalle proprietà elastiche dello stesso;
• le cellule: l’attività cellulare cambia la morfologia e le proprietà del tessuto nel
tempo;
• costituenti: i costituenti della matrice extra-cellulare sono i principali determinanti delle proprietà meccaniche del tessuto.
Gli strumenti in uso clinico sopra citati utilizzano come indice di diagnosi per
l’osteoporosi il valore di densità ossea, senza considerare le proprietà meccaniche del
femore nel loro complesso. Questo vorrebbe dire che tutti i pazienti che presentano la
stessa densità ossea dovrebbero avere lo stesso rischio di frattura. In realtà non è così
perché esiste una grossa serie di fattori specifici al paziente che possono influenzare il
rischio di frattura: il peso, l’età, alcuni fattori di rischio, il controllo neuromuscolare,
la geometria e le proprietà meccaniche del femore stesso, per citarne alcuni. A questo
proposito la FRAX si distingue dalla DXA per un questionario dicotomico che introduce informazioni personali su alcuni fattori di rischio per l’osteoporosi (ad esempio
fumo, alcool, utilizzo di particolari farmaci, inclinazione genetica e altri), ma tali informazioni non sono sufficienti a caratterizzare in modo specifico il paziente, poiché se
due pazienti presentano le stesse caratteristiche di età, peso e sesso, oltre che di densità
ossea, hanno di nuovo lo stesso rischio di frattura.
Molti ricercatori stanno cercando di realizzare degli strumenti paziente-specifici
attraverso alcuni modelli a elementi finiti FE per calcolare il rischio di frattura del
xxi
paziente con una maggiore accuratezza. Lo scopo di questo lavoro è realizzare un predittore per il rischio di frattura del collo femorale di pazienti osteoporotici attraverso
un modello FE specifico del paziente. Per raggiungere il nostro scopo siamo partite
da una corte di 92 pazienti osteoporotici; per ognuno abbiamo ricostruito il femore a
partire dalle immagini da tomografia computerizzata CT tramite un pre-processo di
segmentazione con il software ITK-SNAP; successivamente abbiamo costruito le mesh
sulle geometrie specifiche tramite il metodo del Morphing [GHS+ 11] e assegnato i materiali con il software BoneMat_V3 [TSH+ 07] implementato nel software LHP_Builder.
Abbiamo utilizzato questi modelli FE per applicare due diversi scenari di carico: il
primo simula i diversi casi fisiologici di carico ai quali è sottoposto il femore durante
lo svolgimento delle attività quotidiane. L’implementazione dei carichi per il primo
scenario è stata fatta in locale con il software ANSYS. In questo caso i carichi vengono
applicati al femore isolato. Il secondo scenario simula la caduta laterale considerando
un modello total body. In questo secondo scenario si tiene conto della presenza di
alcuni fattori di smorzamento che diminuiscono il carico che è applicato sul femore, ad
esempio il contributo dei tessuti molli interposti oppure la possibilità che il soggetto
adotti degli espedienti durante la caduta e che quindi il contatto non sia solo sull’anca,
come l’appoggio delle mani o l’appoggio del ginocchio. Questo scenario si inserisce con
il nostro modello in un ipermodello che considera le caratteristiche fisiche del paziente;
tale ipermodello rientra nel progetto internazionale VPHOP e in particolare per questo
lavoro c’è stata una collaborazione tra la University of Sheffield e l’Istituto Ortopedico
Rizzoli di Bologna.
L’analisi dei risultati è stata fatta considerando il carico di frattura per il modello
fisiologico, calcolato dalla deformazione principale, e il rischio di frattura per la caduta
laterale ottenuto come percentuale di rischio di frattura probabilistico a 10 anni: i
risultati sono stati analizzati con degli strumenti statistici, in particolare test di MannWhitney, regressione logistica e curva ROC, implementati nel software SPSS; per la
realizzazione delle curve ROC abbiamo utilizzato anche il software MATLAB.
È risultato che i modelli paziente-specifici da noi realizzati hanno una predittività
superiore agli strumenti utilizzati in clinica: l’ipermodello che simula la caduta laterale
ha una predittività del 75% mentre il modello con il carico fisiologico ha una predittività
del 72%. Per la corte di pazienti utilizzata in questo lavoro, la DXA ha una predittività
del 73% in contrasto con i dati ottenuti dalla letteratura, mentre la FRAX ha una
xxii
SOMMARIO
predittività del 64%. Assemblando i due scenari di carico proposti con lo strumento
clinico DXA abbiamo ottenuto un predittore con un’accuratezza dell’83%, aggiungendo
la FRAX abbiamo ottenuto un predatore con un’accuratezza dell’84%. Considerando
invece anche solo i due scenari insieme (escludendo quindi gli strumenti clinici) si ha
una predittività dell’80%, superiore sia a quella della DXA sia a quella della FRAX.
Possiamo quindi ritenere di avere trovato un predittore migliore di quelli attualmente
utilizzati in clinica e con il vantaggio di avere un modello paziente-specifico.
La rilevanza scientifica di questo lavoro sta quindi nella possibilità di prevedere con
una buona accuratezza il rischio di frattura del collo femorale dei pazienti; se questo
strumento fosse applicato in clinica si avrebbe un elevato risparmio dei costi sanitari e
dei costi sociali, oltre che un miglioramento della qualità della vita di molti pazienti,
poiché la frattura del collo femorale è spesso causa del decesso del paziente.
I punti di forza di questo lavoro sono la possibilità di realizzare dei modelli meccanici
paziente-specifici tramite il pre-processing e la possibilità di calcolarne il rischio di
frattura; la segmentazione permette, infatti, un’accurata ricostruzione del femore, il
morphing permette la generazione di una mesh in tempi rapidi e rende possibile il
confronto tra i modelli dei diversi pazienti, mentre il BoneMat assegna i materiali
basandosi sulle immagini CT proprie del paziente.
I limiti maggiori sono legati alla dipendenza ancora troppo elevata che la fase di preprocessing ha dall’utente e dall’elevato costo temporale e computazionale, in particolare
per quanto riguarda lo scenario di caduta laterale.
Questa tesi si apre anche a possibili futuri sviluppi che riguardano appunto sia il
miglioramento della fase del pre-processing che dovrebbe diventare più automatizzata,
che all’implementazione in locale in ANSYS di uno scenario di carico che simuli la
caduta laterale, partendo da alcuni studi sperimentali già presenti in letteratura.
Di seguito è elencata una breve descrizione del contenuto di ogni capitolo della tesi:
• Capitolo 1: questo capitolo contiene una breve descrizione dell’osteoporosi,
la sua eziologia e i costi socio-economici.
Segue una rappresentazione dello
stato dell’arte per quanto riguarda i metodi di diagnosi della patologia e la
presentazione del progetto VPHOP.
• Capitolo 2: in questo capitolo sono elencate alcune definizioni di terminologia medica utilizzate, nozioni fondamentali di anatomia del sistema muscoloscheletrico e meccano-biologia del tessuto osseo. Sono presenti inoltre una descri-
xxiii
zione della biomeccanica e della cinematica dell’anca, un’introduzione all’analisi
del cammino e delle forze che agiscono sull’anca durante le principali attività quotidiane. Infine è presente un paragrafo sugli effetti del controllo neuromuscolare
e alcune considerazioni sul nostro modello.
• Capitolo 3: in questo capitolo viene presentata la corte di pazienti coinvolta nel
nostro studio, oltre a una chiara spiegazione delle fasi del nostro lavoro. Infine
sono descritti i due scenari di carico studiati.
• Capitolo 4: risultati.
• Capitolo 5: discussione dei risultati.
• Capitolo 6: futuri sviluppi.
Chapter 1
Introduction
1.1 Osteoporosis: social and economic burden
Osteoporosis is a metabolic disease that leads to an increased risk of bones fracture.
In osteoporosis, the bone mineral density (BMD) is reduced, bone microarchitecture
deteriorates, and the amount and probably the nature of proteins in bone are altered
because of a multiple pathogenic mechanisms. The disease may be classified as primary
or secondary type. Primary osteoporosis is the one related to menopause or the senile
age. Secondary osteoporosis is caused by other medical conditions, or by prolonged use
of certain medications such as glucocorticoids.
The pathogenic mechanisms that can lead to skeletal fragility are:
• failure to produce a skeleton of optimal mass and strength during growth;
• excessive bone resorption resulting in decrease bone mass and micro-architectural
deterioration of the skeleton [Rai05].
The other factors that can increase the risk of osteoporotic bone fracture are:
• female gender;
• advancing age;
• ethnicity;
• previous fracture;
• oestrogen deficiency;
1
2
CHAPTER 1. INTRODUCTION
• calcium deficiency;
• vitamin D deficiency;
• low body weight;
• alcoholism;
• medications;
• smoking;
• in the end, also propensity to fall, poor physical function, impaired vision and
environment can increase the risk of fall and , as a consequence, also the risk of
fracture [Ric03].
Around the world, 1 in 3 women and 1 in 5 men are at risk of osteoporotic fracture.
It is estimated that over 200 million people worldwide have osteoporosis. In European
Union, the number of osteoporotic fractures was estimated around 4, 000, 000 that
cost to Europe e30, 000, 000, 000. The costs to health care service and the number of
osteoporotic fractures are considerable and are predicted to double by the 2050 [RB06].
Among all the osteoporotic fractures, the hip fractures are the ones that are most
disabling and costly, can be very severe for a long period and can reduce in the long
term the quality of life of individuals and increase the mortality. In fact, women who
have suffered a hip fracture have a 10% to 20% higher mortality than would be expected for their age. Osteoporotic fragility fractures impose a considerable financial
burden on health service due to reduce mobility, hospitalization, and nursing home
requirement.In most developed countries, it is important and recommended that postmenopausal women at high risk should be screened for osteoporosis and a 10-year
probability of fracture assessed to determine intervention threshold. However, most of
the times osteoporosis is not diagnosed until the first fracture occurs [RB06]. Many
factors are responsible for under-diagnosis; firstly, osteoporosis is a silent disease, which
has no obvious symptoms. Secondary, primary care physicians are often overburdened
with clinical, administrative and regulatory responsibilities that leave little time to
consider a silent disease that increases the risk of an event that may occur far in the
future. Acute fractures are often treated by an orthopaedist or emergency department
specialist who is not responsible for long-term care and prevention of future fractures;
1.2. THE CLINICAL STATE OF THE ART
3
Figure 1.1: Image of the DXA of a femur.
as a result of this, too frequently even after the first osteoporotic fracture the patient
is not properly referred to the osteoporosis clinic [LEW09].
1.2 The Clinical State of the Art
In osteoporotic patients the risk of fracture is high but often it is no possible to have an
accurate diagnosis to prevent the risk of fracture. In clinical practice dual-energy x-ray
absorptiometry (DXA) is used to predict the risk of fracture to quantify the patient’s
bone mineral density (BMD). DXA uses x-rays at two energy levels and subtracts the
difference of absorption between soft tissue and hard tissue to determine the bone
mineral content (Figure 1.1). The outputs of DXA are T-score and Z-score.
T score is the difference, in standard deviations, between the patient’s bone mineral
density and the mean bone mineral density of young adult white woman. The World
Health Organization (WHO) established the following values of T score to classify the
level of pathology:
• normal (T-score ≥ 2.5);
• osteopenia (−2.5 ≤ T-score < −1.0);
• osteoporosis (T-score < −2.5 );
• severe osteoporosis (T-score < −2.5 with a fragility fracture).
4
Figure 1.2: FRAX interface
The Z-score compares the patient’s BMD with the mean value in a population of
similar age, sex and height. This information is useful in determining the probability
of secondary osteoporosis.
We didn’t find in literature agreement on what is the predictive accuracy of the
risk of bone fracture using DXA-measured BMD; however, many studies suggest it
is between 55 − 65%. The patients that have a low value of T-score or Z-score are
treated with alendronate, risedronate, calcitonin, raloxifene, oestrogen or parathyroid
hormone. The use of these therapies is useful to reduce the fracture risk.
Another method to estimate the risk of fracture is through the Fracture Risk Assessment Tool FRAX1 ; it is an electronic clinical tool that has been developed at the
University of Sheffield under the direction of Professor John Kanis. FRAX estimates
the 10-year probability of fracture on the basis of clinical risk factors for fracture and
the BMD of the femoral neck. The combination of BMD and clinical risk factors
predicts fracture risk better than either alone.
FRAX is based on an algorithm that has as inputs the BMD of the femoral neck
calculated by DXA, the patient’s age, sex, height, weight and seven clinical risk factors
(previous fracture, having a parent who had a hip fracture, current smoking, glucocorticoid use, rheumatoid arthritis, secondary osteoporosis, and ingestion of three or
more units of alcohol daily). The most important factor that FRAX neglects is the
1 http://www.shef.ac.uk/FRAX/
1.2. THE CLINICAL STATE OF THE ART
5
propensity to fall. The authors motivate this as follow: falls are not considered a risk
factor because the following reasons: first, existing data are not of adequate quality to
incorporate quantitative adjustment to FRAX at the present time. Information on falls
was available in a minority of cohorts used to derive or validate FRAX. Second, fall’s
risk is inherently taken into account in the algorithm, though not as an input variable.
Thus, the fracture probability given for any combination of risk factors assumes that
the fall’s risk is that observed (but not documented) in the cohorts used to construct
FRAX. Third, the interrelationship of fall’s risk with the other FRAX variables has
been inadequately explored on an international basis. Fourth, the relationship between
the risk variable and mortality needs to be accounted for, but there are no data available [LEW09]. In addition, the FRAX models are calibrated for different countries
using country-specific fracture and mortality rates”.
The advantages to use this tool are:
• it can be used in women and men from age 40 to 90, although the National
Osteoporosis Foundation guidelines suggest to use it only to make treatment
decisions in untreated postmenopausal woman and men age 50 and older with
osteopenia who do not otherwise qualify for treatment;
• it is free and is available on the website http://www.shef.ac.uk/FRAX/ (Figure 1.2).
The disadvantages are:
• the FRAX was not validated on treated patients, children and women and men
out of the age range indicated before;
• it can be used only for four different phenotypic groups (white, black, Hispanic
and Asian);
• the responses to the seven risk factors are yes or no and their severity or dose is
not considered, whereas sometimes the risk belong to them;
• there are other fracture risks that are not considered, such as falling, rate of bone
loss and high bone turnover.
These limitations of FRAX could cause underestimation or overestimation of actual
fracture risk: as for DXA, we didn’t find a unique value of the predictive accuracy of
FRAX; it is reported in most studies in the range 65 − 75% [LEW09], [KHC+ 11].
6
Figure 1.3: The graph shows fracture rates per 100 person-years by phenotypic and
T-score according to the National Osteoporosis Risk Assessment Study. Across all
phenotypic groups, low BMD is a consistent risk factor for fracture.
Osteoporotic fractures are not the only kind of fracture and this could make confusion in the definition of the patient’s diagnosis and therapy. Risk factors for fracture
include BMD, bone geometry (that are different for phenotypic groups), age, fall rates,
fracture history and medication use. It is known that a high level of BMD reduces risk
fracture, for example Africans have a high BMD and a lower risk fracture as shows in
Figure 1.3. Most fractures occur because of falls (but we don’t know anything about
ethnic difference in fall rate). Three risk factors common to all groups are older age,
positive history of prior fracture and positive history of two or more falls. The Figure 1.3 represents the fracture rates per 100 person years by ethnicity and T score
according to the National Osteoporosis Risk Assessment Study [Cau11].
1.3 The VPHOP project
Our work was part of the Virtual Physiological Human Osteoporotic project VPHOP2
(Figure 1.4) that was completed in January 2013. This European project involved 21
partner institutions (between universities and Companies). During this project some
informatics platforms have been developed to support all the people involved in the
project and not.
The first platform is Biomed Town3 that is an on-line community open and free to
2 http://www.vphop.eu/
3 https://www.biomedtown.org/
1.3. THE VPHOP PROJECT
7
Figure 1.4: Image representation of the VPHOP project.
anyone has a professional or educational interest in biomedical research and practice
(Figure 1.5). BiomedTown is a virtual town made of Buildings and Squares. Buildings
host virtual organizations such as research consortia, companies, institutions, interest
groups, etc. Squares are where the people meet, discuss, and exchange experiences;
they represent the communities that populate the city, but they are not specifically
represented by an organisation. In this portal you can find a lot of information about
the VPHOP project, all the things that have been developed, all the news and the
events of the project, and you can also download some program developed during the
project.
Another platform is PhysiomeSpace4 that is the digital library service designed to
help researchers to share their biomedical data and models (Figure 1.6).
4 https://www.physiomespace.com/
8
Figure 1.5: Biomed Town interface.
Figure 1.6: Physiomspace interface.
Another platform is OpenClinica5 that is the world‘s first commercial open source
clinical trial software for Electronic Data Capture (EDC) and Clinical Data Management (CDM). In this platform you can find all the clinical data that you need rapidly
and easily (Figure 1.7).
The Virtual OsteoPorosis (VOP) platform integrates OpenClinica, PhysiomeSpace, and the VPH Hypermodelling framework into and end user environment for
the request of individualized multiscale simulations for the enrolled patients http:
5 https://openclinica.com/
1.3. THE VPHOP PROJECT
9
Figure 1.7: OpenClinica interface.
Figure 1.8: VOP Hypermodel.
//webapp.physiomespace.com/. The Hypermodel is a composition by many sub models, each describing the relevant phenomena taking place at one of the many dimensional
scales involved, as represents in Figure 1.8. The VPHOP project has developed a web
interface to run and configure Hypermodel simulations to predict the risk of fracture
on osteoporotic patients.
The VOP interface is composed by different sections:
• OpenClinica: provide access to the data cohort available in VPHOP. In this
section the user can select the specific information about a patient to be visualized/edit;
10
• risk: the user can run one of the available VPHOP clinical workflows for the
calculation of the personalized risk of fracture selecting it from the list;
• dashboard: the user can submit a workflow or a single service for execution
and/or monitor the status of the workflows already launched;
• service: this page provides information on the available services, modules and
workflows.
The last platform is LHP_Builder that is a program used to develop the file to
upload on Physiomspace portal. This program executes on the user‘s PC and makes
possible to import any kind of biomedical data, interactively visualize them, and fuse
them into a coherent representation, which can then be uploaded and shared. Into this
program there are some software used to developed specific functions like BoneMat
software, used to assign the material properties to the geometry. Every member of
BiomedTown can use his/her credentials to login in PhysiomeSpace, and upload or
