Resonant Ultrasound Spectroscopy for the Viscoelastic Characterization of Cortical Bone Simon Bernard

Resonant ultrasound spectroscopy for the viscoelastic characterization of cortical bone Simon Bernard

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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

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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 : Marc Bonnet Directeur de recherche, CNRS, ENSTA Marc Deschamps Directeur de recherche, CNRS, Université Bordeaux 1 Examinateurs : Philipp Schneider Reader, University of Southampton (UK) François Coulouvrat Directeur de recherche, CNRS, Université Paris 6 Michal Landa Senior researcher, Institute of Thermomechanics (Czech Republic) Directeur de thèse : Pascal Laugier Directeur de recherche, CNRS, Université Paris 6 Co-encadrant : Quentin Grimal 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 prevents a direct measurement of the resonant frequencies. To overcome the diﬃculty, adaptations of all the steps of RUS – measurement, signal processing and inverse estimation 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 complex 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 ii

Résumé

Une meilleure compréhension des relations entre la structure complexe de l’os cortical 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 ultrasonore (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 Introduction1 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 iv Contents

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 organized 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 than bone quality and it is admitted that DXA has only a moderate 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 sensitive 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- 2 Chapter 1. Introduction

Figure 1.1: Two tissue types can be distinguished at the macroscale: trabecular bone (top) and cortical bone (bottom). From Granke(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 characterization 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 characterization 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. From Rho 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) and Rho et al.(1998). The bone structure is illustrated on Fig. 1.2. The nanoscale is the length scale of the basic constituents of bone: mineral 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 approximately 1 − 1.5 nm in diameter and 300 nm long. To form the so-called mineralized collagen fibrils, collagen 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 about 3−7 µm thick called lamellae. In cortical bone, the lamellae are arranged concentrically 4 Chapter 1. Introduction around the Haversian canals (∼ 50 µm) 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) and Royer and Dieulesaint(2000).

1.3.1 Mechanical testing

Hooke’s law for continuum media is a tensorial relation between the stress tensor σ, 1 the linear strain tensor ε, and the fourth-order stiﬀness tensor C

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 where ε is related to the displacement u 1 ∂uk ∂ul εkl = + . (1.2) 2 ∂xl ∂xk Due to the thermodynamics of reversible deformations and to symmetry of the stain and stress tensors, C has at most 21 independent coefficients. In a reduced two index notation called Voigt notation, Hooke’s law (1.1) can be expressed as       σ11 C11 C12 C13 C14 C15 C16 ε11       σ22  C22 C23 C24 C25 C26  ε22        σ33  C33 C34 C35 C36  ε33    =   ·   . (1.3) σ   C C C  2ε   23  44 45 46  23       σ13  Sym. C55 C56 2ε13 σ12 C66 2ε12 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   C11 C12 C13 0 0 0    C22 C23 0 0 0     C33 0 0 0  [Cij] =   . (1.4)  C 0 0   44     Sym. C55 0  C66 If the material is transversely isotropic (one axis of cylindrical symmetry) only five

coefficients remains independent, as the following relations hold: C11 = C22, C13 = C23, C44 = C55, and C12 = 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  1 − ν12 − ν13 0 0 0  E1 E2 E3 − ν21 1 − ν23 0 0 0   E1 E2 E3   ν31 ν32 1  −1 − E − E E 0 0 0  [Cij] =  1 2 3  . (1.5)  0 0 0 1 0 0   G23   0 0 0 0 1 0   G13  0 0 0 0 0 1 G12 6 Chapter 1. Introduction

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 particular 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 specimens 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 2 2 ∂ ui ∂ uk ρ 2 = Cijkl , (1.6) ∂t ∂xj∂xl where ρ is the mass density. Assuming a time harmonic plane wave at pulsation ω and phase velocity V propagating in direction n (unit vector) h n x i u = U · exp iω j j − t , (1.7) i k V the equation of motion (1.6) yields the Christoﬀel’s equation

2 Γik − ρV δik · 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 eigenvalues ρV 2 and eigenvectors U (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), equation (1.8) yields three solutions. For example, for propagation along the first axis 2 3 2 the solutions write ρV = C11 with polarization vector (1, 0, 0) , ρV = C66 with 4 2 4 polarization vector (0, 1, 0) and ρV = C55 with polarization vector (0, 0, 1) . De- noting Vij the velocity of a plane wave propagating along direction i with displacement in the direction j, six relations between the diagonal elastic coefficients and velocities are obtained

2 Cii = ρVii (i = 1, 2, 3), (1.9) 2 2 C44 = ρV23 = ρV32, (1.10) 2 2 C55 = ρV13 = ρV31, (1.11) 2 2 C66 = ρV12 = ρV21. (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 opposite 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 transducers 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 directions. 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. 8 Chapter 1. Introduction

Figure 1.3: Cubic specimen with additional 45◦-oriented faces for the measurement of the oﬀ-diagonal elastic constants. From Rho(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 2 the material (e.g. ρV11 = E1). Lang(1969) was the first to introduce ultrasound bulk wave velocity measurements 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 cortical 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 accurate 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 coefficients 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 precision 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 properties 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 methods on the same specimens. However, two recent studies reach similar conclusions (Grimal et al., 2009; Berteau et al., 2014): 1) elastic properties measured by mechanical 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 ultrasound 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 10 Chapter 1. Introduction

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 requiring 45◦-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 section 1.4.2), 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 by Migliori and Sarrao(1997).

Measurement of the resonant frequencies

The basic setup for a RUS experiment is very simple: it consists of a pair of transducers, 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 before 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 (Chapter2), 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. 12 Chapter 1. Introduction

Computation of the resonant modes

The natural free-vibration modes of a solid occupying the region Ω bounded by the surface S are the pulsations ω and displacement ﬁelds u that solve the harmonic equations of motion6 2 2 ∂ uk ρω ui + Cijkl = 0 in Ω, (1.13) ∂xj∂xl subject to the stress-free boundary conditions

∂uk Cijkl nj = 0 on S. (1.14) ∂xl 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 L of the body, i.e. they must satisfy Z 1 2 1 ∂ui ∂uk δL = δ ρω uiui − Cijkl dΩ = 0, (1.15) Ω 2 2 ∂xj ∂xl 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 2 Z 2 ∂ uk ∂uk ρω ui + Cijkl δui dΩ − Cijkl nj δui dS = 0. (1.16) Ω ∂xj∂xl S ∂xl

