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DISSERTATIONS | CHIBUZOR ENEH | PULSE-ECHO ULTRASOUND ASSESSMENT OF CORTICAL BONE... | No 23
CHIBUZOR ENEH
PULSE-ECHO ULTRASOUND ASSESSMENT OF PUBLICATIONS OF
THE UNIVERSITY OF EASTERN FINLAND
The measurement of cortical bone thickness using pulse-echo ultrasound has been proposed for use in clinical diagnostics of osteoporosis. The ultrasound methods
investigated in this thesis aimed at simultaneous measurement of site-specific cortical thickness and porosity. These methods
are a step in the direction to improve the clinical estimation of fracture risk and facilitate monitoring the effectiveness of osteoporosis therapies also at the primary
healthcare level.
CHIBUZOR ENEH
CHIBUZOR ENEH
Pulse-Echo Ultrasound Assessment of Cortical
Bone Thickness and Porosity
Publications of the University of Eastern Finland Dissertations in Forestry and Natural Sciences
No 235
Academic Dissertation
To be presented by permission of the Faculty of Science and Forestry for public examination in the Auditorium SN200 in Snellmania Building of the University of
Eastern Finland, Kuopio, on September, 30, 2016, at 12 o’clock noon.
Department of Applied Physics
Editor: Prof. Jukka Tuomela, Prof. Pertti Pasanen Prof. Pekka Toivanen, Prof. Matti Vornanen
Distribution:
University of Eastern Finland Library / Sales of publications P.O. Box 107, FI-80101 Joensuu, Finland
tel. +358-50-3058396 http://www.uef.fi/kirjasto
ISBN: 978-952-61-2224-3 (Print) ISSNL: 1798-5668
ISSN: 1798-5668 ISBN: 978-952-61-2225-0 (pdf)
ISSNL: 1798-5668 ISSN: 1798-5676
Author’s address: University of Eastern Finland Department of Applied Physics P.O.Box 1627
70211 KUOPIO FINLAND
email: chibuzor.eneh@uef.fi/ chibuzor.eneh@gmail.com Supervisors: Dean Jukka Jurvelin, Ph.D.
University of Eastern Finland Department of Applied Physics Kuopio, FINLAND
email: jukka.jurveline@uef.fi Professor Juha Töyräs, Ph.D.
University of Eastern Finland Department of Applied Physics Kuopio, FINLAND
email: juha.toyras@uef.fi Janne Karjalainen, Ph.D.
University of Eastern Finland Department of Applied Physics Kuopio, FINLAND
email: janne.karjalainen@boneindex.fi
Reviewers: Reinhard Barkmann, Ph.D.
Universitätsklinikum Schleswig-Holstein
Klinik für Diagnostische Radiologie und Neuroradiologie Kiel, GERMANY
email: barkmann@rad.uni-kiel.de
Assistant Clinical Professor Jonathan J. Kaufman, Ph.D.
The Mount Sinai School of Medicine Department of Orthopaedics
New York, NY, USA
email: jjkaufman@cyberlogic.org Opponent: Professor Brent K. Hoffmeister Ph.D.
Rhodes College Department of Physics Memphis, TN, USA
email: hoffmeister@rhodes.edu
ABSTRACT
Osteoporotic bone is typically characterized by its reduced cortical thickness. Consequently, an assessment of cortical thickness by pulse- echo (PE) ultrasound may be helpful in the diagnosis of osteoporosis.
This thesis evaluated the unknown effect of cortical bone porosity on radial speed of sound (SOS) in PE assessment of cortical thickness.
Additionally, the possibility to improve the assessment of cortical thickness by using subject specific radial SOS was investigated.
A finite difference time domain (FDTD) simulation was performed to study how simultaneous changes in elastic properties and the microstructure of human cortical bone affect the variation present in the measurements of radial SOS. The uncertainty associated with the use of a constant radial SOS in PE cortical thickness assessment was investigated both numerically and experimentally. Finally, PE ultrasound backscatter was used to predict cortical porosity. This information was then utilized to estimate radial SOS in cortical bone and to enhance the accuracy in cortical thickness assessment.
A simultaneous increase of calcified matrix elastic coefficients and cortical bone porosity reduced the age dependent variation in the radial SOS. However, application of a constant radial SOS resulted in an error of over 6% in the in vitro assessment of cortical thick- ness. Conversely, a sample specific SOS estimated based on cortical porosity derived from PE ultrasound backscatter data subjected to multivariate analysis reduced the inaccuracy in the PE assessment of cortical thickness from 0.223 mm down to 0.042 mm.
Cortical bone thickness and porosity contribute to the mechanical competence and fragility of the whole bone. Therefore, simultaneous measurement of site-specific cortical thickness and porosity may im- prove the clinical estimation of fracture risk which would facilitate
monitoring the effectiveness of osteoporosis management with drug therapies. In principle, the ultrasound methods investigated in this thesis may be developed further and exploited in basic healthcare.
Medical Subject Headings:Bone and Bones; Bone Density; Bone Matrix;
Porosity; Elasticity; Biomechanical Phenomena; Ultrasound Waves;
Ultrasonics; Numerical Analysis, Computer-Assisted; Multivariate Analysis; Osteoporosis/diagnosis
Yleinen suomalainen asiasanasto:luu; luuntiheys; paksuus; huokoisuus;
kimmoisuus; joustavuus; biomekaniikka; ultraääni; ultraäänitutkimus;
numeerinen analyysi; monimuuttujamenetelmät; osteoporoosi
To my Family
Acknowledgments
At the outset of my doctoral studies, I was told that a PhD does not make one an outstanding researcher any more than a driver’s license signifies a superb driver.
Rather, throughout the process, I would learn essential elements and skills required in research and in becoming a competent researcher - persevering with experiments, manuscripts, and grant proposals to meet stringent deadlines without evading re- jection of my work. Crucially, the process has taught me that imperfections do not limit anyone, rather a lack of the will to proceed despite them.
“For I am confident of this very thing, that He who began a good work in you will perfect it until the day of Christ Jesus.” -Philippians 1:6
This study was carried out during the years 2011-2016 in the Depart- ment of Applied Physics at the University of Eastern Finland. I have been fortunate to have such excellent supervisors as Dean Jukka Ju- rvelin, Ph.D., his professional and understanding leadership showed me what a prestigious scientist with a gentlemanly character looks like; Prof. Juha Töyräs, Ph.D., who gave me lessons in “blood, toil, tears and sweat”; Janne Karjalainen, Ph.D., who helped me to ap- preciate the impact of my research. I would also like to thank my
co-authors Markus Malo, Ph.D., Jukka Liukkonen, Ph.D., and Isaac Afara, Ph.D., for their crucial input to the studies in this thesis. I owe much gratitude to Senior Scientist Reinhard Barkmann, Ph.D., and Assistant Clinical Prof. Jonathan Kaufman, Ph.D., for acting as official Reviewers of this thesis, as well as Prof. Brent Hoffmeister, Ph.D., for agreeing to act as the opponent for my public examination.
