HAL Id: tel-01427122
https://tel.archives-ouvertes.fr/tel-01427122
Submitted on 5 Jan 2017
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
characterization of cortical bone
Simon Bernard
To cite this version:
Simon Bernard. Resonant ultrasound spectroscopy for the viscoelastic characterization of cortical bone . Acoustics [physics.class-ph]. Université Pierre et Marie Curie (Paris 6), 2014. English. �tel-01427122�
Laboratoire d’Imagerie Biomédicale
Université Pierre et Marie Curie, Paris 6
École doctorale : Sciences mécaniques, acoustique, électronique et robotique de Paris
Thèse de Doctorat
Spécialité : Acoustique Physique
Présentée par
Simon Bernard
Resonant ultrasound spectroscopy for the viscoelastic characterization of cortical bone
Spectroscopie par résonance ultrasonore pour la caractérisation viscoélastique de l’os cortical
Soutenue le 03/12/2014
Jury :
Rapporteurs : MarcBonnet Directeur de recherche, CNRS, ENSTA
MarcDeschamps Directeur de recherche, CNRS, Université Bordeaux 1 Examinateurs : PhilippSchneider Reader, University of Southampton (UK)
FrançoisCoulouvrat Directeur de recherche, CNRS, Université Paris 6
MichalLanda Senior researcher, Institute of Thermomechanics (Czech Republic) Directeur de thèse : PascalLaugier Directeur de recherche, CNRS, Université Paris 6
Co-encadrant : QuentinGrimal Maître de conférences, Université Paris 6
Abstract
Deep understanding of the structure-function relationships of cortical bone in the context of bone quality assessment is still missing. Currently available methods to measure millimeter-scale elasticity ex vivo have limitations arising from the anisotropy of the tissue, its heterogeneity, and the small size of the specimens.
Resonant ultrasound spectroscopy is particularly suitable for the measurement of small anisotropic specimens. This method estimates elasticity from the free resonant frequencies of a specimen, and damping from the width of the resonant peaks. Its application to cortical bone was considered challenging because of the high damping of the vibrations modes, which causes overlapping of the resonant peaks and pre- vents a direct measurement of the resonant frequencies. To overcome the difficulty, adaptations of all the steps of RUS – measurement, signal processing and inverse es- timation of the material properties – have been introduced. Validation of each step of the procedure has been achieved by application to several test samples, including a cortical bone specimen and bone-mimicking composite and polymer specimens.
RUS was shown to be precise and accurate, with the advantage of providing the complete stiffness tensor from the measurement of a single specimen. Additionally, an original Bayesian formulation of the inversion provides an automated solution to a problem that was previously solved by tedious trial-and-error procedures or com- plex additions to the basic experimental setup. Finally, the application to a large collection of human tibiae specimens demonstrates that RUS can be considered a routine method to characterize the viscoelasticity of bone.
Keywords
Resonant ultrasound spectroscopy; cortical bone; viscoelasticity; anisotropy; inverse problem; vibrations
Résumé
Une meilleure compréhension des relations entre la structure complexe de l’os cor- tical et ses propriétés mécaniques est nécessaire à l’évaluation de la qualité osseuse.
Les méthodes conventionnelles ex vivo de mesure de l’élasticité à l’échelle du mil- limètre ont des limitations liées à l’anisotropie du tissu, à son inhomogénéité et à la petite taille des échantillons. Au contraire, la spectroscopie par résonance ultra- sonore (RUS) est bien adaptée à la mesure de petits échantillons anisotropes. Cette méthode estime l’élasticité à partir des fréquences de résonance de l’échantillon, et l’amortissement à partir de la largeur des pics de résonance. Son application à l’os était considérée difficile, du fait de l’amortissement important des modes de vibration, qui induit un recouvrement des pics de résonance et complique la mesure des fréquences. Pour surmonter cette difficulté, des adaptions de la méthode – dans la mesure, le traitement du signal et l’estimation des propriétés du matériau – ont été proposées. Elles ont été validées sur de l’os cortical et sur des échantil- lons de polymère et de matériau composite imitant l’os. La précision de la méth- ode a été démontrée, ainsi que sa capacité à mesurer tous les termes du tenseur d’élasticité à partir d’un seul échantillon. De plus, une nouvelle formulation Bayési- enne de l’inversion apporte une solution automatique à un problème qui nécessitait une stratégie fastidieuse d’essai-erreur ou de complexes modifications du dispositif expérimental. Finalement, l’application à une grande collection d’échantillons de tibias humains démontre que la méthode RUS pourrait être utilisée en routine pour la mesure de la viscoélasticité de l’os.
Mots-clés
Spectroscopie par résonance ultrasonore ; os cortical ; viscoélasticité ; anisotropie ; problème inverse ; vibrations
Contents
1 Introduction 1
1.1 Context and motivation . . . 1
1.2 Hierarchical structure of bone and anisotropy . . . 3
1.3 Assessment of cortical bone elasticity at the millimeter-scale . . . 4
1.3.1 Mechanical testing . . . 4
1.3.2 Ultrasonic waves methods . . . 7
1.4 Resonant Ultrasound Spectroscopy . . . 10
1.4.1 Basic principles . . . 10
1.4.2 Historical review . . . 14
1.4.3 RUS and attenuative materials such as bone . . . 16
1.5 Outline of the thesis . . . 18
2 RUS measurement of cortical bone elasticity: a feasibility study 21 2.1 Introduction . . . 21
2.2 Materials and Methods . . . 23
2.2.1 Specimen . . . 23
2.2.2 Resonant Frequencies Calculation . . . 24
2.2.3 Resonant Frequencies Measurement. . . 25
2.2.4 Signal Processing . . . 26
2.2.5 Elastic Constants Estimation . . . 28
2.2.6 Uncertainty on the elastic constants . . . 29
2.2.7 Engineering moduli. . . 30
2.2.8 Ultrasonic Velocities Measurement . . . 30
2.3 Results. . . 30
2.4 Discussion . . . 32
2.5 Conclusion. . . 35
3 RUS for viscoelastic characterization of anisotropic attenuative solid materials 37 3.1 Introduction . . . 37
3.2 Resonant frequencies computation . . . 39
3.2.1 Rectangular Parallelepiped . . . 40
3.2.2 Cylinder . . . 40
3.3 Samples and measurement setup . . . 42
3.3.1 Samples . . . 42
3.3.2 Setup for the measurement of the frequency responses . . . . 43
3.3.3 Additional elasticity measurements . . . 44
3.4 Processing of the measured spectra . . . 46
3.4.1 Estimation of the resonant frequencies in time domain . . . . 46
3.4.2 Non-linear fitting in frequency domain . . . 48
3.5 Elastic constants estimation . . . 49
3.5.1 Bayesian formulation of the RUS inverse problem . . . 50
3.5.2 Automated pairing of the resonant modes . . . 51
3.5.3 Implementation . . . 52
3.6 Damping factors estimation . . . 54
3.7 Results. . . 55
3.8 Discussion . . . 59
3.9 Conclusion. . . 63
4 Bayesian mode identification and stiffness estimation using Markov Chain Monte Carlo 65 4.1 Introduction . . . 65