Because δui is arbitrary in Ω and on S, 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 X ui = ai,λφλ. (1.17) λ The Lagrangian can then be expressed in a matrix form 1 1 L = ω2atMa − atKa (1.18) 2 2 6A rectangular coordinate system is assumed here for simplicity. A cylindrical or spherical coordinate system can be more adapted to some shapes. See Chapter3 for the cylindrical case. 7Integration by part followed by the use of the divergence theorem, and discarding the second and higher order terms in δui. 1.4. Resonant Ultrasound Spectroscopy 13 where the vector a contains the coefficients of the expansion (1.17), and M and K are 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 Z Miλ,i0λ0 = δii0 ρ φλφλ0 dΩ, and (1.20) Ω Z ∂φλ ∂φλ0 Kiλ,i0λ0 = Ciji0j0 0 dΩ. (1.21) Ω ∂xj ∂xj 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 approximated on the polynomial basis. Fig. 1.5 represents 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 Chapter3).

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

N exp. comp. 2 X fn − fn (Cij) F = exp. , (1.22) fn n=1 where N 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, or pairing, of the resonant modes. Indeed, the measured 14 Chapter 1. Introduction

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 = 2 mm, 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 √ p f = 1/( 2l) × C44/ρ. The approximation is accurate to ten digits for that mode. resonant frequencies are not informative of which predicted vibration mode they correspond 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 LeCraw used 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 in Ohno’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 by Migliori et al. in 1993. Visscher et al.(1991) extended further the work of Demarest and Ohno to 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 science, 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- 16 Chapter 1. Introduction 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 be 0.0002±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 by Ogi 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 formulation 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 attenuative for mechanical waves at ultrasonic frequencies, resulting in important damping of the resonant modes. The few studies in which RUS was applied to highly attenuative 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 factor Q, 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: Q factor measured using RUS for some materials, from very weakly attenuative to highly attenuative. Some results of this thesis are included.

Material Q factor (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)

Q=500

Q=200 10

Q=50

Ampltiude (log scale) Q=20 1 155 160 165 170 175 Frequency (kHz)

Figure 1.6: Simulated example: as the Q factor 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 Q factor 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 Q of about 150 by Ulrich et al.(2002). In this last study, authors investigated the side-length ratios 18 Chapter 1. Introduction 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 Q factor 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. Chapter2 is a research article published under the tile Accurate 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 as Bernard 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. Chapter3 is a research article published under the title Resonant ultrasound spectroscopy for viscoelastic characterization of anisotropic attenuative solid materials 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 coefficients, 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. Chapter5 presents the application of the developments introduced in the preceding 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 spectroscopy in 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 Chapter3.

2.1 Introduction

Cortical bone elasticity is anisotropic at the millimeter length scale and shows important 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 measurements 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 elastic 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 (∼ 5 mm). 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 22 Chapter 2. RUS on bone: a feasibility study 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 measurements (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 mechanical resonances of a freely vibrating specimen are measured, and an iterative numerical procedure is used to adjust elastic constants until the calculated spectrum 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 propagation 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 Q of the order of 150 (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 a Q factor 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) measured Q in 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 developments 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 by Landa et al.(2009) for highly anisotropic materials. Our objective is to show that RUS is suitable for the measurement 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, Buehler R 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 24 Chapter 2. RUS on bone: a feasibility study

2 mm

Figure 2.1: 3D reconstruction of the cortical bone specimen from SR-µCT. The sample’s faces are oriented according to the radial (axis 1), circumferential (axis 2) and axial (axis 3) directions of bone. be ρ = 1948 kg/m3 from a mass measurement (M = 0.4588 g). Dimensions were measured using a digital caliper (accuracy: ±0.02 mm). The accuracy of RUS critically depends on the specimen shape which should be close to a perfect parallelepiped. The faces of our specimen were parallel and perpendicular within 0.5 ◦.(Migliori and Sarrao, 1997) gave a detailed discussion of the acceptable tolerance on sample shape; they found that errors of a few tenth of a degree usually have a negligible inﬂuence on the resonant frequencies. The specimen was imaged using 3D synchrotron radiation microtomography (SR-µCT) with a 10 µm spatial resolution (Granke et al., 2011). The specimen porosity (4.3%) and tissue bone mineral density (1.05 g/cm3) were evaluated from the reconstructed 3D image (Fig. 2.1). Qualitatively, it could be seen on the 3D image that the distribution of the porosity in the specimen is homogeneous (no porosity gradient, no big pores).

2.2.2 Resonant Frequencies Calculation

RUS is based on a comparison of calculated and measured resonant frequencies. We used a well-documented method to calculate the frequencies of mechanical resonance for a solid of given elasticity and rectangular parallelepiped geometry (Migliori and Sarrao, 1997). Briefly, the resonant angular frequencies ω were found by searching the stationary points of the Lagrangian Z 1 2 2 ∂ui ∂uk L = ρω ui − Cijkl dV, (2.1) 2 V ∂xj ∂xl where V and ρ are respectively the specimen’s volume and mass density, Cijkl are the stiffness constants and u is the displacement field. By expanding the displacement field in a set of polynomial functions (Rayleigh-Ritz method), the stationary 2.2. Materials and Methods 25

Receiver

Charge Amplifier

In 5 mm Network Computer Analyser Out Emitter

Figure 2.2: Photograph of the bone specimen inserted between the two transducers and block diagram of the experimental setup. equation ∂L = 0 was written as a generalized eigenvalue problem

ω2Ma = Ka, (2.2)

where M and K denote respectively the mass and stiﬀness matrices of the vibration problem. Eq. (2.2) was solved numerically, giving the eigenvalues ω2 and the eigenvectors a, which contain the coeﬃcients of the polynomial expansion.