I feel so honored to be evaluated by such leading experts in the field of bone ultrasound research. I also would like to acknowledge Ewen MacDonald, D. Pharm., not only for his linguistic review, but also for an excellent sense of rhythm that has enhanced the readability of this thesis.
Many thanks to the current and emeritus members of the BBC group. James Fick, Ph.D., encouraged and mentored me in the dif-
Ph.D., Xiaowei Ojanen, Ph.D. Student, Hans Linder, tech wizard, and Juho Ala-Myllymäki, B.Sc. The help of the staff of the Department of Applied Physics, SIB Labs, and that of the Faculty of Forestry and Natural Sciences has been invaluable. I was initially inspired about physics during my undergraduate studies in Oulu by Uni- versity Lecturer Kari Kaila, Ph.D., later both Prof. Miika Nieminen, Ph.D., and Chief Physicist Antero Koivula, Ph.D., motivated me to pursue Medical Physics. I am grateful for all the support I have received from my friends (IFG, Eelim, footballers, Image, ‘Kaasukuo- lio’, blood bro’s, Oulu bro’s ‘n’ sis’s, 3D animation bro and many others) and extended family for giving me something else to think about outside of work during the past years.
The following sources of funding are acknowledged - the strate- gic funding of the University of Eastern Finland, Kuopio University Hospital (EVO 84/2012, EVO/VTR 5031342 and 5041741 PY210 Di- agnostic Imaging Center), the Doctoral Programme in Science, Tech- nology and Computing (SCITECO), Emil Aaltonen Foundation, the International Doctoral Programme in Biomedical Engineering and Medical Physics (iBioMEP).
I am deeply indebted to my parents, Basil and Oge Eneh, for being role models and believing in me, my siblings: Caleb, Pamela- Rose and Sandra, along with her family, for their continuous love, support and encouragement. What words can I write to express my deepest gratitude to you beloved Stralina for your understanding up to the point of bringing me food and sitting with me in the office into the early hours of the morning? Even if a pound of flesh to you I would pay, it would mean nothing, ifI love youwould not be the melody to which it plays.
Kuopio, August 2016
Chibuzor Eneh
ABBREVIATIONS
2D two-dimensional
3D three-dimensional A/D analog to digital
AIB apparent integrated backscatter ANOVA analysis of variance
Ant anterior
BMD bone mineral density BMU bone multicellular unit BSU bone structural unit BV/TV bone volume fraction
CT X-ray computed tomography
CV coefficient of variation/ or cross-validation DA degree of anisotropy
DXA dual energy X-ray absorption FDTD finite difference time domain FRAX fracture risk assessment tool
FSAB frequency slope of apparent backscatter
HR-pQCT high resolution peripheral quantitative computed tomography HSD honest significant difference
IRC integrated reflection coefficient
Lat lateral
MBD mean of the backscatter difference
Med medial
micro-CT micro-computed tomography MRI magnetic resonance imaging n.s. not significant
NHANES National Health and Nutrition Examination Survey PBS phosphate buffered saline
PDE partial differential equations
PE pulse-echo
PLS partial least squares
Po.N pore number Po.Sp pore separation Post posterior
QCT quantitative computed tomography QUS quantitative ultrasound
RMSECV root mean square error of cross-validation SAM scanning acoustic microscopy
SBD slope of mean of the backscatter difference SD standard deviation
SOS speed of sound TMD tissue mineral density TOF time of flight
TSAB time slope of apparent integrated backscatter
US ultrasound
VOI volume of interest
WHO World Health Organization
SYMBOLS AND NOTATIONS
A amplitude
A(f) amplitude spectrum a time constant
Cii elastic stiffness coefficient in direction ii c speed of sound
cl longitudinal speed of sound cp phase velocity
cu particle velocity
D time interval for which simulated Gaussian function is de- fined
d distance/ or thickness E Young’s modulus
F force/ or value of F-test (statistics) f frequency
h spatial increment in a finite difference approximation of a partial differential equation
I acoustic intensity/ or second moment of inertia K bulk modulus
L length
M bending moment n number of samples
p sound wave pressure for plain waves/ or statistical signifi- cance
R correlation coefficient R2 coefficient of determination RC reflection coefficient
S surface area
T period
TC transmission coefficient
t time
u particle displacement
u0 maximum displacement amplitude
w displacement field in the Cartesian coordinate system
yi measured response ˆ
yi theith predicted response during cross-validation Z acoustic impedance
α attenuation coefficient
δmax maximum deflection of bending beam λ first Lamé constant
λw wave length ρ mass density
µ second Lamé constant (shear modulus) ν Poisson’s ratio
ω angular frequency of the wave φ phase angle
e strain σ stress
θ angle
∂ partial difference operator
∇ gradient operator
∇· divergence operator
LIST OF PUBLICATIONS
This thesis consists of the present review of the author’s work in the field of medical physics. The following selection of the author’s publications is referred to by Roman numerals:
I C. T. M. Eneh, J. Liukkonen, M. K. H. Malo, J. S. Jurvelin and J. Töyräs, “Inter-individual changes in cortical bone three- dimensional microstructure and elastic coefficient have oppo- site effects on radial sound speed,” Journal of the Acoustical Society of America6, 3491–3499 (2015).
II C. T. M. Eneh, M. K. H. Malo, J. P. Karjalainen, J. Liukkonen, J. Töyräs and J. S. Jurvelin “Effect of porosity, tissue density, and mechanical properties on radial sound speed in human cortical bone,”Medical Physics43, 2030 (2016).
III C. T. M. Eneh, I. O. Afara, M. K. H. Malo, J. S. Jurvelin and J.
Töyräs, “Porosity predicted from ultrasound backscatter using multivariate analysis can improve accuracy of cortical bone thickness assessment,”Submitted(2016).
The original articles have been reproduced with kind permission of the copyright holders.
The publications in this dissertation are original research papers on pulse-echo ultrasound assessment of cortical bone thickness. The author was involved in the study design and planning of each paper.
In paperI, the author conducted micro-computed tomography imag- ing, image processing, as well as finite difference time domain mod- eling and all the data analysis. The scanning acoustic microscopic evaluation from which the elastic coefficients were derived was con- ducted by M. K. H. Malo. The author was the principal author of the manuscript.
In paper II, the author conducted sample preparation, ultrasound measurements, mirco-computed tomography imaging, image pro- cessing, finite difference time domain modeling, carried out all of the data analysis, and was the principal author of the manuscript.
