Electrical and Dielectric Characterization of Trabecular

Transcription

KUOPION YLIOPISTON JULKAISUJA C. LUONNONTIETEET JA YMPÄRISTÖTIETEET 214
KUOPIO UNIVERSITY PUBLICATIONS C. NATURAL AND ENVIRONMENTAL SCIENCES 214
JOANNA SIERPOWSKA
Electrical and Dielectric Characterization
of Trabecular Bone Quality
Doctoral dissertation
To be presented by permission of the Faculty of Natural and Environmental Sciences
of the University of Kuopio for public examination in Auditorium L3,
Canthia building, University of Kuopio,
on Friday 24 th August 2007, at 12 noon
Department of Physics
University of Kuopio
JOKA
KUOPIO 2007
Distributor :
Kuopio University Library
P.O. Box 1627
FI-70211 KUOPIO
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Tel. +358 17 163 430
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http://www.uku.fi/kirjasto/julkaisutoiminta/julkmyyn.html
Series Editors:
Professor Pertti Pasanen, Ph.D.
Department of Environmental Sciences
Professor Jari Kaipio, Ph.D.
Author’s address:
P.O. Box 1627
FI-70211 KUOPIO
FINLAND
Tel. +358 17 162 342
Fax +358 17 162 255
E-mail: [email protected]
Supervisors:
Professor Reijo Lappalainen, Ph.D.
Docent Juha Töyräs, Ph.D.
Department of Clinical Neurophysiology
Kuopio University Hospital
Professor Jukka Jurvelin, Ph.D.
Reviewers:
Professor Stig Ollmar, Ph.D.
Division of Medical Engineering
Karolinska Institute
Stockholm, Sweden
Professor Timo Jämsä, Ph.D.
Department of Medical Technology
University of Oulu
Opponent:
Docent Harri Sievänen, Sc.D.
Bone Research Group
UKK Institute
Tampere, Finland
ISBN 978-951-27-0692-1
ISBN 978-951-27-0787-4 (PDF)
ISSN 1235-0486
Kopijyvä
Kuopio 2007
Finland
Sierpowska, Joanna. Electrical and Dielectric Characterization of Trabecular Bone Quality.
Kuopio University Publications C. Natural and Environmental Sciences 214. 2007. 92 p.
ISBN 978-951-27-0692-1
ISBN 978-951-27-0787-4 (PDF)
ISSN 1235-0486
ABSTRACT
Electrical stimulation has been used for decades to stimulate bone growth. However,
the relationships between bone structure, mechanical properties, organic composition
and electrical properties are still not fully understood. Better understanding of these
relationships could enable more accurate estimation of electrical fields in bone during
electrical stimulation. Furthermore, measurements of electrical properties may provide
diagnostically valuable information about bone quality.
In this thesis, bovine and human trabecular bone were investigated using electrical
impedance spectroscopy (EIS) with a parallel-plate capacitance cell in a wide range
of frequencies (20 Hz - 5 MHz) (data published as original articles I-IV). These results were supplemented with high frequency measurements (300 kHz - 3 GHz) which
were published in this thesis but were not included in the studies I-IV. The additional
experiments were conducted with an open-ended probe. The sample preparation procedure and the measurement protocol were designed to preserve the physiological state
of the tissue throughout the investigation. As a reference, mechanical properties, microstructure, density and organic composition of trabecular bone were determined.
Strong significant frequency-dependent relations between the electrical characteristics and the density, composition, structural and mechanical properties of trabecular
bone were found. The dissipation factor and especially the relative permittivity at
frequencies between 100 kHz and 5 MHz were found to be sensitive to variations in
human bone quality. Further, these parameters were able to detect mechanically induced damage in trabecular architecture. The surface-specific parameters of trabecular
structure were found to be the main determinant of relative permittivity, while the
shape and number of trabeculae related primarily to other electrical measures. It was
demonstrated that fat and water content can be estimated by a simple measurement of
relative permittivity and conductivity at two separate frequency bands. Additionally,
bone tissue possessing a healthy dense trabecular structure showed lower conductivity
and higher relative permittivity at frequencies between 100 kHz and 5 MHz than tissue
with sparse structure.
To conclude, the EIS showed potential as a tool to determine bone characteristics.
The relationships presented in this study enable the evaluation of bone electrical properties for the purpose of current and field distribution calculations during stimulated
osteogenesis. The method may be feasible during some special cases of open surgery or
used in vitro for the quantitative evaluation of bone grafts. It could provide a basis for
future improvement of existing electrical impedance tomography techniques towards
clinical in vivo bone diagnosis. However, further theoretical, experimental, clinical
and technical developments of the method are needed to confirm the present findings
and to fully reveal its potential, especially at radiofrequencies.
Universal Decimal Classification: 537.311.6, 621.3.011.21, 621.317.33
National Library of Medicine Classification: QT 36, WE 141, WE 200
INSPEC Thesaurus: biological tissues; bone; characteristics measurement; electric impedance;
electric impedance measurement; electric properties; dielectric properties; permittivity; electrical conductivity; computerised tomography; mechanical properties
ACKNOWLEDGMENTS
This study was carried out during 2000-2006 in the Department of Physics,
University of Kuopio.
I owe my deepest gratitude to my head supervisor, Professor Reijo Lappalainen, for the opportunity of working in his research group and guidance.
His support and encouragement during this work are greatly appreciated.
I wish to thank my supervisor Docent Juha Töyräs, for his extensive collaboration and for the practical supervision, especially for his constructive suggestions and critical reading of the manuscripts.
I am grateful to my supervisor Professor Jukka Jurvelin, for fruitful discussions and professional guidance.
I express my gratitude to the official pre-examiners Professor Stig Ollmar,
and Professor Timo Jämsä, for their constructive criticism, which helped to
improve this thesis. I am also grateful to Vivian Michael Paganuzzi for revising
the language of the thesis.
I owe my thanks to the Delphin Technologies Ltd company for providing the
equipment for high frequency measurements. I am especially grateful to Jouni
Nuutinen, for valuable discussions.
I would like to thank all my friends in the Biomaterial research group: Arto
Koistinen, Hannu Korhonen, Jari Leskinen, Mikko Selenius, Laura Tomppo,
Markku Tiitta, Sami Myllymaa, Siru Turunen, Hanna Matikka, and Esa Miettinen. I am grateful to Juhani Hakala for his technical assistance in constructing
the measurement setups. I give special thanks to Ritva Sormunen for the assistance in sample preparation and invaluable companionship. The atmosphere in
our group has been inspirational and supportive through these years. I would
also like to acknowledge my co-workers in the Biophysics of Bone and Cartilage
(BBC) group, especially Mikko Hakulinen for extensive cooperation.
I am indebted to the secretaries of the Physics Departments, especially to
Tarja Toivanen for administrative support. Additionally, I would like to thank
Heikki Väisänen for the assistance with computer troubles and keeping up my
spirits.
I owe my gratitude to Kati Niinimäki and Anna Ruuskanen for companionship and morale support.
I wish to thank Atria Lihakunta Oyj, Kuopio, and their personnel for providing material for the part of the study involving bovine bones.
I am grateful to the Centre for International Mobility (CIMO) and the Graduate School of Musculoskeletal Diseases and Biomaterials Graduate School for
their financial support.
Finally, I wish to thank my parents, Bożena and Zenon, friends and relatives
for their encouragement. I owe my warmest thanks to my loving Risto for his
understanding and faith during this work.
Kuopio, August 2007
Joanna Sierpowska
NOTATION
A
[A]
C
D
d
E
F
f
G
j
K∗
m
n
L
PL
Pp
p
p
R
r
S
t
tan δ
U
u
X
x
x
Y∗
y
Z
Zsp
Z0
Z∗
α
β
Γ∗
γ∗
δ0
δ1
δ2
²
ε∗
ε, ε0
ε00
εr
area
tensor
capacitance
surface charge density
distance
Young’s modulus
force
frequency
conductance
imaginary unit
complex cell factor
unit vector
number of specimens
length
number of intersections
areal density
statistical significance
structure point
resistance
correlation coefficient
cross-section area
time
dissipation factor
resilience
amplitude of voltage
reactance
variable of integration
center of sphere
complex admittance
mean
impedance
specific impedance
characteristic impedance
complex impedance
distribution parameter of Cole-Cole model
attenuation coefficient
complex reflection coefficient
complex propagation constant
number of bone particles
connectivity
number of cavities
strain
complex permittivity
permittivity (real part)
loss factor (imaginary part)
relative permittivity
ε0
η
θ
κ
λ
σ, σ 0
τ
υ
ϕ
χ
ω
Ω
permittivity of vacuum
trabecular thickness
phase angle
stress
wavelength
conductivity (real part)
relaxation time
standard deviation
angle
Euler number
angular frequency
structure volume
ABBREVIATIONS
2D
3D
AC
BMD
BS
BS/BV
BV
BV/TV
CV
CT
DA
DC
DXA
DNA
EDTA
EIS
FC
FG
FLC
FMC
FTM
GAG
max
min
MIL
NMR
PC
PCA
PBS
Res
ROI
sCV
SMI
two dimensional
three dimensional
alternating current
bone mineral density
bone surface
bone surface-to-volume ratio
bone volume
trabecular bone volume fraction
coefficient of variation
computed tomography
degree of anisotropy
direct current
dual energy x-ray absorptiometry
deoxyribonucleic acid
ethylenediaminetetraacetic acid
electrical impedance spectroscopy
femoral head
femoral groove
femoral lateral condyle
femoral medial condyle
femoral greater trochanter
glycosaminoglycans
maximum
minimum
mean intercept length
nuclear magnetic resonance
principal component
principal component analysis
phosphate buffered saline
resilience
region of interest
standardized coefficient of variation
structure model index
Tb.Th.
Tb.Sp.
Tb.N.
TMP
TV
US
trabecular thickness
trabecular spacing
trabecular number
tibial medial plateau
tissue volume
ultimate strength
LIST OF ORIGINAL PUBLICATIONS
This thesis is based on the following original articles, which are referred in
the text by their Roman numerals (I-IV):
I Sierpowska J., Töyräs J., Hakulinen M.A., Saarakkala S., Jurvelin J.S. and
Lappalainen R. Electrical and dielectric properties of bovine trabecular
bone-relationships with mechanical properties and mineral density.
Physics in Medicine and Biology 48: 775-786, 2003
http://www.iop.org/journals/pmb
II Sierpowska J., Hakulinen M.A., Töyräs J., Day J.S., Weinans H., Jurvelin
J.S. and Lappalainen R. Prediction of mechanical properties of human
trabecular bone by electrical measurements.
Physiological Measurement 26: S119-S131, 2005
http://www.iop.org/journals/pmea
III Sierpowska J., Hakulinen M.A., Töyräs J., Day J.S., Weinans H., Kiviranta I., Jurvelin J.S. and Lappalainen R. Interrelationships between electrical properties and microstructure of human trabecular bone.
Physics in Medicine and Biology 51: 5289-5303, 2006
http://www.iop.org/journals/pmb
IV Sierpowska J., Lammi M.J., Hakulinen M.A., Jurvelin J.S., Lappalainen
R. and Töyräs J. Effect of human trabecular bone composition on its
electrical properties.
Medical Engineering and Physics, 29: 845-852, 2007
http://www.elsevier.com
The original articles have been reproduced with permission of the above copyright holders.
Contents
1 Introduction
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2 Structure and composition of trabecular bone
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2.1 Trabecular bone microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Trabecular bone composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Mechanical properties of trabecular bone
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3.1 Mechanical testing of bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Determinants of trabecular bone mechanical properties . . . . . . . . . . . . . . 26
3.3 Damage of trabecular structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Computed tomography of bone structure
5 Electrical impedance spectroscopy (EIS) of
5.1 Introduction to the relaxation theory . . . .
5.2 Microwave frequency measurements . . . . .
5.3 Limiting Factors of EIS . . . . . . . . . . .
5.4 Electrical characteristics of trabecular bone
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6 Aims of the present study
7 Materials and methods
7.1 Sample preparation . . . . . . . . . . . . . . . . . . .
7.2 EIS methods . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Low frequency impedance spectroscopy . . .
7.2.2 Microwave frequency impedance spectroscopy
7.3 Bone mineral density measurements . . . . . . . . .
7.4 Mechanical testing . . . . . . . . . . . . . . . . . . .
7.5 Microstructural analysis . . . . . . . . . . . . . . . .
7.6 Compositional analysis . . . . . . . . . . . . . . . . .
7.7 Statistical analysis . . . . . . . . . . . . . . . . . . .
8 Results
8.1 Variation
8.2 Relations
8.3 Relations
8.4 Relations
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and frequency dependence of electrical properties
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between electrical and structural properties . . .
between electrical and compositional properties .
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9 Discussion
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10 Summary and conclusions
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References
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Erratum
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Appendix: Original publications
Chapter I
Introduction
Bone is a connective tissue which extracellular components are calcified yielding
a hard and rigid substance. Its constituents are organized on all hierarchical
levels to provide elasticity and high tensile and compressive strength together
with minimal weight and use of material [1]. As such, bone provides mechanical
support and protection for the vital organs of the body, including the brain,
spinal cord and structures within the thoracic cavity. Furthermore, bone enables
locomotion by serving as levers for the muscles and tendons. In addition to its
mechanical function, bone acts as a store of inorganic salts that can be easily
released to maintain balanced concentration of certain ions in body fluids. In
particular, the skeleton stores about 99% of the body calcium [2]. Moreover,
bone houses marrow tissue, which is a hemopoietic organ.
Wolff’s law [3] states that ”under the direction of functional pressure the
given form of bone changes by placing or displacing its elements” [4] and gives
the theoretical basis of bone remodeling. The adaptation of a bone to external
forces requires a strict control system [5]. The discovery of the piezoelectric
properties of bone by Fukada and Yasuda [6] led to speculation that remodeling
could be governed by endogenous electrical potentials. ”Yasuda’s hypothesis”
suggests that the cellular response leading to remodeling is caused by electrical
potentials generated in the tissue as a consequence of applied stress. Further,
the hypothesis suggests that osteogenesis is stimulated by negative potentials
while positive potentials result in bone resorption. This theory led to three main
branches of research.
The first branch concentrated on investigating whether cellular responses
leading to osteogenesis can be stimulated by externally applied electrical potentials without simultaneous application of mechanical stress. This research
resulted in a number of clinical applications of the theory, including treating
nonunions [7, 8], promoting spinal fusions [9, 10] and distraction osteogenesis
[11, 12]. Investigations of the mechanisms underlying electrical and electromagnetical stimulations are Currently underway [13, 14], especially at the cellular
level [15, 16, 17, 18].
The second branch focused on the validity of Yasuda’s hypothesis. Some
investigators (e.g., [19]) argue that endogenous electrical currents are not involved in bone remodeling but this process is directly modulated by mechanical
strains. Further, it has been proposed that the effect of exogenous electromagnetic fields on bone remodeling is secondary. It might modulate the activity of
15
16
1. Introduction
other factors, including mechanical strains. However, to the present date, the
physiological significance of endogenous electrical signals is not fully understood
[19]. Proof of the hypothesis would require a difficult separation of electrical
and mechanical stimuli and should demonstrate that cellular response is caused
by electrical potentials experienced by cells during mechanical stress but is not
caused by the stress itself [20].
The third branch is the most fundamental, uniting the other two. It focuses
on determining and understanding the electrical properties of bone. By construction of an advanced theoretical model of bioelectric phenomena in bone
that could enable objective investigation of the ”Yasuda’s hypothesis,” the electrical environment of bone cells must be understood and values of electrical and
magnetic fields and current densities in the vicinity of cells must be known. This
knowledge could also clarify the uncertainties in the electrical characteristics of
the bone-electrode interfaces and in the current paths between electrodes during
stimulated osteogenesis [21, 22].
The fast development of techniques that expose humans to electromagnetic
radiation has led to the foundation of electromagnetic dosimetry. This new
branch of biomedical science deals with the calculation of internal fields within
exposed tissues. A three-dimensional anatomical model of a human body has
been created to calculate specific absorption rate [23], electrical fields at the
extremely low frequencies [24, 25] and to assess the effect of electrostatic discharges on the human body [26]. As the permittivity and conductivity of a
tissue play a dominant role in the overall consideration of interactions between
field and matter [23], electromagnetic dosimetry requires knowledge of the exact values of the electrical properties of the various tissues at all frequencies
to which a human is exposed [27, 23]. The electrical properties of bone have
been of particular interest in investigations of the possible interrelation between
leukemia and exposure to strong electrical fields [26].
Similarly, during the image reconstruction of electrical impedance tomography, a non-invasive and inexpensive imaging method [28], all tissues must be
accurately characterized electrically in order to obtain the maximum sensitivity.
Electrical properties of bone are of major importance in brain imaging, as the
electrodes are placed on the skull (see e.g., [29]). However, a strong variability
or the lack of appropriate data along with the systematic errors associated with
the measurement technique make both the dosimetry model and accurate image
reconstruction difficult to realize.