download datasets using the dedicated application called PS_Loader, or other enabled
applications such as LHP_Builder.
The aim of the VPHOP project is to develop a multiscale modelling technology
based on conventional diagnostic imaging methods that creates a model that is able to
predict for each patient the strength of bones, how this strength is likely to change over
time, and the probability that the patient will overload his/her bone during daily life.
In this way, the evaluation of absolute risk of fracture will be more accurate than the
prediction with the current clinical practice that are based on oversimplified mechanisms that not take into account the features of the specific patient, like musculoskeletal
anatomy and neuromotor control or the tissue morphology. The predictions based on
the strength can be used to improve the diagnostic accuracy in current clinical practice
and to provide a basis for a prognosis. All the modeling technologies developed during
the project are validated with in vitro experiments or on animal models, and in term
of clinical impact and safety also on small cohort of patients enrolled at four different
clinical institutions. The model developed for osteoporosis patients is:
• predictive: the multiscale model, representing the musculoskeletal mechanobiology, simulates different loading in various condition and predicts bone failure;
1.4. THE AIM OF OUR WORK
11
• personalized: the multiscale model uses the patient’s information. More information are available, more personalized becomes the model.
The VPHOP project has developed personalized computer models that simulate
the daily loading during normal activities, and during the side fall. The technology
developed during the VPHOP project will not replace the current clinical tool, but it
can be used, in a patient with higher risk of fracture, to improve the accuracy of the
prediction and to give the clinicians more detailed information on the region of the
skeleton that are at high risk of fracture. This method will help the clinicians to better
personalize the treatments and recommendations. The VPHOP aim is to deliver to
clinical service a technology that will help to significantly reduce osteoporotic fractures
in the near future.
1.4 The aim of our work
Osteoporosis is a disease that increases the risk of fracture during normal activities
and during sideway falls, especially in elderly people. Subject-specific finite element
(FE) models of bones generated from the patient’s CT data have been proposed to
improve the fracture risk prediction [TSH+ 07], as they take into account the structural
determinants of bone strength and the variety of external loads acting on bones.
The aim of our work is to create a predictor of the fracture risk that improves
the actual clinical methods based on DXA imaging and the FRAX epidemiological
risk predictor. Such predictor should be patient specific, based on the mechanical
properties of the patient’s femur. In order to assess the predictive accuracy of the FEbased predictor we developed, we used data of a retrospective clinical study involving a
cohort of 92 patients, 48 of them with a femoral neck fracture and 44 control patients,
who have been computed tomography scanned (CT-scanned). For each of them, we
extracted the geometry of their femurs; then we meshed the specific geometry applying
the morphing method described in [GHS+ 11] and assigned the material properties using
the Bonemat_V3 software [ZMV99]; in this way we built the patient specific models.
We calculated the risk of fracture as a function of the bone strength, under two different
loading scenarios: the first scenario represents the physiological case of the gait, in
which some forces are applied on the head of the femur along different directions. The
second scenario, proposed in the VPHOP project, models side fall using pre-computed
12
loads trough whole body dynamics model and a probabilistic formulation of the force
damping due to the soft tissues. Finally we calculated the fracture risk composing
the results obtained by the different scenarios with the ones obtained by DXA and
by FRAX using the Mann-Whitney test, logistic regression, and the ROC curve. We
used SPSS6 software to calculate Mann-Whitney test and the logistic regression and
MATLAB7 to calculate the ROC curve.
6 http://www-01.ibm.com/software/it/analytics/spss/
7 http://www.mathworks.it/products/matlab/
Chapter 2
Anatomy and Mechanics
2.1 Body’s reference system and planes
In gait and motor control analysis the human body could be identified as a whole
of segments or geometries defined as rigid bodies. This assumption is made in many
laboratories to simplify the description and the study of the body’s kinematics and
dynamics. When modeling the dynamic behavior of the human body, because of the
significantly higher stiffness of the bones compared to any other tissue, and because
bone deformations under physiological loads are very small, it is normally assumed
that bones are infinitely rigid bodies, and the body compliance all takes place at the
joints that link these rigid bodies.
To identify univocally the position of a rigid body in the space is necessary to
know the number of degrees of freedom of that body because the number of degrees
of freedom of a system is equal to the number of coordinates you need to describe it.
Because of the high number of degrees of freedom of the human body, it is necessary to
define a reference position called anatomical position: the posture is erect, the superior
limbs are stretched along the hips, the palms forward facing, the head erect looking
forward, the inferior limbs stretched and close to each other, the feet laying to the
ground and parallel to themselves.
In anatomy, three main planes are defined:
• sagittal or medial plane: it identifies the major symmetry and divides the right
part by the left one;
• coronal or frontal plane: it is the vertical, perpendicular to the medial one and
13
14
CHAPTER 2. ANATOMY AND MECHANICS
Figure 2.1: Body planes.
passing through the body’s barycenter;
• axial or transverse plane: it is the horizontal plane passing through the body’s
barycenter.
The body planes described are clearly represented in Figure 2.1.
2.2 Relative position
To describe the position of a part of the body respect to another one, the following
terms are used:
2.3. MUSCULOSKELETAL ANATOMY
15
• proximal/distal: these adjectives are used to say that a part is nearer/further to
the barycenter than the other one. For example, the femur is proximal to the
tibia but is distal to the hip;
• medial/lateral: these adjectives are used to say that a part is nearer/further to
the sagittal plane than the other one. For example, the neck is medial to the
shoulder;
• superior/inferior: these adjectives are used to say that a part is up/down to the
other one;
• anterior/posterior: these adjectives are used to say that a part is up ahead/behind
to the other one.
2.3 Musculoskeletal anatomy
The skeletal system is important to the body both bio-mechanically and metabolically,
is made up of individual bones and the connective tissue that joins them. Bone is
characterized by rigidity and hardness as results from inorganic salts impregnating the
matrix, which consists of collagen fibers, a large variety of non-collagenous protein
and mineral. The rigidity and hardness of bone enable the skeleton to maintain the
shape of the body, to protect soft tissues (i.e. pelvic cavities, cranium), to supply the
framework for bone marrow and to transmit the force of muscular contraction from
different parts of the body during movements. The skeleton has been divided into
axial and appendicular region. Composition and functions are different for axial and
appendicular components.
2.3.1
Anatomy and physiology of bone
Bones of different shapes compose the skeleton. Generally the bones can be classified
according to their shape, in:
• long bones (i.e. femur, humerus);
• short bones (i.e. vertebras);
• flat bones (i.e. cranium, scapula).
Each group has its own features. In particular long bones can be divided in:
16
Figure 2.2: Structure of long bone: external structure, epiphysis and diaphysis and a
section of bone with a view of internal structure and component.
• diaphysis, or shaft, is the long tubular portion of long bones. It contains bone
marrow in the central marrow cavity and adipose tissue;
• epiphysis is the rounded end of a long bone. It is filled with red bone marrow which produces erythrocytes (red blood cells); at the joints the epiphysis is
covered with hyaline cartilage;
• metaphysis is the area where the diaphysis meets the epiphysis. It includes the
epiphyseal line, a section of cartilage from growing bones (Figure 2.2).
The diaphysis is composed mainly of cortical bone, while the epiphysis and metaphysis contain mostly cancellous or trabecular bone with a thin shell of cortical bone
(Figure 2.3). The epiphysis is separated from the metaphysis by a plate of hyaline cartilage known as the growth plate-metaphyseal complex, or epiphyseal plate. This plate
and the adjacent cancellous bone constitute a region where cancellous bone production
17
Figure 2.3: Distribution of compact and cancellous bone in the upper part of the femur.
and elongation of cortex occur. On the articulating surface at the ends of long bone,
the shell is covered with a thin layer of specialized hyaline cartilage, the articular cartilage. The outer surface of bone is covered by periosteum, a sheet of fibrous connective
tissue and an inner cellular or cambium layer of undifferentiated cells. The periosteum
has the potential to form bone during growth and fracture healing. The marrow cavity
and the cavities of cortical and cancellous bone are covered with a thin cellular layer
called endosteum.
There are two different kinds of bone:
• cortical or compact bone is a dense type of bone, in which the structure is solid,
compact, with high mechanical properties and low porosities and is able to sustain high loads thanks to the optimal spatial disposition of the cylindrical units
called osteons. Each osteon contains concentric lamellae (layers) of hard, calcified collagen matrix with osteocytes (bone cells) lodged in lacunae (spaces)
between the lamellae. Smaller canals, canaliculi, radiate outward from a central
canal, which contains blood vessels and nerve fibers. Osteocytes within an osteon
are connected to each other and to the central canal by fine cellular extensions.
Through these cellular extensions, nutrients and waste are exchanged between
18
the osteocytes and the blood vessels. Perforating canals provide channels that
allow the blood vessels that run through the central canals to connect to the
blood vessels in the periosteum that surrounds the bone. The external part of
almost every bone is composed by this kind of bone;
• trabecular or cancellous bone composes the metaphysis, the epiphysis and the
medullary part of bones, and is characterized by high porosity between 75 − 95%
and consists of thin, irregularly shaped plates called trabeculae. The pores are
interconnected and filled with bone marrow.
Approximately 80% of the skeletal mass in the adult human skeleton is cortical bone;
instead the remaining 20% of the bone mass is cancellous bone (Figure 2.4). But the
distribution of cortical and cancellous bone varies greatly between individuals.
Bone is a plastic and alive structure that responds to mechanical stimuli. Every
bone has the shape that is the best solution for the loads is exposed to during its
life. The first law that expresses the correlation between loads and human skeleton
is Wolff’s law: if loading on a particular bone increases, the bone will remodel itself
over time to become stronger to resist that sort of loading. The internal architecture
of the trabeculae undergoes adaptive changes, followed by secondary changes to the
external cortical portion of the bone. The inverse is true as well: if the loading on bone
decreases, the bone will become weaker due to turnover. For example trabecular system
of femur is particularly resistant to compressive loading thanks to the disposition of
his trabeculae along line-force of the most common stresses, and also the thickness of
the structure varies under different loading condition (Figure 2.5).
2.3.2 The pelvis
The pelvis is the lower part of the trunk, between the abdomen and the lower limbs
(legs). The pelvis includes several structures: sacrum and coccyx that represent the
caudal portion of the skeleton and the hip bones each composed by three bones, ileum,
ischium and pubis (Figure 2.6). Its primary functions are to bear the weight of the
upper body when sitting and standing, transfer that weight from the axial skeleton to
the lower appendicular skeleton when standing and walking and provide attachments
for the muscles of the lower limbs and of the trunk.
19
Figure 2.4: Structure of the cancellous and cortical bone: a representation of trabeculae
and of osteons.
An ascending force and some descending forces act on pelvis. The first one is the
ground reaction force while the second are gravity and body weight. The gravity centre of the body is located in front of the sacrum. The weight of the upper part of the
body is divided in two equal forces: each of them passes trough sacrum, sacrum-iliac
articulation, ilium, acetabulum, pubis and ischium. The descending force discharges
on the acetabulum, in the upper portion of the femur, while the ground reaction force,
caused by the support on feet on the ground, consists in force ascending lines transmitted from femur to acetabulum, as represented in Figure 2.7. The physiological role
of sacroiliac joint and pubic symphysis is to dump the ascending and descending loads
with ligament system and micro-movements.
20
Figure 2.5: Maximum strength with minimum weight.
Figure 2.6: Structure of the pelvis.
2.3.3 The femur
The femur is the most proximal and also the longest (40cm − 50cm ) and the biggest
bone in human skeleton. The femur is composed by a central part, the diaphysis,
and by two irregular extremities, the epiphyses that belong respectively to hip and
knee joint. The proximal extremity, closed to the trunk, contains the head, neck, little
trochanter, big trochanter and the adjacent structures (Figure 2.8).
21
Figure 2.7: A representation of forces that act on the pelvis from femur and from pelvis
to femur.
The head of femur, which articulates with the acetabulum of the pelvic bone, composes two-thirds of a sphere with approximately 20mm of radius. It has a smooth
surface except for a small groove, or fovea, connected through the round ligament to
the sides of the acetabular notch. The head of the femur is connected to the shaft
through the neck, or collum. The neck is 4 – 5cm long and the diameter is smallest
front-to-back and compressed at its middle. The neck forms an angle with the shaft
of about 130 degrees. This angle is highly variant: an abnormal increase in the angle is known as coxa valga (angle > 130°) and an abnormal reduction is called coxa
vara (angle < 130°). The body of the femur is almost cylindrical in form, is a little
broader above than in the centre. It is slightly arched, so as to be convex in front, and
concave behind, where it is strengthened by a prominent longitudinal ridge, the linea
aspera, which diverges proximal and distal as the medial and lateral ridge. The distal
extremity of the femur (or lower extremity) is larger than the proximal extremity. It
is somewhat cuboid in form, but its transverse diameter is greater than its anteroposterior (front to back). It consists of two oblong eminences known as the condyles.
Anteriorly, the condyles are slightly prominent and are separated by a smooth shallow
articular depression called the patellar surface. Posteriorly, they project considerably
22
Figure 2.8: Anterior and posterior view of the femur.
and a deep notch, the intercondylar fossa of the femur, is present between them. Its
front part is named the patellar surface and articulates with the patella. The distal
part of the femur articulate with the upper parts of two other long bones to realized
the knee joint.
2.3.4 The hip joint and angle
The joints of the lower limbs are the following:
• sacroiliac joint is the joint in the bony pelvis between the sacrum and the ilium
of the pelvis, which are joined by strong ligaments;
23
• pubic symphysis joint is the midline cartilaginous joint (secondary cartilaginous)
uniting the superior rami of the left and right pubic bones;
• coxofemoral joint is formed by the articulation of the rounded head of the femur
and the cup-like acetabulum of the pelvis. It forms the primary connection
between the bones of the lower limb and the axial skeleton of the trunk and pelvis.
Both joint surfaces are covered with a strong but lubricated layer of articular
hyaline cartilage. The acetabulum grasps almost half the femoral ball, a grip
augmented by a ring-shaped fibrocartilaginous lip, the acetabular labrum, which
extends the joint beyond the equator. The acetabulum is oriented inferiorly,
laterally and anteriorly, while the femoral neck is directed superiorly, medially,
and anteriorly (Figure 2.9).
The angles of the hip are the following:
• the transverse angle of the acetabular inlet can be determined by measuring the
angle between a line passing from the superior to the inferior acetabular rim and
the horizontal plane; an angle which normally measures 51° at birth and 40° in
adults, and which affects the acetabular lateral coverage of the femoral head and
several other parameters;
• the sagittal angle of the acetabular inlet is an angle between a line passing from
the anterior to the posterior acetabular rim and the sagittal plane. It measures
7° at birth and increases to 17° in adults;
• Wiberg’s centre-edge angle (CE angle) is an angle between a vertical line and a
line from the centre of the femoral head to the most lateral part of the acetabulum,
as seen on an anteroposterior radiograph;
• the vertical-centre-anterior margin angle (VCA) is an angle formed from a vertical
line (V) and a line from the centre of the femoral head (C) and the anterior (A)
edge of the dense shadow of the subchondral bone slightly posterior to the anterior
edge of the acetabulum;
• the articular cartilage angle (AC angle, also called Hilgenreiner angle) is an angle
formed parallel to the weight bearing dome, that is the acetabular sourcil, and
the horizontal plane, or a line connecting the corner of the triangular cartilage
and the lateral acetabular rim.
24
Figure 2.9: The coxofemoral joint.
• the femoral neck angle is an angle between the longitudinal axes of the femoral
neck and shaft, called the caput-collum-diaphyseal angle or CCD angle, normally
measures approximately 150° in new born and 126° in adults (coxa norma). An
abnormally small angle is known as coxa vara and an abnormally large angle as
coxa valga. Changes in CCD angle are the result of changes in the stress patterns
applied to the hip joint. The Figure 2.10 represents the different shapes of the
CCD angle.
2.4. BONE TISSUE
25
Figure 2.10: CCD-angle in three configuration: coxa norma, coxa vara and coxa valga.
2.4 Bone tissue
2.4.1
Composition of bone
Bone is a connective tissue that is composed by cells and extracellular matrix; the
extracellular matrix is the part of the tissue that provides structural support to the
cells. It is made of an inorganic and an organic part. The inorganic part consists in
crystals of hydroxipatite (CA10 (P O4 )6 OH2 ) that is a form of calcium phosphate. The
organic part is secreted by bone cells and it is composed of collagenous and other kind of
proteins; it is called osteoid. The osteoid can be deposited without a particular direction
or in an organized way, producing lamellar bone. Once the osteoid is apposed, it starts
to mineralize; inorganic materials precipitate on it, stiffening the tissue considerably.
The major cellular elements of bone include osteoclasts, osteoblasts, osteocytes, bonelining cells, the precursors of these specialized cells, cells of the marrow compartment,
and an immune regulatory system that supplies the precursor cells and regulates bone
growth and maintenance.
In the follow paragraph we will focus on osteoclasts, osteoblasts and osteocytes,
which are the cells that are involved in bone resorption , generation and regulation.
26
2.4.2 Bone Cells
Osteoclasts, the bone-resorbing cells, are multinucleated giant cells that contain from