4.2 Method . . . 67
4.2.1 Forward problem . . . 67
4.2.2 Bayesian formulation of the inverse problem . . . 68
4.2.3 Gibbs sampling . . . 69
4.3 Application 1 - Data from Ogi et al., 2002 . . . 76
4.3.1 Prior distributions . . . 77
4.3.2 Results and discussion . . . 77
4.3.3 Additional results. . . 82
4.4 Application 2 - Data from Bernard al., 2013 . . . 83
4.4.1 Prior distributions . . . 83
4.4.2 Results and discussion . . . 85
4.5 Conclusion. . . 85
5 Application to a collection of human tibial cortical bone specimens 89 5.1 Introduction . . . 89
5.2 Materials and methods . . . 90
5.2.1 Specimens . . . 90
5.2.2 Measurement setup and signal processing . . . 91
5.2.3 Estimation of the elastic properties . . . 92
5.2.4 Viscoelasticity . . . 98
5.3 Results. . . 101
5.4 Statistical analysis . . . 104
5.5 Discussion . . . 106
5.6 Conclusion. . . 112
Contents v
Summary and conclusion 115
A Appendix: Transformed elastic parameters 119
A.1 Isotropic symmetry . . . 119
A.2 Cubic symmetry . . . 119
A.3 Transversely Isotropic symmetry . . . 120
A.4 Orthotropic symmetry . . . 120
B Appendix: Cutting protocol for human tibia specimens 123 B.1 Material . . . 123
B.2 Specimen preparation . . . 124
B.3 Transversal cuts. . . 126
B.4 Radial cuts . . . 126
B.5 Axial cuts . . . 127
B.6 C+2/3 specimen . . . 128
Bibliography 131
Chapter 1
Introduction
1.1 Context and motivation
Bones have remarkable mechanical properties that play a role in support and move- ment of the body as well as in protection of the other organs (among other functions not related to mechanics). These mechanical properties result from a highly orga- nized hierarchical structure, from the smaller constituents of the tissue (collagen molecules and mineral crystals) to the whole organ (Rho et al.,1998).
The first motivation for the mechanical study of bone material is related to bone diseases, such as osteoporosis, which affects tens of millions of people across the world. Bone diseases degrade bone quality, a concept encompassing constitution, structure, and mechanical properties of the tissue as well as the presence of dam- age (Wallach et al.,1992; Seeman and Delmas, 2006). This degradation entails an increased risk of fracture. The state-of-the-art clinical method for the diagnosis of osteoporosis and fracture risk prevention is the assessment of bone mineral density (BMD) using dual-energy X-ray absorptiometry (DXA). Although BMD is related to bone mechanical properties such as strength (Ammann and Rizzoli,2003), it reflects bone quantity more thanbone quality and it is admitted that DXA has only a mod- erate success in bone fracture prediction (Schuit et al.,2004;Keaveny et al.,2010).
Alternative diagnosis tools were developed to improve fracture risk prediction, and particularly quantitative ultrasound (QUS) methods (Laugier, 2008; Laugier and Haïat, 2011). Indeed, ultrasound waves are mechanical waves, intrinsically sensi- tive to bone mechanical properties and structure. Nevertheless, QUS methods have failed to emerge as an alternative to X-ray in clinical practice, perhaps due to a lack of understanding of the interactions of sound waves with the complex structure of bone (Laugier and Grimal, 2014). A better understanding of the relationships between the structure and composition of bone tissue and its mechanical properties is thus required.
Other motivations for the study of bone tissue mechanics include orthopedic applications (implant design), engineering of bio-inspired materials (Fratzl, 2007), and computational studies of the biomechanical behavior of bone at organ or body scale, which usually require tissue-level properties to feed the numerical models (e.g. Duda et al.,1998). Bone structure-function relationships are also interesting from biological and physiological points of view.
Two types of bone tissues are observed: trabecular bone, the inner part of bones and cortical bone, the outer shell (Fig.1.1). Cortical bone is a stiff and dense ma-
Figure 1.1: Two tissue types can be distinguished at the macroscale: trabecular bone (top) and cortical bone (bottom). FromGranke (2012).
terial compared the highly porous and softer trabecular bone (Cowin,2001). After having been for a long time focused on trabecular bone, efforts on ultrasound charac- terization of bone are moving to cortical bone (Raum et al.,2014). Indeed, cortical bone has been shown to play a key role in bone mechanical functions at specific sites. For example, Holzer et al. (2009) have recently shown that the contribution of the trabecular part to bone strength at the hip is marginal.
Cortical bone elastic properties at the millimeter-scale (sometimes referred to as the mesoscale) are of interest as they depend on tissue properties at all the smaller length scales (nano and microscales) and have a direct impact on the mechanical behavior of bone at the macroscale. Moreover, clinical applications of ultrasound on bone are typically in the range of 100 kHz−2 MHz (Laugier,2008), and then probe elasticity at the millimeter-scale.
Assessment of bone elasticity at this length scale is challenging, even ex vivo, mainly because of anisotropy and spatial variability. Indeed, the elastic properties vary not only between individuals, but within the same bone (Espinoza Orias et al., 2008;Grimal et al.,2009;Rudy et al.,2011) due to large spatial variations in tissue properties (Rohrbach et al., 2012). Hence, a proper anisotropic elasticity charac- terization should measure all the stiffness coefficients on the same small volume of material to avoid any effect of specimen variability. Another difficulty is that only small specimens are available. For women, the cortical shell thickness is in the range 2−5.5 mm at the tibia and 1.5−3.5 mm at the radius (Sievänen et al.,2001).
Hence, the largest specimens are at most a few millimeters large in one or two of their dimensions. This last issue is even more critical for the study of small animal models.
Conventional methods for bone elasticity assessment, namely mechanical testing and wave velocity methods (both reviewed in a following section), do not completely satisfy to the requirement of a small and unique specimen. The main purpose of this PhD work was to introduce resonant ultrasound spectroscopy (RUS) as an
1.2. Hierarchical structure of bone and anisotropy 3
Figure 1.2: Hierarchical organization of bone from the mineral crystals and collagen molecules to the whole organ. FromRho et al.(1998).
alternative method for cortical bone elasticity assessment at the millimeter-scale.
Indeed, RUS is particularly suitable to the assessment of anisotropic elasticity from a single small specimen.
1.2 Hierarchical structure of bone and anisotropy
This section briefly reviews the hierarchical structure of bone across the scales, and its consequences on the elastic anisotropy, mainly based on Weiner and Wagner (1998) andRho et al.(1998). The bone structure is illustrated on Fig. 1.2.