In paperIII, the author conducted sample preparation, ultrasound measurements, micro-computed tomography imaging, image pro- cessing, carried out all the data analysis and was the principal au- thor of the manuscript.
In all papers the cooperation with the co-authors has been signif- icant.
Contents
1 INTRODUCTION 1
2 BONE 5
2.1 Bone: function, formation and composition . . . 5
2.2 Trabecular bone . . . 8
2.3 Cortical bone . . . 8
2.4 Aging and osteoporosis . . . 11
3 ULTRASOUND ASSESSMENT OF BONE 17 3.1 Basic physics of ultrasound . . . 18
3.2 Acoustical properties of cortical bone . . . 20
3.3 Pulse-echo methods for assessing cortical bone . . . . 21
3.4 Numerical simulation of wave propagation . . . 23
4 AIMS OF THE PRESENT STUDY 27 5 MATERIALS AND METHODS 29 5.1 Sample preparation . . . 29
5.2 Ultrasound measurements . . . 31
5.3 Imaging . . . 32
5.3.1 Bone microstructure and mineral density . . . 32
5.3.2 Microscopic elastic coefficients . . . 33
5.4 Numerical simulations . . . 34
5.5 Multivariate analysis . . . 36
5.6 Statistical methods . . . 39
6 RESULTS 41 6.1 Radial SOS and elastic coefficients of the calcified ma- trix (simulation study) . . . 41
6.2 Constant radial SOS, cortical porosity, and inaccuracy in cortical thickness assessment . . . 43
7 DISCUSSION 51 7.1 Opposite effects of elastic coefficients and porosity of
cortical bone on radial SOS . . . 51 7.2 Estimation of cortical porosity from ultrasound backscat-
ter information . . . 53
8 SUMMARY AND CONCLUSIONS 57
REFERENCES 59
1 Introduction
Bone is a living tissue that alters its shape and adapts its structure to changes in physical loading through a process known as remod- eling (Wolff’s law) [1, 2]. Because of this active remodeling process, the skeleton of the average adult is completely renewed about every 10 years [3,4]. During a lifetime, bone mass increases gradually until about the age of 25, after which it begins to decline [5–9]. Osteoporo- sis is attributed to an accelerated decline of bone mass that reduces the mechanical competence of the whole bone and is frequently fol- lowed by a fracture [10]. Osteoporosis is a skeletal disease, affecting a growing number of elderly people all around the world [11–14].
For example, Caucasian (white) women have a 1.5 times greater lifetime risk of sustaining a hip fracture than developing breast can- cer [15]. Osteoporosis is insidious because it does not evoke early symptoms and it is often diagnosed too late only after a subject has sustained a fracture [16]. In the European Union, the costs of osteo- porosis were approximately 36 billione/year in terms of healthcare
services in 2007 [17] and these expenditures are projected to increase 25% by 2025 [18]. Therefore, there is a growing incentive for early diagnostics, prevention and treatment to effectively manage osteo- porosis.
The current gold standard in osteoporosis diagnostics is axial dual energy X-ray absorptiometry (DXA) of the hip (femoral neck and/or total hip) to measure bone mineral density (BMD) [19]. Osteo- porosis is defined as a BMD value of 2.5 standard deviations below
the mean value of a healthy young reference population [10, 20].
However, DXA exposes patients to ionizing radiation and is not typically available at the primary healthcare level due to the size and cost of the device. Furthermore, as the decrease in BMD makes only about a 30% contribution to all low energy fractures [21], it is important to identify additional indicators of bone strength. These indicators would take into account bone geometry, microstructure
and organic composition [22, 23].
Ultrasound is potentially a non-invasive, non-ionizing and cost- effective modality to improve fracture risk assessment by probing the mechanical and structural properties of bone [24–28]. Cortical bone comprises approximately 80% of human bone mass and ∼80%
of all fractures occur mainly at cortical sites [29]. Cortical thickness is a good surrogate for bone strength, whereas cortical porosity also contributes to the strength and fragility of the whole bone [30]. Thin- ning of the cortical bone layer due to osteoporosis can be detected in the shafts of long bones by conducting diagnostic pulse-echo (PE)in vivomeasurements at the tibia and radius [31]. In fact, the measure- ment of cortical thickness using PE ultrasound has been proposed for clinical screening of osteoporosis [32–34]. As both thickness and porosity of the cortical bone contribute to its strength, and poros- ity simultaneously affects both the density and elastic properties of cortical bone [35, 36], it is reasonable to speculate that changes in porosity will inevitably influence speed of sound (SOS) in cortical bone.
Speed of sound in cortical bone is determined by its tissue elastic properties, density and porosity. These properties change with age, anatomical location and disease. Furthermore, tissue elastic proper- ties also change with the orientation relative to the bone axis [36–38].
Speed of sound in the direction along the long axis of the bone has been reported to decrease with increasing cortical porosity [39–41].
In contrast, elastic coefficients of the calcified matrix increase with aging as well as with increased porosity [37, 42], which in theory, should increase SOS. The effects of simultaneous changes in the me- chanical properties, porosity, and mineral content of cortical bone on radial SOS are not totally clear. In fact, there are no published reports on pulse-echo ultrasound measurements of cortical porosity on its own, or simultaneously with cortical thickness in an individ- ual patient. Considering the effects of varying cortical porosity, one could argue that inclusion of the knowledge of how elastic prop- erties and mineral density influence radial SOS could improve the accuracy of ultrasound cortical thickness assessment and ultimately
Introduction
improve the technique’s diagnostic reliability.
The aim of this thesis is to investigate the effects of cortical bone porosity on radial SOS in PE assessment of cortical bone thickness and to explore whether the assessment is improved by application of a subject specific radial SOS estimated from cortical porosity. In studyI, the effects of concurrent inter-individual changes in corti- cal bone elastic properties and microstructure on radial SOS were simulated. In studyII, the measurement inaccuracy associated with the use of a constant radial SOS in PE-cortical thickness assessment was evaluated by means of controlledin vitroandin silicostudies, us-
ing human cortical bone samples. In studyIII, partial least squares (PLS) regression was employed to predict cortical porosity from ul-
trasound backscatter signals. Subsequently, the predicted cortical porosity was used to adjust the value of radial SOS for calculating
cortical thickness.
2 Bone
From the smallest known vertebrate (a New Guinean frog, Pae- dophryne amanuensis) with an average body size of 7.7 mm to the largest existing one (blue whale, Balaenoptera musculus), which is over 3000 times longer than that frog [43, 44], skeletons of ver- tebrates come in different shapes and sizes but they all consist of bone. Bone is a very unique biomaterial as it possesses the same strength as cast iron but is as light as wood [45]. For example, the bones in the lower limb need to be stiff, strong and able to withstand compressive loads exerted on them by muscles and gravity [46, 47].