2.2.3 Resonant Frequencies Measurement

Transducers were built from small piezoceramic elements (diameter of 3 mm and length of 2 mm) polarized in compression and suspended to copper thin films. This construction reduces the transmission of energy to the mechanical holder and then prevents any unwanted system resonance (Migliori and Sarrao, 1997). Support films were glued to steel cylinders connected to the ground, which provided isolation from electromagnetic perturbation and cross-talking between transducers. With these transducers the measurement system was non-resonant in the frequency range of interest. The specimen was placed between the two transducers, held by two opposite corners to be as close as possible to stress-free boundaries conditions (Fig. 2.2). A Vectorial Network Analyzer (Bode 100, Omicron electronics GmbH, Klaus, Austria) was used to record the frequency response of the specimen between 100 kHz and 280 kHz with 50 Hz resolution. The receiver output signal was preconditioned before recording by a broadband charge amplifier (HQA-15M-10T, Femto Messtechnik GmbH, Berlin, Germany). Phase and magnitude of the frequency response were saved for further signal processing. Preliminary measurements showed that the number of resonant frequencies which can be retrieved may vary from one measurement to another on a same specimen. This is expected due to the difficulty to excite the vibrational modes which are 26 Chapter 2. RUS on bone: a feasibility study associated to a motion of the specimen’s corner nearly parallel to the surface of the transducers. This difficulty is well known for low damping material (Stekel et al., 1992) and is even more critical in the case of high damping, where the signal is weak and the modal overlap is strong. Although in previous preliminary tests we observed that it was possible to determine the full stiffness tensor of a bone-mimicking material based on a single positioning of the specimen (Bernard et al., 2011), the quality of the measurement can be increased by combining a few measurements of the specimen. Indeed, intermediate repositioning introduces small variations of the transducer-specimen coupling which in practice increases the probability of exciting and detecting weakly excited modes. Accordingly, the measurements were repeated, removing and replacing the specimen six times. This limited the number of missed modes in the frequency range of interest. Repeated measurements also provided a way to estimate uncertainty on the resonant frequencies.

2.2.4 Signal Processing

When RUS is applied to low damping materials, resonant frequencies can simply be picked as the sharp local maxima of the spectrum. Each resonant peak can also be ﬁtted to a Lorentzian lineshape if the damping factors are to be studied (Leisure et al., 2004). In the context of bone, peaks are broad and strongly overlap each others. In this case, resonant frequencies rarely correspond to local maxima of the spectrum and peaks cannot be ﬁtted individually. A dedicated signal processing method must be used for the estimation of resonant frequencies from the spectrum. We used a method initially developed for the ultrasonic spectroscopy of large rocks specimens, which show an important damping, described in (Lebedev, 2002) and (Lebedev et al., 2003) and summarized below. The frequency response FR of the specimen in the frequency range of interest can be modelized as a sum of M Lorentzian lineshapes:

M X ak FR(f) = (2.3) (f 2 − f 2) + i(f f/Q ) k=1 k k k with ak the complex amplitudes, fk the resonant frequencies and Qk the resonant quality factors. Equivalently, the response can be represented in time domain by the impulse response, which is a sum of damped exponentials involving the same parameters. The problem of estimating the signal parameters from discrete samples of the impulse response has been addressed in the signal processing literature. A solution using a linear prediction filter proposed by Kumaresan and Tufts(1982) has been introduced by Lebedev in the context of RUS and was used in this study. In a first step, the samples of the experimental frequency response were converted in time domain by inverse Fourier transform. Then, the best linear filter predicting the 2.2. Materials and Methods 27

−2 10 Amplitude (a.u)

130 140 150 160 170 180 Frequency (kHz)

Figure 2.3: Typical measured spectrum (plain line) and reconstruction from the linear prediction method (dot line). The spectrum reconstructed after the non-linear optimization step is indistinguishable from the measured spectrum. •: Resonant frequencies obtained from linear prediction. +: Resonant frequencies after non-linear optimization.

time domain data from the L ≥ M ﬁrst samples was constructed. In the particular

case L = M, the frequencies fk and quality factors Qk are given by the positions of the zeros of the constructed predictive ﬁlter. Finally, the complex amplitudes ak were obtained by the resolution of a linear system in the frequency domain. Note that the number M does not need to be known a priori, but was determined from the inspection of the eigenvalues spectrum of the autocorrelation matrix of the time domain signal.

Because we found slight discrepancies between the experimental response and the response reconstructed from Eq. (2.3) using the estimated parameters, we add a non- linear optimization step to the signal processing procedure presented above. The determined parameters were used as initial parameters of a non-linear optimization algorithm (Levenberg-Marquardt), searching the set of parameters for which the diﬀerence between Eq. (2.3) and the experimental frequency response was minimal in a least-square sense. In order to reduce the number of independent parameters and then the complexity of the optimization step, signal processing was performed in small frequency bands containing only a few resonant frequencies (typically 6-8). In practice, we found that the agreement between experimental and reconstructed spectra was improved by the non-linear optimization step (Fig. 2.3) and that the repeatability of the parameters estimation was improved. 28 Chapter 2. RUS on bone: a feasibility study

2.2.5 Elastic Constants Estimation

Evaluation of the elastic constants in RUS is based on an optimization process, which usually consists in minimization of the function

exp cal 2 X fn − f (Ck) F (C ) = n , (2.4) k f exp n n where Ck stands for the set of independent elastic constants (the number of Ck depends on the assumed elastic symmetry). This optimization process requires association of each measured resonant frequency to the corresponding calculated frequency. This association can be diﬃcult for two reasons:

1. Elastic constants used as initial values for the calculation of the resonant frequencies may be far from the actual values, so that measured and calculated frequencies do not appear in the same order (frequency crossing).

2. Some of the resonant frequencies can be missed during the measurement, as already mentioned.

These two difficulties were overcome by the use of an iterative procedure for simultaneous association of frequencies and constants estimation, as proposed by Landa et al.(2009) for application of RUS to ferroelastic materials. Because modal density is low for the first vibration modes, the probability of frequency crossing is low and a missed frequency can easily be identified. Thus, for a given reasonable initial guess of the elastic properties, the n first measured frequencies could be confidently associated with their corresponding calculated frequencies. A new ∗ set of independent elastic constant Cl was then defined, which has an orthogonal relationship with the original set:

∗ Ck = βklCl , (2.5)

t where [β1l ··· βql] is the lth eigenvector of the sensitivity matrix G G, computed from the partial derivatives of the n associated resonant frequencies with respect to each independent constant:   ∂f1 ... ∂f1 C1 Ck  . . .  G =  . .. .  . (2.6)   ∂fn ··· ∂fn C1 Ck ∗ The combinations Cl could be sorted by their sensitivities v u 2 X ∂fj p S = u β = λ , (2.7) l t kl ∂C l j=1...p k 2.2. Materials and Methods 29

t where λl denote the lth eigenvalue of the matrix G G. The sensitivities Sl reﬂect the quantity of information contained in the n associated frequencies about each ∗ combination Cl , and then sort the combinations from the most determinable to the less determinable. A minimization of

exp cal ∗ 2 X fn − f (C ) F (C∗) = n l , (2.8) l f exp n n

∗ was done (Levenberg-Marquardt method), with the combinations Cl presenting an important sensitivity (S > 1 kHz/GPa) as free parameters, while the other combi- u nations were kept fixed. Updated constants Ck were computed back from Eq. (2.5) at the end of the fit and used to calculate updated resonant frequencies. Because the u updated constants Ck obtained by processing the first n frequencies are closer to the actual elastic properties of the specimen than the initial guess, calculated frequencies are also closer to experimental frequencies and the association can be extended to the n0 > n first frequencies. The procedure was repeated iteratively, using more combinations in the minimization as more resonant frequencies could be associated without ambiguity, until all measured frequencies were associated. Finally, all the ∗ combinations Cl were optimized simultaneously. The iterative process was completed assuming an orthotropic symmetry (nine independent constants), which is the most general elastic symmetry usually assumed for bone. Note however that the method can assume material symmetries of lower order. The initial values for the optimization problem were mean values of bone elasticity of aged women from (Granke et al., 2011): C11 = 19.3, C22 = 19.8, C33 = 29.2, C44 = 5.8, C55 = 5.6 and C66 = 4.2 (all values in GPa). Since this last study did not provide the values of off-diagonal elastic constants, we used

C12 = C13 = C23 = C11 − 2 × C66 = 10.9 GPa.

2.2.6 Uncertainty on the elastic constants

Following Migliori et al.(1993), we used the curvature of the cost function F around the minimum to estimate the uncertainty on each particular constant. The method consists in ﬁnding the largest possible changes on the constants which lead to an increase of 2% in F . The uncertainty on the measured resonant frequencies also provide a way to estimate uncertainty on the elastic constants by the mean of Monte-Carlo simulation. We generated 100 random sets of frequencies from independent normal distributions centered on the experimental mean values and using experimental standard deviations. This 100 sets were used to estimate the distribution of the elastic constants values. The mean and standard deviation on each constant was then calculated. 30 Chapter 2. RUS on bone: a feasibility study

Table 2.1: Elastic constant of the bone specimen evaluated from RUS and BWV, with their uncertainties (all values in GPa).

C11 C22 C33 C12 C13 C23 C44 C55 C66 RUS 20.3 20.2 31.7 10.7 13.4 13.4 6.38 6.32 4.80 (0.6) (0.6) (0.8) (0.6) (0.7) (0.7) (0.02) (0.03) (0.02) BWV 21.6 21.4 31.3 - - - 6.5 6.5 4.8 (0.7) (0.7) (1.0) - - - (0.3) (0.3) (0.2)

2.2.7 Engineering moduli

From the measured elastic constants the 6 × 6 stiﬀness matrix was constructed and numerically inverted to obtain the compliance matrix (Auld, 1990). From the latter, engineering moduli (Young’s moduli, shear moduli and Poisson’s ratios) were calculated (Bower, 2009). This was repeated for the 100 sets of constants obtained from Monte-Carlo simulation, allowing to estimate the uncertainties (standard deviation) on the engineering moduli.

2.2.8 Ultrasonic Velocities Measurement

For comparison purpose, we repeated BWV measurements as described by (Granke et al., 2011). Briefly, the elastic constants on the diagonal of the stiffness tensor were deduced from velocities of longitudinal and shear ultrasonic waves propagating in the principal directions of the specimen. Velocities were obtained from the time- of-flight of an ultrasonic pulse propagating trough the specimen between a pair of transducers contacting two opposite faces. Central frequencies of 2.25 MHz and 1 MHz were used for longitudinal and shear wave velocities measurements respectively (V105RM and V152RM, Panametrics Inc., Waltham, MA). Signals were acquired using an oscilloscope (TDS 2012, Tektronix Inc., Beaverton, OR) and processed with MatLab. Repeatability of the method, evaluated to be 3.2% and 4.7% for longitudinal and shear elastic constants respectively (Granke et al., 2011), was used to evaluate uncertainty on the elastic constants measured from BWV.

2.3 Results

Between 14 and 17 resonant frequencies were estimated from each frequency response measurement. The amplitude of a typical frequency response, on which 16 frequencies were obtained, is plotted in Fig. 2.4. Upon combining six measurements, 20 distinct resonant frequencies were retrieved and used for elastic constants estimation. Note that each of those resonances were present in at least two frequency responses. The mean value and standard deviation on each resonant frequencies was 2.3. Results 31 280 260 240 220 200 Frequency (kHz) 180 ) and a vertical dot line if not. Phase information (not plotted) was also used during M ) and a vertical line if the calculated frequency was associated to a frequency measured in 160 N 140 120 A typical measured frequency response of the bone specimen between 100 kHz and 280 kHz. The 16 frequencies extracted 100 −2

10 Amplitude (a.u.) Amplitude signal processing. this particular spectrum, and by an hollow triangle ( Figure 2.4: from the signal processingsymmetry step are are represented represented by by a plain cross triangle (+). ( Calculated frequencies for the optimized elastic constants with orthotropic 32 Chapter 2. RUS on bone: a feasibility study calculated from 2 to 6 values depending on the frequency. All standard deviations were below 0.5% and the mean value of standard deviation was 0.23%. In the first step of the elastic constant estimation procedure, the three first measured frequencies were associated to the three first predicted frequencies. After six iterations of the procedure, each of the 20 frequencies was confidently associated to one of the 30 first calculated frequencies in the frequency range 110-280 kHz. The elastic constants obtained from RUS are compared to the constants measured with BWV in Table 2.1. Frequencies calculated for the optimized elastic constants are located on the experimental frequency response (Fig. 2.4) and compared to experimental frequencies in Table 2.2. The root-mean-square error between calculated and experimental frequencies was 0.3%. Measured quality factors are also given in Table 2.2. The two methods used to estimate uncertainties on the RUS measurement gave very close results, although systematically slightly higher (∼ 10%) for Monte-Carlo simulation. Consequently, only uncertainties evaluated from this last method are displayed in Table 2.1. Engineering moduli and their uncertainties obtained on our specimen are presented in Table 2.3 beside values from the literature.