The nanoscale is the length scale of the basic constituents of bone: min- eral crystals (mainly hydroxyapatite), collagen molecules (type I), non-collagenous proteins, and water. The mineral crystals form plate-like structures of about 50×25×1.5−4 nm called platelets. Collagen molecules are made of three long amino acids chains organized in a triple helix and are approximately1−1.5 nm in diameter and 300 nm long. To form the so-called mineralized collagen fibrils, col- lagen molecules align along a given direction, and mineral crystal grow in between the collagen molecules with their crystallographic c-axis – the stiffest direction – aligned with the collagen molecules. Mineralized collagen fibrils then assemble into arrays or bundles called fibers. The mineral part provides stiffness and strength to the more compliant collagen part.
At the microscale, fibers arrange parallel to each other in layers of about3−7µm thick called lamellae. In cortical bone, the lamellae are arranged concentrically
around the Haversian canals (∼50µm) to form cylindrical layered structures called osteons (∼ 100- 300 µm in diameter). The Harvesian canals are aligned with the long axis of bone and contains a blood vessel and nerve fibers. Inside a lamella, the main direction of the fibers is tilted. Several patterns of tilting angle along the radius of the osteons have been described in the literature, with different expected consequences on anisotropy (Reisinger et al., 2011a). However, Reisinger et al.
(2011b) observed using nanoindentation that osteons are generally stiffer in the axial direction. The space between osteons is filed with the interstitial matrix, which is made of remnants of osteons.
At the mesoscale (millimeter scale), which is the intermediate scale between the highest level of microstructure (lamellae and osteons) and the whole organ, bone tissue can be seen as a two phase composite: long cylindrical soft fibers (Haversian canals) are aligned in a hard tissue matrix (osteons and interstitial tissue). Bone is stiffer in the axial direction at this length scale, due to a cumulative effect of the preferred orientation of the Harversian porosity and the preferred direction of alignment of the mineralized collagen fibrils inside the matrix.
1.3 Measurement of cortical bone elasticity at the millimeter-scale
This section reviews state-of-the-art methods for the measurement of cortical bone elasticity on millimeter-sized specimens and discusses their respective advantages and shortcomings. These methods can be classified in two categories: 1) mechanical testing methods, in which elasticity is obtained from the stress-strain relationship in low frequency or quasi-static regime and 2) ultrasonic methods, based on the fact that, under some particular conditions, the velocity of a wave propagating in an elastic medium can be directly related to a stiffness coefficient.
For both approaches, basic equations are recalled. An extensive discussion of the elasticity of anisotropic media, and particularly of wave propagation, can be found in textbooks, such as Auld(1990) andRoyer and Dieulesaint (2000).
1.3.1 Mechanical testing
Hooke’s law for continuum media is a tensorial relation between the stress tensorσ, the linear strain tensor ε, and the fourth-order stiffness tensor C1
σij =Cijklεkl, (1.1)
1Throughout this chapter, the conventional summation rule of repeated indices is used. For all implicit summations the indices run over the set{1,2,3}.
1.3. Assessment of cortical bone elasticity at the millimeter-scale 5
whereεis related to the displacement u εkl= 1
2 ∂uk
∂xl + ∂ul
∂xk
. (1.2)
Due to the thermodynamics of reversible deformations and to symmetry of the stain and stress tensors,Chas at most 21 independent coefficients. In a reduced two index notation called Voigt notation, Hooke’s law (1.1) can be expressed as
σ11 σ22
σ33 σ23 σ13
σ12
=
C11 C12 C13 C14 C15 C16 C22 C23 C24 C25 C26
C33 C34 C35 C36 C44 C45 C46
Sym. C55 C56
C66
·
ε11 ε22
ε33 2ε23 2ε13
2ε12
. (1.3)
where pairs of subscripts for the stiffness tensor in Hooke’s law (1.1) have been mapped to a single subscript: 1⇔11;2⇔22;3⇔33;4⇔23;5⇔13;6⇔12.
If the material possesses symmetries, this relation further simplifies. For an orthotropic material (three orthogonal mirror symmetry planes) the stiffness tensor has only nine independent non-zero coefficients in the coordinate system defined by the symmetry directions
[Cij] =
C11 C12 C13 0 0 0 C22 C23 0 0 0 C33 0 0 0 C44 0 0
Sym. C55 0
C66
. (1.4)
If the material istransversely isotropic (one axis of cylindrical symmetry) only five coefficients remains independent, as the following relations hold: C11=C22,C13= C23,C44=C55, andC12=C11−2C44 (for symmetry around axis 3). Finally, if the material is isotropic, only two coefficient remains independent as C11=C22=C33, C44=C55=C66, and C12=C13=C23=C11−2C44.
Hooke’s law (1.1) can be inverted to yield ε = C−1 : σ, where the inverse of the stiffness tensor is called the compliance tensor, and can be expressed using the engineering moduli (Young’s moduli E, shear moduli G, and Poisson’s ratios ν).
For orthotropic materials
[Cij]−1 =
1
E1 −νE12
2 −νE13
3 0 0 0
−νE21
1
1
E2 −νE23
3 0 0 0
−νE31
1 −νE32
2
1
E3 0 0 0
0 0 0 G1
23 0 0
0 0 0 0 G1
13 0
0 0 0 0 0 G1
12
. (1.5)
The basic principle of mechanical testing is to infer an elastic coefficient from a stress-strain curve measured in a low frequency of quasi-static experiment. Usually, a stress field is imposed and the strain measured on the sample using strain gauges or optical methods. If the stress and strain remains sufficiently small, they are linearly related trough Hooke’s law (1.1) and the slope of the curve is equal to the stiffness.
However, due to the tensorial nature of the stress-strain relationship, inferring a par- ticular moduli in a straightforward way requires particular stress and strain fields.
For example, measuring the Young’s modulus in a given material direction requires a uniaxial stress field along that direction. Hence, specific specimen shapes are dedicated to the measurement of different coefficients. This usually prevents form assessing completely the anisotropy form a single specimen, although some speci- mens allow the joint measurement of several coefficients (e.g. two Poisson’s ratios can be obtained jointly to the Young’s modulus in uniaxial compression/tension if the lateral strains are measured). This a important limitation of mechanical testing.
Mechanical testing has been employed on cortical bone in numerous studies. For example,Shahar et al.(2007) used compression tests on 6 mm-sized cubes of equine bone in the three orthogonal directions to determine three Young’s moduli and six Poisson’s ratios2. Ohman et al. (2011) used compression tests as well to measure a Young’s modulus on child and adult bones, but on small cylindrical samples (d=2-3 mm, h=14-18 mm). Due to the natural cylindrical beam shape of long bones such as the femur, mechanical tests are sometimes applied on whole bones (Brodt et al., 1999, torsion and bending) or segments of bones (Chattah et al.,2009, compression) instead of samples prepared to a specific regular shape. This is particularly the case in small animal studies for which the preparation of samples can be challenging.