Measuring electrical properties could provide valuable diagnostic information about bone quality, too. Impedance measurements have been shown to be
efficient in certain cases of open surgery, where traditional techniques are not
feasible [30, 31]. For example, during the limb lengthening with the Ilizarov
technique both the frequency and rate of distraction have to be controlled in
order to avoid premature fusion or insufficient bone formation [32, 33]. The
rate of new bone formation cannot be controlled by plain radiography as it is
not sensitive enough to detect the small changes in the newly formed bone. In
contrast, electrical impedance measurements have been shown to be sensitive
to changes occurring during the bone healing process [31]. By investigating
the interrelations between the electrical properties and mechanical or structural
characteristics of bone, it might be possible to further develop this technique
towards more comprehensive monitoring of the tissue quality and clinical applications.
17
In this thesis, the electrical and dielectric properties of trabecular bone were
measured using a wide frequency range (20 Hz - 3 GHz) and compared with
systematically determined mechanical, microstructural and compositional parameters. The main aim was to clarify the effect of the mechanical, structural
and compositional characteristics of bone tissue on its electrical and dielectric
properties. We aimed at gathering the relevant data in order to contribute to
the understanding of Yasuda’s hypothesis, to provide the information needed for
modeling and to test the sensitivity of the impedance technique in determining
bone characteristics.
18
1. Introduction
Chapter II
Structure and composition of trabecular bone
Long bones consist of a shaft, i.e., diaphysis, located between two heads, i.e.,
epiphysis [34]. The epiphysis is covered by hyaline cartilage forming an articular
surface. The middle portion of the shaft, called the medullary cavity, is filled
with bone marrow. Short bones (e.g., the carpal bones of the wrist) are approximately cube-shaped with nearly equal vertical and horizontal dimensions. Flat
bones are plate-like, flat and thin, e.g., the bones forming the skull cap. Bones
with shapes that do not fit into any of above mentioned groups, such as bones
that contain air spaces, e.g., ethmoid bones, are called irregular. Sesamoid
bones (e.g., patella), embedded within certain tendons, reduce the ability of
tears to propagate across the tendon and improve the mechanical performance
of the muscle across a joint.
Two different types of bone tissue can be distinguished. The spongy-like
meshwork of the epiphysis and the inner part of the medullary cavity and short
and flat bones is called trabecular bone (Figure 2.1). The dense structure forming a cortex around the trabecular tissue and the diaphysis of long bones is
called compact bone. The distinction between the two tissues is made on the
basis of the porosity of the structure. Compact bone has a porosity of 5% - 10%,
while the porosity of a trabecular bone is 75% - 95% [35]. However, as bone
is a dynamic porous structure, its porosity may change. A change in porosity
can result from a pathological situation or from a normal adaptive response to
a mechanical or physiological stimulus [1].
All bone surfaces are covered by a thin layer of connective tissue called
periosteum, except at places where muscle and tendons insert the bone [2] or
where they articulate with another bone. The periosteum consists of outer layer
of collagen fibers and fibroblasts and the inner cellular layer [36]. When the bone
is actively growing, the inner layer consists of osteoprogenitor cells, whereas
when there is no active bone formation, the inner layer is not well developed. It
consists of few cells capable of dividing and differentiating into osteoblasts [34].
The periosteum is attached to the bone by the bundles of periosteal collagen
fibers, called Sharpey’s fibers, that penetrate the bone matrix. The endosteum,
composed of a small amount of connective tissue and a single layer of flattened
osteoprogenitor cells, covers the internal cavities of bone, including the marrow
spaces within trabecular bone. Under an appropriate stimulus, the cells of
endosteum can differentiate into osteoblasts [34]. The main function of both
19
20
2. Structure and composition of trabecular bone
Trabecular bone
Bone marrow
Compact bone
Proximal femur
Lamellae Osteocytes
Figure 2.1: Bone is a connective tissue with calcified extracellular components. The shaft is formed by a dense compact bone. A spongy-like meshwork
of trabecular tissue, arranged in the form of plates or struts called trabeculae,
occupies the epiphysis. Trabeculae consist of parallel layers of an anisotropic
matrix called lamellae. Lamellae, in turn, contain uniformly spaced cavities
that encompass bone cells, or osteocytes. Bone marrow can be found in the
medullary cavity and in the spaces between trabeculae.
membranes is to nourish the bone tissue and provide it with osteoblasts, cells
necessary for the repair and growth [36].
Bone is a dynamic structure. It is removed and formed by means of specialized cells, osteoclasts and osteoblasts, throughout life. This process of constant
modification of tissue architecture to meet physical stress, to repair microscopic
damage and prevent the accumulation of fatigue damage is called remodeling
[37, 1]. It has been estimated that the total remodeling of bone takes around 4
months, with resorption lasting about 3 weeks and formation about 3 months
[1]. Osteoclasts and osteoblasts function on the tissue surface, as they are the
size of trabeculae and thus cannot tunnel within trabeculae without damage [1].
Trabecular bone turnover, or the rate of remodeling, is about 25 % a year. It
varies, however, with skeletal site and the person’s age.
2.1
Trabecular bone microstructure
In trabecular bone, the calcified tissue is arranged in the form of plates or struts
called trabeculae, around 200 µm thick [1], creating numerous interconnected
cavities. Those cavities are filled with bone marrow.
Trabecular bone matrix
Trabeculae are composed of a mosaic of lamellar bone pieces [38], which is a
highly organized material. It consists of parallel layers of an anisotropic matrix
composed of mineral crystals and collagen fibers, called lamellae. Lamellae
2.1 Trabecular bone microstructure
21
contain uniformly spaced cavities, lacunae, that encompass osteocytes. Those
cavities are spread in all directions, penetrate the lamellae and connect with
neighbouring lacunae. Calcified bone matrix is impermeable and osteocytes
cannot be nourished or can metabolites be removed through diffusion. These
processes occur through thin cylindrical spaces, called canaliculi, that perforate
the matrix and extend to the endosteal surface of the trabeculae. All nutrients
are supplied from the marrow cavity. However, a sufficiently thick trabeculae
can contain osteons [34]. An osteon is a structural unit consisting of concentric
lamellae of bone matrix surrounding the central osteonal canal, which contains
nerves and blood vessels.
Cells of bone tissue
There are four types of bone cells: osteoprogenitor cells, osteoblasts, osteoclasts,
and osteocytes.
Osteoprogenitor cells are located in the periosteum, endosteum and osteonal
canals. In mature bone with no active remodeling processes, osteoprogenitor
cells become flattened and spread closely on the bone surface. However, in
growing bone or during periods of high tissue turnover these cells become much
larger and more numerous. Osteoprogenitor cells are derived from embryonic
mesenchyme [2] and, as stem cells, are capable of differentiating into specialized
bone-forming cells, i.e., osteoblasts [37]. Moreover, it has been suggested [34]
that under certain stimuli these cells can modify their morphological characteristics and function to differentiate to adipose cells, chondroblasts or fibroblasts.
Osteoprogenitor cells take part in the healing processes of fractured bones and
are involved in creating cartilage and connective tissues in the callus [34].
Osteoblasts are differentiated bone-forming cells that constitute a closely
packed sheet on the surface of the bone. Active osteoblasts have cuboidal or
columnar shapes. As their activity declines, they become flat and resemble osteoprogenitor cells. Bone remodeling and changes in the growth of the tissue are
activated by osteoblasts as a response to mechanical stimuli [34]. The cells secrete collagen and the ground substance that constitute the unmineralized bone,
or osteoid. Initially, osteoid forms a layer that separates osteoblast from calcified bone, but as the secreted matrix accumulates, osteoblasts become buried
within and transform into osteocytes. This process results in the formation of
a lacuna, a cavity occupied by an osteocyte, its extensions and a small amount
of noncalcified matrix. The calcification process, or the deposition of calcium
salts into osteoid, completes the bone formation. The apposition rate, or the
speed of osteoid formation, is about 1 µm/day [1].
Osteocytes are derived from osteoblasts and are situated in lacunae within
calcified matrix. The cavity of lacunae conforms to the shape of the cell [34]. Osteocytes communicate with neighbouring cells by cytoplasmic processes housed
in canaliculi. Cells receive nutrients through the extracellular substance located
between osteocytes, cytoplasmic processes and the bone matrix. The main function of osteocytes is to maintain the bone matrix. They also help to maintain
homeostasis of blood calcium by synthesizing and resorbing bone matrix. The
resorption of the bone by osteocytes that release calcium ions is called osteocytic
osteolysis. Because of the small size of osteocytes, lacunae and canaliculi, they
occupy only about 1% of bone volume. However, there are about 15000 lacunae
per cubic millimeter of bone and the surface area of canaliculi in an adult male
22
is about 1200 m2 (trabecular bone surface area is about 9 m2 ) [1].
Osteoclasts are large cells, 150 µm in diameter [2], with multiple nuclei.
They are responsible for bone resorption, which occurs at the rate of tens µm
per day [1]. The cells first demineralize adjacent bone matrix with acids and
subsequently dissolve collagen with enzymes [1]. Osteoclasts can be found in
enzymatically etched depressions called resorption bays. The resorption of the
bone occurs through finger-like dynamic processes called ruffled borders. They
are connected to the portion of the cell which creates a microenvironment for
bone resorption called a clear zone [36]. Ruffled borders reflect the activity of
the osteoclasts and thus bone resorption rate. The lack of ruffled borders can
lead to osteopetrosis, a disease characterized by dense bone [36].
2.2
Trabecular bone composition
As an organ of the skeletal system, bone consists of calcified bone tissue, blood
vessels, hemopoietic, fat and nerve tissues. In the synovial joints, articular
cartilage covers the heads of bones.
Osteoid, i.e., the bone extracellular matrix, may be lamellar, i.e., deposited
in regular parallel sheets, or woven, i.e., deposited haphazardly. Bone formed
from lamellar osteoid is strong and healthy. In contrast, tissue synthesized on
the basis of woven osteoid is comparatively weak and pathological [37]. Osteoid is composed of organic and inorganic components. These two components
comprise 35% and 65% of bone dry weight, respectively [38, 2]. Collagen embedded in a supporting gel containing specific proteins constitutes the organic
phase. Deposition of the inorganic phase in the form of mineral salts into this
gel provides bone with rigidity and functional strength [37].
Collagen type I constitutes 90% of the organic phase of bone matrix. It
is highly cross-linked and organized into strong fibres typically 50 − 70 nm in
diameter [38, 2]. It also provides substrate for bone mineral crystals nucleation
[38]. The gel, or ground substance, wherein the collagen is embedded, contains
proteoglycan aggregates and several specific glycoproteins. Glycoproteins are
covalently bound to short proteoglycan cores and are responsible for promoting
bone matrix calcification [34, 37, 36].
Calcium and phosphorous, along with other components such as magnesium,
sodium and potassium, constitute the inorganic phase of bone matrix. Calcium
and phosphorous form hydroxyapatite crystals Ca10 (P O4 )6 (OH)2 in a rod-like
form with a length 40 nm, width of 25 nm and thickness of 1.5 − 3 nm. The
crystals reside mainly in gaps roughly 40 nm long and 25 nm wide formed by
collagen fibers [38]. Additionally to crystal hydroxyapatite, calcium phosphate
is also present in amorphous form. In adults, the mineral phase constitutes
about 65% of the fat-free dry weight. However, in individuals suffering from
e.g., osteomalacia, which is characterized by poorly calcified pathological bone,
the mineral content may be as low as 35% [38].
About 60% of the water in the calcified bone matrix is free and about 40%
is bonded to molecules [1]. The surface ions of hydroxyapatite are hydrated,
facilitating the exchange of ions between the crystals and the body fluids [36].
Nonmineralized spaces within trabecular bone contain marrow, which is
highly cellular soft tissue consisting of the precursors of blood cells, macrophages
and adipose tissue [38]. The main function of marrow is to generate blood cells
[1]. In infants, all bones contain red hemopoietically active marrow. At the age
2.2 Trabecular bone composition
23
of 4-5 years the number of blood-forming cells decrease with a simultaneous increase in adipose cells. As the amount of adipose tissue grows, the colour of the
marrow changes from red to yellow [38]. Under an appropriate stimulus, such
as extreme blood loss, yellow marrow can be converted into red [34]. In adults,
red marrow can be found in the proximal ends of the humerus and femur, and
in the vertebrae, ribs, sternum and ilia of the pelvis [38].
24
Chapter III
Mechanical properties of trabecular bone
3.1
Mechanical testing of bone
In three dimensions, stress κik on the plane i along the direction k can be defined
as follows:
dFi =
N
X
κik dAk ,
(3.1)
k=1
where dFi is the force acting on a small element represented by a vector dAk
which has a length equal to the area of the element, is perpendicular to it and
faces outwards. Stress can be categorized, based on the manner how the force is
applied, into compressive, tensile or shear. Compressive stresses are developed
as material is compressed along its length. A deformation of an object in such
a way that one region is shifted parallel with respect to the adjacent one will
generate a shear stress.
Strain ² is defined as a relative change in the sample dimension. When
bone is deformed in length, its width will normally change: for instance, when
compressively loaded, the tissue will bulge. The ratio of transverse to axial
strain is called Poisson’s ratio.
A compressive test is a common way to determine the mechanical properties
of a material. In this test a sample is compressively loaded, and the strain is
recorded as a function of the stress applied, κ(²) (Figure 3.1). The stress-strain
curve can be divided into two parts: the elastic and plastic deformation region.
Within the elastic, or linear, region, the deformation increases linearly with
increasing load, and when the load is released the sample retains its original
shape. However, due to the presence of the fluids in the bone matrix, some of
the elastic energy is lost, causing viscous effects during the deformation [39].
The slope of the linear response is known as Young’s modulus. The area
under the linear region of the stress-strain curve is called resilience, U :
Z ²y
U=
κ(²)d².
(3.2)
²0
Resilience is a measure of the capacity of a material to store energy while elastically deformed and can be interpreted as the resistance of an elastic material
to deform under an applied force. At yield point the elastic deformation ends
25
26
3. Mechanical properties of trabecular bone
k
Yield point
k( )
'
kmax
ky
E
plastic region
elastic
region
y
'
'
'
0
Figure 3.1: Stress-strain (κ – ²) behavior of trabecular bone. The curve is
divided by a yield point into elastic and plastic strain regions. Yield stress,
κy , and yield strain, ²y , are recorded at the yield point. Young’s modulus, E,
is defined as the slope of the linear part of the stress-strain curve. Resilience
represents the amount of energy stored during elastic deformation. Ultimate
strength, κmax , marks the point where the fracture of bone occurs. The ²0
denote the zero strain.
and a material starts to deform plastically. Strains within the plastic strain
region are called postyield strains. They represent permanent deformations of
bone structure induced by trabecular microfracture, crack growth, slip at cement lines or a combination of these [39, 40]. Ductility represents the amount
of postyield strain that a material is able to sustain before fracture. In contrast
to a ductile material, a brittle material can support no or only minor post-yield
strain and, thus, is liable to fracture.
If the load continues to increase, the maximum stress a material can withstand, or ultimate strength, is reached. As such, strength is an intrinsic property
of bone, i.e., its values do not depend on the size and shape of a sample [39]. The
ultimate strength of trabecular bone on the tissue level varies from 1 MPa to 20
MPa and is strongly dependent on apparent density and trabecular orientation
[41, 39]. As the load continues to increase, bone does not fracture into pieces
and the stress does not fall to zero, but is followed by a plateau. Bone marrow
spaces collapse and trabeculae are compacted against one another. When all
void spaces are filled with fractured bony matrix, the stress rises again.
3.2
Determinants of trabecular bone mechanical properties
The mechanical properties of trabecular bone are determined by the material
properties of the tissue in the individual trabeculae, tissue apparent density and
the anisotropy of the trabecular structure [42, 1].
3.3 Damage of trabecular structure
27
The material properties of the tissue on the trabecular level, i.e., the degree
of mineralization of the extracellular matrix and the variations in canalicular
porosity, influence the mechanical performance of whole bone [43]. Even though
trabecular and compact bone are considered to consist of the same tissue [3, 44],
there are some differences in the elastic modulus. The largest value of Young’s
modulus, 14.8 GPa, reported in the literature for individual trabeculae is smaller
than the frequently quoted modulus for dense cortical bone, E = 17.1 GPa
[43, 45].
Young’s modulus and strength depend strongly on the apparent density [46]
and thus on the bone volume fraction. The dependence between Young’s modulus and density has been reported to be cubic [44] or squared [47] for multiple
anatomical sites analyzed together, or linear for a single site when a sample is
loaded along the main trabecular orientation [48, 49, 50]. These differences may
be a consequence of the imperfections introduced during mechanical testing or
of the neglect of the importance of anisotropy. However, bone volume fraction alone cannot completely explain the variation in elastic properties. Both
Young’s modulus [51, 52] and the strength [50, 51] of trabecular bone have been
found to depend on the orientation, spacing, number and size of trabeculae
[53, 46] as well as on the anisotropy of these variables [1]. Consistent with
Wolff’s law [54] anisotropy is a form of adaptive response to functional loading.