Additionally they need to be light to ensure efficient mobility. This is well illustrated in triple jumpers, whose lower limbs experience forces that are proportional to over 15 times their body weight at the step phase of the triple jump execution [48].
2.1 BONE: FUNCTION, FORMATION AND COMPOSITION In addition to supporting the body, bones provide a framework to which muscles and tendons can be attached in order to enable loco-
motion. They also protect internal organs, provide an environment for marrow where blood cells are produced and they are involved in mineral homeostasis [49, 50]. Throughout life, bone grows via bone formation, modeling and remodeling whereby it adapts its mass and morphology according to the mechanical requirements placed on it [3, 51–53].
Bo n e f o r m a t i o n, m o d e l i n g a n d r e m o d e l i n g
There are three cell types involved in forming, modeling and remod- eling bone: osteoblasts, osteocytes and osteoclasts. Bone is formed by osteoblasts that synthesize organic proteins (organic matrix) and con- trol their calcification, also described as mineralization (deposition of inorganic minerals and salts) [54]. This leads to the production of a composite material, i.e., the calcified matrix (or bone matrix). Os-
teocytes are osteoblasts that become entrapped within the calcified matrix that they were constructing. The entrapment sites are called lacunae [55, 56]. Osteocytes residing within lacunae make contact with each other through dendritic cell processes called canaliculi [57].
When osteoblasts deposit new matrix on existing bone and remain on the surface of the newly formed calcified matrix, they are known as bone lining cells. Bone resorption involves the degradation of the organic and inorganic components of bone by osteoclasts [58]. Osteo- clasts operate in conjunction with osteoblasts in a bone multicellular or metabolic unit (BMU) to resorb and replace bone matrix at the same location, a process referred to as remodeling [59].
Bo n e c o m p o s i t i o n a n d s t r u c t u r e
In general, bone is about 60% inorganic, 30% organic and 10% water, on a weight basis. However, in terms of volume, the proportions are 40%, 35% and 25%, respectively [9, 45, 60]. More specifically, the calcified material of bone is primarily (50-70% by weight) hydroxya- patite (Ca10(PO4)6(OH)2) [61]. This confers on bone material its stiff- ness and compressive strength. The hydroxyapatite forms crystals that are embedded in an organic arrangement consisting of mainly type I collagen molecules (∼90%), noncollagenous proteins (∼5%), lipids (∼2%) and water, by weight [62]. The collagen molecules form triple-helical fibrils that are stabilized and modified by cross- links. The cross-links result from enzymatic (lysil oxidase-mediated) or non-enzymatic (glycation-induced) processes [63]. The organic substance provides the bone with its tensile strength and flexibility, allowing it to stretch, bend, twist and deform (absorb energy) to avoid fracturing [23]. The calcified cross-linked collagen fibrils form bone lamellae that, depending on their organization, make up the elements of either cortical or trabecular bone.
Bone
Figure2.1:Mainanatomicalstructuresoftrabecularandcorticalbone.(Figureismodifiedfrom[65]withpermissionfromWiley& Sons,Ltd.)
2.2 TRABECULAR BONE
Irregularly arranged lamellae organized as “lamellar packets” (Fig.
2.1) comprise individual rod- and plate-like trabeculae (100-200µm in diameter). These trabeculae are the calcified matrix element of a network structure resembling a sponge with 50-90% porosity [45, 64].
Though the material composition of the calcified matrix is similar in both cortical and trabecular bone (including similar tissue mineral density) [64], the Young’s modulus for trabecular calcified matrix is generally lower [47].
Trabecular bone comprises about 20% of calcified matrix volume in the skeleton. It is found within the medullary cavity at the epiphy- ses and metaphyses of long bones as well as in vertebral bodies [66].
The orientation of trabeculae generally depends on the load distri- bution in the bone. Therefore, its architecture is optimized for load transfer. In the vertebral bodies, trabecular bone is the main load bearing structure [67, 68]. The surface area of trabeculae exposed to the marrow is large and therefore readily accessible to remodeling.
Consequently, adaptations to changes in mechanical loading may be detected earlier in trabecular than in cortical bone [5, 45].
2.3 CORTICAL BONE
Lamellae that are assembled around a Haversian canal (Fig. 2.1) form concentric cylindrical units that run roughly in parallel to the long axis of bone making up the Haversian systems (200-250µm in diameter) [69]. Haversian canals contain bone cells, blood vessels and nerves and are connected to each other by Volkmann’s canals that run transversally to the long axis of bone ensuring radial blood flow. Haversian systems are more commonly known as osteons. The boundary between the osteon and interstitial bone is known as the cement line, a thin layer of mucopolysaccharides with a low min- eral and collagen content [57, 70]. Osteons form the bone structural unit (BSU) of cortical bone [24]. The periosteum covers the outer surface of bones, while the endosteum lines the inner surface. Im-
Bone
mediately beneath the periosteum and endosteum, the lamellae are arranged in parallel to the circumference of bone; these are called cir- cumferential lamellae. The spaces between adjacent osteons enclose interstitial lamellae, which are remnants of previously remodeled osteons. Over time, the osteons can be removed completely, leaving behind resorption cavities that are refilled by new osteons [57].
Cortical bone porosity is determined by the number and size of Haversian and Volkmann’s canals (12-200µm in diameter) as well as resorption cavities (200-300µm in diameter) [7, 71–74] excluding osteocyte lacunae and canaliculi [9, 75]. Average cortical porosity at typical fracture sites such as the radius, tibia and the femur in a healthy individual is low, ranging from 3% to 28% [55, 74, 76–78], which is why cortical bone is also referred to as compact bone. The porosity increases from the periosteum to the endosteum and forms a gradient with changing mechanical properties [35, 36, 38, 79]. The interface between the calcified matrix and porous network consti- tutes an internal surface with three compartments: the endocortical surface between the cortical bone and the medullary canal, the tra- becular surface, and the lesser-known intracortical surface (Haver- sian and Volkmann’s canals). Bone modeling and remodeling take place on these surfaces. As the surface area per calcified matrix vol-
ume is lower in cortical bone than in trabecular bone, metabolic activity is lower in cortical bone. Therefore, trabecular bone loss has been the main focus of research into bone fragility for the past 70 years [80]. However, a transitional zone that resides at locations comprising both cortical and trabecular bone has been recently iden- tified as a site with vigorous remodeling activity [81]. It has been reported that accurate segmentation of this transitional zone in X-ray based imaging methods minimizes errors associated with misclassifi- cations, e.g., ‘trabecularized’ areas of cortical bone being considered as the medullary compartment or as trabecular bone [81]. This has led to the appreciation that most bone loss occurs through remodel- ing on the intracortical surfaces after the age of 65 years, rather than by remodeling on trabecular surfaces [80].