This reduces accuracy, as the equations usually assume regular shapes (van Lenthe et al.,2008). Additionally, the spatial variation of bone tissue properties should also be taken into account to improve accuracy (Kim et al.,2012).
Some of the studies cited above (Chattah et al.,2009;Ohman et al.,2011) used an optical method called electronic speckle pattern interferometry to measure the strain on the tested specimen, instead of using more conventional strain gauges. In addition to being a non-contact method, the main advantage of this increasingly popular optical measurement is that the field strain is measured on a surface of the sample, instead of at a single point.
2Because the Poisson’s ratios and Young’s moduli are not all independent, six independent coefficients were obtained. This seems to be highest number of coefficients that can be measured from mechanical testing on a single sample.
1.3. Assessment of cortical bone elasticity at the millimeter-scale 7
1.3.2 Ultrasonic waves methods
Neglecting body forces and using the stress-strain (1.1) and strain-displacement (1.2) relations, the equation of motion for a continuum media is
ρ∂2ui
∂t2 =Cijkl ∂2uk
∂xj∂xl
, (1.6)
whereρ is the mass density. Assuming a time harmonic plane wave at pulsation ω and phase velocityV propagating in directionn (unit vector)
ui = Uk·exp h
iω njxj
V −t i
, (1.7)
the equation of motion (1.6) yields the Christoffel’s equation Γik−ρV2δik
·Uk = 0, (1.8)
with the Christoffel’s matrix Γik =Cijklnjnl and the Kronecker’s delta δik (= 1 if i=k, 0 otherwise).
Christoffel’s equation (1.8) is an eigenproblem for the eigenvaluesρV2and eigen- vectorsU (polarization of the plane wave). For waves propagating in the principal direction of an orthotropic material, i.e. n = (1,0,0), (0,1,0) or (0,0,1), equa- tion (1.8) yields three solutions. For example, for propagation along the first axis the solutions write ρV2 = C11 with polarization vector (1,0,0)3, ρV2 =C66 with polarization vector(0,1,0)4 and ρV2 =C55 with polarization vector(0,0,1)4. De- noting Vij the velocity of a plane wave propagating along direction iwith displace- ment in the direction j, six relations between the diagonal elastic coefficients and velocities are obtained
Cii=ρVii2 (i= 1,2,3), (1.9)
C44=ρV232 =ρV322, (1.10)
C55=ρV132 =ρV312, (1.11)
C66=ρV122 =ρV212. (1.12)
Given a rectangular parallelepipedic sample whose faces are aligned with the material principal directions, the six diagonal coefficients can be obtained from the velocities of waves propagating between a pair of transducers contacting two oppo- site faces, or from a single transducer using the reflection of the waves. Compression transducers must be used for the three first coefficients [Eq. (1.9)] and shear trans- ducers for the three last [Eqs. (1.10), (1.11), and (1.12)]. The off-diagonal coefficients C12,C13, and C23 are related to the velocities of waves propagating in other direc- tions. Their measurement requires additional faces not aligned with the principal direction of elasticity, usually prepared at 45◦ (Fig. 1.3).
3Pure longitudinal wave, propagation and displacement directions are colinear.
4Pure shear wave, propagation and displacement directions are orthogonal.
Figure 1.3: Cubic specimen with additional 45◦-oriented faces for the measurement of the off-diagonal elastic constants. FromRho(1996)
.
The above equations were derived for propagation in an infinite medium (bulk wave propagation), but still hold if the wavelength is sufficiently smaller than the cross-section of the sample. If not, waves are perturbed by the boundaries of the domain and the stiffness coefficients are no longer simply related to the velocities. In the opposite case (wavelength much larger that the cross section), waves propagate in a bar wave regime, in which the propagation is guided by the sample shape. In that case, longitudinal wave velocities are directly related to the Young’s moduli of the material (e.g. ρV112 =E1).
Lang (1969) was the first to introduce ultrasound bulk wave velocity measure- ments in the context of bone research as an alternative to mechanical tests. He measured five independent elastic coefficients (transversely isotropic symmetry) on bone samples from the time-of-flight of ultrasonic pulse waves. The method was subsequently used by other researchers to study bone elastic anisotropy. Yoon and Katz (1976a,b) argued for the transversely isotropic symmetry of human cortical bone and measured the five elastic coefficients on dried specimens. Van Buskirk et al. (1981) find significant deviations from transverse isotropy for bovine corti- cal bone and argued for orthotropy. On human tibial cortical bone, Rho (1996) also observed slightly orthotropic material properties. Espinoza Orias et al. (2008) studied the variation in magnitude and anisotropy of the elastic coefficients along the human femur and found that, while the transverse isotropy assumption was ac- curate for the mid-diaphysis, tissue at extreme (distal and proximal) positions of the diaphysis showed moderate orthotropy. Applying the same method to human femoral bone samples (mid-diaphysis),Granke et al.(2011) also observed transverse isotropy. It is noteworthy that only the diagonal elastic coefficients were measured in this two last studies, due to the difficulty of cutting samples with 45◦-oriented faces.
In consequence, the Young’s moduli and Poisson’s ratios could not be evaluated.
Ashman et al.(1984) introduced a slightly different method, based on the use of continuous waves instead of pulses. They also discussed the sample size constraints to respect for bulk wave propagation. They showed that the transition from bulk
1.3. Assessment of cortical bone elasticity at the millimeter-scale 9
waves to bar waves occurs for a product wavenumber×cross-section inferior to 10.
For the wavelength of 2.25 MHz compression wave in bone5, the minimal sample size is 2.5 mm. This size is comparable or even larger than the cortical thickness in most of the bones of the human body, and obviously much larger that samples from small animals. This constraint cannot be simply overcome by scaling up the frequency because the wavelength must remain much larger that bone heterogeneities. At 2.25 MHz the wavelength is about 1.6 mm, close to 1.5 mm, the minimal length over which human cortical bone can be consider homogeneous with respect to elasticity (Grimal et al.,2011). Moreover, the attenuation of ultrasound waves increases with frequency. For these reasons, wave velocity measurements have been limited to large human bones (tibiae and femur) and bones from large animals (mostly bovines and equines).
Granke et al. (2011) estimated the repeatability of the measured elastic coeffi- cients to be 3.4%and 4.9%for longitudinal and shear coefficients, respectively. The lack of repeatability in velocity measurements is in part caused by the limited preci- sion of the time-of-flight measurements. Indeed, due dispersion of ultrasound waves in bone, the shape of the transmitted signal changes during propagation. Moreover, the signal is also very sensitive to the transducers delicate positioning and coupling with the sample (Maynard,1992;Grimal et al.,2009).
Mechanical tests vs ultrasound velocity
The equations of linear elasticity introduced for both methods hold only for small strains. When stress and strain increase above a certain threshold, they are no longer linearly related as the material enters regions of non-linear elasticity and plasticity (permanent deformation) before eventually fracturing. Mechanical prop- erties beyond the elastic region can be measured using mechanical testing, but not using ultrasonic methods. This is a considerable advantage of mechanical methods.