1 to more than 50 nuclei and range in diameter from 20µm to over 100µm. Their role
is to resorb bone. Actively resorbing osteoclasts are usually found in cavities on bone
surfaces, called resorption cavities or Howship’s lacunae, as shows in Figure 2.11. Osteoclasts adhere to the bone surface by filaments positioned on the border which acts
as a permeable seal to maintain the microenvironment needed for bone resorption to
occur. The ruffled border secretes products leading to bone destruction. The osteoclasts solubilize both the mineral and organic component of the matrix. Osteoclasts
can be either active or inactive and, when active, they are polarized and exhibit ruffled
borders. The signals for the selection of sites to be resorbed are unknown. Bone lining
cells may contract and dissolve the protective osteoid layer to expose the mineral. The
mechanism of attachment of osteoclasts to the bone surface is not known. It is thought
that cell membrane receptors (α2 β 1 and αv β 3 ) which are called integrins expressed
by osteoclasts and which interact with extracellular matrix protein, are involved; the
α2 β 1 interacts with collagen and αv β 3 associates with vitronectin, osteopontin, and
bone sialoprotein through an arginine–glycine–aspartic acid (RGD) sequence.
Osteoblasts are bone-forming cells that synthesize and secrete demineralized bone
matrix (the osteoid), participate in the calcification and resorption of bone, and regulate
the flux of calcium and phosphate in and out of the bone (Figure 2.12). Osteoblasts
occur as a layer of contiguous cells which in their active state are cuboidal (15µm to
30µm thick). The osteoblasts function is to synthesize and to secrete the demineralized
bone matrix or ground substance of bone. The bone matrix consists of 90% collagen
and about 10% non-collagenous protein.
Bone formation occurs in two stages: matrix formation and mineralization. Matrix
formation, which precedes mineralization by about 15 days, occurs at the interface
between osteoblasts and osteoid. Preosteoblasts have been present for 9 days before
matrix synthesis occurs. Extracellular mineralization occurs at the junction of osteoid
and newly formed bone; this region is known as the mineralization front. Because mineralization occurs some time after matrix production, a layer of demineralized matrix
called the osteoid seam remains. Bone consists predominately of type I collagen with
traces of type III, V, and X collagen. These trace amounts may be present during
certain stages of bone formation and may regulate collagen fibril diameter. Collagen
2.4. BONE TISSUE
27
Figure 2.11: Light micrograph of osteoclasts (arrows). Typical multinucleated osteoclast nestled in its Howship’s lacuna. Bone (B), calcified cartilage (CC). Decalcified,
methylene blue, and azure II stained section; original magnitude X 800.
fibers constitute the shape-forming structural framework of bone in which the hydroxyapatite is inserted. The hydroxyapatite confers rigidity on the collagen framework.
The non-collagenous proteins (NCPs) include glycoaminoglycan-containing proteins,
glycoproteins, gamma carboxyglutamic acid (GLA) -containing proteins and other proteins. The role of the non-collagenous proteins is not yet well understood. These NCPs
are thought to play an important role in the calcification process and the fixation of
the hydroxyapatite crystals to the collagen. Control of bone mineral crystal growth
and proliferation is governed by the spatial limitation of the collagen fibrils, as well as
by the absorption of matrix proteins. The growth of bone mineral crystals is governed
in part by the constraints of the collagen matrix on which the mineral is deposited
and species introduced through the diet, given therapeutically. Many impurities thus
introduced tend to make crystals smaller, more imperfect, and more soluble. Bisphosphonate, a type of anti-remodeling agent used in the treatment of osteoporosis, binds
to the surface of apatite crystals and thereby is believed to be one of its mechanisms
28
Figure 2.12: Light micrograph of osteoblasts. Spicule of calcified core line with osteoblasts (Ob) and thin osteoid (arrows). Osteoprogenitor cells (Opc) are located
between osteoblasts and blood vessel. Original magnification X600.
of action in blocking dissolution. Although bisphosphonate-treated crystals are not altered in size, they tend to increase the bone mineral content (BMC). Cellular activity
can influence mineral properties such as in hypophosphatemic rickets (retarded deposition), osteopetrosis (small crystal persists), osteoporosis (larger crystal persists), and
fluorosis (larger crystals formed). The size and distribution of mineral crystals may
influence bone mechanical properties.
Osteocytes are the most abundant cell type in mature bone with about ten times
more osteocytes than osteoblasts in normal human bone. During bone formation some
osteoblasts are left behind in the newly formed osteoid as osteocytes when the bone
formation moves on. The embedded osteoblasts in lacunae differentiate into osteocytes
by losing much of their organelles but acquiring long, slender processes encased in the
lacunar–canulicular network that allow contact with earlier incorporated osteocytes
and with osteoblasts and bone lining and periosteal cells lining the bone surface and
the vasculature. The osteocytes are the cells best placed to sense the magnitude and
distribution of strains. They are strategically placed both to respond to changes in
2.4. BONE TISSUE
29
mechanical strain and to disseminate fluid flow to transduce information to surface
cells of the osteoblastic lineage via their network of canalicular processes and communicating gap junctions. Gap junctions are transmembrane channels, which connect
the cytoplasm of two adjacent cells that permits molecules with molecular weights of
less than 1 kDa such as small ions and intracellular signaling molecules (i.e., calcium,
cAMP, inosital triphosphate) to pass through. The functions of osteocytes are stabilize bone mineral by maintaining an appropriate local ionic milieu, detect microdamage
and respond to the amount and distribution of strain within bone tissue that influence
adaptive modelling and remodelling behaviour through cell–cell interaction. Thus, osteocytes play a key role in homeostatic, morphogenetic, and restructuring processes of
bone mass that constitute regulation of mineral and architecture.
2.4.3
Bone mechanobiology
Bone is a dynamic tissue that continuously replaces its cells and extracellular matrix or changes its architecture and density. The term bone remodeling indicates the
metabolic process that constantly destroys and regenerates the bone extracellular matrix in bone individuals. This process can produce no variation of the total skeletal
mass (homeostasis) or variation of the total skeletal mass. The term bone adaptation
indicates the process that produces variations of the total skeletal mass due to changes
in the environmental conditions, thus implying that the changes in bone mass are an
adaptation to such environmental variations.
The bone remodelling cycle starts at a certain location of the extracellular matrix
surface during the activation phase. A portion of the matrix is destroyed during the
resorption phase. The resorption cycle is terminated by a reversal phase, followed by
a formation phase.
The activation phase is referred to the conversion of quiescent bone surface to
resorption activity. Since few years ago the factor that initiates this process was unknown, but activation was believed to occur partly in response to local structural or
biomechanical requirements. All we knew was that bone matrix was covered by a layer
of resting osteoblasts (lining cells) attached to a thin layer (0.1µm − 0.5µm ) of demineralized, collagen-poor connective tissue called the endosteal membrane. At activation,
the lining cells digested this membrane and detached from the matrix surface, opening
the way to osteoclasts, and then to active osteoblasts.
30
Figure 2.13: Diagrammatic representation of working hypothesis of bone resorption.
A typical resting bone surface is lined by a thin demineralized layer (OO), a lamina
lamitans (LL), and a flat bone-lining cells (BLC).
In a recent work Ellen M. Hauge and colleagues at the Department of Pathology
of Aarhus University defined the concept of Bone Remodeling Compartment (BRC)
as a cells unit that wraps sites at the matrix surface where bone active remodeling
is taking place. Today we can confirm that bone remodeling activation starts with
the lining cells covering the portion of matrix surface to be remodeled lifting from
their attachment, to form a closed sac around the site, which is then connected with
capillaries (Andersen et al., 2009). The BRC is formed of tightly packed cells, which
are positive to osteoblast markers, and remain connected at the periphery of the BRC
to the layer of lining cells, actually sealing the BRC from the marrow space.
During the resorption phase, the BRC creates a conduit between the capillary system and the bone matrix surface, which is not fully exposed. Osteoclast and osteoblast
precursor reach the BRC space and is only through an intense signaling between these
cells, the resting osteoblasts forming the BRC canopy, and other cell types present
in the BRC, such as macrophages, that the resorption process starts. Where osteoclasts come in contact with the surface of bone, they begin to erode the bone, forming
cavities referred to as Howship’s lacunae, in cancellous bone, and as cutting cones or
resorption cavities, in cortical bone (Figure 2.13). The cutting cone elongates with a
typical speed of resorption of 20 − 40 micron/day, and expands radially with a speed
of 5 − 10 micron/day. The resorption cycle typically lasts 1 − 3 weeks, leaving cutting
cones 100µm deep in cortical bone, and 60µm deep in cancellous bone.
2.4. BONE TISSUE
31
After the resorption there is the reversal phase (coupling) that is the period of 1 − 2
weeks between the end of resorption and the beginning of the formation during which
the cutting cone shows no osteoclasts, but various mononuclear cells of unclear origin.
The cellular and hormonal mechanisms involved in coupling are unclear. One suggestion is the growth factor concept of coupling in which osteoblast-stimulating factors
(IGF I and II, TGF-, and FGFs) are released from bone matrix and stimulate osteoblast
activity for new bone formation. Another is that bone surfaces other than resorbing
surfaces are populated by osteoblast-lineage cells because of cell-surface molecules.
Finally, after that the osteoclasts retracted from the bone surface and the macrophages removed the by-products of their resorptive activity, the resorption space is
invaded by active osteoblasts, which start to secrete collagen and other protein to form
new osteoid. The osteoid seam will reach a level of approximately 70% of its final
mineralization after about 5 to 10 days. Complete mineralization takes about 3 to 6
months in both cortical and trabecular bone. This is called formation phase.
2.4.4
Effects of underloading and overloading
The skeleton undergoes a continuous cycle of resorption and regeneration, and adapts
its shape and density to the bio-mechanical environment. This is called bone adaptation. A very large number of experiments have been conducted on both animal and
human models to understand if and how the reduction of mechanical loading reduces
the skeletal mass. If the load acting on the skeleton is drastically reduced, bone mass
decreases over time with a negative asymptotical trend toward a lower limit.
Two kind of overloading experiments have been made from different researcher: the
first one consisted in doing different activities increasing the load, and these works
have produced results that showed a correlation between increase physical activity
and increase in bone mass but we don’t know this relationship. The second type of
experiment consisted in applying a direct force to a segment of the skeleton and in
a quantification of how its bone mass adapts to changes in the loading regime. This
second kind of experiment gave us a true quantitative relationship between the load
and the bone adaptation that it produces, but there are a lot of contradictory results
so it is not possible to rule the bone adaptation in overloading conditions.
32
Figure 2.14: Hip joint lateral view.
2.5 Hip Biomechanics
The hip joint is one of the biggest and most stable joint of the body. It has an intrinsic
stability because of hinge structure and huge mobility that permits many activities like
walking, the execution of the daily activities and more complex actions.
Hip joint (Figure 2.14) is composed by the femoral head and by the acetabulum.
It has a capsule wrapped by some big and strong muscles; its state of integrity depends by the articular surfaces alignment and by the control of the relative position
of these surfaces. A lot of ligaments take part in hip stability: iliofemoral ligament
and pubicfemoral ligament reinforce anteriorly the capsule; the ischialfemural ligament
has the same function posteriorly. Inside the capsule, the round ligament binds the
femoral head with the acetabulum. Many synovial sacs are dislocated inside the tissues
permitting a good lubrication.
2.5. HIP BIOMECHANICS
2.5.1
33
Hip muscles
Flexion: the muscles that permit hip’s flexion pass anteriorly, and six of them are
very important: iliac and major psoas (iliopsoas), pettineo, rectus femoris, sartorius
and tensor fasciae latae. The iliopsoas is the muscle that has the most important role
in flexion; it is a monoarticular muscle that has proximal insertions in the pelvis and
on the vertebral column. The rectus femoris is a biarticular muscle with the distal
insertion on the tibia that permits simultaneously the leg flexion and knee extension.
The sartorius is inserted in iliac spine and on the top of the tibia and is the longest
muscle of the body.
Extension: the extension muscles are gluteus maximus and hamstrings (femoral
biceps, semimembranosus, long and short semitendinosus). The gluteus maximus is
active when the hip is flexed (stair climbing, cycling) or when it is necessary to contrast
a flexion generated by the hip. The hamstrings are biarticular muscles that permit leg
extension and knee flexion. They are active in maintaining erect posture, walking and
running.
Abduction: the gluteus intermedius, which is assisted by the small gluteus, is the
biggest hip abductor. These muscles stabilize the pelvis during monopodalic standing
and walking and running.
Adduction: the adductor muscles pass through the medial side of the hip and include lungus adductor, adductor magnus, adductor minimus, adductor brevis, pectineus
and gracilis. Of these muscles, only the gracilis is biarticular, and all of them permit
the flexion and internal rotation of hip, especially when the femur is external rotated.
External and internal rotation: although a lot of muscles permit the rotation
of the femur, four of them have only this function: piriformis, externus and internus
obturators and quadratus femoris. During daily activities their function is to adapt the
femur position to pelvis typical rotation. The small and medium gluteus (with minimus
and magnus adductors) permit the internal rotation of the femur. The internal rotation
of the femur is not a movement that needs to win high resistance, so the muscles that
permit this action can produce only the third part of the force that can be produced
by the external rotation muscles (Figure 2.15).
34
Figure 2.15: Representation of the main hip muscles.
2.5.2 Hip kinematic
Although the movements of the femur depend at first by the hip rotations, the pelvis
has the important function of placing the joint respect to the external environment, in
order to optimize the efficacy of the movements of the inferior limbs. This is possible
because the pelvis rotations allow the acetabulum to be dislocated to guarantee the
best kinematic and dynamic efficacy of the distal segments.
The posterior inclination of the hip supports the hip flexion, the anterior inclination
supports the hip extension; in the same way on the frontal plane, the elevation of the
pelvis supports the abduction of the same side, its dip supports the adduction. The hip
mobility is major in sagittal plane where the flexion can be of 140° and hyperextension
of 15°. In frontal plane abduction can be of 30° while adduction is less than 25°.
Because of the presence of biarticular muscles, the flexion-extension of the hip has
to be related with knee flexion. The femur rotations, when the hip is fixed, change
between 90° (external rotation) and 70° (internal rotation). When the hip is extended,
some rotations are not permitted because of soft tissue. Hip mobility is very important
for many daily activities: in sagittal plane, the hugest angle variations are requested
for activities like binding shoes with both feet on the ground or raising an object from
the floor (122°-124°). Raising an object from the floor requests also a good mobility in
the frontal plane (28°) while binding the shoes with crossed legs requires the highest
2.6. DEFINITION AND BASIC INFORMATION
35
Figure 2.16: Principal movements’ angles
level of rotation (33°).
In general we can say that for doing daily activities 120° of flexion and 20° of
abduction and internal rotation are required.
The principal movements’ angles of the hip are represented in Figure 2.16.
2.6 Definition and basic information
The motion analysis is a useful instrument that permits to obtain quantitative information about the motor action. The study and the analysis of the forces that act on
the whole body or on an organ like the femur, permits to understand and predict its
behavior during daily activities like walking and in pathologic case, like sideway fall.
Motion analysis is based on the non-invasive measurement of some physical time
dependent quantities like spatial coordinate of landmark points of the body, reaction
forces, electromyography signals and other complementary signals (i.e. on-off contact
signals between some parts of the body and the external environment). In order to
obtain the variables of interest it is necessary to elaborate these measured quantities
and to do some simplifying hypotheses about musculoskeletal system. It is necessary
to adopt a model of the system we are examining. In general the modeling consists in:
• defining the anatomical segments and their geometrical, structural and inertial
properties;
• defining the binding joint between the segments and their kinematic properties;
• defining the kind of interaction between anatomic segments.
36
In general in the kinematic and dynamic description of the human body, it is used
to attribute at the anatomic segments the property of ‘rigid body’, and consequently
the definition of anatomical segments consists in identifying the parts of the body that
have this behaviour. A rigid body is identified by a long bone and is delimitated by
one or two joints. For a kinematic analysis of a body, it is necessary to define a fixed
coordinate system (OX, Y, Z) corresponding to the laboratory, and a local coordinate
system (O′ x, y, z) corresponding to the rigid body. The local coordinate system is
useful to describe the position of the rigid body in the global coordinate system; the
rigid body has six degrees of freedom in space and to individuate it respect to the
global reference system it is necessary to have information about its position respect
to the origin and about its orientation respect to the axes. The position of the body is
instantly individuated by the three components of the vector (O′ −O); its orientation is
individuated by the direction cosines of the x, y, z axes respect to the global coordinate
system. The total of the linear and angular coordinates of all rigid bodies with which
the human body has been modeled and the velocities and accelerations, represents the
kinematic description of the motion. It is known that the coordinates are correlate to
the internal and external forces and moments applied to the structure; in general the
relationships between kinematic variables and forces are defined by the equations of
dynamic equilibrium (second principle of the dynamic):
dΓ ∑
=
M
dt
dQ ∑
=
F
dt
∑
dΓ
is the derivative of the moment of the whole system respect to a point,
M
dt
dQ
is the sum of all internal and external moments applied to the system,
is the
dt
∑
derivative of the momentum of the system and