Cortical bone covers most of the human skeleton and comprises
80% of its mass. It exclusively forms the diaphysis of long bones and the thin shells that surround the metaphyses [45]. Most long bones are hollow with a medullary cavity that contains marrow. The corti- cal shell essentially provides strength to resist compression, tension, bending, torsion and shearing. Its structure is optimized to favor stiffness while being as light as possible, leading to thicker cortices and wider diameters when more strength is needed [4, 68, 82–84].
Bi o m e c h a n i c a l p r o p e r t i e s o f c o r t i c a l b o n e
Long bones are often considered as levers with movements that are controlled through contraction and relaxation of muscles. A balance between strength and stiffness is required so that the bone may maintain its shape and resist deformation, while still being able to deform reversibly to absorb energy. These properties are commonly investigated by applying a load on a bone specimen and measuring its deformation (displacement) [47, 68]. To examine the properties of bone tissue, i.e., to eliminate the influence of the geometry into which the tissue is arranged, load and deformation may be converted to stress and strain, respectively. Here, the stiffness determined from the linear part of the stress-strain curve is referred to as the modulus of elasticity (Young’s modulus, E) [85, 86]. Bone’s Young’s modulus may depend on the rate of strain (change in strain divided by the time over which deformation occurs), doubling when the strain rate increases by a factor of 106 from extremely slow to high impact strain rates (0.001 to 1000 s−1) [87, 88]. This is one manifestation of viscoelastic behavior. The end of the linear part of the stress-strain curve is known as the yield point. After this point, increasing strain does not result in a linear stress response and bone is unable to return to its original shape. The bone finally fractures at a peak stress value, this is called the ultimate stress. The area under the stress-strain curve represents the amount of energy needed per unit volume to cause bone failure which is commonly described as its toughness.
The stress-strain curve yields the value of Young’s modulus in one specific direction, but due to the variation in osteonal organi-
Bone
zation of cortical bone [35, 36, 47], the modulus may be different in other directions. When the properties of a material are different in various directions, it is considered to be anisotropic. Analogically, the Young’s modulus may be thought of as the normalized propor- tionality constant in Hooke’s law. Hooke’s law can be expanded to three dimensions to relate stress, strain and the mechanical proper- ties of cortical bone. This representation includes six components of stress (three normal and transverse/shear stresses) and, likewise, six components of strain.
For a completely anisotropic material, 21 independent elastic con- stants are needed. In practice, however, cortical bone in the diaph- ysis of long bones has been successfully characterized by transverse isotropy (five independent elastic constants) [89–93]. This means that its mechanical properties can be considered to be equivalent in two directions (in a plane), and different along the third perpendicular di- rection. In the case where the mechanical properties are considered the same in all directions, only two independent elastic constants are required, namely Young’s modulus and Poisson’s ratio (ν). Poisson’s ratio describes the resulting ratio of strains parallel and perpendicu- lar to directions of uniaxial compression or tension [24, 47, 94].
2.4 AGING AND OSTEOPOROSIS
Bone ages as a result of the metabolism occurring on its surfaces by BMU’s. Peak bone mass, structure and composition are initially deter- mined by genes [97, 98]. However, the composition and structure of bone adapts to prevailing loads as dictated by Wolff’s law [1, 2, 9, 47].
For example, periosteal bone formation is increased in areas where mechanical stresses are highest. This ultimately increases the sec- ond moment of areaI(the periosteal and endosteal walls are further displaced from the central axis of the bone [68]). During growth, a positive balance between the formation and the resorption of bone acts towards establishing the skeleton’s peak bone strength. With epiphyseal closure and the end of longitudinal growth, the balance shifts toward an equilibrium where the amount of bone formed
Table 2.1:Characteristic equations to describe mechanical parameters for elastic isotropic material [24, 85, 95, 96]
Parameter Equation
Young’s modulus (Pa) E=σ
e = µ(3λ+2µ) λ+µ
Bulk modulus (Pa) K= E
3(1−2ν) =λ+2 3µ
Poisson’s ratio (-) ν=−etransversal
eaxial = λ
2(λ+µ)
Maximum deflection of bending beam (m) δmax= ML
2
8EI
Stress (Pa) σ= F
S
Strain (-) e= ∆L
L0
First Lamé constant (Pa) λ= Eν
(1+ν)(1−2ν) Second Lamé constant (shear modulus) (Pa) µ= E
2(1+ν)
M= bending moment,L= length,I= second moment of area,F= force,S= surface area,∆L= change in the length andL0= original length.
and resorbed is equal. Throughout this phase, bone strength is main- tained by replacing old or damaged calcified matrix by new calcified matrix [68, 99]. Hence, bone tissue age is determined by its distance from active bone metabolic surfaces [100]. As the calcification of new osteons is slow, lasting even months, the process may result in varying degrees of mineralization (tissue mineral density) of osteons in any given bone cross-section [57, 101]. However, after skeletal mat- uration, tissue mineral density reaches a relatively stable value for older tissue (i.e., calcification does not continue indefinitely) [57,102].
This is apparent when comparing the mineralization or mechanical
Bone
properties between osteonal (degree of mineralization of bone, DMB:
1.06 g/cm3, elastic moduli: 21.8 GPa) and interstitial (DMB: 1.16 g/cm3, elastic moduli: 24.1 GPa) tissue [96, 103].
Im b a l a n c e i n b o n e m e t a b o l i s m
Four abnormalities have been identified to develop in bone model- ing and remodeling due to aging [66]. Firstly, there is the abrupt and rapid decline in periosteal apposition after completion of longi- tudinal growth. The second and third abnormalities are associated with BMU imbalance; ultimately less bone is deposited than is be-
ing resorbed, although the amounts of deposited and resorbed bone both decrease. In addition to increasing porosity, repetitive loading due to exercise and high impact, e.g., from a fall or some other phys- ical trauma, can induce cracks in the bone. These cracks that are initiated at the sub-micron level may eventually develop into visible cracks. The inability to replace the damaged bone as a result of the BMU imbalance can lead to its failure [9, 104]. The fourth abnormal- ity occurs in women, where increased remodeling, especially after menopause, accelerates bone loss and a deterioration in the quality of the bone structure.
Tr e b e c u l a r v s c o r t i c a l b o n e
Remodeling causes trabeculae to become thinner, leads to loss of trabecular connectivity, and ultimately to the complete loss of trabec- ulae. The loss of trabecular mass through a total loss of trabeculae compromises bone strength more than loss through trabecular thin- ning [66, 105]. The remodeling of cortical bone enlarges the canals and resorption cavities that eventually results in their coalescence.