Trabecular bone has been reported to display orthotropic symmetry1 [55, 56], or
in some cases transverse isotropy2 [57]. For instance, the mean value of Young’s
modulus of human vertebral bone in the superior-inferior direction is 3.4 times
higher than that in the transverse directions [51, 46]. Remarkably, bone volume
fraction together with trabecular orientation and anisotropy is able to explain
about 90% of the variation in mechanical properties [58].
In addition, organic composition was found to be imbalanced in certain
diseases such as osteoporosis [59, 60, 61]. Thus, it has a significant impact
on tissue mechanical properties. In particular, collagen has been shown to
be a significant determinant of bone toughness [62]. The orientation of the
collagen fibers relates to the tissue strength in such a way that longitudinal
fibers promote strength in tension, while transverse fibers promote strength in
compression [1, 63, 64].
3.3
Damage of trabecular structure
During the course of daily activities skeletal tissue is subjected to repetitive
cyclic loading. A fatigue failure may occur if this loading is of sufficient magnitude or duration. Compressive stress fractures in the proximal femur, calcaneus,
and proximal tibia are the most common fatigue fracture sites observed clinically
within trabecular bone [65]. The physical damage that occurs with overloading will not start with a fracture of entire trabeculae, but with subtle damage
within them [46, 66, 67]. Cracks extending only over a part of trabeculae are
called microcracks [65] and can be as small as 10 µm in length [68]. They can
be repaired by microcallus formation. On the other hand, fatigue microfractures, or complete fractures of individual trabeculae, are confined to elements
1 An orthotropic material is an anisotropic material that possesses different properties in
each of three perpendicular directions [39].
2 Transversely isotropic material is an orthotropic material whose properties are the same
in two of the three directions, i.e. in a plane [39].
28
3. Mechanical properties of trabecular bone
oriented transversely to the loading direction. They originate from age-related
femoral neck fractures, aseptic necrosis of the femoral head or degenerative joint
diseases. They might trigger trabecular bone remodeling.
Fractured trabeculae are likely to be completely resorbed by osteoclasts [4,5]
and may not be replaced during tissue repair [4,6,7]. Hence a microfracture
weakens the tissue by decreasing the load-carrying capacity of the trabecular
structure [69] and has a permanent detrimental effect on bone volume fraction,
microarchitecture, and its mechanical properties [70]. The resistance of trabeculae to fracture is determined by their ductility which, in turn, is governed by
collagen cross-linking [71, 70].
Chapter IV
Computed tomography of bone structure
Modern high resolution computed tomography (microCT) methods enable the
gathering of three-dimensional images of trabecular bone with micrometer resolution. Based on the cross-sections of an object obtained by means of X-ray
radiation, a virtual 3D model of the specimen is reconstructed nondestructively.
In the following, a microCT technique and computational methods for determining quantitative structural parameters are introduced.
The analysis of the 3D images begins with separating hard tissue from soft
tissues. For this purpose, a segmentation of the original gray-scale data set is
introduced and a single CT number is selected with a histogram technique [72].
Voxels with a CT value higher than the predefined threshold are interpreted as
bone whereas the other voxels are defined as soft tissues. The morphological
parameters can be determined from such a segmented data set using various
algorithms [72, 73, 74, 73, 75, 76]. For instance, a ”marching cubes” method has
been suggested for modeling the bone surface using continuous triangles (surface
triangulation) [77, 78]. As the segmentation method affects significantly the
calculated values of structural parameters [72, 74, 79], it has been suggested that
using local threshold values instead of global ones may improve the segmentation
[80].
The tissue volume (TV) is the volume of the entire sample [81]. It can be
calculated by multiplying the total number of voxels within a sample, both solid
and void, by voxel volume. Bone volume (BV) is calculated similarly to TV,
but only voxels of the solid phase are counted. BV can be normalized with TV
to obtain the bone volume fraction (BV/TV).
Bone surface (BS) is defined as the total area of the triangles. In order
to enable the comparison of surfaces among samples with different sizes, BS
can be normalized with BV to obtain a surface-to-volume ratio, BS/BV. This
parameter characterizes the complexity of bone structures.
The degree of anisotropy, DA, reflects the orientation of trabeculea, their
main direction and the degree of dispersion around it [82]. It is a dimensionless
quantity defined as a ratio of eigenvalues of primary and tertiary direction [82,
58].
The degree of anisotropy can be determined using mean intercept length
(MIL) analysis [83]. A circular region of interest (ROI) is defined on each 2D
microCT slice. The areal density, Pp , of the ROI is calculated as the ratio of
29
30
4. Computed tomography of bone structure
the number of bone pixels to the total number of pixels in the analysis region
[58]. A reference direction is defined. A grid of parallel test lines is sent through
the ROI at an angle ϕ with respect to a reference direction. For each test
line the number of intersections between the hard and soft tissues is counted.
Subsequently, the number of intersections per unit length of test line, PL (ϕ) is
calculated by dividing the total number of intersections for all test lines by the
total lines length. Thus the MIL for trabecular bone is defined as follow:
M IL(ϕ) =
2Pp
[58].
PL (ϕ)
(4.1)
As the grid of test lines is rotated of ∆ϕ, the MIL can be determined as a
function of the angle MIL(ϕ), where 0◦ 6 ϕ < 180◦ . A polar plot of MIL(ϕ)
versus the angle ϕ can be fitted to an ellipse from which the number of trabeculae
in a certain direction can be found.
For 3D analysis, a spherical ROI is defined and a grid of lines is superimposed over this volume in a large number of planes with different angles.
The MIL for each plane is calculated and a polar plot of MIL versus angle is
constructed. The uneven length of the MIL will result in an elongated, not
spherical, line distribution. Using a statistical fit, an equation for the ellipsoid
can be determined:
1
= mT [A]m,
M IL2 (m)
(4.2)
where m is the unit vector in the direction of interest, and [A] is the orthogonal
second rank symmetric anisotropy tensor formed from the coefficients of the
elipsoid equation (Figure 4.1). The principal structural directions and magnitudes can be determined by solving the eigenvalue problem for the tensor [A].
The degree of anisotropy is defined as the maximum (MILmax ) to minimum
(MILmin ) MIL ratio:
DA =
M ILmax
.
M ILmin
(4.3)
The mean trabecular thickness, (η), can be determined as an average of the
local thickness of the individual trabeculae [84, 73]:
Z Z Z
1
η=
η(x)d3 x,
(4.4)
V (Ω)
Ω
where
Z Z Z
d3 x
V (Ω) =
(4.5)
Ω
and Ω is the set of points in the trabecular structure under investigation. The
local thickness for a point in solid is defined as the diameter of the largest sphere
which contains the point (p) and fits completely inside the trabeculae [73] (Figure 4.2). Trabecular bone volume can also be characterized as a distribution of
thicknesses. For instance, for a specimen showing a large variation of trabecular
thickness, i.e., having thick plates interconnected with thin rods, the thickness
distribution will be widespread. In contrast, a homogenous structure with plates
and rods of almost constant thickness will yield a narrow distribution [73].
31
vector 1
vector 2
vector 3
Figure 4.1: Using a statistical fit, an ellipsoid can be fitted to the 3D distribution of mean intercept length.
P
d
x
W
Figure 4.2: Local thickness for a point p in solid structure Ω is defined as
the diameter d of the largest sphere with the center in x, which contains the
point p and fits completely inside the trabeculae.
The trabecular separation is the thickness of the marrow spaces between
trabecular structures. It can be calculated by applying the same technique as
used for the trabecular thickness calculation to the non-bone parts of the 3D
image [84].
The trabecular number, T b.N., is defined as a number of intersections with
solid per unit length of a linear path through the trabecular structure. It can
be calculated from the following equation:
T b.N. =
BV /T V
.
T b.T h.
(4.6)
32
4. Computed tomography of bone structure
Connectivity reflects the degree to which a structure is interconnected [85].
It is an important parameter, reflecting tissue integrity. For instance, a decrease in connectivity in osteoporotic bone may reflect a structure for which the
restoration of original architecture will be difficult [86]. Trabecular bone can
be represented as a node-and-branch network, where more than one path exists
between any two nodes (unlike in the case of a tree). A number of branches
may be cut in this structure without separating the network. In contrast, in
the case of a tree, cutting one branch will split the unit into two parts [76].
The maximum number of trabecular connections that can be ruptured without
dividing the structure is defined as connectivity [87]. It can be interpreted as
the number of trabeculae minus one [76].
The Euler number χ is a determinant of connectivity:
χ = δ0 − δ1 + δ2 ,
(4.7)
where δ0 is the number of disconnected bone pieces, δ1 is the connectivity, and
δ2 is the number of marrow cavities enclosed in bony matrix [87]. Equation
(4.7) can be simplified, assuming that the trabecular bone matrix is one interconnected structure (δ0 = 1), and that the marrow is placed in one main marrow
space, and thus no marrow cavities are enclosed within the bone (δ2 = 0) [76]:
χ = 1 − δ1 .
(4.8)
Equation (4.8) holds for all cases except bone during formation and healing,
where isolated tissue parts may exist [85].
The structure model index (SMI) reflects shapes in the trabecular structure,
i.e., plate-like, rod-like or a combination of the two. An ideal plate structure
displays the SMI value of zero, while for an ideal rod structure the value is three.
The intermediate values depend on the rods-to-plates volume ratio [88]. Osteoporosis of trabecular bone is characterized by a transition of trabeculae from
plate-like to rod-like shapes [75]. Determination of SMI requires an artificial
addition of one voxel of thickness to all binarised object surfaces, or dilation. It
can be calculated as follows:
µ
¶
BS 0 · BV
SM I = 6
,
(4.9)
BS 2
where BS is the trabecular surface area before dilatation, BS 0 is the change in
the surface area after dilatation and BV stands for the undilated volume of the
trabeculae [89].
Chapter V
Electrical impedance spectroscopy (EIS) of bone
The bulk electrical properties of tissues determine the pathways of current flow
through the body [90]. They are essential in research on the interaction mechanisms of electromagnetic waves with biological systems [90], especially in calculating the specific absorption rate and its distribution in animal and human
models [91]. They enable the measurement of physiological parameters using
impedance techniques. Further, the theoretical description of electrocardiography, muscle contraction, nerve transmission and electrosurgery would not be
feasible without detailed knowledge of the electrical parameters [90].
5.1
Introduction to the relaxation theory
If a material is placed in the parallel-plate capacitor of a plate area A and
separation d, then an electrical field between the plates will induce a charge
density. The passive electrical properties are characterized by capacitance C
and conductance G that can be calculated as follows:
G=
Aσ
d
(5.1)
and
Aε
Aεr ε0
=
.
(5.2)
d
d
Conductivity σ describes the readiness of charged particles to move through a
material under the influence of an electrical field. The factor ε0 is the permittivity of free space (8.854 · 10−12 F m−1 ), ε is the permittivity of a material and
εr is the permittivity of a material relative to the permittivity of free space.
Permittivity indicates the extent to which charge distribution can be polarized
when an electrical field is applied. In biological materials the charges are associated with either polar molecules or electrical double layers near membrane
surfaces or solvated macromolecules [92].
Each polar or polarisable particle in a material exhibits characteristic responses to the applied electrical field. The field disturbs the system. After the
excitation has ceased, the system relaxes to a new equilibrium. Relaxation in the
time domain can be described by a relaxation time τ . This parameter depends
on the polarization mechanisms in a material. The shortest τ within times of
C=
33
34
5. Electrical impedance spectroscopy (EIS) of bone
a picosecond is connected to electronic polarization. On the other hand, the
polarization of large organic molecules such as proteins can take up to seconds.
The Debye relaxation model describes a simple case. If a material excited by
a step voltage at time t = 0 possesses only one relaxation process with a single
characteristic time constant τ , then the surface charge density on a capacitor
plates D is given by:
D = D∞ + (D0 − D∞ )(1 − e−t/τ ).
(5.3)
The instantaneous response of a system is included in D∞ . This parameter is a
surface charge density just after the step voltage has been applied (t = 0+ ) and
arises from electronic polarization. In contrast, D0 is the charge density long
after the application of the step, when a system has obtained a new equilibrium.
The response of this system in a frequency domain can be obtained by a Laplace
transform:
ε∗ = ε0 − jε00 = ε∞ +
εs − ε∞
.
1 + jωτ
(5.4)
es
80
60
es - e
8
Relative permittivity
The factor ε∞ denotes permittivity measured at a high frequency where the
polar or polarisable particles are not able to respond to the applied electrical
field. Static permittivity εs is the limiting low-frequency permittivity where
the polarization is fully manifested (Figure 5.1A). It must be emphasized that
parameters ε∞ and εs are used at frequency extremes, and thus are real: ε∞ =
ε0∞ and εs = ε0s [93]. The factor ω is the angular frequency of the electrical field.
40
20
8
e
0
109
1010
Frequency (Hz)
1011
80
8
s
60
40
s - ss
8
Conductivity (S m-1)
A. 108
20
B.
0
ss
8
10
109
1010
Frequency (Hz)
1011
Figure 5.1: Dielectric dispersion of pure water at 20◦ C [92].
5.1 Introduction to the relaxation theory
35
The real part of the complex permittivity ε0 , corresponding to the permittivity in equation (5.2), can be expressed in the form:
ε0 = ε∞ +
εs − ε∞
.
1 + (ωτ )2
(5.5)
Dielectric loss, or the imaginary part of the complex permittivity ε00 , reflects
the losses associated with the movement of charges in phase with the electric
field. It is related to the in-phase (lossy) component of the conductivity:
ε00 =
σ0
(εs − ε∞ )ωτ
=
.
ωε0
1 + (ωτ )2
(5.6)
The characteristic relaxation frequency fc corresponds to the maximum value
of the dielectric loss factor ε00 (ωτ = 1) and is given by:
1
.
(5.7)
2πτ
Using an appropriate model for the Debye single relaxation model, the complex conductivity can be determined. In particular, the real (lossy) part is given
by:
fc =
σ0 =
(εs − ε∞ )ω 2 τ ε0
(σ∞ − σs )(ωτ )2
=
.
2
1 + (ωτ )
1 + (ωτ )2
(5.8)
The above equation follows from the fact that the limit values εs , ε∞ , σs and
σ∞ (Figure 5.1B) are interrelated as follows:
(εs − ε∞ )ε0
= 2πfc (εs − ε∞ ).
(5.9)
τ
Equation (5.9) is a special case of the Kramers-Kronig relation that interrelates permittivity and losses in the system. It states that the permittivity and
conductivity of a material cannot vary independently: any decrease in permittivity with increasing frequency yields an increase in conductivity. This relation
can be explained by the fact that for a given voltage the total energy of the electric field is constant. It is either stored or dissipated by the material the field is
interacting with. The real part of a complex permittivity (ε0 ) expresses storing
of the energy, while the imaginary part (ε00 ) reflects the losses. Equation (5.9)
holds only for a system with a single relaxation time. However, any material
with a relatively small spread of relaxation times can be approximated by this
simple form of the Kramers-Kronig relation.
The ionic dc conductivity σdc is often added to the equation (5.8) even
though it does not represent a relaxation mechanism [93]:
σ∞ − σs =
σ 0 = σdc +
(σ∞ − σs )(ωτ )2
.
1 + (ωτ )2
(5.10)
A Cole-Cole plot or a plot of ε00 against ε0 with frequency as a parameter
yields a semicircle with a center on the ε0 axes (Figure 5.2). It intersects the ε0
axes at points ε∞ and εs . A similar plot can be obtained for conductivity. The
maximum values of ε00 and σ 00 correspond to the characteristic frequency fc .
In contrast to the simple Debye model of a material with a single relaxation
time, biological systems are characterized by a multiple relaxation processes
36
(corresponding to charge densities D1 , D2 , ...), usually each with different relaxation times (τ1 , τ2 , ...):
D = D∞ + D1 (1 − e−t/τ1 ) + D2 (1 − e−t/τ2 ) + ...
(5.11)
0
Also, the center of the Cole-Cole semicircle is shifted below the ε axes (Figure
5.2). To meet this problem, a simple modification in Debye’s equation has been
made by Cole and Cole [94]:
ε∗ = ε∞ +
εs − ε∞
.
1 + (jωτ )1−α
(5.12)
The α-parameter in the above equation is purely experimental and thus lacks
a proper physical or molecular basis [92]. For most biological materials the αparameter falls in the range between 0.3 and 0.5 [90]. For compact bone this
parameter was found to be 0.6 [95].
wt=0
-je
w
e
e
8
es
p(1-a)
2
Figure 5.2: The complex permittivity plot of the Cole-Cole equation (5.12)
[90, 93]. The dotted line corresponds to the single Debye relaxation.