F is the sum of all internal and
external forces applied to the system.
There are two different approaches to the motion analysis; if the forces , the moments and the inertial properties of the rigid bodies (masses and inertial moments)
that compose the system are known, it is possible to calculate the kinematic variables
and determine the motion of the system: this is known as direct dynamic problem.
If the kinematic variables and the inertial properties of the anatomical segments are
37
Figure 2.17: Representation of anatomical segments of human body and global and
local coordinate systems.
known it is possible to calculate the forces and moments connected to the motion: this
is known as inverse dynamic problem (Figure 2.17).
2.6.1
The importance of Gait analysis for this study
In this work we calculated the strains of particular geometries using the direct dynamic
problem; for choosing the applied forces and their directions we started from the study
of Bergmann et al. [BDH+ 01] who have used instrumented implants to define the
typical loading scenario. The aim of this study was to create a unique database of
hip contact forces and simultaneously measured gait data for future improvements of
hip implants. For this purpose measurements were taken in four patients during nine
heavy-loading and frequent activities of daily living. A new mathematical averaging
procedure was developed to calculate ‘typical’ results from the data of various trials
and patients. From the average data of the individual patients, data for a ‘typical
patients NPA’ were calculated. NPA is representative for the investigated group of
individuals. The combination of average activity numbers with the typical hip contact
forces and joint movements presented in this study could serve to test the strength of
bone and of the hip implants more realistically than before.
The hip contact forces were measured using instrumented hip implants with telem-
38
Figure 2.18: Coordinate system for measured hip contact forces. The hip contact force
vector –F and its components –Fx , −Fy , −Fz acts from the pelvis to the implant head
and is measured in the femur coordinate system x, y, z.
etric data transmission; the titanium implants had an alumina ceramic head and a
polyethylene cup. The contact force with the magnitude F and the components –Fx ,
−Fy , −Fz were measured in the femur coordinate system x, y, z (Figure 2.18). It was
transmitted by the acetabular cup to the implant head; the angles of inclination of F
in three planes were denoted as Ax, Ay, Az. The force F caused an implant moment
M around the intersection point NS of shaft and neck axes of the implant.
Nine different activities were investigated which were assumed to cause high hip
joint loads and occurred frequently in daily living and most exercises were performed
4−6 times (trials) for each patient. These activities were: slow walking, normal walking,
fast walking, up stairs, down stairs, standing up, sitting down, standing on 2−1−2 legs
and knee band. A system with six cameras and a sampling rate of 50Hz was used to
measure the positions of body markers and two plates measured the ground reaction
39
Figure 2.19: Joint centres, reference points and coordinate system for gait analysis.
Table 2.1: Peak loads of single and average patients, cycle times and body weight for
average patient.
forces; all data from gait analysis and the readings from the instrumented implants
were synchronized using a common marker signal (Figure 2.19). The coordinates of
external markers at legs and pelvis as well as the ground reaction forces were recorded
in a fixed coordinate system.
The contact forces of the typical patient NPA and their components are charted in
Figure 2.20 for the nine investigated activities. The peak value Fp of the individual
and average patients are listed in Table 2.1.
The vectors of the contact force F from the typical patient NPA as seen in frontal
and transversal plane of the femur are assembled in figure2.21. The force directions
in the frontal plane were very similar during all activities and their variation was
remarkably small. Small forces acted more from medial than large ones. The indicated
40
Figure 2.20: Contact force F of typical patient NPA during nine activities. Contact
force F and its components –Fx , −Fy , −Fz . F and –Fz are nearly identical.
Figure 2.21: Contact force vector F of typical patient NPA during nine activities. The
z-scales go up to 300% BW. Upper diagrams: Force vector F and direction Ay of F
in the frontal plane. Lower diagrams: Force vector F and direction Az of F in the
transverse plane.
angle Ay of the peak force Fp was in the extremely small range of 12°-16° for all
activities except standing on one leg when it is 7°. The angle Az in the transverse plane
varied more than Ay . During activities, which caused high forces, i.e. for standing, level
and staircase walking, Az increased with the magnitude of F. The indicated directions
Az of the peak force Fp were in the range of 28°-35° when standing, walking and going
41
downstairs. For walking upstairs Az =46° was larger.
2.6.2
Effect of sub-optimal neuromotor control
A total body model is a very complex system with a lot of factors to consider, like
neuromuscular control, musculoskeletal system, age and health condition of the patient.
Furthermore the risk of fracture that a given subject faces while performing a given
motor task depends not only on the specific bone strength, but also on the internal
forces that a physical activity induces on our skeleton through the joints, the ligaments
and the muscle insertions.
The problem is affected by a dramatic indeterminacy: in effect, even if we model the
skeleton as a mechanism made of idealized joints, represent each major muscle bundle
with a single actuator, and impose all physiological limits to the force expressed by
each actuator, the resulting mathematical problem has more unknown than equations
[MTC+ 11]. The best solution, when the kinematic of each segment has been measured
experimentally, is to postulate that the neuromotor control activates the muscle fibres
ensuring the instantaneous equilibrium while minimizing the cost function (Collins,
1995; Menegaldo et al., 2006; Praagman et al., 2006). The assumption that in healthy
subjects the neuromotor control works in fairly optimal conditions seems reasonable.
Indeed, when applied to volunteers, this approach predicts muscle activation patterns
in good agreement with electromyography (EMG) recordings (Anderson and Pandy,
2001). Also the predicted intensity of the hip load is comparable to that recorded
with telemetric instrumented prostheses (Heller et al., 2001). This approach presumes
that the neuromotor control chooses, among the infinite available solutions, the muscle
activation pattern that optimizes a certain cost function, always the same one. But
this assumption seems unrealistic for the following reasons:
• there is a large variability of the internal forces in a single subject through several
repetitions of the same motion task [BDH+ 01];
• while we move, our goal is dependent on a number of factors: specific activation
patterns were found in case of patella-femoral pain (Besier et al.,2009), in unstable
conditions (Bergmann et al., 2004), in sudden motion tasks (Yeadon et al., 2010)
and different muscles controls were found during the execution of precise and
power activities (Anson et al., 2002);
42
Figure 2.22: Comparison of the predicted pattern of the hip load (solid black line) with
the variability of the hip load magnitude (grey band) measured on 4 subjects through
an hip prosthesis instrumented with a telemetric force sensor.
• the way we move is also affected by emotions. Depression has been found a
co-factor for the risk of falling in elders (Skelton and Todd, 2007), whereas somatization, anxiety and depression were found intrinsic co-factors in non-specific
musculoskeletal spinal disorders (Andersson, 1999);
• even if the optimal control assumption is acceptable for normal subjects, it has
been demonstrated that it is not true for model specific patients that are known
to have neuromotor deficiencies (Liikavainio et al., 2009).
Martelli et al. [MTC+ 11] demonstrated that when a subject-specific model is solved
imposing the optimal control condition, the hip load is predicted in a reasonably good
agreement with reported measurements [BDH+ 01] (Figure 2.22). In fact, the walking
dynamics of the body-matched volunteer of this study induced a peak load on the
recorded ground reaction of 1, 33 BW while this value was always approximately 1
BW on the reference study [BDH+ 01]. But when the sub-optimal neuromotor control
conditions are allowed, it was showed that the hip load intensity could drastically increase up to approximately 9 BW. In epidemiology studies on spontaneous osteoporotic
fractures, there is a fraction of the population for which the decrease of bone density
appears insufficient to explain the fracture event (Yang et al., 1996). But this observation can be easily explained if we accept that the degradation of the neuromotor
control not only increase the risk of falling, but also produces overloads during normal
physiological activities.
2.6.3
43
Hypothesis of our model
In our work we imposed a physiological load scenario to a cohort of osteoporotic patients. Since the aim of the work was to find a predictor of the risk of femoral neck
fracture, we adopted the worst case: the absence of muscle forces and neuromuscular
control.
• We applied net forces on the head of the femur without considering opposing
muscle forces or cost function.
• We used a linear stress-deformation relationship so that applying a force of 1 BW
permits us to estimate the risk of fracture.
• We used an organ model that consists in only the proximal part of the femur:
this estimates the patient’s instantaneous fracture risk and has a very lower computational and time cost than other more complex models.
Chapter 3
Material and methods
3.1 Patients’ cohort and CT scanning
The cohort of the patients we used in our work is composed of 92 women, between 54, 8
and 91 years. All the patients were subjected to Dual energy X-ray Absorptiometry
(DXA) to evaluate the T-score, Z-score and e bone mineral density (BMD). All the
femurs fell in the range from osteopenia to osteoporosis with the following values:
• the T-score medium value is −1.58 in the range between −5.0 ÷ 1.4;
• the Z-score medium value is 0.22 in the range between −2.8 ÷ 3.3;
• the BMD medium value is 0.75 in the range between 0.3 ÷ 1.1.
The patients were also CT-scanned (QCT) with the following protocol:
• Patient setup: position the patient in the correct way on the CT table; orient
the “head” end of the calibration phantom to correspond with the patient; verify
that the table is set to the correct height. The patient should be positioned feet
first and supine in the scanner. Check that the patient’s greater trochanter is
centered over the length of the phantom. In the end, elevate the patient’s legs
(Figure 3.1).
• Scanner setup: use always the same voltage, 120kV and amperage changes between a minimum of 150mA and a maximum of 170mA to ensure that the QCT
scan gives the patient the lowest possible dose.
45
46
CHAPTER 3. MATERIAL AND METHODS
Table 3.1: CT scan parameters and setting.
• Densitometric calibration: originally the CT scans were performed with an in-line
calibration pillow that embedded three liquid tubes at different concentration
of potassium salts. However, in the work done at the Rizzoli Institute, more
accurate calibration was obtained using the European Spine Phantom, a solid
HA densitometric phantom. This ESP phantom was scanned with the same
machine used to scan the patients, and the range of energies used in the clinical
imaging; given very small differences were produced, the values averaged over the
different energies were used to calibrate the CT image into HA equivalent ash
density. For all the patients, two reconstructions will be performed: one for the
entire volume at 512 × 512 × 1mm including phantom, the other one 15 × 15cm
field of view of just the right hip (or left hip if the right hip had a hip replacement
or fracture). The slice thickness is 1mm.
• Data archive: the QCT data will be double copied in uncompressed DICOM
format on CD-ROM and archived at the Clinical Trial Unite, Metabolic Bone
Centre, at the Northern General Hospital in Sheffield, UK.
All the setting parameters are summarized in Table 3.1.
The patients were subjected to the DXA analysis and to CT-scans for their hip
fracture at Northern General Hospital, Sheffield. They also completed a basic questionnaire to collect some information about their life style and bone health. For this
study, 48 postmenopausal women who have suffered a recent hip fracture due to lowenergy trauma and 44 postmenopausal age matched women as controls were recruited,
3.2. MATERIALS
47
Figure 3.1: Phantom and patient set-up during CT-scan.
but all the patients were subjected to these analysis for diagnostic reasons. The resolution of the CT-scan were 300µm that is the clinical resolution for the analysis. Subjects
must meet the inclusion and exclusion criteria, matching criteria, and have signed informed consent prior to any study procedures being undertaken. All the women have
more or less the same age, the same weight and the same height, and differ markedly
only for BMD. This cohort of women was chosen appropriately in this way for the
study to investigate the influence of BMD on fracture risk without the influence of the
other parameters. Patients underwent DXA scans of their hips on a Hologic Discovery
DXA machine using a new 3D imaging technique called 3DHipT M . This machine is
owned by the University of Sheffield. Patients underwent CT-scan at Northern General
Hospital, Sheffield.
3.2 Materials
In this section we explain all the materials we used. First of all we generated the
geometry of the femur from CT data images using ITK-SNAP1 , then we realized the
mesh of the model with the Morphing method, we converted the elements of the mesh
with ANSYS 14.02 , then we assigned the material properties to the bone using BoneMat software implemented in LHP_builder program. Finally we completed the file in
LHP_builder with all the specific information in order to obtain an appropriate output
that we upload on the VOP platform to be processed.
1 http://www.itksnap.org/pmwiki/pmwiki.php
2 http://www.ansys.com/
48
3.2.1 Segmentation and morphing
The image segmentation is the process of partitioning a digital image into multiple
segments (sets of pixel). The goal of segmentation is to simplify or change the representation of an image into something that is more meaningful and easier to analyze.
The first step of our work was the segmentation of a CT_data set of our cohort of
patients. The software we used to segment is ITK-SNAP, free software that provides
semi-automatic segmentation using active contour methods, as well as manual delineation and image navigation.
The phases of the segmentation process were:
• to import TAC images as DICOM images;
• to individuate the region of interest by an intensity region filter: this phase
permitted us to visualize only the cortical bone fixing the lower threshold in a
range value between 450 − 600 and the upper threshold major than 1200;
• automatic segmentation: to create the snake evolution. This phase consisted
in putting some “bubbles” in the region of interest. The word snake indicates
a closed curve that evolves from an approximate definition of the anatomical
structure defined by the bubbles to a new one where the region of interest is
occupied by the anatomical structure;
• manual segmentation: in this phase we corrected and finished the contour created
during the manual segmentation on the basis of pixel’s grey level. The white
regions meant that there was cortical bone, the grey ones meant trabecular bone
and the more pixels were dark the more the tissue was soft. In this step we had
to take care that the contours were as continuous as possible, not to include soft
tissue in our geometry but only the bone and to create a regular anatomy femur;
• when the manual segmentation finished, we saved the image as a VTK format
and then we exported the image as a surface mesh and saved it in STL format.
Sometimes the visualized image from ITK-SNAP was not clear, so it was useful to
import the STL in a software that permitted us to compare and overlap the created
geometry and the CT_data set. The software we used is LHP_Builder, a software tool
developed by the VPHOP project. When the geometry was definitely matched with
the TAC, we smoothed the surface and saved it.
3.2. MATERIALS
49
The next step was meshing the geometries; in order to do this, we used a Morphing
method developed in the VPHOP project [GHS+ 11]. Creating a mesh of each femur
without using morphing has some limits: the first one is automation, since it is often
user-intensive and time consuming. A second one is flexibility, since it does not permit
fast mesh adaptation and transposition between subjects and it cannot be easily used to
define an indexation of the population variability in terms of both anatomical parameters and material properties distribution to generate collections of synthetic models and
define response surfaces. Morphing is a technique that consists in deforming a template
geometry onto a target one; mesh morphing consists in adapting a template mesh onto
a subject specific geometry from CT images. Morphing of subject-specific models of
bone segments permit us to define indexation of bone shape or material properties on
a population. Moreover it permits to fast re-mesh when conducting sensitive studies,
easily compare results from more meshes and improve the speed and automation of
subject-specific FE model generation.
The morphing method that we used is based on an algorithm that morphs a volumetric template mesh onto a faceted 2D specific geometry, producing a volumetric
mesh of the specific geometry considered. The inputs are a femoral faceted geometry
and a set of 8 or more landmarks corresponding to some relevant points of the template
mesh. The template mesh was generated on a femur, which geometric characteristics
correspond to the average values using ICEM ANSYS3 software. A tetrahedral mesh was
automatically generated by the Octree meshing method. The resulting mesh has an
excellent element quality: average aspect ratio (AR) 1.55 and a maximum volumetric
skewness of 0.60.
The main steps of the morphing algorithm are summarized below.
• Pre-processing of the template mesh to be aligned to the specific geometry, by:
– translating the template mesh by aligning its centroid onto the STL centroid;
– rotating the template mesh around its shaft axis in order to put the landmarks of the head and greater trochanter on the corresponding landmarks
of the STL;
– scaling the bounding box of the template mesh to the bounding box of the
STL;
3 http://www.ansys.com/
50
– extracting the surface mesh of the template.
In case a left femur is morphed on a right femur or vice versa, a mirror transformation is preliminary applied, followed by a reorientation of the mesh elements.
• Surface morphing of the template surface mesh on the specific geometry, using
the defined landmarks as constrains and interpolating the motion of all surface
nodes on the motion of the landmarks. The motion of a node close to a relative
landmark is similar to the motion of that landmark, while the motion of nodes
far from landmarks is smoothly interpolated from the motion of all landmarks.
A method based on radial basis functions (RBF) was chosen in order to obtain
this behaviour:
– each landmark corresponds to the centre of a basis function k, solving a
linear equation system;
– the linear system constrains the influence of motion between reference points
and minimizes the deformation close to the constrained points.
If we call pi for i = 1, . . . , n the landmarks and xi for i = 1, . . . , n the nodes, the
following equation describes the motion of the nodes xi like the weighted sum of
that of all the landmarks:
xnew = f (xold ) +
∑
f (x) = xold +
n
∑
k(xold , pi )wi
i=1
The coefficients wi have been found imposing the condition:
f (pi ) = p′i
pi is the initial position of the landmark on the template mesh; p′i is the final
positions of the corresponding landmark on the target geometry.
This leads to the matrix system:

 
p′i − pi
k11

 
..

  ..

= .
.

 
′
pn − pn
kn1
  
k1n
w1
  
..   .. 
· 
. 
  . 
. . . knn
wn
...
..
.
3.2. MATERIALS
51
The matrix form is P = K W , where wi are the unknowns of the linear system.
Having as many basis functions as constraints, matrix K is a square matrix and
can be inverted to obtain W = K −1 P .
In this algorithm inverse multi-quadratic RBF were used, defined by equation:
2
k(x, p) = (x − p + c2 )β
where p is the centre of the basis function corresponding to a landmark, c is a
coefficient that controls the radius of the basis function (a low c value results in
a high local deformation and could generate distort elements; for a high c value
the deformation is distributed over a larger region) and β is a coefficient that
controls how strongly the nodes outside the radius are weighted (−1 ≥ β ≥ 0).
The values of c and β were chosen empirically: for each landmark, c was set
to the distance between its position in the template mesh and its correspondent
position in the target STL, β was set to −0.15.
• Projection of the resulting surface mesh on the STL geometry. Each node of the
morphed surface mesh is perpendicularly projected on the centroid of the closest
triangle in the STL to adjust its poor recovery.
• Laplacian smoothing. Generally meshes with high AR and intersecting triangles
are obtained; to adjust this, a smoothing based on the Laplacian operator is done.
This operator consists in replacing each node of the mesh with the centroid of its
neighbouring nodes. The Laplacian operator is applied twice and is followed by
a re-projection of the resulting surface mesh on the STL geometry to correct the
shrinkage usually induced by Laplacian smoothing. The whole process is iterated
three times.
• Morphing of the template volume mesh. The same method is applied in 3D using
the nodes of the morphed mesh as contour. In this case the functions used are
the Gaussian RBF, defined by equation:
( x − p2 )
k(x, p) = exp −
2σ 2
where σ is a coefficient controlling the radius of the function and its value is 0.1;
it depends by the mesh resolution and by the number of the nodes [GHS+ 11].
52
In order to obtain a good accuracy a control was introduced to ensure that the
average and the maximum distance between the morphed surface mesh and the
original STL is lower than 0.1 mm and 1 mm respectively. The morphing method
was created to study long femurs, while our work is about short femurs. Because
of this, we needed to create a new mesh template to implement this method in
our work.
We have generated a mesh template using ICEM ANSYS based on a short femur
chosen in a femurs’ database with geometric characteristics corresponding to
the average of our cohort. The mesh has solid elements type 200, with 53388
elements, maximum skewness value of 0.599 and maximum AR value of 4.7. The
skewness is a measure of the asymmetry of the probability distribution of a realvalued random variable; the aspect ratio is the ratio of the width of a shape to
its height when the width is larger than the height. They are both indices of the
mesh quality. When the mesh template was completed, we picked the elements
corresponding to the landmarks in order to visualize their ID number; these
numbers have been written in a xml file. Xml file is a file read by the Morpher
that combines the picked elements of the template mesh with the landmarks of
the STL geometry.
After completing this preliminary phase, we imported our STL femurs into the
LHP_Builder software and picked some landmarks on their surface; the landmarks are placed on the head, on the big trochanter, under the big trochanter,
on the little trochanter and on the four edges where the femurs have been cut by
the TAC: the posterior one, the anterior one, the medial one and the lateral one,
as shown in Figure 3.2 and in Figure 3.3. After placing all the landmarks, we
exported their coordinates and insert them in the xml file. Finally, we launched
the file in the Morpher by the command prompt. The process produces two cdb
files, one is a parametric surface file and the other one is a parametric volume
file.
When the iteration is completed we control the quality of the mesh using ANSYS:
the maximum skewness has to be lower than 0.995 and the maximum AR lower
than 30. If the mesh doesn’t respect these parameters, we change the position of
the landmark where these values are bad or add someone else till obtaining a good
mesh. When the mesh has good values, we import it into ANSYS Mechanical
3.2. MATERIALS
53
Figure 3.2: Points of morphing:head, greater trochanter, under below greater
trochanter, and lesser trochanter.
to convert the element type from 200 to solid 187, which are 10 nodes tetrahedral elements that have quadratic displacement behaviour and are well suited
to model irregular meshes. Finally, we import the mesh into LHP_Builder to
assign the material properties with Bonemat_V3 and to complete the msf file.
The morphing mapping method is graphically represented in Figure 3.4.
3.2.2
BoneMat software
It is known that the behaviour of bone structures depends on their shape, size and
mechanical properties of the material of which they are composed. Some of the major
problems related with 3D modelling are, in fact, the assessment and measurement of
the geometric and mechanical properties of bones.
In this paragraph we will discuss in particular about the assignment of the mechanical properties to the bone model obtained with the segmentation described before. In
our work femurs were scanned by computerized tomography (CT) with the protocol
54
Figure 3.3: Points of morphing:the four points on the basis of the femur: anterior,
posterior, medial and lateral.
reported in paragraph 3.1; CT images provide accurate information about geometry
and mechanical properties of bones. The radiographic density (RD) reported in CT
images can be related to mechanical properties of bone. CT data provide quantitative
information on the attenuation coefficient of the bone tissue that can be related to its
density. The attenuation is the gradual loss in intensity of any kind of flux trough a
medium. The relationship between CT numbers, Hounsfield Units (HU), and tissue
density is monotonic and linear, as first approximation, and the process of translating
the CT number into the density of biological tissue is called calibration of CT date
set. The density can be then related to the mechanical characteristic of the bone tissue using one of the many experimental relationships available in literature [ZMV99],
[TSH+ 07].
In order to choose the relationship between the mechanical and physical properties,
we have to account mechanical properties of both cortical and trabecular bone that are
correlated significantly to tissue mineralization. Linear or power relations are reported
by this general equation:
E = a + bρcapp
where E is the material Young modulus, ρapp is the apparent density and a, b and
3.2. MATERIALS
55
Figure 3.4: Steps of morphing algorithm: (a) original template mesh, (b) original STL
mesh, (c) result from morphing the template mesh on the STL using RBF method, (d)
results after projection (c) on (b), (e) result from the Laplacian smoothing, (f) final
result represented in a high quality mesh of the STL geometry.
c the model parameters.
A CT data set represents a volume sampled at points of a regular grid. All the grid
cells are axis-aligned cubes and radiographic density information is associated with each
point of the grid, the cube vertex. In our work we used a method to assign the material
to the model in which the local stiffness is evaluated by considering all the CT grid
points which fall inside the model element and preserving all the density information
provided by the data set. The software that we used to assign the mechanical properties
is software routine BoneMat, developed and tested for the VPHOP project. The aim
of the program is to relate 3D Finite Element model (FEM) mesh with the bone
radiographic density information available in the corresponding CT data set. The FE
model is generated from CT data set, then an elastic modulus is assigned to each
model element by the BoneMat routine. The modulus is derived from the apparent
bone density at the element location. The apparent density is evaluated considering
all CT grid points located inside each model element.
The BONEMAT program can be divided in four steps as followed:
• input of the model geometry and CT data set.
56
In this first step the program reads the model geometry (STL file) created with
the segmentation from a Neutral file format and the CT data set corresponding.
• Input of the data set calibration values and of the empirical model parameters.
In this step we use a data set calibration file as an input. This file contains the
linear relationship between the HU numbers and the bone ash density. To obtain
the parameters of the linear regression a calibration phantom with different densities was used, embedded in a water-equivalent resin-based plastic (The European
Spine Phantom). The input file containing the data set calibration and empirical
model parameters, and a linear calibration between two points specified by the
user is provided. Being (RD1, ρ1) and (RD2, ρ2) the apparent and radiological
density measured in two regions of the CT data set, the ρapp is calculated in each
point of the data set:
ρapp (x, y, z) = ρ1 +
ρ2 − ρ1
[RD(x, y, z) − RD1 ]
RD2 − RD1
• Evaluation of the Young modulus of each model element.
After have obtained ρapp , the Young modulus of each element is calculated using
one of the many relationships that have been published in literature that express
the Young modulus E as a function of bone tissue density.
• Output of the complete model.
In this step we obtain the FE mesh provided with the assigned material properties
[TSH+ 07].
The version of the software that we used is the third version Bonemat_V3 developed
during the project. This version of the program first transform the HU number into
Young modulus value then performs the numerical integration over the element’s volume. Model V 3 is characterized by 388 different materials and automatically mapped
the inhomogeneous material properties onto the FE models. In this version of the
algorithm the HU uniform values assigned to each finite element of the mesh was determinate with a numerical integration of the HU field as follows:
∫
HUn =
Vn
HU (x, y, z)dV
∫
=
dv
Vn
∫
Vn′
HU (x, y, z) det J(r, s, t)dV ′
Vn
3.2. MATERIALS
57
where Vn indicates the volume of the element n, (x, y, z) are the coordinates in
the CT reference system, (r, s, t) are the local coordinates in the element reference
system, and J represents the Jacobian of the transformation. In this way the HU value
calculated for each element is accurate [TSH+ 07].
Then the calibration equation used to calculate the uniform density assigned to the
element n is the following:
ρn = α + βHUn
where ρn is the uniform density assigned to the element n of the mesh, HUn is the
uniform CT number and α and β are the calibration coefficients.
As mentioned above there are various model that can be used to calculate Young
modulus starting from the apparent density calculated with the calibration (ρn ). Among
these various relationships the following equation was chosen as it was obtained with
a very robust experimental protocol that minimizes random errors [STM+ 07]:
E = 6.950ρ1.49
n
A linear regression between experimental and predicted strain was performed to
quantify the prediction accuracy and the root mean square (RMS) error and the peak
error were calculated. This equation was chosen among the various relationships because it showed the highest correlation between experimental and predicted strain.
The slope and intercept value of the regression line were found to be not significantly
different from unity and zero, respectively; also the RMSE had a very low values
(RM SE < 10%) and the regression line parameter R2 was equal to 0.911.
We assigned the material properties using the software LHP_builder in which BoneMat is implemented. Once the mesh is generated and imported on the geometry, in
order to map the material properties on the model, we picked the femur and selected
the operation BoneMat. Then we chose the CT data set and the calibration file as
inputs and the program executed the assignment of the mechanical properties to the
mesh. From this operation we expected to obtain maximum values of Young modulus between 18000MPa and 22000MPa for cortical bone [ZMV99]. At the end of the
operation we checked the quality of the mapping materials and we evaluated the distribution of Young modulus on the whole femur, on the diaphysis, on the neck and on
the trochanteric area.
58
Figure 3.5: Representation of the regression line between experimental and predicted
strain [STM+ 07].
The BoneMat operation essentially allows the user to assign the material properties
to each finite element of a bone mesh as described above. The input data are represented in Builder with data objects of type VmeVolume and VmeMesh (VME stands
for Virtual Medical Entity). The output that we obtain is an updated VmeMesh, in
which an elastic modulus value is assigned to each element.
3.2.3 Structure of msf file
In these files we added all the necessary information to run the simulations. The final
structure of the file is the following (Figure 3.7 ) (i.e. Hip076):
Under the main root named Hip076 we have already imported the CT scan (patient_CTscan), the STL geometry obtained in the segmentation (femur_STL) to which
the points of morphing’s cloud and the Bonemat file (Bonemat) are associated. The
msf file was created with the following steps:
• add anatomical points cloud under femur_STL: one point in the middle of the
neck (TF), one in the centre of the basis of the femur (IF), one in the centre of the
head (CH) and another one between the little trochanter and the big trochanter
(NL); copy CH and translate it along z-axis onto the superior surface of the head
(project of CH). In order to place the NL point we took the measures of the
distance between BA_LT and the one between BA_NL and we moved the NL
point until the measure BA_NL was half of BA_LT. We measured the radius
3.2. MATERIALS
59
Figure 3.6: BoneMat of the femur, with the range of Young’s modulus between
18000MPa and the maximum value obtained from the software (19832 MPa).
of the head expanding CH point until the surface of the head was completely
covered.
• Add constraint cloud under femur_STL: copy IF and call it z_min.
• Add ZLT cloud under femur_STL: copy of LT.
• Create REF_SYS group under the root; we created two reference systems: the
first is the ref_sys_TF_IF_CH that has the origin in TF, the x-axis passes
through IF and y-axis through CH. Then freeze VME not to lose the information during the final upload. Copy NL point under anatomical points and
call it project of NL (PNL) and set the z-coordinate to 0 respect to the frozen
ref_sys_TF_IF_CH. The second is the Charité reference system (CHA_ref_sys)
that has the origin in CH, the x-axis passes through project of CH and the yaxis passes through project of NL (PNL). Then freeze VME. The check of the
direction of the frozen reference systems is the following:
60
Figure 3.7: final structure of VME data tree (final structure of msf file).
– ref_sys_TF_IF_CH: x-axis points to the basis of the femur, z-axis points to
anterior side for right femur and to posterior side for left femur (Figure 3.8).
– CHA_ref_sys: x-axis points to anterior side, y-axis points to medial side
and z-axis point to top for the right femur; x-axis points to anterior side,
y-axis points to lateral side and z-axis points to top for the left femur (Figure 3.9).
• Add keypoints ansys cloud under CHA_ref_sys: referring to CHA_ref_sys point
3.2. MATERIALS
61
Figure 3.8: TF_IF_CH reference system.
0 has coordinates (0, 0, 0), point 1 has coordinates (50, 0, 0) and point 2 has
coordinates (0, 50, 0) (Figure 3.10).
• Create planes group under root; we created two planes: the first is the frontal
plane that passes through IF, TF, CH. We obtained this plane selecting the
CT_scan, setting IF as reference system and fixing z-coordinate to 0.5. Then we
rotated the plane to make it pass through the points TF and CH. The second one
is the transversal plane that we obtained making a copy of frontal plane; rotating
it of 90° around x-axis and translating along z-axis up to the beginning of the
neck where the its edges are parallel. Frontal and transversal planes are shown
in Figure 3.11.
• Realize the measurements of the neck length (neck_length) and of the varo/valgo
angle (ccd_angle). The measure neck_lenght is the distance between CH and
project of NL. The ccd_angle is identified by three points CH, IF and project of
NL that is the vertex of the angle. This angle is calculated between the head and
the shaft of the femur and identifies the deformity of the hip. This angle is used
to calculate the direction of the muscular strength.
At the end of the msf file we imported under the root some files, in the extension .lis,
in which we wrote all the requested information about the femur. These files are the
62
Figure 3.9: CHA reference system; blue line is z-axis, red line is x-axis and green line
is y-axis.
Figure 3.10: Disposition of the Ansys key-points on CHA reference system.
following:
• anteversion.lis: information not available;
• ccd_angle.lis: entire number of the angular amplitude;
• CH.lis: coordinates of the CH point relative to the root;
3.2. MATERIALS
63
Figure 3.11: Frontal plane and transversal plane.
• Head_Radius.lis: entire number of the head radius;
• IF.lis: coordinates of the IF point relative to the root;
• keypoints_ansys.lis: coordinates of the keypoints relative to the root;
• Neck_Length.lis: entire number of the measure of the neck;
• NL.lis: coordinates of the NL point relative to the root;
• PNL.lis: coordinates of the PNL point relative to the root;
• side_femur.lis: set 0 for right femur and 1 for left femur;
• Z_LT.lis: z-coordinate of copy of LT relative to CHA_ref_sys;
• Z_LT_NMS.lis: z-coordinate of copy of LT relative to the root;
• z_min.lis: z-coordinate of z_min relative to the root.
Finally we exported the Bonemat file as an Ansys Input file (.inp) and imported it
into ANSYS Mechanical to create a cdb file containing the assigned properties of the
material. When the file has been generated, we imported it under the root with the
name femur.cdb.
64
3.3 The mechanical load scenarios
Hip fracture due to osteoporosis is a severe health care problem with an high social and
economic impact. These kind of fractures can occur spontaneously without trauma or,
in an estimated 76 − 97% of cases, can be the result of impact from a fall. In both
situations, force can be applied to the femur in a variety of directions. Identifying the
load directions under which the proximal femur is most likely to fracture would improve
our understanding of the mechanisms of hip fracture and may aid the development of
methods for preventing hip fracture.
The clinician diagnose osteoporosis using DXA, which considers only the proximal
femur’s BMD to identify the presence or not of the pathology; an alternative new tool
is FRAX, that considers BMD and seven risk factors, but not the risk of fall. The
finite elements analysis is a method that permits to predict and to know the strength
and deformation of the bone applying loads in different directions.
In this work we estimated the fracture load using two different load scenarios and
used them to create a new predictor for the fracture risk. The first scenario we used
represents the physiological case and the second one is a total body model suggested
in VPHOP project called “femur-fall Charité database”.
3.3.1 Strength loading scenario
This configuration represents the loads applied to the hip during normal daily activities.
The aim of the work is to estimate strength and deformation values on the femoral
neck; in particular we considered the load of fracture to estimate the patient’s fracture
risk. In the pre-processing phase we started from the femur STL; we defined a local
coordinate system in LHP_Builder to apply the force along a single axis: the forces we
wanted to apply have different directions and decomposing each single force into the
three components x, y and z for each femur may have a high user cost. To solve this
problem we decided to define a local coordinate system and to apply a force along a
single axis; the main advantage of defining a local coordinate system is that it identifies
the position and orientation of a body integral to it relative to the global coordinate
system: since it is identified by three keypoints for which the coordinates values are
known, it is possible to detect a local system for each applying load direction operating
with the rotation matrix:
3.3. THE MECHANICAL LOAD SCENARIOS
65