Due to the difficulties linked to sample geometry for both mechanical testing and wave velocity measurement, very few studies compared the results of the two meth- ods on the same specimens. However, two recent studies reach similar conclusions (Grimal et al., 2009; Berteau et al., 2014): 1) elastic properties measured by me- chanical tests and ultrasonic wave velocities are only moderately correlated (R=0.63 - 0.76) and 2) the mechanical Young’s modulus is significantly lower than the ultra- sound modulus (e.g. for Grimal et al.: Emech. = 17.3±1.3 and Eac. = 22.1±0.9).
Although Grimal et al. have shown that the lack of correlation in their results may be entirely explained by the uncertainties in both methods, the measured bias seems to be physically relevant. A possible cause is bone viscoelaticity, which entails a variation of elastic modulus with frequency, but this is still an open question.
5Assuming a velocity of 3600 m.s−1
1.4 Resonant Ultrasound Spectroscopy
Summarizing the particularities of bone elasticity assessment and the limitations of the conventional methods described above, the three main requirements for an enhanced method are:
• The method should be able to completely characterize the anisotropic elasticity of a material from a single specimen, without requiring45◦-oriented faces. This is linked to the spatial variability of bone elasticity.
• It should be fully applicable on small specimens (of the order of a few mm3), since the thickness of the cortical shell of bones limits the specimen size.
• It should be as accurate and reproducible as possible. Hence, it should not assume idealized state of stress and strain but rather take into account the complete stress-strain relationship. Moreover, delicate contact or bounding between the sample and the apparatus should be avoided.
Resonant Ultrasound spectroscopy (RUS) is a method that has been developed to satisfies these requirements, originally for the characterization of geological materials (see the historical review in section1.4.2), and that could therefore be beneficial for the study of bone mechanics. The following section introduces the basic principles of RUS, followed by a brief historical review. Finally, the difficulties of the application of RUS to bone tissue samples are introduced.
1.4.1 Basic principles
The basic principle of RUS is to use the frequencies of mechanical resonance of a sample of material to infer its stiffness tensor. Indeed, the resonant frequencies of an elastic body depend only on its geometry, mass density and stiffness tensor. If the two first parameters are directly measured, elasticity can then be estimated by matching model-predicted frequencies to the experimental results (inverse problem approach). This requires a setup for the measurement of the resonant frequencies, a method to compute the vibration modes of a sample given its elasticity and finally a procedure to solve the inverse problem. These elements are described in a generic way in the next paragraphs, mostly based on the book byMigliori and Sarrao(1997).
Measurement of the resonant frequencies
The basic setup for a RUS experiment is very simple: it consists of a pair of trans- ducers, a signal generator working at ultrasonic frequencies (usually from about 100 kHz up to a few MHz) and an apparatus to record the output signal (Fig.1.4).
The sample is gently held by points of low symmetry (e.g. corners for a cube) between the transmitting and receiving transducers. This position is expected to
1.4. Resonant Ultrasound Spectroscopy 11
Figure 1.4: Left: the basic setup of a RUS experiment. Right: A portion of a typical spectrum measured for a metallic sample (copper).
maximize the number of measured frequencies because these points are never nodes of the vibration modes (i.e. the displacement in never null). Another reason is that the model assumes stress-free boundary conditions, obtained if the contacting forces on the sample are small and punctual. It is important that the holding forces are as small as possible, as a force of the order of 0.01 N produces a measurable shift of the resonant frequencies (Yoneda,2002). A precise orientation of the sample is however not important, in contrast with wave velocity methods and mechanical testing.
The frequency response of the sample is usually recorded using a stepwise sweep- ing over the frequency band of interest: the sample is excited at a given frequency using a steady signal and the phase and amplitude of the response are recorded be- fore stepping to the next frequency. The main advantage against methods based on the Fourier transform of a temporal signal is that the signal to noise ratio is usually higher. The steady signal can be run for as long as desired and efficient filtering methods such as homodyne or heterodyne detection can be used. In that work, a network analyser was used to perform the frequency sweep as well as the recording of the frequency response (chapters2 and3).
The transducers must be broadband, or at least have no sharp resonances in the frequency band were those of the sample are expected. In the first part of that work (Chapter 2), home made transducers were firstly built following recommendations of Migliori and Sarrao (1997): a small piece of piezoelectric material is suspended to a thin film and put directly in contact with the sample. That way, the system is resonant-free up to the fundamental mode of the piezoelectric piece. Commercial shear transducers were later found out to produce stronger signals (Chapter3). This due to the fact that many resonant modes have a larger displacement in a direction colinear to the surface of the transducers.
Computation of the resonant modes
The natural free-vibration modes of a solid occupying the region Ωbounded by the surface S are the pulsations ω and displacement fields u that solve the harmonic equations of motion6
ρω2ui+Cijkl ∂2uk
∂xj∂xl = 0 inΩ, (1.13)
subject to the stress-free boundary conditions Cijkl∂uk
∂xlnj = 0 onS. (1.14)
No complete analytical solution exist for that problem except for the isotropic sphere. For some other simple shapes, partial solutions exist. For example, some modes of a rectangular parallelepiped with specific side-length ratios, called the Midlin-Lamé modes, have closed form expressions (Mindlin, 1956). For a general solution, the analysis must use energetic principles and numerical approximations.
According to Hamilton’s principle, the solutions of equations (1.13) and (1.14) must be stationary points of the Lagrangian Lof the body, i.e. they must satisfy
δL=δ Z
Ω
1
2ρω2uiui−1
2Cijkl∂ui
∂xj
∂uk
∂xl
dΩ
= 0, (1.15)
with δ referring to a variation due to a small perturbation δui of the displacement field and where the first and second terms of the integrand are respectively the kinetic and potential energy. Indeed, some manipulations7 yields
Z
Ω
ρω2ui+Cijkl ∂2uk
∂xj∂xl
δui
dΩ− Z
S
Cijkl∂uk
∂xlnj
δui
dS= 0. (1.16) Becauseδuiis arbitrary inΩand onS, the two terms in parenthesis in the integrands must vanish independently everywhere over their domain of integration, implying equations (1.13) and (1.14).
The Rayleigh-Ritz method expands the displacement field as a finite sum of known basis functions φλ with yet unknown coefficients ai,λ
ui =X
λ
ai,λφλ. (1.17)
The Lagrangian can then be expressed in a matrix form L= 1
2ω2atMa−1
2atKa (1.18)
6A rectangular coordinate system is assumed here for simplicity. A cylindrical or spherical coordinate system can be more adapted to some shapes. See Chapter3for the cylindrical case.
7Integration by part followed by the use of the divergence theorem, and discarding the second and higher order terms inδui.