After the age of 60, cortical bone loss is more prevalent than trabec- ular bone loss, because cortical bone comprises about 80% of the skeleton [45, 80]. This can be understood as an acceleration in corti- cal bone loss with increasing surface area, while trabecular bone loss decelerates due to decreasing surface area (Table 2.2). Consequently, increases in cortical porosity rather than trabecular bone porosity compromise bone strength with more than ∼80% of all fractures
occurring at cortical sites [80].
Table 2.2:Comparison between trabecular and cortical bone.
Trabecular bone Cortical bone Lamellae arrangement Irregular Concentric
“lamellar packets” around Haversians
Unit Rod or plate-like Osteon
trabeculae
BV/TV Low (10-50%) High (>70%)
Mass of skeleton 20% 80%
During aging Bone loss Bone loss
declines accelerates
BV/TV= bone volume fraction
Ag i n g a n d m e c h a n i c a l p r o p e r t i e s o f c a l c i f i e d m a t r i x Increase of longitudinal elastic coefficients of the calcified matrix has been associated with increasing age in men [37]. Increased elastic coefficients affect the bone such that it becomes more brittle and less tough with aging [106,107]. This may account for a greater part from changes in the organic phase of bone, since differences in degree of bone tissue mineralization contribute very little to the differences observed in tissue elastic properties [108–110]. These changes con- sist of several processes: variation in the co-alignment and orienta- tion of mineral crystals with the fiber axis of collagen [9, 104, 110];
changes in the relative amounts of enzymatic and non-enzymatic cross-links [104, 106], where an increase in the amounts of the latter make collagen more brittle [106,107]; and a decrease of water bound to the bone matrix plays a role in modifying the elastic properties of collagen [111].
Bone
Os t e o p o r o s i s
Osteoporosis is essentially a skeletal disease where compromised bone strength is followed by an increased fracture risk (particularly of the hip, spine or forearm as a result of minimal trauma, e.g., in conjunction with routine daily activity). Conceptually, bone strength can be thought of as the product of BMD, bone material properties, thickness and porosity of both cortical and trabecular bone [13, 18].
However, an areal BMD value (grams per square centimeter) of 2.5 or more standard deviations at the femoral neck and/or total hip below the young female adult mean value (T-score) as measured with a dual X-ray absorbtiometry (DXA) device indicates osteoporo-
sis. The reference range is based on femoral neck measurements of Caucasian women aged 20 to 29 years (the Third National Health and Nutrition Examination Survey, NHANES III, database). The reference values also apply to men aged 50 or more because the age-adjusted risks of hip fracture for a given BMD value are similar for both sexes [11]. For women prior to menopause and men below the age of 50, BMD is compared to the corresponding value in an age, gender and ethnicity matched healthy reference population. In practice, for men under 50 years of age, a BMD measurement on its
own is not sufficient to diagnose osteoporosis [112].
There are many different causes of osteoporosis; the disease can be hereditary, mechanical (extended periods of unloading), hor- monal, and nutritional. These factors can act independently or syner- gistically to compromise bone strength [113]. In Finland alone, 400 000 people are estimated to have osteoporosis and approximately 30 000-40 000 fractures per year are attributed to the deterioration of bone [114]. The prevention of osteoporosis mainly focuses on sufficient mechanical stimuli, a balanced diet, preventing falls, and stopping smoking. Drug therapy is commonly directed to individ- uals with the highest risk of fracture [114]. Individuals referred to drug therapy and those already under treatment are recommended to be monitored with DXA [19]. DXA scans conducted every 1-2 years serve as a means to verify the response to treatment in clinical practice. Due to the two-dimensional (2D) nature of the measure-
ment, areal BMD may overestimate the fracture risk for individuals with a smaller body size (lower areal BMD) than on average. Ad-
ditionally, variations in bone marrow composition and structures overlying central sites such as the spine (e.g., aortic calcification) and hip can affect the determined BMD value [19, 66].
Wh e n D X A i s n o t ava i l a b l e
When BMD-measurements are not available, the presence of one or more low-impact fragility fractures is considered as a sign of osteo- porosis. This can also be the case even if an individual does not have an osteoporotic BMD value [21, 115]. This discrepancy has led to the development of fracture prediction tools that take into account clin- ical risk factors (fracture history, smoking, alcohol, glucocorticoids, and rheumatoid arthritis) with age, weight, height, ethnicity and sex to estimate an individual’s 10-year probability of fracture (e.g., WHO’s FRAX tool) [11,29,116]. Additionally, in Finland according to
the recommendations of the International Society for Clinical Densit- ometry (ISCD), individuals may be referred to pharmacological ther- apy on the basis of peripheral ultrasound measurements [117, 118].
Co m p l e m e n t a r y d i a g n o s t i c s
Much effort has been invested in providing quantitative measures of status of bone in addition to that provided by BMD. Potentially, they could also be adopted to monitor outcomes of pharmaceuti- cal trials. There are some widely used radiologic modalities, i.e., quantitative computed tomography (QCT) [119], high-resolution pe- ripheral quantitative computed tomography (HR-pQCT) [120], mag- netic resonance imaging (MRI) [121] and quantitative ultrasound (QUS) [25, 32, 118]. This thesis is mainly concerned with recent ad- vances in QUS. Therefore, in the following section, the principles of ultrasound physics will be discussed, and some current approaches to assess bone with ultrasound will be introduced.
3 Ultrasound assessment of bone
Bone mass, shape and qualities (i.e., geometry, microstructure and mechanical properties) are factors that account for the strength of the whole bone; of these, bone qualities are the most difficult to assess. One modality proposed for the evaluation of bone qualities is non-ionizing quantitative ultrasound (QUS) since it is cost-effective [26]. Currently established QUS parameters, such as speed of sound and attenuation, have displayed statistically significant correlations with BMD and fracture risk, in both trabecular and cortical bone [26–28]. Additionally, QUS has been used with variable success to explore bone elastic properties [36, 91, 122], microstructure [36, 123–
127], matrix constituents (i.e., organic and mineral phases) [128–131]
and microdamage accumulation [132, 133].
Ultrasound measurements on cortical bone have been applied since the mid twentieth-century [134, 135]. However, only after the introduction of the first clinical QUS parameters in 1984 [25], was ul- trasound proposed for screening and diagnosing osteoporosis [136].
Nevertheless, widespread clinical exploitation of QUS methods is still lacking due to the low agreement between peripheral QUS pa- rameters and BMD at the proximal femur [26, 28, 32]. On the other hand, recent progress in pulse-echo ultrasound technology has re- vealed diagnosticin vivoinformation relevant for severe fracture sites, such as the proximal femur [31]. Therefore, even though not yet fully clarified, QUS possesses substantial potential, e.g., in evaluation of fracture risk, osteoporosis screening and monitoring the effects of drug therapies for osteoporosis (Axial Transmission: [137–141], Through Transmission: [142], Pulse-Echo: [31–34]).