2.4 Discussion

The present study is the first, to our knowledge, to measure enough resonant frequencies to perform a reliable estimation of the anisotropic elastic constants of bone or any other anisotropic material with comparable damping (Q ≈ 25). Use of a dedicated signal processing method allowed to overcome the difficulties encountered by previous authors (Lee et al., 2002). Resonant frequencies associated to more than half of the predicted vibrational modes could be measured with a repeatability better than 0.5% (standard deviation) despite the strong modal overlap. However, 10 modes could not been observed. These correspond to vibration modes which are weakly excited in our experimental configuration and masked by strongly excited modes. The use of the iterative procedure for simultaneous frequency association and elastic constants estimation proposed by Landa et al.(2009) is original in the context of this study. Here, this method allows to perform modes association when as many as one third of the modes are missing. It is usually assumed that a minimum of five times more resonant frequencies than the number of unknown elastic constants must be measured for a precise assessment of elasticity (Migliori and Sarrao, 1997). However in this study we achieved a full characterization of the orthotropic elastic tensor with only 20 measured frequencies. Because the association of experimental and calculated frequencies is made progressively, the outcome of the optimization 2.4. Discussion 33

Table 2.2: Experimental Q factors and resonant frequencies, compared to calculated frequencies for the optimized elastic constant.

Mode Qexp fexp(kHz) fcal (kHz) % err 1 25.8 113.67 113.50 -0.15 2 24.2 140.19 139.97 -0.16 3 21.8 148.75 148.11 -0.43 4 22.6 159.01 159.14 0.08 5 21.6 163.75 164.34 0.36 6 28.2 170.75 171.00 0.14 7 24.8 178.25 178.11 -0.08 8 25.5 181.24 180.98 -0.14 9 29.9 185.35 185.08 -0.14 10 - - 192.20 - 11 24.9 193.86 194.02 0.08 12 24.8 197.19 197.59 0.21 13 13.6 206.31 206.76 0.22 14 24.8 215.05 214.32 -0.34 15 - - 219.30 - 16 - - 220.35 - 17 27.3 223.00 222.48 -0.23 18 23.8 231.20 229.49 -0.74 19 - - 236.40 - 20 - - 243.48 - 21 - - 245.04 - 22 24.4 247.77 247.41 -0.14 23 - - 249.65 - 24 - - 250.22 - 25 - - 253.08 - 26 21.2 256.22 257.71 0.58 27 24.6 263.00 263.64 0.24 28 - - 269.44 - 29 21.2 273.24 274.13 0.33 30 33.0 279.20 280.01 0.29 34 Chapter 2. RUS on bone: a feasibility study

Table 2.3: Young’s moduli (E), shear moduli (G) and Poisson’s ratios (ν) of the bone specimen determined from RUS measurement and values from the literature. All values in GPa, except Poisson’s ratios (dimensionless).

This study Ashman et al.(1984) Rho(1996)

E11 12.8 (0.1) 12.0 11.7 E22 12.8 (0.1) 13.4 12.2 E33 20.3 (0.1) 20.0 20.7 G12 4.80 (0.02) 4.53 4.1 G13 6.32 (0.03) 5.61 5.17 G23 6.38 (0.02) 6.23 5.7 ν12 0.348 (0.009) 0.376 0.420 ν13 0.276 (0.007) 0.222 0.237 ν23 0.274 (0.006) 0.235 0.231 ν21 0.348 (0.010) 0.422 0.435 ν31 0.434 (0.011) 0.371 0.417 ν32 0.421 (0.008) 0.350 0.390 procedure is not very sensitive to the choice of the initial values. The latter can be set to values available in the literature (Granke et al., 2011). The final RMS error is quite low (0.3%). Migliori and Sarrao(1997) stated that a fit with a RMS error of about 0.1-0.2 % can be considered as good in the context of homogeneous, low damping materials. Ulrich et al.(2002) considered a fit as valid when RMS error was below 1% for rock specimens. Moreover, the RMS error is only slightly higher than the mean uncertainty on the measured resonant frequencies (0.23%). It suggests that the model used to calculate the frequencies is successful to predict the resonant behavior of the bone specimen, although several simplifying assumptions were made to model the bone specimen as a homogeneous viscoelastic solid. In particular any frequency dependence of the viscoelastic behavior is neglected. The validity of the homogeneity hypothesis follows from the fact that the wavelengths involved in vibrational motion of the specimen are much larger than bone tissue heterogeneities. Furthermore, it has been shown that the resonant frequencies are not sensitive to a weak gradient in elastic constants and mass density (Seiner et al., 2012). Elastic constants measured with RUS and the popular BWV method agree well. The discrepancies between the two methods may be due to measurement uncertainties but also to the frequency-dependent behavior of bone (Katz and Yoon, 1984) (RUS operates in the range 100-280 kHz; BWV method operates at 2.25 MHz and 1MHz). All the engineering moduli are in agreement with values from the literature, particularly the Young’s moduli. Slight discrepancies on shear moduli and Poisson’s ratios could be explained by the fact that the literature values are mean values obtained for many specimens while our values stand for a single specimen. 2.5. Conclusion 35

The evaluated uncertainties on the elastic constants are approximately 0.4%, 3% and 5% for shear, longitudinal and off-diagonal constants respectively. This sorting of the uncertainties associated with each type of constants is typical in RUS (Migliori and Sarrao, 1997), and originates in the fact that (1) many of the first vibrational modes of a weakly anisotropic solid with small aspect ratios are shear modes; (2) the relative weight of off-diagonal stiffness constants in the measured frequencies is small. Since the shear moduli and the shear stiffness constant are equal in the orthotropic elastic symmetry (Bower, 2009), their uncertainty are equal (0.4%). However, uncertainties on the Young’s moduli and Poisson’s ratios are respectively about 0.6% and 2.5%, which is significantly less than the uncertainties on the longitudinal and off-diagonal stiffness constants. As a consequence of the physics of waves propagation involved in vibrational motion, resonant frequencies are more related to ratios of stiffness constants than to the constants themselves. Accordingly RUS can perform more precise measurements of these ratios. Since the expressions of Young’s moduli and Poisson’s ratios mainly involve ratios of stiffness constants (Bower, 2009), these are determined with a lower uncertainty. However, because we measured only one specimen and only six spectra, our uncertainty values have to be confirmed with repeated measurements on several specimens. In its current development stage, our implementation of the RUS method requires several repositioning of the measured specimen to limit the number of missed modes and to obtain a satisfactory measurement precision. This repositioning could be automated or avoided by the use of different transducers types. Using an automated identification procedure, combined with a random testing of many sets of initial values, it should be possible to improve the robustness of the method. In this paper we have presented results of the measurement of a unique specimen for the purpose of illustrating the method. Another bone specimen and a sample of bone-mimicking material have also been successfully measured (results not shown).