[
cos αx
] 

Roo′ =  cos βx

cos γx
cos αy
cos βy
cos γy
cos αz


cos βz 

cos γz
This matrix contains all parameters (director cosines) that describe the orientation of the local coordinate system relative to the global one. New keypoints in each
direction are calculated with the operation:
 
 
x
Xk
  [ ]  k


 
 Yk  = Roo′  yk 
 
 
zk
Zk
The local coordinate system that we used has the origin in CH, x axis passing
through Project of CH and y axis passing through Project of NL; we defined three
keypoints 0, 1, 2, on this local system, which permit us to identify it in ANSYS
Mechanical APDL; the keypoints have coordinates (relative to local coordinate system):
• K 0 (0, 0, 0)
• K 1 (50, 0, 0)
• K 2 (0, 50, 0)
The local coordinate system and the keypoints are showed in Figure 3.12. The directions along which we decided to apply the load are (Figure 3.13 and Figure 3.14):
• nominal: along z axis of local coordinate system. This load simulates standing
up.
• 18° around x axis: this load simulates hip flexion.
• 12° around y axis: this load simulates hip adduction.
9° around x axis: this load simulates hip flexion.
• −3° around x axis: this load simulates hip extension.
13.5° around x axis: this load simulates hip flexion.
66
Figure 3.12: Femur’s local coordinate system and keypoints.
8° around x axis: this load simulates hip flexion.
We chose these directions referring to the study of (Bergmann et al., 2001) [BDH+ 01].
In the processing phase we imported the femur in ANSYS Mechanical APDL and subsequently:
67
Figure 3.13: Representetion of the nominal direction of load.
Figure 3.14: Representetion of all direction of load.
• fix key-points in order to generate the local coordinate system;
• constrain the femur with a joint on the diaphysis base; for constraining the femurs
we wrote the z value of IF for each case, and we imposed a null displacement in
all directions for all nodes with an inferior z value;
• rotate the femur in the new local reference system in order to apply the force
along one axis;
• apply a load of 1BW along each direction in the corresponding local coordinate
system. We created a boundary condition input macro to apply all the loading
68
forces consecutively. For each condition we changed the reference system and we
rotated the femur accordingly the local reference system; to catch the application node for the load we used a function that is implemented in ANSYS, the
GET function, which finds the highest node on the head surface along a specify
direction. After that we applied the force on the preselected node and with the
intensity of 1BW along z-axis. This procedure permitted us to run the entire
simulation quickly.
3.3.2 Femur-fall Charité database
The Charité loading spectra database includes femoral and spinal loading spectra for
various physiological activities, computed assuming optimal neuromuscular control.
Preliminary results suggest that none of these loads can produce femoral bone fracture, no matter how low is the bone mass. The database also contains deterministic
predictions of loads acting on the femur during side fall. Preliminary data suggest that
these loads produce bone fracture in most if not all cases. However, the whole body fall
model predicts the force that the ground transmits to the body under the worst-case
scenario of fully un-moderated side fall. In most cases, the actual load that is transmitted to the greater trochanter will be only a fraction of this ground force, due to
the attenuation of the soft tissues wrapping the hip, and due to the fact that in many
cases the impact is somehow moderated by factors reducing the acceleration (grabs,
counterbalancing movements, etc.) or by multiple impact sites (hands, back, knee,
etc.). Another factor contributing to the impact force reduction might be the landing
surface (from cement to grass or sand). Since these moderating factors are totally
randomized, we must assume that the side fall load returned by the Charité database
defines a loading spectrum where the moderation fraction is normally distributed.
The workflow is composed by:
• the set of MT required to generate the organ data;
• a query to the Charité database module which returns the side fall load for the
patient’s body height and weight;
• a module which generates the probabilistic loading spectrum by assuming a normally distributed moderation fraction;
69
• a Personalized fracture risk module, which loops over time up to 10 years and for
each year call the resorption modules, samples the scaled loading spectrum, run
the organ-level model, collect fracture/no-fracture predictions, and compute the
Personalized Fracture Risk;
• a phenomenological remodelling module that computes for the given resorption
rate, a modified organ data set at given time;
• an Ansys organ-level model that reads the organ data, a loading case, the resorption indices, and return the fracture/no fracture prediction.
Chapter 4
Results
In this work we obtained the results through two different methods for the different
loading scenarios, and then we combined them together in statistical analysis. For the
physiological loading scenario we preselected in ANSYS a specific region of interest
(ROI) of the template femur that included only the surface nodes of the femoral neck
and of the greater trochanter, and then we generated stress and strain tensor for that
ROI. We processed nodal principal stress and deformation data and then, for each
node, we averaged the tensors on a circular area with a diameter of 3mm. For doing
this, we implemented an algorithm that measures the distance between a referent node
m and all the other ones j through the relation:
√(
dist(j) =
)2 (
)2 (
)2
x(m) − x(j) + y(m) − y(j) + z(m) − z(j)
The algorithm takes the nodes with a distance inferior or equal to 3mm, and makes
an average between their deformations on the circle area.
We considered only the deformation along principal directions 1 and 3 since we
supposed a plane stress state: minimum values of ϵ1 and ϵ3 were used to define nodal
fracture risk. Deformation values were then multiplied by a strain-rate factor and the
fracture risk calculated as:
risk_e1 =
ϵ1 × f actor
0.0073
risk_e3 =
ϵ3 × f actor
0.0104
71
72
CHAPTER 4. RESULTS
Figure 4.1: Visualization of the distribution of principal deformation on the femur
(anterior view). The highest deformation is located on the top of the neck.
where 0.0073 and 0.0104 are tensile and compressive factors [BMN+ 04]. Finally the
fracture load was calculated as:
f racture_load =
BW
maximum_risk
This algorithm was implemented for each load direction, but in a last analysis we
considered only the minimum value of fracture load, because it is the critical load that
causes the patient’s fracture. We checked that the fracture point was on the femoral
neck and not in other areas to validate the accuracy of the model and of the loading
scenario (Figure 4.1 and Figure 4.2). The highest fracture load is 8240N and the lowest
is 3140N with a mean value of 4667.9N for the fracture group of patients, while for the
control group of patients the highest fracture load is 7830N and the lowest is 1510N
with a mean value of 3807.5N.
Regarding to the results from workflow 2, the probabilistic loading conditions, as
well as the post-processing criteria were defined by the VPHOP consortium, and exposed for use to us; however, at the time of this writing, the details of such boundary
conditions were not disclosed to us, except that they simulated the loading occurring
during side fall. We could submit our cases to this workflow, and then we could accede
with our personal account in PhysiomeSpace and download for each uploaded femur
the percentage of fracture risk in 10 years. The highest fracture risk is 100% while
73
Figure 4.2: Visualization of the maximum strain point in posterior view of the femur.
Table 4.1: Descriptive Statistic of control group.
the lowest is 11.91%; both these values belong to control group, since the lowest value
of fracture risk is 59.75% and the highest value is 99.16% in fracture group. For the
control group the mean fracture risk value is 70.6% while for the fracture group it is
87.9%.
We analyzed the results making a statistic study that includes descriptive statistics,
Mann-Whitney test (chosen because of a non-normal distribution), box plot, logistic
regression and ROC curve. We did the descriptive statistics distinguishing fracture
patients from control ones; the descriptive statistics are reported in tables Table 4.1
and Table 4.2.
We performed a Mann-Whitney test to see the variable’s significance; the null
hypothesis H0 of the equivalence of the groups is rejected when p < 0.05. The results
are reported in Table 4.3. When the null hypothesis is rejected it means that there
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CHAPTER 4. RESULTS
Table 4.2: Descriptive Statistic of fracture group.
is significant difference between the patients for that variable and the variable is a
discriminant for the considered population.
The groups are different only for Strength, fracture risk calculated with WF2,
FRAX, BMD.
The mean values of the descriptive statistics compared with p-value of MannWhitney test are reported in Table 4.4. There is no statistically significant difference
between the fracture and the control groups for age, height and weight (p < 0.05). A
Mann Whitney test shows that there is a statistically significant difference (p < 0.001)
between cases and controls in terms of aBMD. We did descriptive statistic also for
fracture load of physiological scenario, WF2 and for the FRAX percentage fracture
risk; the results are reported Table 4.5.
The box plot diagrams show the distribution of the values predicted by the four
predictors (aBMD, FRAX, Strength and WF2): there are some femurs that are significantly different from the mean value, especially using FRAX predictor. The Box plot
are reported in Figure 4.3, Figure 4.4, Figure 4.5 and Figure 4.6.
To see how much a predictor is able to distinguish fracture by non-fracture and
which is its prediction power, we performed a ROC curve analysis. ROC curve is a
fundamental tool in diagnostic test evaluation. In a ROC curve the true positive rate
(Sensitivity) is plotted as a function of the false positive rate (1 − Specif icity) for
different cut-off points of a parameter. Each point on the ROC curve represents a
sensitivity/specificity pair corresponding to a particular decided threshold. The area
under the ROC curve (AUC) is a measure of how well a parameter is able to distinguish
75
Table 4.3: Results of Mann-Whitney test. This test show which are the significant
values.
between two diagnostic groups (diseased/normal). If we consider our predictors singularly and separately, the predictor that has the best prediction power is WF2 with
a AUC of 0.75, but it is not significantly different from Strength and aBMD which
have respectively AU C = 0.72 and AU C = 0.73, respectively, while it is significantly
different from FRAX that has AU C = 0.65; this result appears contradictory because
the FRAX risk index includes the aBMD information. ROC curves are represented in
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CHAPTER 4. RESULTS
Table 4.4: Descriptive Statistic: the mean value and the standard deviation for Age,
Weight, Height, Femoral neck BMD and total BMD.
Table 4.5: Descriptive Statistic: the mean value and the standard deviation for risk of
fracture of WF2, strength and risk of fracture of FRAX.
Figure 4.7, Figure 4.8, Figure 4.9 and Figure 4.10.
Finally we performed a logistic regression analysis from whom we obtained the
ROC curves to test the prediction power of our combined predictors. We chose a
logistic regression and not a linear regression because the dependent variable has a
dichotomous and qualitative value, that is the membership or not to a group (fracture
or non-fracture). The variables in the equation of the logistic regression are reported
in Table 4.6, Table 4.7, Table 4.8, Table 4.9, Table 4.10 Table 4.11 follow by the
relative ROC curves (Figure 4.11, Figure 4.12, Figure 4.13, Figure 4.14, Figure 4.15
and Figure 4.16).
We have the best predictions when we combine all the variables (Strength, RF_WF2,
N_BMD, TH_BMD, FRAX_HIP_fracture_Ten_years) obtaining AUC of 0.84 and
77
Figure 4.3: Box plot of BMD.
Figure 4.4: Box plot of FRAX.
when we combined the mechanical predictors with TH_BMD and N_BMD obtaining
AUC of 0.83. We also have good result (AU C = 0.80) when we combine only RF_WF2
with Strength or RF_WF2 with Strength and FRAX and it is not significantly different. Combining only one of the mechanical predictor with FRAX we obtain a prediction
better than FRAX itself (AU C = 0.65) but equal to using the mechanical predictors by themselves; infact AU C_F RAX_Strength = 0.74, AU C_Strength = 0.72,
78
CHAPTER 4. RESULTS
Figure 4.5: Box plot of Strength.
Figure 4.6: Box plot of WorkFlow 2.
AU C_F RAX_W F 2 = 0.76 and AU C_W F 2 = 0.75.
79
Figure 4.7: ROC curve of BMD. AUC is of 0.73.
Figure 4.8: ROC curve of FRAX. AUC is of 0.64.
80
CHAPTER 4. RESULTS
Figure 4.9: ROC curve of strength. AUC is of 0.72.
Figure 4.10: ROC curve of WF2. AUC is of 0.75.
81
Table 4.6: In this table are reported all the variables in the equation obtained with the
logistic regression of strength, WF2, N_BMD, TH_BMD and FRAX.
logistic regression of strength and FRAX.
logistic regression of WF2 and FRAX.
logistic regression of strength, WF2 and FRAX.
82
CHAPTER 4. RESULTS
Table 4.10: In this table are reported all the variables in the equation obtained with
the logistic regression of strength and WF2.
Table 4.11: In this table are reported all the variables in the equation obtained with
the logistic regression of strength, WF2, TH_BMD and N_BMD.
Figure 4.11: ROC curve of strength, WF2, N_BMD, TH_BMD and FRAX. AUC is
of 0.84.
83
Figure 4.12: ROC curve of strength and FRAX. AUC is of 0.74.
Figure 4.13: ROC curve of WF2 and FRAX. AUC is of 0.76.
84
CHAPTER 4. RESULTS
Figure 4.14: ROC curve of strength, WF2, and FRAX. AUC is of 0.80.
Figure 4.15: ROC curve of strength and WF2. AUC is of 0.80.
85
Figure 4.16: ROC curve of strength, WF2, N_BMD and TH_BMD. AUC is of 0.83.
Chapter 5
Conclusion
We undertook this study to find a an individualized predictor of the risk of femoral
neck fracture in osteoporotic patients more accurate than the currently available predictors based on statistical regressions of epidemiological data (FRAX). We started
from a patient-specific model of the organ and we imposed a load calculated from the
weight of each patient. We decided to investigate two loading scenarios: the first one
simulates physiological activities like normal walking, standing up, sitting down and
other activities, and it is applied to an organ model; the second one simulates the side
way fall and it is applied to a total body model. In the first scenario we applied a
load of intensity 1BW with different inclinations respect to the longitudinal axis, and
each of them represents a specific physiological activity. The second scenario takes into
account all the patient’s information, an optimal neuromuscular control and a damping
coefficient with a Gaussian distribution which considers the presence of the attenuation
of the soft tissues wrapping the hip and of other factors reducing the acceleration or
of multiple impact sites (hands, back, knee, etc.). In this way the load transferred to
the hip is lower than the one transferred in the worst loading scenario without any
attenuation.
The first scenario has a predictivity of 72% (expressed as the area under the ROC
curve); the second scenario has a predictivity of 75%. In both cases this represents a
significant improvement over the current standard of care (FRAX) that over the same
cohort showed a predictivity of 64%. Health Technology Assessment experts suggest
that an improvement of 10% is usually considered sufficient to justify a change in the
clinical practice. The statistical analysis suggested that in spite the individualized risk
87
88
CHAPTER 5. CONCLUSION
indicators include the bone density distribution, the areal density at the trochanter
and at the neck remained independent predictors. The predictivity of a model built
by including in a logistic regression both the individualized risk factors and the two
aBMD was found to be up to 80%. In conclusion, the best predictors we found are
the one that combines Strength, WF2, TH_BMD and N_BMD and the one that
combines Strength, WF2, TH_BMD, N_BMD and FRAX. Adding also FRAX we
obtained the same result. The strengths of our work are the following: first of all we
created a patient-specific model, starting from patient’s CT scan and assigning the
material properties based on an automatic algorithm that combines the meshed femur
geometry with the grey’s level of the CT scan. Second, we started to consider the side
way fall. In our knowledge only few works considered this loading scenario. This kind
of study is controversy because in the most of the cases the patients, especially elderly
people, can’t understand the exactly moment in which the fracture occurs. So it is
hard to understand if the fracture is due to a side way fall or the fall is due to the
fracture because the two events happen in the same time. In our work we missed this
information, even thought would be little reliable; however, the combination of both a
physiological loading and a fall loading risk fracture account for both scenarios.
The principal limitations of our work are the following:
• the calculated risk of fracture calculated with WF2 needs a very high computational power; the process to create the femur geometry is long and operatordependent and the simulation takes some hours to calculate the stress and strain
fields on a femur; despite the advantage to consider the side way fall, the predictivity is not higher enough in comparison to clinical instruments to justify such
a long procedure, thus resulting incompatible with clinical applications.
• The calculated risk of fracture calculated in physiological loading scenario needs
a lower computational cost than WF2, but the process is once more longer than
clinical instruments.
• Another critical point concerns the combination of the results; the first loading
scenario gives an actual risk of fracture while the second loading scenario gives a
probabilistic risk of fracture in ten years.
Despite these limitations the predictivity of each model is higher than the clinical
instruments and the combination of the two models increases predictivity up to 80%.
89
In addition, we can focus on each step we have done to prepare the model. The
cohort of the patients has been selected from a clinical database with some inclusion and
exclusion criteria. All the patients have the same weight, height and age, and they differ
only for different BMD values. This assumption could explain the high predictivity of
the BMD (73%) to calculate the risk of fracture in this cohort of patient. The CT
scans are performed in order to minimize the dosage given to the patient varying slice
by slice the intensity of the incident beam. In this way the analysis was safer for
the patient but less clear for the operator who uses the CT scan to build the STL
geometry. Segmentation is an operator-dependent process and the low quality of the
CT scan doesn’t help the operator to obtain high quality in the model. In addition all
the patients are osteoporotic and in some cases present on the greater trochanter some
holes or calcification due to an excessive bone adsorption or abnormal bone deposition.
Another critical point is the morphing phase. The principal advantage of the morphing
process is that the resulting meshes have the nodes in the same position on all the
femurs, they are enumerated with the same number and this process was born to
automatize the meshing phase. Actually the meshes that we have obtained were often
with high values of aspect ratio and skewness because we adapted a template mesh on
different femur geometries: in some cases it took a lot of time to improve the quality
of the meshes. We have obtained meshes with aspect ratio’s maximum values between
7 and 20, and skewness’s maximum values in the range between 0.94 and 0.995. The
method we used to assign the material properties to each femur is Bonemat_V3. This
method assigns the material properties with a pre-implemented algorithm in which
a value of Young modulus is assigned to each element of the mesh. The modulus is
derived from the apparent bone density at the element location. The apparent density
is evaluated considering all the CT grid points located inside each model’s element. A
CT number HU is assigned to each CT grid point, and the apparent density is calculated
by the integration of all the HU located in the element. The principal advantage of this
method is that the assignment of the material properties is automatized and permits
to create a model with anisotropic properties. Furthermore the properties assign to the
femur are patient-specific because the assignment is based on the patient’s CT scan. In
some cases the assignment of the material to an element can be critical, in particular on
trochanteric area where holes and calcification are frequent especially in osteoporotic
patients. These areas have different material properties and their inclusion in the femur
90
CHAPTER 5. CONCLUSION
model could modify the Young modulus assigned to an element of the mesh, and the
resulting mesh has a lower quality. We obtained Young modulus values in the range
between 200MPa to 23000MPa.
The most important innovation in this study is the creation of a patient-specific
model that predicts the risk of fracture in osteoporotic patient and increments the
predictivity and the reliability of the clinical instruments FRAX and BMD. Another