3.1 BASIC PHYSICS OF ULTRASOUND
A vibrational wave propagating in matter (gas, fluid, solid or plasma) at a frequency > 20 kHz, i.e., above the upper audible limit of an av- erage healthy young human, is known as ultrasound (US). In solids, US waves can propagate in four principle modes, i.e., longitudinal waves, shear waves, surface (Rayleigh) waves and plate waves in thin materials. Longitudinal waves are most commonly used in assessing biological tissues [24, 143–145]. Longitudinal waves can propagate in all kinds of media, and result from particle motion in parallel to the direction of wave propagation (forming compressional waves).
When the particle motion is small (linear propagation), wave prop- agation speed (i.e., speed of sound, SOS) is independent of the am- plitude of the particle motion and is influenced by the density as well as the geometric and elastic properties of the medium. Typical relationships characterizing the linear propagation of ultrasound in an elastic solid are presented in Table 3.1.
In summary, an examination of the equation concerning particle displacement reveals that a wave radiated by a source reproduces the motion of that source. For a plane wave in a non-attenuating medium, the product of a medium’s density and SOS is commonly known as the medium’s characteristic acoustic impedance (or spe- cific acoustic impedance). It describes the resistance an ultrasound wave encounters as it passes through the medium. Moreover, it de-
termines the acoustic transmission and reflection at the boundary of two materials with different characteristic acoustic impedances and contributes to the attenuation of sound in a medium. Attenuation results from the combined effects of wave spreading (decreasing of wave amplitude with distance), scattering (i.e., deviation of sound energy from the initial propagation direction) and absorption (the conversion of sound energy to other forms of energy, e.g., heat) and depends on the sound frequency and characteristics of the medium.
Attenuation in a homogeneous medium is typically observed as the (exponential) decrease of the initial amplitude of the wave.
In addition to the effect of impedance mismatch, reflection at
Ultrasound assessment of bone
Table 3.1: Basic equations describing the interaction of planar ultrasonic waves with media [24, 143, 144, 146]
Parameter Equation
Particle displacement [m] u=u0sin(ωt−φ) Angular frequency of particle [rad/s] ω=2πf
Wavelength [m] λw= cp
f =cpT
Acoustic impedance [rayl] Z=ρcl
Longitudinal speed of sound in isotropic elas- tic solid [m/s]
cl=
r E(1−ν)
ρ(1+ν)(1−2ν)
Sound wave pressure for plain waves [Pa] p=ρclcu
Snell’s law [-] sinθc 1
1 =sinθc 2
2 = sinθc 3
3
Reflection coefficient [-] RC= ZZ2cosθ1−Z1cosθ2
2cosθ1+Z1cosθ2
Transmission coefficient [-] TC= 4Z1Z2cos2θ1
(Z2cosθ1+Z1cosθ2)
Attenuation law [-] p(d) =p0e−αdandI(d) =I0e−2αd
u0 = maximum displacement amplitude,t= time, φ= phase angle, f = fre- quency,cp= phase velocity,T= period,E= Young’s modulus,ν= Poisson’s ratio,ρ= mass density,cu= particle velocity,d= distance,α= attenuation coef- ficient,p0andI0are the initial pressure and intensity atd= 0, respectively.θ1
andθ2are the angles of the incidence and refraction, respectively. Subscripts 1, 2, and 3 refer to the first and second medium and shear wave, andlto the
longitudinal speed of sound.
the boundary of two materials occurs only if the object’s boundary is thicker and the dimensions of its surface are much smaller than the wavelength of the incident wave. The angle and fraction of the reflected wave pressure amplitude can be derived from the princi- ples of Snell’s law, continuity of particle velocity and local pressures
across the boundary. Acoustic scattering occurs as a result of the interaction between the primary wave and the rough surface of a het- erogeneous object with different density and/or elastic properties than those of the surrounding medium. Scattering is most likely to take place when the dimensions of the surface of the object (scatterer) are similar to or smaller than the wavelength of the incident wave.
When the wavelength is much greater than the dimensions of the scatterer, it is known as Rayleigh scattering. In Rayleigh scattering, sound is uniformly scattered in all directions and the incident wave undergoes a minor deflection at the edges of the object. When the wavelength and dimensions of the scatterer are comparable, the scat-
tering is more complex and may be estimated by Faran scattering models as is the case for trabecular bone [123, 147].
3.2 ACOUSTICAL PROPERTIES OF CORTICAL BONE
For cortical bone, the structural heterogeneities at different length scales (collagen content, fiber orientation; mineralization, size and ori- entation of hydroxyapatite crystals; water content; non-collagenous proteins; and volume fraction of canals and cavities) determine its elastic and acoustic properties. The concept of an effective elastic modulus (Ee) and density (ρe) has been developed in order to de- scribe the relationship between SOS and these properties [24, 144].
The effective modulus depends on the wave type (e.g., bulk compres- sion, bulk shear, and guided wave) and its interaction with cortical bone on a particular length scale, determined by the wavelength. Sim- ilarly, effective density is the density measured for the homogenized material with which the wave is interacting,
cCB= sEe
ρe, (3.1)
wherecCB is the longitudinal SOS in cortical bone. For wavelengths on the millimeter scale, cortical bone can be described as a two- phase composite material consisting of calcified matrix and pores
Ultrasound assessment of bone
(Haversian and Volkmann’s canals and resorption cavities contain- ing bone fluid) [148, 149]. On this length scale, porosity mainly af- fects bone elastic properties, because both the elastic coefficients and the tissue mineral density of the calcified matrix are relatively constant [36, 131]. Cortical bone porosity has also been postulated to influence sound attenuation [150–152], scattering [153] and disper- sion [154, 155]. On the other hand, the calcified matrix is considered to be transversely isotropic since the calcified collagen fibrils are predominantly oriented in parallel to the osteons [131]. Viscoelastic-
ity of the calcified matrix, thought to arise from the bound water in collagen bonds [35, 156, 157], may also contribute to attenuation through absorption.
3.3 PULSE-ECHO METHODS FOR ASSESSING CORTICAL BONE
Pulse-echo ultrasound techniques are promising for screening of osteoporosis in primary healthcare facilities [31,32]. The PE measure- ment involves a single transducer that both transmits and receives an ultrasound pulse. This approach can be used to evaluate the proper- ties of bone in a quantitative manner (Table 3.2) via parameters such as the integrated reflection coefficient (IRC) and apparent integrated backscatter (AIB) for trabecular bone [129,130,158–160]; and cortical bone thickness [31, 33, 34]. More recently, the time slope of appar- ent backscatter (TSAB); the frequency slope of apparent backscatter (FSAB) and the mean of the backscatter difference spectrum (MBD)
have also been adopted [159–162].