The general form of a Kramers-Kronig relation for a material with multiple
relaxation times is given by:
Z
2 ∞ xε00 (x)
dx
π 0 x2 − f 2
Z
σs
−2f ∞ ε0 (x) − ε∞
ε00 (f ) −
=
dx,
ωε0
π
x2 − f 2
0
ε0 (f ) − ε∞ =
(5.13)
where x is the variable of integration. This relation holds for dielectric materials
with any distribution of relaxation times, even if the system displays a resonance
behavior [90].
5.1 Introduction to the relaxation theory
37
The electrical relaxation in tissues can be described in terms of only three
parameters: the dielectric increment εs −ε∞ , the relaxation time τ (alternatively
characteristic frequency fc ) and the α-parameter [92].
Dispersion is the corresponding concept to relaxation in the frequency domain [93]. Electrical permittivity in biological tissues decreases with increasing
frequency. It is a consequence of different electrical charges not being able to
follow the changes in the applied electrical field. Three major drops in the permittivity value, or dispersions, are observed, named consecutively the alpha,
beta and gamma dispersions (Figure 5.3). In addition, a tissue may exhibit a
small dispersion between 0.1 and 3 GHz called a delta dispersion [90]. Clearly
separated single dispersions are present typically in cell suspensions. In contrast, tissues display much broader and overlapping dispersions, occasionally in
the form of a continuous fall almost without plateaux and over a wide range of
increasing frequencies [93].
108
a
5
10
b
102
g
102
106
Frequency
1010
Figure 5.3: The value of permittivity decreases with increasing frequency.
Three major drops are distinguished: the alpha (α), beta (β) and gamma (γ)
dispersions.
In the alpha dispersion region, relative permittivity attains values as large as
106 − 107 [90, 93]. This can be explained by e.g., counterion effects or charging
of intracellular organelles [90] and does not mean that the capacitive properties
of living tissue are dominant [93]. On the other hand, the alpha dispersion is
hardly noticeable in the conductivity. Tissue conductivity is small compared
with background conductivity. Theoretically the increase in conductivity at 100
Hz associated with the alpha dispersion would be about 0.005 Sm−1 , provided
that the value of permittivity is around 106 . However, the ionic conductivity of
typical tissues is about 200 times higher. It follows that at low frequencies, despite the enormous permittivity values, the impedance of most tissues is resistive
[90]. For beta and gamma dispersions the associated increase in conductivity is
more pronounced (Table 5.1).
It is worth emphasizing that dispersion is related to the electrical and physical behavior of molecules, as opposed to such analytical techniques as nuclear
38
Table 5.1: Dielectric dispersions [90, 93].
Type Characteristic
frequency
α
mHz − kHz
β
0.1 − 100M Hz
γ
0.1 − 100GHz
δ
0.1 − 3GHz
Mechanism
Counter ions effects,
Ionic diffusions,
Active membrane effects,
Charging of intercellular structures
Capacitive charging of
cellular membranes,
Dipolar orientation of
tissue proteins
Dipolar mechanisms in
polar media such as water
salts and proteins
Dipolar relaxation of water,
Rotational relaxation of
polar side chains,
Counterions diffusion along
small regions of charged surfaces
Increase in
conductivity
∼ 0.005 Sm−1
∼ 0.4 Sm−1
∼ 70 Sm−1
∼ 0.5 Sm−1
magnetic resonance (NMR), which examine molecular structure [93]. These
analytical methods are based on resonance phenomena. In contrast, dielectric
spectroscopy below 1 GHz deals with relaxation, routinely presented as a nonresonance event. Of course, if the frequency is sufficiently high, a resonance
might be observed. For instance, the resonance frequency for deoxyribonucleic
acid (DNA) molecules is in the lower gigahertz region [93].
5.2
Microwave frequency measurements
A high inductance of electrode leads makes the use of a parallel-plate measurement cell at frequencies above 10 MHz problematic. Although a method
allowing an accuracy of the order of 1% up to frequencies of 100 MHz has been
designed [96], it is accepted that the classical techniques should not be used for
high radiofrequencies and microfrequencies [97].
Measurements at above 10 MHz are based on the transmission-line theory.
A transmission line is characterized by its characteristic impedance Z0 . It is
defined as a ratio of the longitudinal current to the transverse voltage in the
line. The voltage and current waves move along the line in opposite directions
with a complex propagation constant γ ∗ :
γ ∗ = β + j2π/λ,
(5.14)
where β is the attenuation coefficient and λ is the wavelength. When the transmission line is terminated with a load that has impedance different from Z0
then the waves will be reflected at the point of the discontinuity. A complex
reflection coefficient (Γ∗ ) can be defined as the ratio of the amplitude of the
incident voltage (ue ) to the the amplitude of the reflected voltage (u0e ) at the
reflection point:
5.2 Microwave frequency measurements
Γ∗ =
39
ue
.
u0e
(5.15)
The impedance of the load, Ze can be expressed in terms of the characteristic
impedance of the line and the reflection coefficient:
Ze =
Ue
ue + u0e
Z0 (1 + Γ∗ )
= u
=
,
0
ue
e
Ie
1 − Γ∗
Z0 − Z0
(5.16)
where Ue and Ie are the voltage and current through the load, respectively. It
follows from equation (5.16) that it is possible to determine the impedance of
the load from the measurement of the reflection coefficient, Γ∗ .
Conductors
Insulator
Coaxial
probe
Cf
e*C0
Coaxial
cable
B.
A.
Figure 5.4: The microwave frequency measurements are conducted with a
coaxial probe consisting of one layer of insulating material between two conductors. The probe is connected to the network analyzer through a coaxial
cable (A). A coaxial probe can be represented electrically as a parallel connection of two capacitors (B).
The function of an open-ended coaxial line at frequencies where its inner and
outer diameters are small compared with the wavelength of the electromagnetic
field, can be represented as a parallel connection of two capacitors (Figure 5.4):
CT = Cf + ε ∗ C0 .
(5.17)
In the above equation, CT is the total capacitance, which is independent from
the electrical properties of the load. The capacitance Cf is called the fringe
capacitance and represents the energy of the field inside the line. The terminal
capacitance, ε∗ C0 , represents the energy inside the load, or the material under
investigation. The term C0 is the terminal capacitance of the line in the air
and ε∗ is the complex permittivity of the dielectric material. On the basis of
the model depicted in Figure 5.4 and on equation (5.16), ε∗ can be determined.
The complex admittance can be written:
Ye∗ =
1
= jω(Cf + ε∗ C0 ).
Ze∗
(5.18)
Hence, when the reflection coefficient is measured and C0 and Cf are determined, e.g., from the calibration measurements, the complex permittivity of a
material can be calculated from:
40
ε∗ =
1
1
1 − Γ∗
Cf
(
− Cf ) =
−
.
∗
C0 jωZe
jωZ0 C0 (1 + Γ∗ ) C0
(5.19)
When a lossy material, e.g., trabecular bone, terminates the coaxial line,
then the term ε∗ C0 also contains a conductance representing the dielectric losses
according to the equation (5.6):
ε∗ = ε0 − jε00 = ε0 −
jσ 0
.
ωε0
(5.20)
Thus, the conductivity can be obtained from:
σ 0 = ε00 ε0 ω.
(5.21)
The measurements can be performed in the frequency or time domain. In
the former case, the excitation is produced by a radiofrequency generator. In the
latter case a pulse generator with a very short rise time is used for excitation.
The reflection coefficient is measured at the input of the line by a network
analyzer.
5.3
Limiting Factors of EIS
The major limiting factors of impedance spectrometry include the polarization
impedance of the measurement electrodes, the geometry of the measuring cell
and the material used for the electrode.
Polarization impedance. During the measurement, the acquisition of the
potential and injection of the current into the sample occurs through the electrodes. However, due to the finite impedance of the electrode material, complex
electrochemical reactions can occur at the interface between the tissue and electrode, leading to polarization phenomena. As these phenomena may introduce
errors in the measurements, the polarization impedance and low frequency limit
above which the measurements can be carried out with sufficient signal-to-noise
ratio shall be determined [97]. High polarization impedance is especially pronounced in a system where the same set of electrodes is used to inject current
and measure the voltage.
When a metal is placed into an electrolyte, two current transport mechanisms occur [93]. First, when the charges are moved across the electrode-tissue
interface, a faradic current will result. Second, a non-faradic, or capacitive, current will originate from the electrical double layer that acts as a capacitance.
Due to the molecular scale distance between layers, the capacitance per unit
area is large. The current across the electrode interface will charge this capacitance and results in its polarization. The combination of the faradic and
non-faradic mechanisms acts as a high-pass filter [97]. For signals with relatively
small amplitude and frequencies approximately above 100 Hz, the polarization
impedance Zp can be represented as a resistance R and reactance X in series:
Zp = R − jX = R +
1
.
jωC
(5.22)
The polarization impedance decreases with increasing frequency. Its value depends on the electrode material and the nature of the electrolyte. The series
41
resistance in equation (5.22) decreases as the frequency increases. The magnitude of the capacitance depends on the surface condition of the electrode and
is lowest for polished surfaces and highest for roughened ones [98].
Cell factor. The use of simple mathematical tools for analyzing the measurement system, such as Ohm’s law, is possible when the recorded signals are
determined solely by the input stimulation, the electrical parameters of the investigated material and the geometry of the electrode system. However, this
assumption of linearity is fulfilled only when the electrical field is uniform and
undisturbed by objects placed in the measurement cell [97]. The errors introduced by the measurement cell can be determined by investigated a phantom
material as a function of frequency and the geometry of the cell. The complex
cell factor K ∗ can be represented as
K ∗ = K 0 + jK 00 =
Zpha
,
Zth
(5.23)
where Zth is the theoretical impedance of the phantom and Zpha is the impedance
of the phantom measured in the cell of cross-section S with electrodes separated
by d:
Zpha = [1/σpha ][d/S].
(5.24)
∗
Ztissue
The tissue impedance
can be estimated from the measured impedance
spectrum Z ∗ corrected by the cell factor:
∗
Ztissue
= Z ∗ /K ∗ .
(5.25)
Electrode material. The polarization impedance Zp is strongly affected
by the electrode material [97]. For a given electrolyte, enlarging the electrode
surface area decreases the cut-off frequency of the high-pass filter equivalent
to the polarization impedance. Additionally, the risk of infections or allergies
due to the toxic or catalytic effects of the electrode surface must be taken into
account.
Silver-silver chloride electrodes are the most common in electrophysiological
measurements. They exhibit a stable potential and are easy to manufacture in
any desired shape [97]. However, the potential may change with exposure to
light, and the chloride coating can be easily removed by abrasion.
Stainless steel electrodes are cheap and easy to use. However, they exhibit a
high polarization impedance at low frequency. The frequency characteristic can
be improved by increasing the roughness of the electrode, e.g., by grinding with
sand paper [99, 100]. The roughness of the electrode can disappear with time,
and after a few measurements the polarization impedance approaches that of
polished stainless steal.
The polarization impedance of platinum electrodes is lower than that of
other materials such as stainless steel or silver [101, 102]. If metal electrodes
cannot be used, then porous carbon fiber electrodes have been suggested as an
alternative [103].
5.4
Electrical characteristics of trabecular bone
The frequency dependence of bone electrical properties [104, 105] is comparable
to that of other tissues with low cellular content, e.g., ligaments [104]. The
42
relatively low water content of these tissues might be the main reason for the
differences in permittivity values between tissues with low and high cellular
content [104].
A significant contribution to relative permittivity is expected from the polarization of ions near charged surfaces in the tissue. A large number of interfaces,
e.g., between lamellae, fibrils and in many small capillaries that penetrate the
tissue, affect electrical properties. The polarization of membranes within the
tissue and fluid-solid interfaces has been hypothesized to have a lesser effect
[90].
The electrical properties of compact bone have been widely studied [106,
107, 108, 109, 110, 111, 112, 95, 104, 113, and others]. The tissue has been investigated in a dehydrated [114] state as well as in a range of moisture content
[115]. To obtain results resembling those of in vivo conditions, bone tissue has
been measured wet [116]. Also, there have been some attempts to conduct measurements in vivo [30, 117, 118, 119]. Recently, Skinner et al. [31] reported that
the healing process during corticotomy and distraction in the Ilizarov technique
can be followed by impedance spectroscopy.
In contrast to the extensive data existing on compact bone, few reports
have been published on trabecular bone (e.g., [120, 104, 121]). Moreover, many
studies have just hypothesized the interpretation of results leaving certain phenomena unexplained. Despite the fact that compact and trabecular bone are
quite different in physiological state (due to e.g., differences in porosity and
the inclusion of marrow in trabecular tissue), some findings obtained in investigations of compact bone are applicable to trabecular bone. The conclusions
drawn from studies of dried tissue (e.g., [114]) serve as a good example. The
calcified matrixes of compact and trabecular bone are very similar (see Chapter 2). However, the findings should be carefully interpreted when describing
fluid-saturated matrix. The electrical behavior of hydrated matrix containing
water bound to its molecules differs from the behavior of a dry network. Further, the results obtained in experiments on decalcified compact tissue [122] or
as a function of immersion fluid [112] are useful in analyzing trabecular bone,
remembering that it is the interaction between different tissue components and
their internal microstructure that govern the electrical properties.
The electrical properties of different bone components, such as bone marrow
[123, 21], collagen [124], hydroxyapatite [125] and water [126] have been studied.
The combination of these components is responsible for the overall electrical
properties of trabecular tissue, and thus the data should be used with care
and interactions between the phases should not be omitted. Typical values
of relative permittivity and conductivity for bone and marrow in physiological
state are presented in Table 5.2 and Table 5.3. Due to the large variability in
low-water-content tissues, no single value of permittivity and conductivity can
be presented [123]. A range of values or an approximate number is given, as the
exact composition of the tissue is not known.
Trabecular bone was found to be electrically transversely isotropic. Relations
between the electrical properties in different directions have been studied [104].
In particular, resistivity in longitudinal RL direction was related to resistivity
in anterior-posterior direction RAP as follows:
RAP = −1.70 + 1.23RL .
(5.26)
43
Table 5.2: Typical values of relative permittivity for bone and marrow in
physiological state.†
Type of
tissue
Trabecular
bone
Species
10 kHz 1 MHz 10 MHz
Human distal tibia [121, 104]
600 - 800 50 - 80
Bovine femur [120]
∼ 6000
∼ 100
Ovine skull [127]
∼ 800
∼ 100
Compact
Human distal tibia [104]
∼ 1050
∼ 80
bone
Bovine femur [120, 111, 128, 27] 600 - 1000 60 - 110
Ovine skull [127]
∼ 500
∼ 60
Bone marrow
Calf femur and tibia [123]
∼ 1000
∼ 100
∼ 50
† Values estimated from graphs.
Table 5.3: Typical values of conductivity (Sm−1 ) for bone and marrow in
physiological state.†
Type of
tissue
Trabecular
bone
Species
10 kHz 1 MHz 10 MHz
Human distal tibia [121, 104]
0.2 - 0.5 0.2 - 0.5
Bovine femur [120]
∼ 0.07
∼ 0.07
Ovine skull [127]
∼ 0.09
∼ 0.2
Compact
Human distal tibia [104]
∼ 0.05
∼ 0.06
bone
Bovine femur [120, 111, 128, 27] 0.05 - 0.5 0.08 - 0.6
Ovine skull [127]
∼ 0.3
∼ 0.8
Bone marrow
Calf femur and tibia [123]
∼ 0.2
∼ 0.2
∼ 0.3
† Values estimated from graphs.
Some attempts have been made to explain these variations in terms of trabecular
orientation. It has been hypothesized that an increase in the density of a healthy
bone results in an increase in the area of trabeculae in all directions. However, in
specimens obtained from diseased persons, such as osteoporotic bone, horizontal
trabeculae are selectively resorbed. Thus, a pathological condition of trabecular
bone may be reflected by the altered directional relations in electrical properties
[104].
In the frequency range between 100 kHz and 1 MHz specific capacitance
has been shown to correlate highly and significantly with wet density [121]. De
Mercato and Garcia-Sanchez [120] have suggested that trabecular bone conductivity is a consequence of the tissue microstructure and liquid content. The
permittivity of trabecular bone is additionally affected by bone marrow, which
exhibits its own dielectric behavior. Moreover, the relative contribution of bone
matrix to the total impedance has been found to be more significant at higher
frequencies (i.e., above 100 kHz) [121, 120].
The electrical properties of bone are affected by various factors, e.g., moisture content, temperature, immersion fluid, and the age and disease state of the
person from whom the specimen was obtained, in the following ways.
Moisture content. It has been suggested that the electrical behavior of
a material is connected to the rotational mobility of water molecules [105].