1.4. Resonant Ultrasound Spectroscopy 13
where the vector a contains the coefficients of the expansion (1.17), and M and Kare the mass and stiffness matrices, related respectively to kinetic and potential energy. Requiring δL= 0 now impose ∂L/∂ai,λ = 0, which leads to
ω2Ma−Ka= 0, (1.19)
an eigenproblem for the eigenvalues ω2 and eigenvectors a. Expressions for the elements of the matrices are
Miλ,i0λ0 =δii0ρ Z
Ω
φλφλ0dΩ, and (1.20)
Kiλ,i0λ0 =Ciji0j0 Z
Ω
∂φλ
∂xj
∂φλ0
∂x0j dΩ. (1.21)
For simple geometries and polynomial basis functions, the integrals have simple analytical expressions and the eigenproblem (1.19) can then be numerically solved.
The number of basis function usually required to get satisfactory accuracy is of the order of 1000. An eigenproblem of this size is manageable for even modest desktop computers, particularly when considering the splitting of the eigenproblem in smaller independent sub-problems allowed for specific shapes and elastic symmetries (see Chapter3 and references therein).
The eigenvectors a allow representing the shapes of the normal modes approxi- mated on the polynomial basis. Fig. 1.5represents some of the first normal modes of a 2 mm-sized steel cube, computed from the Rayleigh-Ritz method with Legendre polynomials up to order 10 (basis functions are described in Chapter 3).
Inverse estimation of the stiffness
The objective of the inverse problem is to find the set of elastic coefficients giving the best match between the measurement and the model. This is usually done through the minimization of a quadratic cost function F
F =
N
X
n=1
fnexp.−fncomp.(Cij) fnexp.
2
, (1.22)
whereN is the number of measured resonant frequencies. The Levenberg-Marquardt algorithm is the usual method employed for the minimization. It is an iterative algorithm for non-linear optimization, which mixes search in the direction of the gradient and Gauss-Newton steps (Press et al.,1992). The partial derivatives of the frequencies with respect to the elastic coefficients, required in the inversion process, are obtained after solving the eigenproblem from the derivative of equation (1.19).
The main difficulty of the inverse problem is however not the minimization in itself, but the identification, orpairing, of the resonant modes. Indeed, the measured
f=704.8 kHz f=949.0 kHz f=965.2 kHz f=1096.3 kHz
f=1128.4 kHz f=1182.8 kHz f=1338.9 kHz f=1360.9 kHz
Figure 1.5: The eight first modes of a steel cube (l = 2mm, E=200 GPa,ν = 0.3, ρ = 8000 kg·m−3). Colors map the norm of the displacement vector (red: large, blue: small). The fourth mode is a Midlin-Lamé mode whose exact frequency is f = 1/(√
2l)×p
C44/ρ. The approximation is accurate to ten digits for that mode.
resonant frequencies are not informative of which predicted vibration mode they cor- respond to, and this causes ambiguities in the definition of the cost function (1.22).
Because the frequencies computed for the initial guess of the elasticity may appear in a different order than for the "true" elasticity, and because some predicted mode are potentially not observed in the measurement, there is no straightforward way to correctly index the frequencies, i.e. finding which predicted mode corresponds to the n-th experimental frequency in equation (1.22). Solving this so-called pairing problem was an important part of this work.
It is usually admitted that a precise estimation of the elastic coefficients requires about 10 to 20 measured resonances for isotropic materials, and at least four to five times the number of independent coefficients for anisotropic materials (Migliori and Maynard,2005). However, the present work will show that an accurate and precise RUS measurement can be done with a smaller number of measured frequencies.
1.4.2 Historical review
In this section, the history of RUS is briefly reviewed, from its very beginning to some recent developments. A more extensive review of the early days of RUS can be found in the book by Migliori and Sarrao(1997).
Although the use of the natural modes of a solid or structure to assess some of its properties is certainly very old, the birth year of resonant ultrasound spectroscopy
1.4. Resonant Ultrasound Spectroscopy 15
might be 1964: inspired by the work of geophysicists on the resonant modes of the earth,Fraser and LeCrawused the natural frequencies of a small elastic sphere to assess its stiffness, and called this method the resonant sphere technique. An analytical solution to the vibration problem of the isotropic elastic sphere was known since the works of Lamb (1882) and Love (1944), while no solutions were available for other geometries. The main motivation for the development of such a technique in the late 60’s and early 70’s was the study of geological materials, including lunar rocks from the Apollo 11 and 12 missions (e.g. Schreiber and Anderson,1970).
Two important steps toward the modern form of RUS were successively made by Demarest(1971) and Ohno(1976). Demarest was the first to use the Rayleigh- Ritz method to compute the natural frequencies of cubes of isotropic and cubic elastic symmetry. He called his method the cube resonance method. Ohno then extended that approach to rectangular parallelepipeds of arbitrary anisotropy, and applied it to orthotropic materials (1976) and later to a trigonal crystal (1986). The method was called rectangular parallelepiped resonance. Another important novelty inOhno’s work was the use of an iterative minimization method to obtain the elastic coefficients from the measured frequencies. It was not necessary for isotropic cubes and spheres, as one can find resonant modes depending on a single elastic coefficient and obtain the other coefficient from a graphical method (Fraser and LeCraw,1964;
Demarest,1971).
In the late 80’s, material physicists interested in the characterization of small single crystals started to apply the methods developed by geophysicists (Migliori et al., 1990) and the name resonant ultrasound spectroscopy (RUS) was proposed for the first time in a paper byMigliori et al. in 1993. Visscher et al.(1991) extended further the work ofDemarestandOhnoto more complex geometries by changing the basis functions in the Rayleigh-Ritz method. Instead of using Legendre polynomials, they used simple polynomials of the Cartesian coordinates. That way the method could be applied to a wide variety of shapes (cylinders, ellipsoids, tetrahedron, sandwich composites, ...), the only requirement being the tractability of integrals of the basis functions over the shape. During that time, the well-known minimization procedure called the Levenberg-Marquardt algorithm became the standard method to fit the predicted frequencies (Migliori et al.,1993) and the accuracy and precision of the elastic coefficients estimated through RUS were studied (Renken et al.,1997).
RUS was further developed and applied by many research groups in material science during the 90’s. The method was extended to piezoelectric materials and used to characterize quartz (Ohno,1990) and other materials (e.g.Ogi et al.,2003;
Ledbetter et al.,2004;Delaunay et al.,2008). In both geophysics and material sci- ence, one is often interested in the elastic properties of materials over a broad range of temperatures, and many challenging technical developments were accomplished towards this goal (for a review, see Li and Gladden, 2010). One of the impor-
tant results obtained during these years is the experimental evidence for the elastic isotropy of quasicrystals. Spoor et al. (1995) measured the anisotropy parameter ε= 1−2C44/(C11−C12)of an AlCuLi quasicrystal to be0.0002±0.0007, compared to 0.007±0.002 for tungsten, which has the smallest anisotropy reported in the literature for a cubic crystal. Measurement of the elastic coefficients, including the off-diagonal coefficient C12, with such an high accuracy on sub-millimeter samples was only possible using RUS.