innovation is to consider in addition to the physiological loads, the loads that act on
the femur during a side way fall. In this way we can have a complete scenario of all
the loads that could act on a femur during normal life and also during extraordinary
events.
Chapter 6
Future development
In spite of the possible limitations regarding in particular the operator-dependent nature of the process, we found two predictors with a higher predictivity higher than
clinical instruments. The use of these instruments is incompatible with clinical application because they both take a long time to develop the entire process and to give the
results. In order to improve the predictivity of the model it would be possible to create
a loading scenario for the pathological load in the same way of the one created for
physiological loads. Starting from some experimental studies [GST+ 12] it is possible
to create a finite element model of the femur and constrain and apply some loads on it
to simulate the side way fall. For example we could use the same geometries and place
a hinge distally on the pot, a no-friction slider on the greater trochanter and apply the
load on the head to replicate experimental boundary conditions.
As regard the entire process to obtain the geometry, starting from the segmentation
until the assignment of the material with Bonemat software, the process should be
automatized, even if it would remain an operator-dependent process. Instead of the
morphing, it would be possible to use Ansys ICEM to directly mesh the geometry,
in order to obtain a high quality of the elements. As regard the Bonemat, it would
be possible to assign to an element with a very low Young modulus a default value.
For example if the element’s Young modulus is lower than a critical value, a default
value is assigned to this element (for example 3500MPa, that is the Young modulus of
the cancellous bone, [BMN+ 04]). By implementing these expedients, we think that it
would be possible to increase predictivity of these instruments.
91
Appendix
As in the whole scientific experimental research, also in biomedical research the knowledge of the statistic is very important for management and investigation problem solving. For publishing the results of research on a scientific journal, you have to present
your results through some universal criteria.
If you want to do scientific research in the correct way, that is collect a sample
with a sufficient number of data, considering the population’s conditions and the test
application, is necessary to follow some phases:
• the experimental design is necessary to plan the observations in nature and
reply them in laboratory, in function of the research and of the explicative hypothesis. In this phase is important to know the hypothesis that you want to
verify. With the formulation of the hypothesis, you have to answer to these
questions: “are the differences between the groups or the observations caused by
specific casual factors or only by unknown casual factors? Are these differences
generated by the natural difference in the measures or there is a specific cause
that has determined them?”
• Samples permits to collect data in function of the aim of the research, respecting
the characteristics of the population. A problem of the statistic is to collect a
limited number of data but however to obtain valid and general results.
• Statistical description may permit to verify both the correctness of the experimental design and samples, and the correctness of the analysis done and of the
obtained results.
• Tests have to be planned in the experimental design phase, because samples
depends by them. A test is a logical-mathematical process that conduces to
93
94
APPENDIX
refuse or not some hypothesis of randomness, through the computation of specific
probabilities of doing errors with these statements. The hypothesis that obtained
results are casual is called null hypothesis H0 . With this hypothesis you say
that the differences between two groups or between a group and the attended
value are casual. To come to this you need the inference, that is the possibility
of obtain general conclusions (about the population or the universe) using only
a limited number of variable data.
In this work we used the mean and the standard deviation to describe the population
and box-plot to see data distribution (samples), Mann-Whitney test and ROC curve
to see the predictability of the variables and logistic regression to know the relation
between the variables (inference).
Descriptive statistics
The general purpose of descriptive statistical methods is to organize and summarize a
set of scores. The most common method for summarizing and describing a distribution
is to find a single value that define the average score and can serve as a representative
for the entire distribution. In statistic this concept is called central tendency. The goal
in measuring central tendency is to describe a distribution of scores by determining
a single value that identifies the centre of the distribution. Ideally, this central value
will be the score that is the best representative value for all of the individuals in
the distribution. Statisticians have developed three different methods to calculate the
central tendency: the mean, the median and the mode. They are computed differently
and have different characteristics, and each of them is the best in a particular situation.
In this work to do a descriptive statistical analysis we used the mean.
The mean is commonly known as the arithmetic average and is computed by adding
all the scores in the distribution and dividing by the number of scores. The formula
for the population mean is:
∑
µ=
X
N
where µ in mean for a population, X are the scores in the distribution and N is
the number of scores.
DESCRIPTIVE STATISTICS
95
The most commonly used and the most important measure of variability is the
standard deviation. Standard deviation uses the mean of the distribution as reference point and measures variability by considering the distance between each score and
the mean. It determines whether the scores are generally near or far from the mean.
It says if the scores are clustered together or scattered. In simple terms, the standard
deviation approximates the average distance from the mean. The first step to find
the standard deviation is to determine the deviation, or the distance from the mean,
for each individual score. By definition, the deviation for each score is the difference
between the score and the mean:
deviation = X − µ
The deviation scores add up to zero, because the total of the distances above the
mean is equal to the total distances below the mean. Because the mean of the deviations
is always zero, it is of no value as a measure of variability. The second step is to find
the variance. The variance (SS) is a measure of variability. It is the sum of the squared
distances of data value from the mean divided by the variance divisor. The Corrected
SS is the sum of squared distances of data value from the mean. Therefore, the variance
is the corrected SS divided by N − 1. Finally we can define the standard deviation
σ. The standard deviation is the square root of the variance. It measures the spread
of a set of observations. The larger the standard deviation is, the more spread out
the observations are. A low standard deviation indicates that the data points tend
to be very close to the mean; high standard deviation indicates that the data points
are spread out over a large range of values. The formula to calculate the standard
deviation is:
√
σ=
(X − µ)2
N −1
In descriptive statistics, a box plot is a convenient way of graphically depicting
groups of numerical data through their five number summaries: the smallest observation (sample minimum), lower quartile (Q1), median (Q2), upper quartile (Q3), and
largest observation (sample maximum). A box plot may also indicate which observations, if any, might be considered outliers.
Box plots display differences between populations without making any assumptions
of the underlying statistical distribution: they are non-parametric. The space between
96
APPENDIX
Figure 1: Representation of a box plot. The dots indicate the outliers.
the different parts of the box helps to indicate the degree of dispersion (spread) and
skewness in the data, and identify outliers. An example of box plots are shown in figure
Figure 1.
The Mann-Whitney test
The Mann-Whitney test is designed to use the data from two separate samples to
evaluate the difference between two treatments or two populations. The calculation
for this test require that the individual scores in two samples be rank-ordered. The
mathematics for the Mann-Whitney test is based on the following observation:
• a real difference between the two treatments or populations should cause the
scores in one sample to be generally larger than the scores in the other sample.
If the two samples are combined and the all the scores placed in rank order on
a line, then the score from one sample should be concentrated at one end of the
line, and the score from the other sample should be concentrated at the other
end.
• On the other hand, if there is not treatment difference, then large and small scores
will be mix evenly in the two samples because there is no reason for one set of
scores to be systematically larger or smaller than the other. This observation is
demonstrated in Figure 2.
THE MANN-WHITNEY TEST
97
Figure 2: Representation of Mann-Whitney method. In (a) the different treatment
cause different effects, while in (b) they don’t produce any differences.
Because Mann-Whitney test compares two distributions, the hypotheses tend to
be somewhat vague. We state the hypothesis in terms of consistent and systematic
difference between the two treatments being compared.
• H0 : There is no difference between the two treatments. Therefore there is no
tendency for the ranks in one treatment condition to be systematically higher (or
lower) than the ranks in the other treatment condition.
• H1 : There is difference between the two treatments. Therefore the ranks in one
treatment conditions are systematically higher (or lower) than the ranks in the
other treatment conditions.
The first steps in the calculation of Mann-Whitney test are:
• a separate sample is obtained from each of the two treatments. We use na to
refer to the number of individuals in sample A and nb to refer to the number of
individuals in sample B;
• these two sample are then combined, and the total group of na + nb are rank
ordered.
The remaining problem is decide whether the ranks from the two samples are mixed
randomly or are systematically clustered to opposite end of the scale. This is the
98
APPENDIX
familiar question of statistical significance: are the data simply the results of chance or
has some systematic effect produced these results? To answer we look at all possible
results that could have been obtained. Then, separate these outcomes into two groups:
• those results that are reasonably like to occur by chance;
• those results that are reasonably unlike to occur by chance (this is the critical
region).
For the Mann-Whitney test the first step is to identify each of the possible outcomes.
This is done by assigning a numerical value to every possible set of sample data. This
number is called Mann-Whitney U, whose distribution under the null hypothesis is
known. In the case of small samples, the distribution is tabulated, but for sample
sizes, above 20 approximations, using the normal distribution is fairly good. Some
books tabulate statistics equivalent to U , such as the sum of ranks in one of the samples,
rather than U itself. The U test is included in most modern statistical packages. There
are two ways of calculating U .
For small samples a direct method is recommended. It is very quick, and gives an
insight into the meaning of the U statistic: for each observation in sample A, count
the number of observations in sample B that have a smaller rank (count a half for any
that are equal to it). The sum of these counts is U .
For larger samples, a formula can be used:
• add up the ranks for the observations which came from sample A. The sum of
ranks in sample B is now determinate, since the sum of all the ranks equals
1
N (N + 1) where N is the total number of observations.
2
• U is then given by:
1
Ui = Ri − ni (ni + 1)
2
where ni is the sample size for sample i, and R − i is the sum of the ranks
in sample i. The smaller value of U1 and U2 is the one used when consulting
significance tables.
For large samples, U is approximately normally distributed. In that case, the
standardized value:
THE MANN-WHITNEY TEST
99
z=
U − mU
σU
where mU and σU are the mean and standard deviation of U , is approximately
a standard normal deviate whose significance can be checked in tables of the
normal distribution. mU and σU are given by:
mU =
√
σU =
1
nA nB
2
nA nB (nA + nB + 1)
12
When computing U , the number of comparisons equals the product of the number
of values in group A times the number of values in group B. If the null hypothesis
is true, then the value of U should be about half that value. If the value of U is
much smaller than that, the P value will be small. The smallest possible value of
U is zero. The largest possible value is half the product of the number of values
in group A times the number of values in group B.
Another output of the test is the P value. The P value is a probability, with a
value ranging from zero to one, that answers this question (which you probably never
thought to ask): in an experiment of this size, if the populations really have the same
mean, what is the probability of observing at least as large a difference between sample
means as was, in fact, observed? You can’t interpret a P value until you know the null
hypothesis being tested. For the Mann-Whitney test, the null hypothesis is that the
distributions of both groups are identical, so that there is a 50% probability that an
observation from a value randomly selected from one population exceeds an observation
randomly selected from the other population.
The P value answers this question:
• if the groups are sampled from populations with identical distributions, what is
the chance that random sampling would result in the mean ranks being as far
apart (or more so) as observed in this experiment?
• If the P value is small, you can reject the null hypothesis that the difference is
due to random sampling, and conclude instead that the populations are distinct.
100
APPENDIX
• If the P value is large, the data do not give you any reason to reject the null
hypothesis. This is not the same as saying that the two populations are the
same. You just have no compelling evidence that they differ. If you have small
samples, the Mann-Whitney test has little power. In fact, if the total sample size
is seven or less, the Mann-Whitney test will always give a P value greater than
0.05 no matter how much the groups differ.
ROC curve
The tests are the most used instruments in epidemiology screening for identifying precautionary the presence of a disease. Also in diagnostic routine the tests are used in
the decision process to confirm or to exclude the presence of a suspected disease on the
clinical data. The tests can be divided in two different kinds:
• qualitative test: this test gives a dichotomy response (positive/negative, true/false);
• quantitative test: this test gives a numerical response that can be discrete or
continuous.
For the second kind of test is necessary to establish a cut off point in order to discriminate positive results from negative results. In this way is possible to divide in
positive or negative all the possible results. There is one problem that could cause error during a test: it is the possibility that the two distributions of the two populations
(non-disease/disease) are not completely separated but present an overlapping area. If
the populations are completely separated it is easy to find the cut off points, like is
shown in Figure 3.
Unfortunately in practice there is always an overlapping of the two populations
with an area more or less large as shown in Figure 4.
In this case it is impossible to find a cut off point (or threshold) that cancels the
true negative and the false positive results.
The performance of a test is the capability of the test to give a positive diagnosis in
the patients affected by the disease and negative diagnosis in the patients not affected by
the disease. The performance can be evaluated with a contingency table that compares
the real condition of the patients with the output of the test (Figure 5).
The comparison between the test’s results and the real state of the individuals allows
to obtain two important parameters: the sensitivity and the specificity. The sensitivity
ROC CURVE
101
Figure 3: Gaussian distribution of two population completely separated.
Figure 4: Gaussian distribution of two population with overlapping area.
measures the proportion of actual positives, which are correctly identified as such; the
specificity measures the proportion of negatives, which are correctly identified as such.
Sensitivity relates to the test’s ability to identify positive results.
Se =
TP
TP + FN
Specificity relates to the test’s ability to identify negative results.
Sp =
TN
TN + FP
102
APPENDIX
Figure 5: Contingency table.TP represents the true positive results, TN represents the
true negative results, FP represents the false positive results and FN represents the
false negative results.
It is clear how much is important to choose a correct value of cut off: the cut off
point could influence the Sp and Se value. Usually the best cut off point is the one
corresponding on the abscissa axis to the intersection point of the two distributions:
this is the cut off value that minimizes the classification errors. However the choice of
the cut off point couldn’t be done only on statistical consideration, but it is necessary
to take into account also social, sanitary, epidemiological and economical impact. It
is possible to choose a cut off point in order to privilege Sp or Se accordingly to the
specific problem.
Receiver operating characteristic (ROC) analysis is a graphical plot which
illustrates the performance of a binary classifier system as its discrimination threshold
is varied. It is created by plotting the fraction of true positives out of the positives (TPR
= true positive rate or Se) versus the fraction of false positives out of the negatives
(FPR = false positive rate or 1 − Sp), at various threshold settings. An example of
ROC curve is shown in Figure 6.
The test’s discriminant ability, in other words its inclination to separate “sick” from
“non-sick”, is represented by the area under curve (AUC). The area under ROC curve
ranges from 0.5 and 1. For a perfect test that doesn’t give a false positive and a false
negative AUC’s value is of 1 that corresponds to a probability of 100% of a correct
classification. Instead for a non-significant test the AUC’s value is of 0.5 and the
ROC CURVE
103
Figure 6: Representation of the ROC curve as a function of specificity (Sp) and sensitivity (Se).
representative curve is the diagonal (chance-line) passing trough the origin. The AUC
under an empiric curve, which is a curve obtained from a finite sample, is obtained
connecting the different points of ROC plot with the abscissa axis and summing up
all the polygonal areas generated (Figure 7). This method produce an approximate
AUC’s value, instead the “real” value could be obtained with a calculator using some
statistical or mathematical software (like SPSS or MATLAB).
A classification for AUC’s value that can be used, is the one proposed by Swest
(1998) and is the following:
• AU C = 0.5 not informative test;
• 0.5 < AU C ≤ 0.7 little accurate test;
• 0.7 < AU C ≤ 0.9 fairly accurate test;
• 0.9 < AU C < 1 highly accurate test;
• AU C = 1 perfect test.
104
APPENDIX
Figure 7: Representation of the AUC, the area under ROC curve.
Logistic regression
Logistic regression is a classification where some or all variables are qualitative. The
response variable Y is restricted to two values only, it means that Y is a dichotomy
variable. We can always code the two cases as 1 and 0 (for example we can consider disable/non-disable, true/false, positive/negative). The probability p of 1 is the
parameter of interest. The mean values is:
mean = 0 × (1 − p) + 1 × p = p
and the variance is the following:
variance = 02 × (1 − p) + 12 × p − p2 = p(1 − p)
Instead to model the probability p with a linear model, we can use Logit model.
First we must consider the odds ratio:
odds =
p
1−p
which is the ratio of the probability of 1 to the probability of 0.
LOGISTIC REGRESSION
105
Figure 8: Logistic function with β0 = −1 and β1 = 2.
In logistic regression for a binary variable, we model the natural log of the odds
ratio, called logit(p). Thus:
(
)
logit(p) = ln odds = ln
(
p
1−p
)
The logit is a function of a probability of p. In the simplest model we consider this
relationship:
(
)
logit(p) = ln odds = ln
(
p
1−p
)
= β0 + β1 z
in which log odds are linear in the predictor variable.
(
ln
p
1−p
)
= β0 + β1 z
The logit relationship written in exponential form is:
θ(z) =
p(z)
= exp(β0 + β1 z)
1 − p(z)
p(z) =
exp(β0 + β1 z)
1 + (β0 + β1 z)
which describes the logistic curve. The relation between p and z is not linear but
has an S-shape graph as shown in Figure 8.
β0 gives the value of p when z = 0 instead β1 represents how quickly p changes with
z.
106
APPENDIX
Now we consider the model with several predictor variables. Let zj1 , zj2 , . . . , zjr
are the variables. Yj is Bernoulli and p(zj ) is its success probability.
P (Yj = yj ) = pyj (zj )(1 − p(zj ))1−yj , yj = 0, 1
E(yj ) = p(zj ), V ar(yj ) = p(zj )(1 − p(zj ))
The model we assume is the following:
(
ln
where:
p(z)
1 − p(z)
)
= β0 + β1 z1 + · · · + βr zr = β T zj
 