2.5 Conclusion

This study demonstrates that RUS is suitable for an accurate measurement of cortical bone anisotropic elasticity despite the strong viscoelastic damping. RUS lends itself to a precise and automated measurement of all the stiﬀness constants of a bone specimen. We foresee that the the availability of such method will stimulate investigations of normal and pathological bone material properties. Since in principle RUS does not require a minimum size for the tested specimens, it may be appropriate for the characterization of small bone specimens (∼ 1 mm3) and for bone phenotyping in small animal studies.

Chapter 3 RUS for viscoelastic characterization of anisotropic attenuative solid materials

This chapter is a research article published under the tile Resonant ultrasound spectroscopy for viscoelastic characterization of anisotropic attenuative solid materials in The Journal of the Acoustical Society of America in 2014, and referred to in this manuscript as Bernard et al.(2014). The full text of the article is reproduced here with no addition and no modiﬁcations except in the form. This work presents results obtained on two materials whose damping is close to that of bone: PMMA, an isotropic polymer and a bone-mimicking composite material. Both rectangular and cylindrical specimens are used and the viscoelastic damping of the materials is estimated from the width of the resonant peaks. Descriptions of the technical aspects of RUS for the application to attenuative materials are more detailed than in Chapter2. In addition, a ﬁrst Bayesian framework is introduced for the resolution of the inverse problem. The proposed algorithm for the pairing of resonant frequencies can be seen as an automated version of the procedure proposed in Chapter2 .

3.1 Introduction

Resonant ultrasound spectroscopy (RUS) is an accurate and efficient method to characterize the elastic properties of isotropic or anisotropic solid materials (May- nard, 1992; Migliori et al., 1993; Migliori and Sarrao, 1997; Migliori and Maynard, 2005). In a RUS experiment, the mechanical resonant frequencies of a freely vibrating sample are measured, and the material elastic properties are adjusted until model-predicted frequencies match the measured frequencies (inverse problem approach). Likewise, viscoelastic damping of the material can be inferred from the width of the resonances (Ogi et al., 2003; Leisure et al., 2004). In the present work, we are interested in the application of RUS to strongly attenuative viscoelastic materials. The main advantage of RUS over concurrent elasticity measurement methods is that the full anisotropic elastic tensor can be assessed from a single sample in a single experimental configuration. Quasi-static mechanical tests usually require a dedicated sample for each elastic modulus and wave velocity-based methods requires longitudinal and shear waves propagation in several directions. Another advantage is that RUS is well-adapted to small samples (a few millimeters or less). Velocity-based methods require samples much larger than the wavelength, and this is not always 38 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids possible to achieve for heterogeneous materials by scaling down the wavelength. Indeed, such materials require that the wavelength remains much larger than the inhomogeneities to avoid strong dispersion effects. Finally, RUS is reputed as being very accurate. Since the method minimizes the coupling between the sample and the apparatus it avoids errors due to inaccurate modeling of such problems (Maynard, 1992). Moreover, RUS takes into account the complex vibrations of a finite sample, instead of assuming idealized states of stress and strain.

RUS has been successfully applied to many low damping materials, such as single crystals (Isaak and Ohno, 2003), quasi crystals (Spoor et al., 1995), or metallic composites (Ledbetter et al., 1995), most often on rectangular parallelepipeds, but also on cylinders (Heyliger et al., 1993; Jaglinski and Lakes, 2011). In contrast, its application to attenuative materials has not received much attention. This is due to the diﬃculty caused by the broadening of the resonant peaks in the presence of mechanical damping. Indeed, as damping increases, the quality factor Q = f/∆f of the resonant peaks decreases, peaks overlap, and it becomes complicated to retrieve the resonant frequencies. In geophysics, RUS has been applied to rock samples with a Q about 150 by Ulrich et al.(2002). Lebedev(2002) proposed the use of a signal processing method to retrieve resonant frequencies with Q=50. 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 Q factor about 30. Recently, we applied RUS to a cortical bone sample (Bernard et al., 2013) with a comparable Q, using the processing method proposed by Lebedev.

Although this last study was successful in measuring bone elastic properties, it reported that several predicted resonant modes could not be observed, despite signal processing. Indeed, due to the width of the resonant peaks, weakly excited resonant modes were masked by strongly excited ones (Bernard et al., 2013). This was a source of trouble in the inverse problem, and will be as well for all highly attenuative materials. Missing resonant frequencies means that there is less data available to identify the mechanical properties and that we don’t know which of the overnumerous predicted frequency compare to each measured frequency. For these reasons, the inverse problem could beneﬁt from a probabilistic formulation, in a Bayesian framework, which allows one to conveniently introduce available prior information (Tarantola, 2005) and, as we will see, to deﬁne a quantitative criterion for the pairing of each measured frequency to its corresponding predicted one.

Applicability of RUS to attenuative materials could beneﬁt several ﬁelds. In the context of biological materials (bone, tooth material) there is a lack of accurate methods for anisotropic viscoelasticity complete characterization. Design of appropriate materials for prosthesis and implants also requires characterization methods. Many composites used in industrial applications are anisotropic and viscoelastic, 3.2. Resonant frequencies computation 39 which makes methods for an accurate and complete characterization of their viscoelastic properties of great interest. In this work, our objective is to describe some adaptations of the conventional RUS method, including the mentioned Bayesian formulation of the inverse problem, for the application to attenuative materials. Section 3.2 describes the direct computation of the natural frequencies of a sample of given elasticity. Section 3.3 presents the samples for which the method is illustrated and the measurement of their spectra. The dedicated signal processing method used to retrieve the resonant frequencies of the overlapped peaks is detailed in section 3.4. Inverse estimation of the elastic properties based on a Bayesian framework is presented in section 3.5 and the estimation of the damping properties in section 3.6. Results are presented and discussed in section 3.7 and section 3.8 respectively. Concluding remarks end the paper in section 4.5.