The next ground-breaking step was accomplished byOgi et al.(2002), who used a laser interferometer to measure the modal shapes of the resonant modes in addition to their frequencies, tackling the problem of mode identification presents since the early days of RUS. By measuring the modal shapes (which are also predicted by the Rayleigh-Ritz method), mode identification can be done without ambiguity from images comparison. This methodology, used since its introduction in many RUS studies (e.g. Ogi et al.,2003; Landa et al.,2009), is not without any inconvenient:
it requires a reflective sample with a large (compared to the laser dot) plane face, as well as an optical path to the sample (problematic for studies under extreme temperature). Moreover, it considerably increase the complexity and cost of a RUS experiment.
A new potential extension of RUS has recently emerged: the use of the finite- element method (FEM) to compute the resonant frequencies of arbitrarily shaped samples. The first paper addressing this issues was one by Plesek et al. (2004).
Since the cost of the Levenberg-Marquardt minimization scheme when using FEM is prohibitive, they proposed a fixed-point iterative scheme requiring less forward computations. More recently,Liu and Maynard(2012) proposed a dedicated formu- lation of FEM for use in RUS. In both studies, the method was tested on rectangular specimens. To the best knowledge of the author, no real-world application of RUS requiring FEM as yet been published.
1.4.3 RUS and attenuative materials such as bone
In this section, we discuss the resonant behavior of materials that are highly attenu- ative for mechanical waves at ultrasonic frequencies, resulting in important damping of the resonant modes. The few studies in which RUS was applied to highly attenu- ative materials, including bone, were in purpose excluded from the above historical review and will be considered here.
Many RUS applications, and all that have been cited in the historical review above, were done on materials in which the damping of the mechanical vibrations at ultrasonic frequencies is very low, such as crystals. The quality factorQ, usually introduced to characterize the level of damping, is defined as inverse of the relative width of a resonant peak at half amplitude Q =f /∆f (Lakes, 2009). For a metal
1.4. Resonant Ultrasound Spectroscopy 17
Table 1.1: Qfactor measured using RUS for some materials, from very weakly atten- uative to highly attenuative. Some results of this thesis are included.
Material Qfactor (approximate) Reference
Diamond 1.5×106 Migliori and Maynard(2005)
Niobium (polycrystal) 105 Migliori(2008)
Brass 2000 Lee et al.(2000)
Black gabbro 350 Ulrich et al.(2002)
Sierra white granit 150 Ulrich et al.(2002)
PMMA 40 Bernard et al.(2014) (Chapter3)
Cortical bone (human) 25 Bernard et al.(2013) (Chapter2)
Cortical bone (bovine) 10 - 30 Lee et al.(2002)
155 160 165 170 175
1 10
Frequency (kHz)
Ampltiude (log scale)
Q=500 Q=200
Q=50
Q=20
Figure 1.6: Simulated example: as the Qfactor decreases, the width of the peaks increases and close peaks overlap. The frequency gap between the peaks is 2 kHz (about 1.2% of the resonant frequencies), a situation often encountered in practical applications of RUS.
or for crystal samples, Q is usually well above 1000 (Table 1.1 for some values), producing very sharp peaks in the spectrum at positions of the resonant frequencies (Fig.1.4, right). In these cases, it is not a problem to find the resonant frequencies at the positions of the local maxima of the spectrum, and close peaks are well resolved (Fig. 1.6). As the damping in the material increases, the Q factor decreases, and the peaks broaden. Then, close peaks may overlap and the frequencies may not correspond anymore to maxima of the spectrum (Fig. 1.6). For bone, the Qfactor is of the order of 20 (Table 1.1), which corresponds to the worst case in Fig.1.6.
In geophysics, RUS has been applied to rock samples with a Qof about 150 by Ulrich et al. (2002). In this last study, authors investigated the side-length ratios
of a rectangular parallelepiped that minimizes peak overlapping. In the context of bio-materials,Lee et al.(2002) applied RUS to cortical bone but could not measure all the elastic moduli, due to the low Qfactor in the range 10-30. They were able to find the three first resonant modes and used them to estimate the three shear moduli (as the first resonant modes in RUS are generally mostly dependent on the shear moduli). Lebedev(2002) proposed the use of a signal processing method to retrieve resonant frequencies with Q= 50, based on the fact that the frequency response of a sample has a known functional form: it is a sum of Lorentzian line-shapes. The work of Lebedev, as a possible path to overcome the difficulties reported by Lee et al. in the application of RUS to bone, was the starting point of the present work.
1.5 Outline of the thesis
This first chapter introduced the research context and motivations of the present work. State-of-the art methods for the assessment of cortical bone elasticity were described, and the need for a new method was argued. Basic principles and history of RUS were briefly reviewed. Finally, the difficulties of the application of RUS to bone were presented.
Chapter2is a research article published under the tileAccurate measurement of cortical bone elasticity tensor with resonant ultrasound spectroscopy in The Journal of the Mechanical Behavior of Biomedical Materials in 2013, and referred to in this manuscript asBernard et al.(2013). In this work, the difficulties of the application to bone are tackled, mainly by adequately combining tools available in the RUS literature. This chapter is a demonstration of the feasibility of RUS for cortical bone elastic characterization.
Chapter 3 is a research article published under the title Resonant ultrasound spectroscopy for viscoelastic characterization of anisotropic attenuative solid mate- rials in the Journal of Acoustical Society of America in 2014, and referred to in this manuscript as Bernard et al. (2014). It consolidates the results of chapter 2 and provides more extensive technical details about the method. RUS is applied on parallelepipedic and cylindrical samples of two attenuative materials: PMMA, an isotropic polymer, and an anisotropic bone mimetic material. Results are compared with wave velocity methods and mechanical testing. Precision and accuracy of RUS in that context are quantitatively discussed. The imaginary parts of the stiffness co- efficients, reflecting viscoelasticity, were also inferred from the resonant peak width Additionally, a first Bayesian formulation of the inverse problem and an iterative algorithm solving the pairing problem are proposed.
Chapter4 generalize the Bayesian formulation of Chapter3, and introduce the use of Markov chain Monte Carlo methods for an automated and robust inversion.
The proposed methodology is validated on data from the literature for a weakly
1.5. Outline of the thesis 19
attenuative material (aluminium alloy) and on data from chapter 2 for cortical bone. This chapter is also a research paper, prepared for submission to the journal Mechanics of Materials.
Chapter 5 presents the application of the developments introduced in the pre- ceding chapters to a collection of 59 human tibial bone samples. The Bayesian approach is further extended to handles collection of specimens and the full stiffness tensor of the specimens is obtained under a transverse isotropy assumption, as well as some viscoelastic parameters. An expected strong dependence of bone stiffness to apparent mass density is observed, and some original results about the anisotropy of the viscoelastic properties are presented.