44
An increase in conductivity of almost an order of magnitude per percent of
moisture content has been found [129]. Possibly at 98% relative humidity, only
the small pores of bone are filled with fluid, leaving the large pores empty
[130]. Thus, if the tissue is allowed to dry, only solid phase and perhaps a
thin layer of water absorbed on it is measured. Already a 5-minute exposure of
bone tissue to air already affects significantly the measured electrical values. For
instance, resistance may increase by 92% [105, 131]. On the other hand, in most
measurement geometries the surface conductivity changes due to free liquid can
increase the apparent conductivity of the sample and lead to erroneous results.
Temperature. The electrical properties of bone have been found to be
temperature dependent [104, 92, 132]. In particular, conductivity increases with
increasing temperature [133].
Immersion fluid. The electrical properties of fluid-saturated wet bone are
influenced by the properties of the fluid in the pores [105]. For instance, a
deviation in pH by ±2 from the neutral solution (pH = 7.0) can change the
measured value of resistance by 70% [131, 105]. Moreover, the permittivity of
compact bone has been found to exhibit a dispersion with a relaxation frequency
proportional to the conductivity of the immersion fluid [112].
Age. Little information is available on age-related changes in the electrical
properties of bone [104]. However, it is believed that the chemical changes that
occur in bone with increasing age may affect its electrical properties [104, 108,
134]. Peyman et al. [135] reported a decrease in the values of the dielectric
properties of rat tissue in the microwave frequency range due to changes in the
water content and organic composition of tissue.
Species and bone status. The electrical response is different in human
and in bovine bone. Bovine bone is generally more compact, dense and brittle
than human bone [105]. Various bone diseases affect bone microstructure and
chemical composition, e.g., osteoporosis [136, 137] or osteogenesis imperfecta
[138]. However, at present little information is available on the quantitative
effects of these factors.
Chapter VI
Aims of the present study
The electrical and dielectric properties of bone, especially human trabecular
bone, and their relations with structure, composition and mechanical properties
are poorly known. It has been suggested that electrical stimulation increases
bone growth and repair after fracture. Better knowledge of the determinants
of trabecular bone electrical properties would make it possible to calculate the
electrical field and currents during stimulation. Furthermore, knowledge of these
relations might make it possible to use electrical measurement in the evaluation
of bone composition, structure and mechanical properties in situ.
To clarify these issues, the present study aimed to:
1. Determine the optimal electrical and dielectric parameters for the assessment of bone mechanical, structural and compositional properties and
mineral density.
2. Reveal the most sensitive frequency range for the prediction of the mechanical, structural and compositional properties of human trabecular bone.
3. Investigate the interrelationships between electrical quantities and trabecular bone mechanical properties including evaluation of the EIS potential
to detect bone microdamage.
4. Understand the role of the composition and microstructure of bone in creating its electrical properties as well as describing a number of correlations
between the impedance parameters of bone and a variety of parameters
related to bone structure and composition.
5. Provide a basis for the development of a possible method of bone evaluation in vitro or in situ, as well as for the improvement of the existing EIT
technique towards clinical in vivo bone diagnostics.
45
46
6. Aims of the present study
Chapter VII
Materials and methods
Four studies (I-IV), supplemented with the data obtained from the measurements at frequencies above 5 MHz (not included in the original articles I-IV),
constitute the present thesis work. The study design is summarized in Table 7.1.
7.1
Sample preparation
Specimens for Study I were obtained from bovine proximal and distal femur
within few hours post mortem. After slaughter at the local abattoir (Atria Lihakunta Oyj, Kuopio, Finland), an entire femur was excised. Soft tissues were
removed and the epiphysis was cut out. Forty cylindrical trabecular bone samples (diameter: 25.4 mm, height: 14.2 mm) were prepared from four anatomical
locations: femoral lateral condyle (FLC; n = 9), femoral medial condyle (FMC;
n = 11), femoral head (FC; n = 10) and femoral greater trochanter (FTM;
n = 10) (Figure 7.1). The plugs were drilled in the medio-lateral direction using a hollow drill bit. The faces of the samples were cut parallel in the sagittal
plane using an Isomet low-speed diamond saw (Buehler, Lake Bluff, IL, USA).
After preparation, the samples, immersed in PBS, were sealed in plastic bags,
frozen (−20◦ C) and thawed prior to measurement.
For Studies II-IV, knee joints from thirteen human cadavers (age = 28 − 58
years, 1 female, 12 males) were obtained from Jyväskylä Central Hospital with
the permission of the Finnish National Authority for Medicolegal Affairs (TEO,
1781/32/200/01). After the knees were excised, the soft tissues were removed
and the distal femur and proximal tibia were wrapped in a bandage moistened
with saline, sealed in plastic bags and stored in a freezer (< −20◦ C) for about
12 months. The joints were thawed before sample preparation. Twenty six samples (diameter: 16 mm, height: 8 mm) were drilled from trabecular tissue of the
femoral medial condyle (FMC, n = 10), tibial medial plateau (TMP, n = 10)
and the femoral groove (FG, n = 6) in a direction perpendicular to the articular
surface (Figure 7.1), i.e., along the preferred loading direction. The end-faces
of the resulting cylindrical plugs were cut parallel with a micro-grinding system
(Macro Exakt 310 CP, Exakt, Hamburg, Germany). The samples were subsequently immersed in phosphate buffered saline (PBS) in a plastic tube, frozen
(−20◦ C) and thawed prior to measurement.
For the bovine and human samples, an increase in temperature and drying
were prevented during the cutting process by spraying the sample with PBS.
47
48
7. Materials and methods
Table 7.1: Summary of materials and methods.
Study
Material
n
Electrical
testing
Reference method
and parameters
Other
methods
I
Bovine proximal
and distal femur
40
20 Hz - 5 MHz
ε0 , ε00 , θ, σ 0
Mechanical testing
E, σy , σmax , Res
BMDvol
II
Human distal femur 26
and proximal tibia
50 Hz - 5 MHz
ε0 , ε00 , θ,
σ 0 , Zsp , tan δ
Mechanical testing
E, σy , εy , σmax , Res
BMDvol
III
and proximal tibia
50 Hz - 5 MHz
ε0 , ε00 , θ,
σ 0 , Zsp , tan δ
Structure analysis
BV/TV, SMI, Tb.N.,
Tb.Sp., BS/BV,
Tb.Th., Connectivity,
Anisotropy
-
IV
and proximal tibia
50 Hz - 5 MHz
ε0 , ε00 , θ,
σ 0 , Zsp , tan δ
Composition analysis
Wet and dry density;
Fat, water, GAG
and collagen content
-
1*
Human distal femur 26 300 kHz - 3 GHz
and proximal tibia
ε0 , σ 0
-
-
* Unpublished data
Explanation of the measurement parameters:
BM Dvol
ε0
ε00
θ
σ0
Zsp
tan δ
E
σy
εy
Volumetric bone mineral density n
σmax
Loss factor
Res
Phase angle
BV/TV
Conductivity
SMI
Specific impedance
Tb.N.
Dissipation factor
Tb.Sp.
Young’s modulus
BS/BV
Yield stress
Tb.Th.
Yield strain
GAG content
number of specimens
Ultimate strength
Resilience
Trabecular bone volume fraction
Structure model index
Trabecular number
Trabecular spacing
Bone surface-to-volume ratio
Trabecular thickness
Glycosaminoglycans content
No fixation agent was used because electrical properties are sensitive to fixation treatments. In order to improve the comparison between the samples and
minimize the artifacts originating from preparation, the procedure was carefully
standardized. Samples were harvested from different anatomical sites to obtain
specimens with a variety of structural, composition and mechanical properties.
7.2
EIS methods
Electrical and dielectric properties were measured in a wide frequency range:
20 Hz - 5 MHz for bovine trabecular bone and 50 Hz - 5 MHz for human trabecular bone. A parallel-plate capacitance cell was used in the experiments.
Additionally, the results published in the four original articles (I-IV) have been
supplemented with measurements at frequencies 300 kHz - 3 GHz conducted
only on human trabecular bone. A method using an open-ended coaxial trans-
7.2 EIS methods
49
Distal femur
FTM
FG
FC
FLC
FMC
TMP
Proximal femur
Proximal tibia
Figure 7.1: The bovine bone samples were prepared from four anatomical positions: femoral lateral condyle (FLC), femoral medial condyle (FMC), femoral
head (FC) and femoral greater trochanter (FTM). The samples from human
tissue were obtained from distal femur (femoral medial condyle (FMC) and
femoral groove (FG)) and proximal tibia (tibial medial plateau (TMP)).
mission line was used in this study.
7.2.1
Low frequency impedance spectroscopy
Measurements were made with an LCR meter (HIOKI 3531 Z HiTester, Koizumi,
Japan). The parallel-plate cell consisted of two round stainless steel electrodes
(AISI316L) mounted in insulating supports that allowed adjustment of the distance between the electrodes (Figure 7.2). Measurements of empty and shortcircuited cell allowed corrections for the stray capacitance, the series lead inductance and resistance and shunt conductance arising within the instrument.
Repeated measurements with a sample permitted checking of the reproducibility of the set-up. For the experiments, the samples were removed from the
PBS and the excess liquid was removed with a paper towel, and the sample
was placed in the measurement cell. To avoid drying of the sample, which is
known to significantly affect electrical properties, the time between the removal
of a sample from the PBS and the end of the measurements was minimized and
standardized. The modulus of impedance, parallel resistance, parallel capacitance and phase angle were measured as a function of frequency. From these
measurements relative permittivity, conductivity, loss factor, specific impedance
and dissipation factor were calculated according to equations shown in the Table
7.2.
In Study I electrodes with a 30 mm diameter were used. In order to provide
good contact between the electrodes and tissue, a thin layer of conducting gel
(Spectra 360, Parker Laboratories, New Jersey, USA) was applied to the sample
surface. To assess the effect of gel, a phantom material as well as five samples of
five different lengths (20 mm, 14 mm, 10 mm, 8 mm and 2 mm)were measured
50
Lever
HIOKI 3531 Z
HiTESTER
LCR ANALYSER
Weight
Data to PC
Electrode
Sample
Plastic
box
PBS
Electrode
Figure 7.2: Schematic representation of the low-frequency measurement
setup. A sample was placed in the holder between two stainless steel electrodes and closed in a plastic box containing PBS at the bottom. A small
weight of 300 g was applied to the holder to ensure good contact between the
specimen and electrodes. The data were collected over a frequency range of
50 Hz - 5 MHz with an LCR meter and further processed with MatLab 6.5.1
(The Mathworks Inc., Natick, MA, USA).
with and without gel. To minimize measurement artifacts, the electrical examination was conducted in a Faraday cage. The current was directed through
samples in a direction perpendicular to the in-vivo loading axis.
In Studies II-IV, a current was applied through electrodes 16 mm in diameter. Good contact between sample and electrodes was assured by placing a
weight (300 g) on top of the parallel-plate cell. The electrical parameters measured were found to be practically independent of the amount of weight above a
certain threshold. The current was directed through all samples along the long
axis of the cylindrical sample, i.e., along the in vivo loading axis. A phantom
material (resistor of a known value) was measured before each measurement
in order to ensure the reproducibility and validity of the data. During measurement, the specimens and electrodes were enclosed in a cylindrical plastic
insulating cage of a slightly larger diameter than the samples. PBS was placed
at the bottom of the cage, but not in contact with samples. This measurement
geometry prevented the samples from drying.
Additionally, in Study II, the samples were also measured after mechanical
7.2 EIS methods
51
Table 7.2: Basic equations for low frequency EIS [93, 139, 121].
Quantity
Equation
1
1
Y∗ = ∗ =
+ jωCp
Z
Rp
Y∗
ε∗r = ε0r + ε00r =
jωCe
Cp
0
εr =
Ce
1
00
εr =
jωCe
ε0
σ0 =
Rp Ce
Rsp
Zsp = p
(2πf Rsp Csp )2 + 1
ε00
tan δ = 0
ε
Complex admittance
Complex relative permittivity
Loss factor
Conductivity
Specific impedance
Dissipation factor
Explanation of the parameters:
Z∗
Rp
ω
f
Cp
Complex impedance
Parallel resistance
Angular frequency
Frequency
Parallel capacitance
ε0
Rsp
Csp
Ce
Permittivity of free space
Specific resistivity
Specific capacitance
Capacitance of an empty
measuring cell
destructive testing. The time between electrical measurements was 10 months.
To minimize artifacts and provide straightforward comparison between values in
the parameters before and after micro-damage, the data were normalized. The
values after destructive testing were normalized with values measured before
destructive testing. Normalization was also conducted for control group by
dividing the values from the latter measurement by the values obtained from
the first measurement.
The polarization impedance of the parallel plate measurement cell was studied. Six samples (diameter = 16 mm) of different lengths (L = 55 mm, 35 mm,
24 mm, 20 mm, 16 mm, 14 mm, 10 mm, 8 mm and 2 mm) were investigated
at frequencies 20 Hz - 5 MHz. The measurement protocol was identical to that
used in Studies II-IV.
7.2.2
Microwave frequency impedance spectroscopy (supplementary
data)
Human trabecular bone specimens were also investigated (unpublished data)
with radio-frequencies (300 kHz - 3 GHz) using the reflection principle. Samples were always measured first at low frequencies, followed by high-frequency
investigation. The electromagnetic waves were generated by an HP 8753C Network Analyzer (Agilent Technologies, Palo Alto, CA, USA) and applied through
a coaxial probe to the tissue (Figure 7.3). A probe with an intrinsic impedance
of 50 Ω with an electrical field penetration depth of about 3 mm1 were used. The
coaxial open-ended sensor consisted of two brass conductors with gold-coated
1 Penetration
depth was confirmed in personal communication with Dr E. Alanen.
52
surfaces with diameters of 4 mm (center pin diameter) and 10 mm (inner diameter of the outer electrode) (see Figure 5.4). The space between the conductors
was filled with insulating Teflon. Errors arising from reflections from the probe
insulator and other instrumental artifacts were corrected by the standard calibration procedure employing short, open and 50 Ω terminations (matched load).
Additionally, measurements of a phantom material of known properties (water)
were conducted to assure the validity of the data. As in the low-frequency EIS
protocol, all measurements were conducted at room temperature (22◦ C). A
sample was removed from the PBS and the excess liquid was removed with a
paper towel. The time between the removal of a sample from the PBS and the
end of the measurements was minimized. The end-face of the specimen cylinder
was placed against the probe and the real and imaginary parts of the reflection coefficient were measured with reference to the plane of the sample-line
interface. A specially designed holder ensured good contact between the coaxial
probe and the trabecular bone sample. An open container with PBS at the
bottom was mounted on the sample holder just below the tissue. The liquid
could evaporate during the measurements, which prevented sample drying. As
the measurement of both end-faces were found to yield similar results, only the
end-face of the cylinder closer to the cartilage surface was investigated. Relative
permittivity and conductivity were calculated from equations (5.19) and (5.21),
respectively.
Lever
HP 8753C
300 kHz - 3 GHz
NETWORK ANALYSER
S-PARAMETER
TEST SET
HP 85046A
50W PORT
Coaxial probe
Weight
Sample
Sample holder
Plastic
Coaxial cable
box
with PBS
Data to PC
Figure 7.3: Schematic representation of the microwave measurement setup.
The sample was placed in the holder and in contact with the coaxial probe. A
small weight of 300 g was applied to the holder to ensure good contact between
the specimen and the probe. An open plastic box with PBS on the bottom was
placed below the sample. The data were collected over a frequency range of
300 kHz - 3 GHz with a network analyzer and further processed with MatLab
6.5.1 (The Mathworks Inc., Natick, MA, USA).
7.3 Bone mineral density measurements
7.3
53
Bone mineral density measurements
The bone mineral density (BMD) of the bovine and human specimens was determined using a dual energy x-ray absorptiometry (DXA) technique in a direction
perpendicular to the parallel ends of the cylindrical samples. Volumetric bone
mineral density (BMDvol ) was calculated by normalizing the measured BMD
with the sample thickness determined with a micrometer (Mitutoyo, Kawasaki,
Japan).
In Study I, the investigations were performed with a clinical Lunar Expert
densitometer (Lunar Co. Madison, WI, USA). In order to optimize the measurement accuracy, the soft tissues were simulated by placing samples in a PBS
bath of a constant depth.
In Study II, the BMD of human trabecular bone was measured with a Lunar
Prodigy instrument (GE Medical, Wessling, Germany). The AP spine measurement protocol was used. Samples were immersed in a water bath of a constant
depth simulating soft tissues to improve the accuracy of the density measurements.
7.4
Mechanical testing
A destructive mechanical test was performed on all the bovine and human trabecular bone samples, and Young’s modulus, ultimate strength, yield stress,
yield strain and resilience were determined.