3.2 Resonant frequencies computation

The Rayleigh-Ritz method is an efficient and accurate method to compute the natural vibration frequencies of solids of regular shapes, including rectangular paral- lelepipededs (RP) (Ohno, 1976; Maynard, 1992; Migliori et al., 1993; Migliori and Sarrao, 1997) and cylinders (Heyliger et al., 1993; Heyliger and Johnson, 2003; So and Leissa, 1998). We here recall the basic principles of the method and then give further details for both geometries. We emphasize on the mathematical properties allowing us to increase the efficiency of the computation. Efficiency is important for the probabilistic formulation of the inverse problem (section IV), which implies several thousands of direct computations of the natural frequencies. The harmonic solutions of the equations of motion for a freely vibrating elastic body are the stationary points of the Lagrangian L Z 1 2 2 L = ρω ui − Cijklεijεkl dV, (3.1) 2 V

where V and ρ are respectively the specimen’s volume and mass density, Cijkl are the stiffness constants, ui is the ith component of the displacement field, and εij is a component of the strain tensor. The displacement field components are expanded as finite sums of known basis functions and the stationary equation ∂L = 0 is reduced to a generalized eigenvalue problem:

ω2Mα = Kα, (3.2)

where M and K denote respectively the mass and stiﬀness matrices of the vibration problem and α is a vector containing the coeﬃcients of the expansion. For simple solid shapes and well-chosen basis functions, components of the matrices are easily 40 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids computed and Eq. (3.2) is numerically solved, giving the natural angular frequencies

ωn and the corresponding eigenvectors αn, which represent the displacement patterns of the resonant modes. The partial derivatives of a given resonant frequency with respect to any parameter p can be obtained from

∂ω ∂K ∂M 2ω n = αt − ω2 α . (3.3) n ∂p n ∂p n ∂p n

Partial derivatives of the natural frequencies with respect to the stiﬀness constants are used in the inverse problem.

3.2.1 Rectangular Parallelepiped

The considered RP solid is aligned with a Cartesian coordinate system {x1, x2, x3} and has length Li in the ith direction. Normalized Legendre polynomials of the scaled coordinates are used as basis functions, i.e.

a+b+c≤N X 2x1 2x2 2x3 ui = αi,abcPa Pb Pc (3.4) L1 L2 L3 a,b,c=0 where Pa denotes the normalized Legendre polynomial of order a. Thanks to the orthogonality of the Legendre polynomials over the interval [−1, 1], M is the unit matrix and Eq. (3.2) is a standard eigenvalue problem. Many elements of K are equal to zero, and the matrix turns to be block-diagonal. Eq. (3.2) therefore splits in several independent smaller problems (Ohno, 1976; Migliori et al., 1993; Migliori and Sarrao, 1997). For an orthotropic elastic symmetry, involving three orthogonal symmetry planes, the problem splits in eight smaller ones. Each sub-problem corresponds to vibration modes with different combinations of symmetry or anti- symmetry in the three direction of space. This split not only drastically reduces the computation cost but allows description and labeling of the resonant modes according to their belonging to one of the eight symmetry group. Moreover, the matrix K has a linear dependence to the stiffness constants (Landa et al., 2009). Therefore, in an iterative computation of the resonant frequencies for many different sets of elastic properties, elements of K need not to be computed each time but are linearly obtained. We found, as Migliori and Sarrao(1997), that a maximum polynomial order N = 10 gives sufficient accuracy while keeping the computation time small.

3.2.2 Cylinder Although the case of the cylinder can be treated in a rectangular coordinates system (Visscher et al., 1991), cylindrical coordinates provide improved eﬃciency and accuracy (Heyliger and Johnson, 2003). The considered cylinder has radius R and height 3.2. Resonant frequencies computation 41

H (Fig. 3.1). In the following, we assume that the material has an orthotropic (or higher) symmetry in the cylindrical coordinate system, i.e. the stress-strain relation is (in Voigt notation)

      σrr C11 C12 C13 0 0 0 εrr σθθ C12 C22 C23 0 0 0   εθθ        σ  C C C 0 0 0   ε   zz  13 23 33   zz    =     . (3.5) σθz  0 0 0 C44 0 0  2εθz       σrz  0 0 0 0 C55 0  2εrz σrθ 0 0 0 0 0 C66 2εrθ

Although it is possible to deal with a more general elastic symmetry, this assumption corresponds to many practical cases and allows many simplifications which increase the efficiency of the computation. We denote by u, v and w the displacement components in the r, θ and z directions respectively (Fig. 3.1). For an isotopic cylinder, So and Leissa(1998) expressed the displacement field as: u(r, θ, z) = U(r, z) cos(nθ), (3.6a)

v(r, θ, z) = V (r, z) sin(nθ), (3.6b)

w(r, θ, z) = W (r, z) cos(nθ), (3.6c)

for n = 0, 1, 2, ..., ∞. In Eq. (3.6), the θ coordinate is decoupled from the two other coordinates. This is due to the common circular symmetry of the specimen shape and elastic properties (this is discussed in details by Heyliger and Johnson (2003)). For an orthotropic cylinder, the same symmetry exists and we can use the same expressions. Therefore, the vibration modes involve a single azimuthal wave number n, and modes with diﬀerent wave numbers are uncoupled. The problem then degenerates into a set of smaller problems (one for each n), considerably increasing the eﬃciency of the calculation. In the Rayleigh-Ritz approximation U, V , and W are expanded as sums of simple polynomial functions in r and z directions, e.g.

M N X X α β U(r, z) = aα,βr z . (3.7) α=0 β=0

Using expressions (3.7) and (3.6), the mass and stiﬀness matrices are computed and Eq. (3.2) is numerically solved for each n. It should be noted than another set of solutions than (3.6) exists, replacing the cos(nθ) terms by sin(nθ) and conversely (So and Leissa, 1998). For n ≥ 1 it gives the same natural frequencies, but with modal shapes rotated of angle π/2 (i.e. modes with n ≥ 1 are doubly degenerated). For n = 0 the two solutions do not provide the same frequencies and should be both included in the computation. Natural modes of the type (u = 0, v 6= 0, w = 0) are 42 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids

Figure 3.1: Cylindrical sample geometry in the (r, θ, z) cylindrical coordinate system. The components of the displacement field are denoted u, v and w in the directions r, θ and z, respectively. The material is cylindrically orthotropic so that both the shape and the elastic properties are invariant for any rotation around the cylinder axis. torsional modes while modes of type (u 6= 0, v = 0, w 6= 0) are axi-symmetric modes. For each n the problem can be further split in two uncoupled parts, according to whether the displacement is symmetric or antisymmetric with respect to the plane z = 0 (β is even or odd)(So and Leissa, 1998). In practice, we found that, for the study of the ∼30 first resonant modes, it was sufficient to compute resonant frequencies up to n = 5 and that maximal polynomial orders of 8 in both radial and axial direction (M = N = 8) provided sufficient accuracy. The present approach extends the work of So and Leissa(1998) on isotropic cylinders to orthotropic cylinders while keeping the calculation efficient. However, it is less general than the approach of Heyliger and Johnson(2003), which includes trigonal elasticity but in consequence does not split the eigenvalue problem and is less efficient.