Chapter 2
RUS measurement of cortical bone elasticity:
a feasibility study
This chapter is a research article published under the tile Accurate measurement of cortical bone elasticity tensor with resonant ultrasound spectroscopyin The Journal of the Mechanical Behavior of Biomedical Materials in 2013, and referred to as Bernard et al. (2013) in this manuscript. The full text of the article is reproduced here with no addition and no modifications except in the form.
This Chapter presents the first results that were obtained on a cortical bone specimen and demonstrates the feasibility of RUS in that context. The technical points, notably computation of the resonant frequencies and signal processing, are only briefly described here, but will get a more extensive treatment in Chapter 3.
2.1 Introduction
Cortical bone elasticity is anisotropic at the millimeter length scale and shows im- portant inter-individual and intra-individual variations. This variability is largely determined by a variable volume fraction of Haversian pores (Granke et al., 2011) and to a lesser extent to variable tissue mineral content (Currey,1988b) and average orientation of mineralized fibrils (Deuerling et al.,2009;Zebaze et al.,2011). There is a strong demand of precise and practical measurement methods for document- ing cortical bone elasticity and understanding structure-function relationships. A proper characterization of bone elasticity at the millimeter scale requires measure- ments of all the terms of the elastic tensor on a unique small volume of material to avoid any effect of specimen variability. A method based on the measurement of longitudinal and shear ultrasonic bulk waves velocities (BWV) propagating along various directions of a bone specimen was introduced in the 60’s by Lang (1969) as an alternative to static mechanical techniques , which are ineffective to measure several elastic constants on the same specimen. BWV measurements in the range 1-2 MHz are still the state-of-the-art method to retrieve longitudinal and shear elas- tic constants in the different anatomical directions of a specimen (Espinoza Orias et al.,2008; Granke et al.,2011). However the method has several drawbacks: (1) the specimen must have typical dimensions larger than a few millimeters (∼5mm).
This size limitation is related to the requirement that measured wave velocities be associated to bulk waves and not to bar waves, which propagate when the wavelength is close to the dimension of the specimen (Ashman et al., 1984). This limitation can not easily be overcome by scaling down the wavelength (use of transducer with
higher working frequency) because the wavelength must remain sufficiently large with respect to bone tissue heterogeneities (osteon, remodeling cavities, Haversian pores) in order to minimize the frequency dependence of the wave velocities and to measure bone effective properties. (2) The BWV method involves multiple measure- ments (one for each constant) with specimen repositioning and delicate transducers manipulation, in particular for shear wave measurements. (3) Depending on the degree of anisotropy of the specimen (transversely isotropic or orthotropic) one or several 45 degrees oblique cuts are necessary to retrieve all non-diagonal terms of the stiffness tensor. Because such cuts considerably increase the specimen preparation time and complexity, they were not realized in many of the studies using the BWV method. (4) The precision of the BWV method is determined by the consistency and the repeatability of the shape of the time-domain through-transmitted signal which is in fact highly sensitive to transducer positioning and quality of the acoustic coupling between the transducers and the specimen (Grimal et al.,2009).
In a resonant ultrasound spectroscopy (RUS) experiment, the frequencies of me- chanical resonances of a freely vibrating specimen are measured, and an iterative numerical procedure is used to adjust elastic constants until the calculated spec- trum corresponds to the measured frequencies (Maynard,1992;Migliori et al.,1993;
Migliori and Sarrao, 1997). In this way, all elastic constants are determined from a single measurement configuration, a clear advantage over BWV measurements.
Another advantage is that RUS takes into account all the complexity of wave prop- agation in a finite specimen instead of assuming propagation of pure bulk waves.
Therefore, there is in principle no limitation in specimen size. It is noteworthy that the method has been applied to measure the elasticity of crystalline specimens smaller than 1 mm3 (Spoor et al.,1995).
While RUS has become the gold standard technique to characterize elasticity of small single crystals, its application to biological materials is, to our knowledge, limited to two studies. Kinney et al. (2004) were successful in the characterization of elastic anisotropy of human dentin. In contrast, Lee et al.(2002) showed that the usual implementation of the method, using a commercial device, failed to measure all the anisotropic elastic constants of bone. The failure in Lee’s study was attributed to the high viscoelastic damping of bone, which causes resonant peaks to overlap and prevent the measurement of the resonant frequencies. Indeed, a mechanical quality factor Q of an order of a few hundreds and up is advised in RUS (Migliori and Maynard,2005) since it implies sharp resonant peaks. This is usually the case for crystalline or metallic specimens. In the context of geophysics, RUS has been successfully applied to materials with a smaller Qof the order of150(Ulrich et al., 2002). Lebedev (2002) introduced a signal processing method for the resolution of overlapped peaks in order to apply RUS to materials with aQfactor of the order of 50. This method was applied to simulated (Lebedev,2002) and experimental signals
2.2. Materials and Methods 23
(Lebedev et al.,2003). For bovine cortical bone,Lee et al.(2002) measuredQin the range 10-30; however they did not use any particular method to resolve overlapped peaks and therefore were unable to retrieve more than three resonant frequencies.
In this work we take advantage of recent developments of RUS to overcome the difficulties associated to the high damping of bone, which have until now prevented the measurement of the stiffness tensor of bone samples with RUS. These develop- ments include the processing method for the resolution of overlapping peaks already mentioned, which is completed here by a non-linear least-square optimization step to improve its precision. Another development used here is the iterative procedure for the estimation of the elastic constants developed byLanda et al.(2009) for highly anisotropic materials. Our objective is to show that RUS is suitable for the measure- ment of anisotropic elastic properties of cortical bone, despite the strong damping of bone. We evaluate the precision of the determination of the elastic constants and discuss the applicability of the method for a routine assessment of cortical bone elastic properties.
2.2 Materials and Methods
2.2.1 Specimen
The human cortical bone specimen used in this study was taken from a specimen collection prepared for a previous study (Granke et al.,2011). Ethical approval for the collection of specimens was granted by the Human Ethics Committee of the Centre du don des Corps at the University Paris Descartes (Paris, France). The tissue donors or their legal guardians provided informed written consent to give their tissue for investigation, in accord with legal clauses stated in the French Code of Public Health.
The specimen was a rectangular parallelepiped (dimensions:
5.44x6.58x6.58 mm3) extracted from the lateral quadrant of a transverse cross-section cut in the mid-diaphysis of a female donor femur (age 78 years). The faces of the specimen were oriented according to the radial (axis 1), circumferential (axis 2) and axial (axis 3) directions of bone. The specimen was defatted for 12 h in a chemical bath of diethylether and methanol (1:1). The six faces of the specimen were polished with a hard synthetic cloth using 3 µm polycrystalline diamond abrasive particles followed by a 0.05 µm aluminum oxide suspension (Metadi Supreme and Master-prep, BuehlerR GmbH, Düsseldorf, Germany). The specimen was stored in saline solution at ambient temperature and taken out of the solution only for the time of a measurement. Therefore, all measurements were made on a fully hydrated specimen. The specimen mass density was deduced to