In Study I, mechanical testing was performed with a 12.5 kN servohydraulic
material testing device (Matertest Oy, Espoo, Finland). To stay within the
capacity of the device, plugs of diameter 19 mm were drilled from the original
samples prior to the examination. In order to prevent drying of the tissue, the
samples were immersed in PBS bath during the test. A small 0.13 MPa prestress
was applied to a specimen for 2 min followed by five non-destructive cycles up
to 0.7% strain. This ensured good contact between the loading platens and
sample. After preconditioning, a destructive strain of 4.5% was applied to the
tissue with a strain rate of 0.35 · 10−3 s−1 .
A 200 kN material testing device (Zwick 1484, Zwick GmbH & Co. KG, Ulm,
Germany) equipped with a 10 kN force transducer was used in Study II. Frictionless contact between platens and the tissue was ensured by placing a Teflon
foil on each end-face of the sample. Prior to the destructive testing, a sample
was subjected to a small 25 kPa prestress and subsequently preconditioned with
five consecutive cycles up to 0.5% strain. Finally, samples were destructively
compressed to a destructive strain of 5% at a strain rate of 4.5 · 10−3 s−1 . The
samples were moistened during the mechanical testing to avoid drying.
Young’s modulus was calculated as a slope of linear fit to a stress-strain
curve between 40% and 65% and between 45% and 60% of the maximum stress
for bovine and human bone, respectively. Yield stress was determined with the
offset method [39]. In this technique, a line parallel to the linear portion of
the stress-strain curve but offset by 0.2% is constructed. Yield stress is defined
as an intersect between this line and the stress-strain curve. A tangent to the
stress-strain curve at the point 15% of the yield stress was constructed. The
zero-strain point was defined as an intersect of this tangent and the strain axes.
Yield strain was defined as the difference between the strain corresponding to
yield stress and that at the zero-strain point. The maximum stress detected
54
during the experiment represented ultimate strength. Resilience was calculated
by integrating the stress-strain curve from zero strain to the yield strain.
7.5
Microstructural analysis
In Study III, the microstructure of human trabecular bone was investigated using a high-resolution micro-computed tomography (microCT) system (SkyScan
1072, Aartselaar, Belgium). Each cubic voxel measured 18 x 18 x 18 µm3 .
Accurate 3-D data sets were obtained by segmenting each image with a local
threshold method [80]. Several structural parameters were calculated based on
true and unbiased methods. Firstly, trabecular bone volume fraction (BV/TV)
and the bone surface-to-volume ratio (BS/BV) were quantified. Subsequently,
trabecular thickness (Tb.Th.), spacing (Tb.Sp.) and number (Tb.N.) were calculated using a direct 3-D analysis without any model assumptions of trabecular structure. The degree of anisotropy (DA) was defined as the ratio between
the maximal and minimal eigenvalues of the fabric tensor of the architecture
[76]. Connectivity was determined according to Odgaard and Gundersen [85],
whereas the structural model index (SMI) was determined according to Hildebrand and Rüegsegger [75].
7.6
Compositional analysis
In Study IV, biochemical composition analysis was conducted on human trabecular bone tissue. Wet samples were weighed and their volume determined
using the Archimedes principle. Subsequently, the specimens were freeze-dried
and weighed in order to determine dry weight and water mass. Fat was removed
from the specimens with acetone, and the mass of fat was determined.
Hydroxyproline and uronic acid assays were performed on fat-free bone powder. Approximately 20 mg of each sample was portioned for acid hydrolysis in
5 M HCl at 108◦ C for 16 hours. A microplate assay was used to analyze the
hydroxyproline content in the hydrolysate [140]. The estimates for the total
collagen content can be obtained by multiplying the analyzed hydroxyproline
content by a factor of 7, since the hydroxyproline content of collagen is approximately 14% of the collagen mass [141]. Proteoglycans were extracted from a
20 mg sample of the fat-free bone powder using 4 M GuHCl including 0.2 M
EDTA in 50 mM sodium acetate buffer, pH 6.0, for 70 hours. Uronic acid content was determined with a spectrophotometric assay [142]. The GAG content
was obtained by multiplying the uronic acid content by a factor of 3.2. This
follows from the fact that 31% of chondroitin sulfate weight comes from uronic
acid.
Wet and dry densities were calculated by normalizing wet and dry weight,
respectively, with sample volume. Water, fat, collagen and GAG contents were
computed by normalizing individual weights with corresponding wet masses of
the samples.
7.7
Statistical analysis
In study I, the significance of the variation in the measured parameters between
different anatomical sites was tested with the Kruskall-Wallis H-test. To reveal the relation between electrical and mechanical parameters, linear Pearson’s
7.7 Statistical analysis
55
correlation analysis was used. The reproducibility of the electrical impedance
measurement setup was expressed in terms of standardized coefficient of variation (sCV). This parameter can be calculated from the coefficient of variation
(CV), which can be expressed as a ratio of standard deviation (υy ) to the mean
(y) of repeated measurements of parameter y. For n samples, the CV becomes
a root mean square averaged value CVRM S [143, 144]
CVRM S
v
u n µ
¶
u 1 X υyi 2
=t
.
n i=1 y i
(7.1)
The CV provides only a short-term precision and does not take into account the
biological variation of the parameters measured [144]. The true diagnostic sensitivity of the measurements can be obtained with the standardized coefficient
of variation, determined as follow:
sCV =
µ · CVRM S
,
4σµ
(7.2)
where µ is the average of standard deviation within the population and υµ
is the population standard deviation. A smaller sCV value indicates better
reproducibility, with an extreme case of 0% meaning a perfect reproducibility
[144].
In study II, Pearson’s correlation analysis was used to investigate the relationships between electrical and mechanical parameters. The significance of the
anatomical site-dependent variations of parameters measured was tested with
the Wilcoxon signed rank test. The effect of bone microdamage was studied
using the Wilcoxon signed rank test for two related samples by comparing the
electrical parameters of samples obtained before and after destructive mechanical testing. In order to eliminate possible confounding factors in this comparison, the results were normalized and further tested for variations solely due to
microdamage. The differences in the normalized electrical parameters between
the control and the mechanically tested groups were compared using the MannWhitney U-test. The electrical properties of human and bovine trabecular bone
were investigated with the Mann-Whitney test.
In Study III, the Friedman test for several dependent samples was used
to assess the site-dependent variations in electrical and structural parameters.
Pearson’s correlation analysis was used to determine the linear correlation coefficients between parameters tested. Principal component analysis (PCA) was
used to classify and reduce the number of electrical and structural variables
(Table 7.3). Two PCA approaches were used for electrical parameters. The
frequencies (50 Hz - 5 MHz) used for the measurement of electrical properties
were grouped into the main principal components. Extreme multicollinearity
was avoided by excluding specific frequencies from the analysis. In addition,
PCA was conducted on relative permittivity, conductivity, phase angle, dissipation factor and specific impedance at 1.2 MHz. Loss factor was not included
in the analysis, as it correlates strongly with conductivity. Varimax rotation,
i.e. rotation of the axes to a position in which the sum of the variances of the
loadings reaches the maximum values, was used to improve the interpretability
of the components. The eigenvectors remained orthogonal during the rotation.
Multiple stepwise linear regression was used to reveal whether a combination
56
Table 7.3: Principal components extracted from electrical and structural
parameters.
PC
Name
Field propagation
component
Energy component
Low
Electrical frequency permittivity
principal
Middle
components frequency permittivity
High
frequency permittivity
Low frequency
dissipation factor
High frequency
dissipation factor
Structural
Trabecular structure
principal
component
components
Surface component
Anisotropy component
Explained
variation
68.2%
27.0%
26.3%
46.3%
25.3%
56.6%
39.5%
49.1%
32.5%
14.8%
Composition
θ, σ 0 , Zsp , tan δ
tan δ, ε0
Permittivity at
50 Hz - 1.2 kHz
Permittivity at
300 Hz - 500 kHz
Permittivity at
300 kHz - 5 MHz
Dissipation factor at
50 Hz - 120 kHz
Dissipation factor at
50 kHz - 5 MHz
SMI, Connectivity
Tb.Sp., BV/TV, Tb.N.
BS/BV, Tb.Th.
Anisotropy
PC
ε0
θ
σ0
Zsp
tan δ
Principal component
Phase angle
Conductivity
Specific impedance
Dissipation factor
BS/BV
Tb.Th.
BV/TV
SMI
Tb.N.
Tb.Sp.
Bone surface-to-volume ratio
Trabecular thickness
Trabecular bone volume fraction
Trabecular number
Trabecular spacing
of electrical parameters or principal components could predict bone structure.
Moreover, a combination of structural parameters and their PCs were applied
to linear regression analysis in order to investigate the effect of bone microarchitecture on electrical parameters.
In study IV, the same statistical procedures as in study III were used to
reveal relations between the electrical properties of bone and its composition.
For the EIS at microwave frequencies, Pearson’s correlation analysis was
used to relate relative permittivity and conductivity with bone structure. The
significance of site-dependent variation in the electrical parameters measured
was studied with the Friedman test for several dependent samples.
Chapter VIII
Results
7000
6000
5000
4000
3000
2000
1000
A.
0
0
10
20
30
40
50
60
Polarization impedance (W)
Modulus of impedance (W)
The measurements of the electrical and dielectric properties of bone were found
to be reproducible over a wide frequency range. The standardized coefficient
of variation (sCV) was below 1% for all parameters investigated at a frequency
range 1 kHz - 5 MHz. Below 1 kHz, the sCV of conductivity and loss factor
remained at the level of 1%, while the sCV of relative permittivity and phase
angle increased up to 4.5% and 8.5% at 50 Hz, respectively.
Polarization impedance of the parallel plate cell was investigated (Figure
8.1 A). It was found to decrease with frequency from 211 Ω at 100 Hz to 44 Ω
at 1 MHz (Figure 8.1 B). The effect of contact impedance for 8 mm samples
decreased with increasing frequency from 19% at 100 Hz through 10% at 1 kHz
to 4.5% at 1 MHz. No significant difference in impedance values was observed
between the samples with and without gel applied to the surface (p > 0.05).
Additionally, the gel was found to have no or only minor effect (< 1%) on the
impedance values of the phantom material investigated.
Length (mm)
220
200
180
160
140
120
100
80
60
40
B.
102
103
104
105
Frequency (Hz)
Figure 8.1: The mean modulus of impedance of six bone samples was measured at nine different lengths. Polarization impedance was determined from
these measurements by a linear fit and was found to be e.g., 211 Ω at 100
Hz (a). Polarization impedance decreased with increasing frequency (b). The
error bars indicate standard deviation.
57
106
58
8. Results
8.1
Variation and frequency dependence of electrical properties
The electrical and dielectric properties of bovine and human trabecular bone
were dependent on frequency. The observed dispersions were broad. For instance, relative permittivity dropped steadily over all frequency ranges from
5 · 106 at 50 Hz to 10 at 3 GHz (Figure 8.2 A). Conductivity was almost frequency independent at low frequencies but increased from 0.125 Sm−1 to 0.582
Sm−1 over the frequency range 700 MHz - 3 GHz (Figure 8.2 B). A small shift
in the values of the parameters measured with parallel plate cell and openended cell configurations over the same frequency range (300 kHz - 5 MHz) was
observed (Figure 8.2).
Conductivity (S/m)
7
10
105
3
10
1
10
1
10
A.
10
3
10
5
7
10
Frequency (Hz)
9
10
100
-1
10
10-2
1
10
B.
103
105
7
10
9
10
Frequency (Hz)
Figure 8.2: The spectrum of relative permittivity (a) and conductivity (b) of
human trabecular bone at frequencies 50 Hz - 3 GHz obtained by parallel-plate
(50 Hz - 5 MHz) and reflection methods (300 kHz - 3 GHz). The error bars
indicate standard deviation.
Interestingly, at low frequencies, a strong relation was found between conductivity and relative permittivity. This association decreased with increasing
frequency. For instance, at 100 Hz the correlation was 0.79 (p < 0.01) (Figure
8.3 A), while at 1.2 MHz it was only −0.18 (p > 0.05) (Figure 8.3 B).
In addition, the electrical parameters of bovine trabecular bone showed significant site-dependent variation (Figure 8.4) (p < 0.05). Relative permittivity at 1.2 MHz reached the highest value in femoral medial condyle (FMC)
(79.9 ± 10.8) and the lowest in femoral trochanter major (FTM) (34.0 ± 7.3).
However, the site-dependent variation in human trabecular bone samples was
found to be insignificant (p > 0.05, Figure 8.4).
Statistically significant (p < 0.01) differences in electrical parameters between bovine and human trabecular bone were observed. Relative permittivity
at frequencies of 50 Hz - 30 kHz was higher in human bone, while at frequencies
of 50 kHz - 5 MHz it was higher in bovine bone (Figure 8.5 A). The conductivity
of human samples was higher than that of bovine specimens at all frequencies
(e.g., 0.08 ± 0.03 Sm−1 for human and 0.03 ± 0.01 Sm−1 for bovine at 1.2
MHz) (Figure 8.5 B). The peak value of dissipation factor was seen at a higher
frequency in human bone (120 kHz) than in bovine bone (1 kHz, Figure 8.5 C).
8.2 Relations between electrical and mechanical properties
59
5
3
1
A.
50
6
11 x 10
f = 100 Hz
r = 0.79
9 p < 0.01
n = 26
7
0.02
0.04
0.06
0.08
0.1
Conductivity (S/m)
0.12
45
f = 1.2 MHz
r = -0.18
p > 0.05
n = 26
40
35
30
25
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
B.
Conductivity (S/m)
Figure 8.3: At frequencies below 300 kHz, about 62% of the variation in
relative permittivity can be explained by conductivity (a). The association
between relative permittivity and conductivity gets weaker with increasing
frequency (b).
The electrical properties of human trabecular bone samples were measured
twice. The samples were frozen for about 10 months between measurements.
Among the parameters measured with a parallel-plate cell, significant differences
were observed in the values of permittivity at frequencies of 80 Hz - 2.5 MHz
and in the values of dissipation factor at frequencies of 50 Hz - 50 kHz (p <
0.05, n = 6). The difference observed in conductivity and loss factor was
statistically insignificant (Figure 8.6).
At all frequencies (300 kHz - 3 GHz), relative permittivity and conductivity
measured with the open-ended coaxial probe showed significant differences in
values before and after long-term-freezing (p < 0.05, n = 6) (Figure 8.7).
8.2
Relations between electrical and mechanical properties
Significant but frequency-dependent relationships were found between the electrical and mechanical characteristics of bovine and human trabecular tissue.
Relative permittivity displayed strong linear correlations with ultimate strength
(bovine: r = 0.62, p < 0.01, n = 40, f = 50 kHz; human: r = 0.72, p <
0.01, n = 20, f = 1.2 MHz) and Young’s modulus (bovine: r = 0.55, p <
0.01, n = 40, f = 50 kHz; human: r = 0.71, p < 0.01, n = 20, f = 1.2 MHz)
(Table 8.1).
In bovine bone, correlations between mechanical properties and relative permittivity or phase angle were strongly frequency dependent, being significant
above 6 kHz (Figures 5 and 6 in Study I). Conductivity and loss factor of bovine
bone did not correlate significantly with any of mechanical properties at any frequency (p > 0.05). Phase angle correlated only moderately with all mechanical
properties (r ≤ |0.45|).
In human bone, correlations between mechanical properties and relative permittivity or dissipation factor were strongly dependent on frequency, changing
60
8. Results
human
bovine
0.14
Conductivity (S/m)
100
80
60
40
20
A.
FMC FLC FTM FC FMC TMP FG
0.12
0.1
0.08
0.06
0.04
0.02
B.
2200
1800
Loss factor
Dissipation factor
80
60
40
1400
1000
20
600
0
C.
200
D.
Figure 8.4: The electrical parameters of bovine trabecular bone displayed
statistically significant topographical variation in relative permittivity at 1.2
MHz (a), conductivity at 1.2 MHz (b), dissipation factor at 1.2 MHz (c) and
loss factor at 1.2 MHz (d) (FMC - femoral medial condyle, FLC - femoral
lateral condyle, FTM - femoral trochanter major, FC - femoral caput, TMP tibial medial plateaus, FG - femoral groove). The error bars indicate standard
deviation.
the sign from negative to positive for relative permittivity and from positive
to negative for dissipation factor with increasing frequency. Correlations with
other parameters were almost frequency independent, being either negative or
positive. Conductivity, phase angle and loss factor displayed only weak or moderate relations with mechanical characteristics (r ≤ |0.45|). Specific impedance
showed significant correlation with yield strain (r = 0.65, p < 0.01, n = 20, f =
1.2 MHz).
In human samples, electrical principal components correlated significantly
with mechanical properties (Table 8.2). Relatively strong correlations (r ≥
|0.57|, p < 0.01) were found between ultimate strength, Young’s modulus, yield
stress or resilience and high frequency permittivity, dissipation factor as well as
field propagation component. Low frequency permittivity correlated moderately
with resilience (r = −0.45, p < 0.05). Yield strain correlated only with the
energy component (r = −0.46, p < 0.05). No significant correlations were
found between mechanical properties and relative permittivity or conductivity
8.3 Relations between electrical and structural properties
human
61
bovine
0.14
Conductivity (S/m)
7
10
106
105
104
103
102
0.12
0.1
0.08
0.06
0.04
0.02
1
A.