Pulse-echo assessment of cortical bone thickness has generally been conducted with focused piezoelectric transducers at 2.25 and 5 MHz center frequencies [31–34]. Within this range, excessive at- tenuation in thick (∼6 mm) cortical bone layers is avoided, while sufficient axial resolution for thin (1 mm) cortex is achieved [33, 34].
The spatial resolution for ultrasound measurements is determined laterally and axially. The axial resolution describes the shortest dis- tance between two objects that lie in parallel with the propagating
Table 3.2: Pulse-echo quantitative parameters used for the assessment of bone status [24, 161].
Parameter Equation
Integrated reflection coefficient [dB] IRC= 1
∆f R
∆f20 log10 Asr(f) Are f(f)d f Apparent integrated backscatter* [dB] AIB= 1
∆f R
∆f20 log10 Abs(f) Are f(f)d f Mean of the backscatter difference** [dB] MBD= 1
∆f R
∆f20 log10 Abs2(f) Abs1(f)d f Cortical bone thickness [mm] d= cl∆t
2
∆f = analyzed frequency band (typically refers to the -6dB frequency band- width of the reference spectrum),Asr= amplitude spectrum of the signal gated at the surface reflection,Are f= amplitude spectrum of the perfect reflector,Abs
= amplitude spectrum of the signal gated at the backscatter,∆tdenotes the time difference (in milliseconds) between reflections from the periosteal and endosteal surfaces of cortical bone, or it can be derived from the frequency band of harmonic spectral oscillations [34]. In the MBD, the subscripts 1 and 2 refer to delay at the gated backscatter, where 1 is kept constant and 2 is de- layed.
*Frequency slope of apparent integrated backscatter (FSAB) is determined as a slope of the linear part of the AIB.
**Slope of mean of the backscatter difference (SBD) is determined as the slope of the linear part of the MBD.
wave and can be separated. It has also been defined as the length of the pulse at half maximum [144]. The lateral resolution at a given frequency is determined by the ultrasound beam width and changes with distance [143].
For the PE assessment of cortical bone thickness, the transducer is oriented such that the emitted wave propagates radially with respect to the bone axis. In vitro measurements are conducted in degassed water (or phosphate buffered saline solution), where the specimen is placed in the focal zone of the transducer. For in vivo
Ultrasound assessment of bone
measurements, a standoff pad may be used to place the targeted cortical layer (e.g., tibia or radius) in the focal zone of the transducer.
In this case acoustic coupling is achieved by applying ultrasound gel between the skin-standoff pad and standoff pad-transducer inter- faces [33, 34].
Most approaches for computing the thickness of the cortical layer require strong reflections from the endosteal and periosteal surfaces of the cortical bone. The information regarding the time of flight (TOF) between these reflections obtained by signal processing ap- proaches (autocorrelation, cepstrum, and the envelope method ap- plying Hilberts transform), and a predefined SOS is used to compute
the thickness [31–34]. Speed of sound, however, changes with den- sity, porosity and elastic properties of bone and it is also affected by the direction of propagation relative to the bone axis [36,41,124,154].
3.4 NUMERICAL SIMULATION OF WAVE PROPAGATION Numerical simulations can be used to explore the relationships be- tween different bone properties and ultrasound parameters. More- over, simulations can also help to interpret experimental findings between tissue structure, composition and acoustic properties. A nu- merical simulation is essentially a numerical approach for solving a mathematical problem. Therefore, it is essential to formulate a math- ematical description of wave propagation in bone. For a body with a small volume and surface area, Newton’s second law dictates that the sum of the surface stresses and force (e.g., gravitational, electri- cal, magnetic) acting on the body is equal to the acceleration field experienced by the mass of the body. When the total stress on the surface of the body is considered to be equal to the net force per unit volume of the body, Newton’s law for all continua undergoing small deformations can be expressed as:
∂σij
∂xj +Fi = ρ∂2wi
∂t2 i,j=1, 2, 3. (3.2) In the expressionσij are components of the stress tensor [N/m2], xj
is a component of the displacement vector, Fi represents the compo- nents of the body force (N.B.this is equal to zero for freely vibrating media), ρ is the mass density [kg/m3] of the body, ∂ denotes the partial differential operator, t is time [s] and wi is a component of the body’s displacement field.
In acoustics, the linear ‘spring constants’ pertaining to elastic vibrations with small amplitude can be used to relate the stress and strain present in a particular material by Hooke’s law. In prac- tice, six independent stresses are related to six independent strains through 21 independent constants in a homogeneous (i.e., constants do not depend on space coordinates) anisotropic medium. As a first approximation, the calcified matrix may be considered as a homoge- neous isotropic material. This means that for a given calcified matrix, wave propagation in cortical bone will depend on its microstructure.
This assumption can be useful for determining important trends observed in experimental measurements for wavelengths of the or- der of a millimeter [35, 36]. For an isotropic material, Hooke’s law may be expressed in terms of the first and second Lamé constants [N/m2] λ and µ, respectively. Replacing the stresses in Newton’s law of motion with strains according to Hooke’s law rewrites the equation in terms of displacements
ρ∂2w
∂t2 = µ∇2w+ (λ+µ)∇(∇ ·w), (3.3) where ∇ is the gradient operator, ∇ · is the divergence operator and vectorw(x,y,z,t)is the body’s three-dimensional displacement field.
When the wavelength is smaller than the dimensions of the spec- imen in which it propagates, the wave will not “see” the boundaries of the specimen. Therefore, the wave (i.e., bulk wave) may be de- scribed as propagating in an infinite or unbounded medium. Then the SOS in both longitudinal and transverse directions is estimated as listed in Table 3.1.
The finite difference time domain (FDTD) method can be ap- plied to resolve wave propagation problems in continuous media,
Ultrasound assessment of bone
like bone [93, 124, 153, 163]. In the FDTD method, a material is rep- resented by isotropic grid elements consisting of nodes that are dis- placed, with respect to time, relative to their neighboring nodes. The value at these nodes at a certain time is approximated by replacing the derivatives in eq. 3.3 by finite differences. Three finite difference forms are generally considered in the FDTD method. For a function
f(x)with a continuous derivative, the forward difference is:
d f
dx(x0) = f(x0+h)− f(x0)
h , (3.4)
the backward difference is d f
dx(x0) = f(x0)− f(x0−h)
h , (3.5)
and the central difference is d f
dx(x0) = f(x0+h)− f(x0−h)
2h , (3.6)
wherex0 is the node andhis the spatial increment. Equations eq. 3.4 and eq. 3.5 are first-order approximation of the first derivative, while eq. 3.6 is a second-order approximation of the second derivative. As time increments become smaller, the spatial increments will also decrease until the difference between the partial differential and the difference approximation vanishes.