β0
 
 
β1 

β T zj = 
 .. 
.
 
βr
Bibliography
[BDH+ 01] G Bergmann, G Deuretzbacher, M Heller, F Graichen, A Rohlmann,
J Strauss, and GN Duda. Hip contact forces and gait patterns from routine
activities. Journal of biomechanics, 34(7):859–871, 2001.
[BMN+ 04] Harun H Bayraktar, Elise F Morgan, Glen L Niebur, Grayson E Morris,
Eric K Wong, and Tony M Keaveny. Comparison of the elastic and yield
properties of human femoral trabecular and cortical bone tissue. Journal
of biomechanics, 37(1):27–35, 2004.
[Cau11]
Jane A Cauley.
Defining ethnic and racial differences in osteoporosis
and fragility fractures.
Clinical Orthopaedics and Related Research®,
469(7):1891–1899, 2011.
[GHS+ 11] Lorenzo Grassi, Najah Hraiech, Enrico Schileo, Mauro Ansaloni, Michel Rochette, and Marco Viceconti. Evaluation of the generality and accuracy of
a new mesh morphing procedure for the human femur. Medical engineering
& physics, 33(1):112–120, 2011.
[GST+ 12] L Grassi, E Schileo, F Taddei, L Zani, M Juszczyk, L Cristofolini, and
M Viceconti.
Accuracy of finite element predictions in sideways load
configurations for the proximal human femur. Journal of biomechanics,
45(2):394–399, 2012.
[KHC+ 11] John A Kanis, Didier Hans, Cyrus Cooper, Sanford Baim, John P
Bilezikian, Neil Binkley, Jane A Cauley, JE Compston, Bess DawsonHughes, G El-Hajj Fuleihan, et al. Interpretation and use of frax in clinical
practice. Osteoporosis international, 22(9):2395–2411, 2011.
107
108
[LEW09]
BIBLIOGRAPHY
E MICHAEL LEWIECKI. Managing osteoporosis: challenges and strategies. Cleveland Clinic journal of medicine, 76(8):457–466, 2009.
[MTC+ 11] Saulo Martelli, Fulvia Taddei, Angelo Cappello, Serge van Sint Jan, Alberto Leardini, and Marco Viceconti. Effect of sub-optimal neuromotor
control on the hip joint load during level walking. Journal of biomechanics, 44(9):1716–1721, 2011.
[Rai05]
Lawrence G Raisz. Pathogenesis of osteoporosis: concepts, conflicts, and
prospects. Journal of Clinical Investigation, 115(12):3318, 2005.
[RB06]
Jean-Yves Reginster and Nansa Burlet. Osteoporosis: a still increasing
prevalence. Bone, 38(2):4–9, 2006.
[Ric03]
Bradford Richmond. Dxa scanning to diagnose osteoporosis: Do you know
what the results mean? Cleveland Clinic journal of medicine, 70(4):353–
360, 2003.
[STM+ 07] Enrico Schileo, Fulvia Taddei, Andrea Malandrino, Luca Cristofolini, and
Marco Viceconti. Subject-specific finite element models can accurately predict strain levels in long bones. Journal of biomechanics, 40(13):2982–2989,
2007.
[TSH+ 07] Fulvia Taddei, Enrico Schileo, Benedikt Helgason, Luca Cristofolini, and
Marco Viceconti. The material mapping strategy influences the accuracy of
ct-based finite element models of bones: an evaluation against experimental
measurements. Medical engineering & physics, 29(9):973–979, 2007.
[ZMV99]
Cinzia Zannoni, Raffaella Mantovani, and Marco Viceconti.
Material
properties assignment to finite element models of bone structures: a new
method. Medical engineering & physics, 20(10):735–740, 1999.

POLITECNICO DI MILANO Generation of a patient specific predictor

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