10 1
10
102
103
104
105
106
7
10
B.
Frequency (Hz)
0 1
10
102
103
104
105
106
7
10
Frequency (Hz)
140
Dissipation factor
120
100
80
60
40
20
C.
0 1
10
102
103
104
105
106
7
10
Frequency (Hz)
Figure 8.5: Statistically significant (p < 0.01) differences in electrical parameters between bovine and human trabecular bone were observed: relative
permittivity (a), conductivity (b) and dissipation factor (c). Samples were
obtained from the same anatomical site (femoral medial condyle (FMC)) but
in different direction (bovine: medio-lateral; human: perpendicular to the articular surface). The error bars indicate standard deviation.
of human samples measured with the open-ended coaxial probe.
Relative permittivity and dissipation factor at the frequency range 300 Hz
- 1.5 MHz displayed significant (p < 0.05) differences in their values before and
after destructive mechanical testing as compared with the control group (Figure 8.8 A). The differences in the other parameters measured with a parallelplate cell were insignificant. Normalized conductivity and relative permittivity
determined with an open-ended coaxial line showed small, but statistically insignificant variations between destructively tested and control groups (Figure
8.8 B).
8.3
Relations between electrical and structural properties
The strength and sign of linear correlations between human trabecular microstructure and relative permittivity as well as dissipation factor were strongly
frequency dependent. Relative permittivity correlated significantly with all
62
8. Results
before
7
Conductivity (S/m)
106
10
5
104
103
102
A.
after
0.11
10
1
10
1
10
102
103
104
105
106
Frequency (Hz)
0.07
0.05
0.03
0.01
1
10
7
10
0.09
B.
102
103
104
105
7
106
10
106
10
Frequency (Hz)
7
10
106
60
105
50
Loss factor
Dissipation factor
70
40
30
20
103
102
10
0 1
10
C.
104
1
102
103
104
105
Frequency (Hz)
106
7
10
10 1
10
D.
102
103
104
105
7
Frequency (Hz)
Figure 8.6: Relative permittivity at frequencies of 80 Hz - 2.5 MHz and dissipation factor at frequencies of 50 Hz - 50 kHz measured with the parallel-plate
cell showed significant differences in values before and after long-term freezing:
relative permittivity (a), conductivity (b), loss factor (c) and dissipation factor
(d) (n = 6, samples from FG). The error bars indicate standard deviation.
structural parameters, except for degree of anisotropy (Figure 4 in Study III
and Table 8.1). On the other hand, dissipation factor did not depend on trabecular thickness nor on degree of anisotropy. The significant correlations were
found at low and high frequencies, while at the range 5 kHz - 50 kHz the relations
were insignificant.
The linear correlations between microstructure and loss factor, conductivity,
phase angle as well as specific impedance parameters were almost frequency
independent and exhibited either positive or negative sign. The values of these
correlations were significant with all structural parameters excluding BS/BV,
Tb.Th. and degree of anisotropy.
Importantly, the dense trabecular structure with high BMDvol , BV/TV and
collagen content, but low SMI, fat and water contents showed higher values
of relative permittivity at high frequencies (e.g., 1.2 MHz) but lower at low
frequencies (e.g., 100 Hz), than did more sparse low density trabecular bone
(Figure 8.9). Interestingly, conductivity displayed the opposite behavior, being
8.3 Relations between electrical and structural properties
before
after
10 0
Conductivity (S/m)
10 3
10 2
10 1
A.
63
10 5
10 6
10 7
10 8
10 9
Frequency (Hz)
B.
10 -1
10 -2
10 5
10 6
10 7
10 8
10 9
Frequency (Hz)
Figure 8.7: Relative permittivity (a) and conductivity (b) measured with an
open-ended coaxial probe showed significant differences in values before and
after long-term-freezing. The error bars indicate standard deviation.
Tested
Control
2.5
1.2
**
**
2
1.5
1
0.5
0
A.
Dissipation
Specific
impedance
Relative factor
Conductivity
permittivity
Normalized data
Normalized data
3
1
0.8
0.6
0.4
0.2
0
B.
Relative
permittivity
Conductivity
Figure 8.8: Only relative permittivity and dissipation factor at the frequency
range of 300 Hz - 1.5 MHz were able to distinguish between intact and mechanically damaged trabecular structure. Normalized electrical parameters at
1.2 MHz (a) and 30 MHz (b) for destructively tested and control groups (**
p < 0.01). The error bars indicate standard deviation.
low for dense bone and high for low density bone.
Structural parameters showed significant relations with all electrical principal components, except for middle frequency relative permittivity and low
frequency dissipation factor. Bone surface-to-volume ratio correlated highly
with high frequency permittivity and field propagation component (r > |0.71|,
p < 0.01). High frequency dissipation factor was strongly related to BV/TV
and SMI (r > |0.72|, p < 0.01).
In contrast, anisotropy component correlated with no electrical parameter
measured at 1.2 MHz, and the surface component was strongly related only
with relative permittivity (r = 0.64 p < 0.01). Trabecular structure component
64
8. Results
Table 8.1: Linear correlation coefficient for selected electrical (measured at 1.2
MHz for human and at 50 kHz for bovine tissue) and biomechanical, structural
and compositional parameters in human and bovine bone.
Type of test
Relation
ε0 vs. U S
ε0 vs. E
Mechanical
σ 0 vs. U S
testing
σ 0 vs. E
tan δ vs. U S
tan δ vs. E
ε0 vs. Tb.N.
ε0 vs. SMI
Structural
σ 0 vs. Tb.N.
testing
σ 0 vs. SMI
tan δ vs. Tb.N.
tan δ vs. SMI
ε0 vs. fat†
ε0 vs. collagen†
Compositional
σ 0 vs. fat†
testing
σ 0 vs. collagen†
tan δ vs. fat†
tan δ vs. collagen†
ε0 vs. BMDvol
DXA
σ 0 vs. BMDvol
testing
tan δ vs. BMDvol
† content, ∗ p < 0.05, ∗∗ p < 0.01
Human bone
0.72∗∗
0.71∗∗
−0.47∗
−0.36
−0.61∗∗
−0.57∗∗
0.45∗
−0.59∗∗
−0.64∗∗
0.61∗∗
−0.63∗∗
0.71∗∗
−0.85∗∗
0.64∗∗
0.10
−0.43∗∗
0.42∗
−0.55∗∗
0.67∗∗
−0.50∗∗
−0.61∗∗
Bovine bone
0.62∗∗
0.55∗∗
−0.19
−0.15
−0.50∗∗
−0.44∗∗
0.87∗∗
−0.48∗∗
−0.65∗∗
ε0
σ0
tan δ
US
DXA
E
Conductivity
Tb.N.
Dissipation factor
SMI
Ultimate strength
BMDvol
Dual energy x-ray absorptiometry
Young’s modulus
Trabecular number
Volumetric bone mineral density
Table 8.2: Linear correlation coefficients between the electrical principal components and mechanical properties of trabecular human bone samples (n =
20).
Ultimate Young’s
strength modulus
Permittivity middle frequencies
−0.32
−0.21
Permittivity low frequencies
−0.44
−0.38
Permittivity high frequencies
0.65∗∗
0.61∗∗
Dissipation factor low frequencies
0.32
0.28
Dissipation factor high frequencies −0.64∗∗ −0.57∗∗
Energy component
−0.35
−0.23
Field propagation component
0.68∗∗
0.70∗∗
Yield
stress
−0.31
−0.43
0.65∗∗
0.30
−0.64∗∗
−0.35
0.68∗∗
Yield Resilience
strain
−0.16
−0.33
−0.35
−0.45∗
0.20
0.59∗∗
0.16
0.28
−0.42
−0.63∗∗
−0.46∗ −0.44
0.12
0.56∗∗
8.4 Relations between electrical and compositional properties
0.18
49
f = 1.2 MHz
r = 0.68
p < 0.01
40 n = 26
Conductivity (S/m)
65
30
f = 1.2 MHz
r = -0.59
p < 0.01
n = 26
0.15
0.1
0.05
20
10
sparse
structure
15
20
25
BV/TV (%) dense
structure
10
sparse
structure
15
20
25
BV/TV (%) dense
structure
Figure 8.9: Sparse trabecular bone structure can be distinguished from a
dense structure by measurement of permittivity and conductivity at frequencies > 100 kHz. Human trabecular bone samples with a dense structure display
high relative permittivity and low conductivity. Specimens with a sparse structure exhibit high conductivity and low permittivity. A variation in BT/TV
of 9.4% - 25.7% corresponds to a variation in BMDvol of 0.14 - 0.37, in SMI
ofn 1.52 - 0.53, in fat content of 48.2% - 35.7%, in collagen content of 5.9% 11.2% and in water content of 23.6% - 13.0%.
correlated with all electrical parameters (r > |0.57|, p < 0.01), except for
relative permittivity.
Interestingly, moderate linear correlations were found between the degree
of anisotropy and relative permittivity measured with the open-ended coaxial
probe at frequencies of 36 MHz - 50 MHz and 1 GHz - 2.5 GHz (|0.39| <
r < |0.42|, p < 0.05). Furthermore, conductivity correlated moderately with
anisotropy component at the frequency bands of 300 kHz - 430 MHz and 1.5
GHz -3 GHz.
8.4
Relations between electrical and compositional properties
Strong or moderate correlations were found between the electrical parameters
and densities. Collagen content was strongly related to relative permittivity
at 1.2 MHz (r = 0.64), but only moderately to other parameters (Table 8.1).
Trabecular bone GAG content showed no or only minor impact on electrical
properties at all frequencies. The sign and strength of correlations between
relative permittivity and composition parameters, water and fat content in particular, was dependent on frequency (Figure 8.10).
Fat content was significantly related to dissipation factor and relative permittivity only at some specific frequency bands (> 100 MHz), but showed no
correlations with other electrical parameters at any frequency. Water content
strongly affected all electrical parameters, conductivity in particular (r = 0.79
at 1.2 MHz, p < 0.01, n = 26) (Figure 8.11). The linear combinations of electrical principal components accounted for more than 50% of the variation in
66
8. Results
2.5
2
1.5
1
0.5
0
f = 1.2 MHz
r = -0.85
p < 0.01
n = 26
45
40
35
30
25
20
15
12 14
A.
50
6
4.5 x 10
4 f = 200 Hz
r = 0.83
3.5 p < 0.01
3 n = 26
16
18
20
22
24
30
26 28
Water content (%)
35
40
45
50
Fat content (%)
B.
Figure 8.10: Water and fat content of human trabecular bone may be determined from measurements of relative permittivity at low (a) and high (b)
frequency bands.
Conductivity (S/m)
organic composition or densities. The high frequency permittivity was a strong
predictor of fat content (r2 = 0.72). In contrast, a good predictor of variation in water content was the linear combination of low and middle frequency
permittivity (r2 = 0.66).
0.18
0.16
0.14
0.12
f = 1.2MHz
r = 0.79
p < 0.01
n = 26
0.1
0.08
0.06
0.04
0.02
12 14
16
18
20
22
24
26 28
Water content (%)
Figure 8.11: Conductivity correlated highly significantly with water content.
Chapter IX
Discussion
In present present study the interrelations between trabecular bone composition, structure, mechanical and electrical properties were investigated. Initially,
bovine trabecular bone samples were measured to verify the reproducibility and
reliability of the system as well as to reveal the potential of the EIS technique
to diagnose mechanical properties of the tissue. Later, electrical measurements
of human trabecular bone samples from distal femur and proximal tibia were
conducted over a wide range of frequencies using a parallel-plate capacitance
cell and an open-ended coaxial probe1 . The sample preparation procedure and
the measurement protocol were designed to preserve the physiological state of
the tissue. The bone samples were never allowed to dry and no fixation agent
was used. Bone mineral density was measured with a clinical DXA technique
followed by high resolution microCT investigations and biochemical analysis.
This study protocol provided a unique insight into the electrical properties of
bone. The findings indicate that, when further developed, the EIS may proove
to be a useful technique for the rapid assessment of bone status in vitro or
during certain special cases of open surgery.
The results of the present study would suggest that the interstitial liquid of
trabecular bone determines its conductivity. In particular, the dc conductivity
was found dominating, possibly due to the shunt pathways provided by the
structural organization of trabecular tissue. At frequencies below 100 kHz,
only free, extracellular water, originating from both hard and soft tissues, is
available to current flow [90]. The mineralized bony matrix contains about
25% water, of which 40% is free [1]. Yellow bone marrow contains mostly
lipids, the molecules of non-polar and insoluble in water triglycerides [145]. As
such, they are shielded from water, leaving the tissue with a substantial amount
of unbound liquid [123]. Additionally, the increase in conductivity associated
with the β-dispersion is masked by a high ionic-, frequency-independent dc
conductivity. The mineralized bony matrix contains an insufficient quantity of
cells to significantly increase the amount of current paths2 , while the lipids of
bone marrow are known to be poor conductors at all frequencies. The increase in
conductivity values at frequencies above 10 MHz is associated with δ dispersion.
1 Supplementary
data not included in studies I-IV
are known to posses capacitive properties [93]. If the frequency is sufficiently high
(e.g., above 100 kHz), the cell membranes will be short-circuited, increasing the amount of
current paths and, thus, conductivity.
2 Cells
67
68
9. Discussion
It can be interpreted as a significant contribution of bound water to the current
carriers [90].
A previous study [146] investigated bone conductivity as a function of bathing
solution and found a limiting value of 0.1 mS m−1 as the conductivity of the
medium fell to zero. It has also been reported that the measured values of conductivity of dry bone are higher than those predicted theoretically [115]. Both
of the previous investigations attributed those results to the conductivity of the
matrix, which is strongly controlled by bounded water. This might explain the
strong correlation observed in Study IV between conductivity and dry density
(r < −0.7 for 1 kHz - 5 MHz).
The observed dispersion curve of trabecular bone is broad and appears featureless. This may be due to the superposition of several processes with different
relaxation times [114], originating from both mineralized matrix and marrow.
At frequencies below 300 kHz, about 62% of the variation in relative permittivity
can be explained by conductivity, the movement of free water ions in particular. The unaccounted variation may originate from the diffusion of counterions
near the charged interfaces and the spatial distribution of water within the tissue [112, 111]. This possibility is further supported by the strong correlation
between relative permittivity and the densities or microstructure found in the
present study. The association between relative permittivity and conductivity
diminishes with increasing frequency, possibly due to the polarization processes
in collagen protein [114, 95, 122] and in bone marrow [123]. The collagen bundles in particular appear to be an important source for permittivity, as they are
rich in interfaces. Further, in an aqueous environment, the charged groups of
proteins as well as the charged membrane surfaces electrostatically interact with
water molecules. The non-polar hydrophobic groups of proteins are surrounded
by water molecules, forming hydrogen bonds. This results in the formation of
one or two layers of water molecules near protein or membrane surfaces. This
bound water is characterized by different physical properties than bulk, free water [92]. The rotation of the bound water in both mineralized and soft tissues
determines the δ dispersion.
The frequency spectrum of the dissipation factor is consistent with the results presented by Saha and Williams for human trabecular bone [104]. The
rather high value is probably due to the large amount of water present in trabecular tissue [116] and might serve as further evidence for the significance of
the dc conductivity contribution to the overall dielectric response of the tissue
[147, 148]. The peak in the dissipation factor spectra indicates the Debye-like
relaxation processes [149, 148]. However, the width of the peak suggests the
existence of processes with different relaxation times [148, 150].
The difference in spectra between bovine and human bone can be attributed
to the differences in the tissue structure, composition and the organization of one
phase relative to the other. Bovine bone is known to be denser [151, 152, 153],
and hence contains less bone marrow. The higher amount of liquid in human
bone is directly reflected in its higher conductivity. Water, the universal solvent,
is particularly effective in increasing ionic concentrations and the mobility of
such particles as surface ions of minerals [115]. The association of water with
ionizable constituents within, on the surface of, or on the interface of dielectrics
results in higher values of the peak dissipation factor and relative permittivity at
low frequencies in human bone. Additionally, a certain amount of energy is lost
by particles vibrating in the ac field. The amount of such energy is determined
69
by the number and size of the particles participating in the motion (see e.g.,
[154]) that can be different for bovine and human bones. At frequencies above
100 kHz, the substantial amount of water in human bone prevents polarization of
the tissue by increasing the current flow. At the same time, the higher amount
of collagen in dense, or bovine, bone may significantly increase the values of
relative permittivity.
The electrical properties varied to some extatation between different anatomical sites. Those variations may be attributable to the topographical differences
in the amount and orientation of collagen fibers [114], the microstructure and the
composition between samples. In human bone the variation in collagen content
was statistically insignificant, possibly also reflected by the lack of significant
fluctuations in electrical parameters. In contrast, significant variations in density and mechanical parameters and hence electrical characteristics were found
in bovine bone. It is worth emphasizing that the largest difference in bovine
bone was found between the femoral greater trochanter (FTM) and femoral
groove (FG). However, human bone samples from these locations were not investigated, and this might explain the insignificant differences.
The effect of long-term freezing on human trabecular bone has been investigated. The values of electrical parameters except dissipation factor have been
found to decrease after freezing. However, in previous studies conductivity was
reported to increase after the removal of a tissue from the body, over a period
of time ranging from 50 h up to one month [130, 95, 155]. This clear contradiction may be explained in two ways. First, the significant changes in tissue
occur within a few hours post mortem [156, 157, 90]. For instance, Saha and
Williams [155] observed that the resistivity of bone samples remained similar
after undergoing a certain number of freezing-thawing cycles, while the most
rapid changes occurred during the first days of the experiment. In the present
study, due to the prolonged procedure of harvesting human bones, freezing and
thawing of the specimens prior to measurements was unavoidable. It is possibe
that already during those first cycles cell membranes may have been destroyed,
and the washout of the cellular components and changes in the ionic content
occurred, leading to the increase in conductivity. With time and during further
cycles, the value of conductivity could reach equilibrium. Thus, the changes
observed in this study might be different in nature and cannot be attributed to
the same phenomena as reported earlier. Second, it has been shown that freezing may change the physical state of lipids and the structure of proteins [158] in
such a way that the electrical double layer formations [159] could be impaired.
Further, extened storage of a sample in a freezer affects the properties of cells,
and possibly also the whole tissue, in different ways than doeas a short time
freezing [160]. Thus, it can be hypothesized that long-term freezing reduces the
possible current paths and decreases conductivity, although the precise mechanism remains unclear. The alternations in the protein structure might explain
the observed changes in permittivity. Long-term freezing of the samples for
electrical measurements should therefore be avoided when-ever possible.
Relative permittivity and dissipation factor showed strong associations with
mechanical properties in both human and bovine trabecular bone. Additionally,
in human bone all mechanical properties showed significant correlations with
electrical characteristics at certain frequencies, but in bovine bone conductivity
and loss factor did not correlate significantly with any mechanical parameter.
This may be due to the differences in structure and composition of human and
70
9. Discussion
animal tissues, as discussed above. Dissipation factor and permittivity in the β
relaxation frequency range was demonstrated to be sensitive to variation in bone
strength and, in particular, to its elastic behavior, with the exception of yield
strain. The latter parameter related to specific impedance at all frequencies.
In this study, the significant linear correlations between mechanical characteristics and electrical parameters of bovine trabecular bone, particularly relative permittivity, appeared to be stronger for low rather than for high density
tissue. For instance, a significant difference was found in BMD between femoral
caput (FC) (0.586 g cm−3 ) and femoral greater trochanter (FTM) (0.198 g
cm−3 ). However, the significant correlations between certain mechanical parameters and relative permittivity at specific frequencies persisted only for FTM.
This might explain the stronger associations between mechanical properties and
relative permittivity in human bone, as a low density tissue, when compared
with bovine bone. This hypothesis should be further verified by experiments
with a large number of samples. Some of the correlations in the present study
may appear insignificant due to the small number of specimen within anatomical
sites.
Fractures of trabecular tissue can occur even without a traumatic event and
for this reason they cannot be detected by simple measurements of the amount
of bone [86], such as determining BMD. At frequencies below 5 MHz, relative
permittivity and dissipation factor, i.e., parameters strongly associated with the
postyield behavior of bone, distinguished microfractured and intact trabecular
structure. The microCT investigations revealed the sensitivity of relative permittivity to the processes occurring on the surface of mineralized bony matrix.
This might explain the sensitivity of EIS in detecting the microcracks accompanying microdamage. In contrast, the parameters measured with an open-ended
coaxial probe did not change as the structure fractured, possibly because of the
significance of free and bound water abundant in superficial surface layer. This
water is the main determinant of the gamma and delta dispersions.
In human trabecular bone, strong relations were found between EIS measures and microstructure. However, as structural parameters are strongly interrelated [77, 161], it is difficult to evaluate the effect of individual structural
parameters on electrical properties. Nevertheless, the PCA showed that the
trabecular structure component, reflecting in particular the degree to which the
mineralized matrix is multiply interconnected, the number of trabeculae and
space between them, correlated strongly with the electrical parameters, except
for relative permittivity. Interestingly, after adjustment for these structural
parameters in the regression analysis, the degree of anisotropy provided additional information on the dissipation factor at 1.2 MHz. On the other hand,
the variation in trabecular properties related to surface characteristics, such as
the amount of surface, the thickness and shape of trabeculae were the main determinants of the relative permittivity. Additionally, trabecular orientation has
been reported to be strongly associated with the mechanical behavior of bone
[162], as well as to be the main reason for the existing relationships between
permittivity in different directions [104]. In the present study, despite the wide
variation (2.0◦ - 22.6◦ ) in mean trabecular orientation relative to the axes of
the cylindrical samples, no correlation between the electrical parameters and
trabecular orientation was found. This lack of correlation emphasizes the importance of other differences between the samples. The specimens with different
trabecular orientation were not obtained from adjacent sites of the same bone.
71
The difference in the data obtained with parallel-plate and open-ended probes
(Figure 8.2) may be due to the heterogeneity and anisotropy of the tissue investigated and to the different volume of the sample probed by each technique [123].
The parallel-plate cell measured the tissue in the direction parallel to the main
loading axis of the bone. The electric field fills the volume of the whole sample,
approximately 1.6 cm3 . In contrast, the field from the open-ended probe penetrated a specimen in the medio-lateral direction, or perpendicular to its long
axis. The volume occupied is much smaller, approximately 0.5 cm3 . Moreover,
the superficial surface layer contains more damage and liquid than the bulk
sample. This may lead to higher values of permittivity and conductivity for the
open-ended probe measurements.
The size of the sample was optimized based on the feasibility of all experiments conducted. A compromise was made between e.g., the capacity of the
servo-hydraulic material testing device and the size of the impedance electrodes.
The preferred dimensions of specimens limited the possible electrode configurations. For instance, there were certain practical limitations when introducing
four electrodes big enough for errorless experiment. The feasibility of a measurement setup described by Gandhi et. al. [163] was investigated. The impedance
probe consisted of four stainless-steel electrodes coated with gold placed in one
plane. Contact between the probe and specimen was found to be poor, because
of the small size of the electrodes (2 - 3 mm in diameter). This led to too
high impedance values and not reproducible results. Poor contact may be due
to roughnesses on the surface of the specimen. Flat surfaces can be obtained
by polishing, but this may affect the microarchitecture of the sample. Additionally, due to the small inter-electrode distance (1 - 2 mm) and the presence
of a certain amount of liquid on the surface of a sample, the electrodes easily
become short-circuited, leading to an unreliable measurement protocol. Similar shortcomings were found at microwave frequency range. The results were
found to be erroneous when a probe with dimensions below a certain threshold was used. Further, the size of the sample was also restricted by biological
factors. Due to the limited volume of trabecular bone in the human skeleton,
only relatively small specimens will be homogeneous. Measuring inhomogeneous
samples would result in averaging the mechanical, structural and compositional
properties, impeding the investigation of EIS sensitivity.
Consequently, two relatively large electrodes, ensuring good contact without
short-circuiting, were chosen for the measurements. They probe a sufficiently
big volume of the sample so that the surface condition does not introduce a
significant error, but the sample remains homogenous. As discussed in the 5.3
Section, a two-point method is prone to electrode polarization. However, the
latest research shows that even a four-electrode setup, which is believed to be
suitable for the measurements of biological materials and to be a remedy for the
shortcomings of the two-point technique [98, 164], is not free from inaccuracies
due to polarization impedance, and may be even more susceptible to errors than
a two-point system [165]. In the present study, the error due to the polarization
impedance was found to be 20% - 4.5% in the frequency range of 20 Hz to 5
MHz and below 8% in the most relevant frequency range of this study. This
is of the same order of magnitude as in a previous investigation on bone [123],
so the present results are in line with those of previous studies [121, 104, 27].
Electrical and dielectric properties are sensitive to factors such as the technique
of the sample preparation, quality of the surface, method of preservation or
72
9. Discussion
temperature (see Section 5.4). Even if all these factors are kept constant, the
electrical and dielectric properties may vary from specimen to specimen by as
much as 20%. Consequently, the average error introduced by the polarization
impedance does not significantly increase the uncertainty of the results. Nevertheless, the values of the electrical and dielectric properties obtained in this
study should be used with caution. The main aim of the present study was to
conduct a comparison analysis. For instance, the values of the electrical and
dielectric properties before and after mechanical testing were compared with
each other. Alternatively, the effect of variations in bone composition on the
change in electrical and dielectric properties was investigated. A shift in absolute values will not affect the correlations obtained and the conclusions drawn
remain valid.
Electrode gel was used in Study I to improve the contact between the electrode and the sample, especially at low frequencies (< 1 kHz). However, the
gel may contain long jelly molecules that might interact with the electric field
and might influence the outcome of the investigations. For this reason, the use
of gel has not been recommended in impedance measurements [166]. The results obtained in Study I were analogous to those of previous studies without
gel [27]. Interestingly, previous studies report no significant differences in bone
tissue impedance due to the use of gel (see e.g., [131]). In order to determine
the possible effect of gel on the results, a phantom material was tested with and
without gel rendering indistinguishable results. Moreover, no significant difference was observed in bone samples tested with and without gel. However, the
difference may appear insignificant due to the small number of specimens and
relatively high standard deviation of the bone sample set. The largest variation
between the results with and those without gel among the five tested samples
was 10%. As discussed above, this is less than the variation in electrical and
dielectric properties due to specific biological factors, such as composition or
microstucture, which are often not accounted for in the literature. Further,
only a thin layer of gel was applied to the surface of the sample, just prior to
the experiment. During the short measurement time, it is unlikely that the gel
penetrated the sample and changed its properties significantly.
The results of the present study suggest that a low density trabecular bone
structure can be distinguished from a high density structure by measurements
of permittivity and conductivity at low (< 100 Hz) or high (> 100 kHz) frequency bands. A sparse structure possesses higher relative permittivity below
100 kHz and higher conductivity at all the frequencies considered than does
tissue with more dense architecture. Possibly, high water and fat3 content as
well as low BMD and BV/TV value for sparse structure result in a high DC
conductivity that masks the AC conductivity and increases relative permittivity at low frequencies. On the other hand, the high values of SMI and BMD
for high density bone are associated with high amount of mineralized matrix
and collagen content, increasing the β dispersion for this structure. Interestingly, the differences in electrical parameters between human and bovine bone
showed similar behavior to the sparse-dense structure relationship. This could
possibly result high collagen content and low water content found in bovine trabecular bone. It could also suggest that the basic structural and compositional
3 It has been hypothesized in study IV that the interstitial bone marrow water significantly
contributes to the overall trabecular bone conductivity.
73
properties are the main determinants of electrical characteristics, and that the
other differences between human and animal bone tissue are consequently less
significant.
The findings of the supplementary data suggest that EIS may be feasible
and useful also at microwave frequencies. Measurements at this high frequency
band may provide information on e.g., microstructural anisotropy not obtainable from low-frequency measurements. However, further research is needed
to confirm the findings. Contact between electrode and sample must be optimized, by for instance improving the flatness of the sample surface or employing Micro-Electro-Mechanical System (MEMS) technology when preparing the
probe. Additionally, the small number of samples could be the reason for the
insignificance of certain correlations obtained in this study.
Even though electrical properties, relative permittivity in particular, were
good predictors of trabecular bone BMD as well as other structural, mechanical
and compositional quantities, the applications of EIS techniques in clinical bone
diagnosis are limited. The method may be feasible during some special cases
of open surgery. Further, it may be used for quantitative evaluation of bone
grafts in vitro. Additionally, it might be useful in laboratory practise, providing fast and cheap estimation of bone mechanical, structural and compositional
characteristics. The results presented here might be useful in the development
of electrical impedance tomography methods, where the quality of bone could
be assessed based on the electrical parameters measured. On the other hand,
the detailed electrical characterization of bone is necessary for determining the
relationship between the current and the electrical field, as well as for the current paths during electrical stimulation of bone fractures [130]. For instance,
conductivity should be known if we are to choose the correct magnitude of the
voltage [119]. Thus, if the mineral density of bone is known, the relationships
presented in this study provide a tool for the estimation of electrical or dielectric
properties.
74
9. Discussion
Chapter X
Summary and conclusions
In this study electrical and dielectric properties of trabecular bone have been
characterized. The wide range of frequencies applied improves the significance
and potential application of the results. The quality of bone samples, understood as the architecture of trabecular elements, their connectivity, the amount
of minerals, the mechanical strength, the presence or absence of microdamage
and the chemical composition of the matrix, was evaluated and correlated with
EIS results. The most important results can be summarized as follows:
1. Dissipation factor and especially relative permittivity were sensitive parameters to changes in mechanical, structural or compositional characteristics of trabecular bone. Additionally, conductivity showed a strong
dependence on water content as well as on the shape and amount of trabeculae. Yield strain in trabecular tissue was only predicted by specific
impedance.
2. Due to the frequency dependence of linear correlations between certain
electrical and dielectric parameters and other investigated quantities, frequency bands with the highest potential to estimate bone quality were
determined. In bovine trabecular bone, parameters obtained at frequencies above 50 kHz were sensitive to variation in mechanical characteristics.
In human trabecular bone, however, electrical spectra obtained at frequencies between 100 kHz and 5 MHz contained the most valuable information. Additionally, some parameters, such as water content, dry density
or connectivity, could be characterized more efficiently by measurements
at frequencies between 100 Hz and 1 kHz.
3. In human bone, electrical and dielectric parameters were more sensitive to
variation in mechanical properties than in bovine bone. Especially tissue
strength was reflected by the EIS parameters. Relative permittivity and
dissipation factor were found to be capable of detecting microdamage in
trabecular structure.
4. Microstructural parameters related to the surface of trabecular structure
were found to be the main determinants of relative permittivity, while the
shape and amount of trabeculae were primarily related to other electrical
75
76
10. Summary and conclusions
and dielectric parameters. This suggests that the variation in different
microstructural elements may be detected by various electrical parameters.
Water content was strongly related to all electrical and dielectric parameters, conductivity in particular. It is possible that that interstitial bone
marrow water has a major impact on overall trabecular bone conductivity.
On the other hand, fat content correlated only with the relative permittivity at frequencies higher than 100 kHz. Additionally, collagen content
and the polarization effects on its surface associated with the hydration
layer, were the main determinant of β dispersion.
5. Bone with healthy, dense trabecular structure was found to have lower
conductivity and higher relative permittivity at frequencies between 100
kHz and 5 MHz than tissue with a sparse structure. Moreover, tissue water
and fat contents could be determined by a measurement of relative permittivity and conductivity at two separate frequency bands. This, along with
the ability of EIS to detect microdamage of trabecular structure, suggests
the potential of EIS as a tool to assess bone quality. However, further
theoretical, experimental, and technical developments of the method are
needed to confirm the present findings and to fully reveal the potential of
the method, especially at radiofrequencies. The geometry of the measurement cell at frequencies below 1 MHz could consist of two electrodes in
one plane in order to make the measurements fast and simple.
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BIBLIOGRAPHY
ERRATUM
In the study II the Figure 3C, as well as the values for phase angle at 1.2 MHz,
has been misprinted. The correct figure (Figure E.1) and the values of the parameter (Table E.1) are presented below.
Phase angle (o)
-10
-10
-10
-10
Femur
-1
Tibia
0
1
2
10
1
10
2
10
3
10
4
10
5
10
6
10
7
Frequency (Hz)
Figure E.1: Phase angle as a function of frequency (mean ± υ) for the femur
and tibia.
Table E.1: Mean values (± υ) for the phase angle at 1.2 MHz of human
trabecular bone in femur and tibia.
Femur (n = 10) Tibia (n = 10)
Phase angle (◦ )
−2.0 ± 0.8
91
−1.3 ± 0.5
Additionally, in the study III, the name of the 2nd and 3rd column in the Table
1 has been exchanged. The correct table is presented below (Table E.2).
Table E.2: The variances and coefficients of the principal components for
electrical parameters of trabecular bone samples (n = 26) measured at 1.2
MHz.
Field propagation
Energy
component
component
Phase angle
Conductivity
Specific impedance
Dissipation factor
% of explained variation
68.2%
27.0%
0.985
-0.092
0.982
-0.146
-0.939
0.034
0.770
-0.586
-0.042
0.988
92
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Electrical and Dielectric Characterization of Trabecular

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