3.3 Samples and measurement setup

3.3.1 Samples

To illustrate the adaptations of the RUS method to be exposed in this paper for the application to attenuative materials, we prepared parallelepipedic and cylindrical samples of Polymethylmethacrylate (PMMA). This material is an isotropic polymer, with a strong mechanical damping reflected in a low mechanical quality factor Q ≈ 40. We also prepared two parallelepipedic samples of a composite material made of glass fibers embedded in an epoxy matrix (Sawbones, Pacific Research Laboratory Inc, Vashon, WA). This material is transversely isotropic, highly attenuative (Q ≈ 25) and is a bone mimetic material. Four cylindrical PMMA samples were cut from 3.3. Samples and measurement setup 43

Table 3.1: Dimensions (in mm) and mass (in mg) of the four cylindrical samples an the two RP samples of PMMA.

Material Sample Diameter Height Mass PMMA Cyl. 1 5.15 3.68 90.8 Cyl. 2 5.15 5.67 139.9 Cyl. 3 5.15 5.71 140.9 Cyl. 4 5.15 7.15 176.1

L1 L2 L3 PMMA RP 1 2.92 2.85 3.93 38.5 RP 2 2.38 2.92 3.89 32.0 RP 3 2.45 2.85 3.88 32.1 RP 4 2.38 2.45 3.88 26.9 Glass/Epoxy Comp.1 4.05 7.90 7.36 387.0 Comp.2 4.06 5.23 7.35 256.5 the same long cylinder (diameter 5.15 mm) with diﬀerent lengths comprised between 3 and 8 mm. Four rectangular parallelepipeds were cut from a PMMA plate and two from a glass/epoxy composite plate. Dimensions of all samples are given in Table I. All samples were cut using a linear saw (Isomet 4000, Buehler GmbH, Dusseldorf, Germany). During cutting, care was taken to obtain pairs of faces that were as parallel and perpendicular as possible. In the case of the composite, care was taken to align the ﬁbers axis with one of the axis of the samples. Despite that, all samples have measurable geometrical imperfections. However, these imperfections were small (parallelism and perpendicularity errors smaller than 1 ◦) and were neglected. Mass density of each sample was deduced from a measurement of the dimensions using a digital caliper (accuracy 0.01 mm) and a mass measurement (accuracy 0.1 mg). Mass density was 1.18 g.cm−3 for PMMA and 1.64 g.cm−3 for the composite.

3.3.2 Setup for the measurement of the frequency responses

The mechanical frequency responses of the samples were measured over a frequency range containing approximately the 30 first resonant frequencies. This frequency range was estimated using previously reported stiffness coefficients for the materials. For example, the frequency band for the smallest cylindrical PMMA sample was 140- 420 kHz. For the measurement, a sample was placed with slight contact between a pair of commercial shear ultrasonic contact transducers (V154RM, Panametrics Inc., Waltham, MA). The frequency response was recorded using a vectorial network analyzer (Bode 100, Omicron electronics GmbH, Klaus, Austria) and a broadband charge amplifier (HQA-15 M-10 T, Femto Messtechnik GmbH, Berlin, Germany). The setup is illustrated on Fig. 3.2. 44 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids

Receiver Charge Amplifier

Specimen In Network Computer Analyser Emitter Out

Figure 3.2: Experimental setup illustrated with a cylindrical sample inserted. The setup works as well for RP samples in contact with the transducers by two opposite corners

The use of shear transducers for RUS was proved to produce stronger signal than compressional transducers and to selectively excite different kinds of resonant modes (Wang and Lakes, 2003). Indeed, depending on the orientation of the sample with respect to the polarization direction of the transducers, different resonant modes are excited. This is illustrated for cylinders on Fig. 3.3 but works as well for RP samples. To take advantage of this, we made 6 successive measurements on each samples, rotating the sample of a small angle between each measurement. As a result, we obtained 6 different frequency responses, involving different resonant modes. Fig. 3.3 shows two of these spectra obtained for a PMMA cylindrical sample. One of the difficulties of the application of RUS to attenuative materials is that several predicted vibration modes are weakly excited and cannot be measured because they are masked by strongly excited modes (Bernard et al., 2013). Here, the rotation of the sample varied the relative amplitudes of the vibration modes and then increased the number of resonant modes that could be observed.

3.3.3 Additional elasticity measurements

For comparison purposes, we carried out concurrent elastic properties measurements on the composites samples, based on ultrasonic waves velocities. We followed the protocol described by Granke et al.(2011). Briefly, the stiffness constants on the diagonal of the stiffness tensor were deduced from velocities of longitudinal and shear waves propagating in the principal directions of the samples. Velocities were obtained from the time-of-flight of an ultrasonic pulse propagating through the sample between a pair of transducers contacting two opposite faces. Central frequencies of 2.25 MHz and 1MHz were used for longitudinal and shear waves respectively (V105RM and V152RM, Panametrics Inc., Waltham, MA). 3.3. Samples and measurement setup 45

Figure 3.3: Diﬀerent positions of the sample with respect to the direction of the excitation (arrows) excite diﬀerent vibration modes. In this example on a PMMA cylinder, torsional modes are more likely to be excited in Position 1 and bending modes in Position 2. Frequency response measured in Position 1 (resp. Position 2) contains a large peak at the frequency of the fundamental torsional (resp. bending) mode. 46 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids

3.4 Processing of the measured spectra

Due to the high levels of attenuation considered in this work, retrieving of the resonant frequencies requires some processing of the measured spectrum. Indeed, the resonant peaks are broad and systematically overlap each other, except for the very ﬁrst few modes (Fig. 3.3). In that context, the resonant frequencies do not generally correspond to local maxima in the spectrum. Despite damping, and assuming linear response of the solid sample, its frequency response FR is a sum of individual responses of single degree-of-freedom resonators, described by Lorentzian line shapes, i.e.

M X ak FR(f) = (3.8) (f 2 − f 2) + i(f f/Q ) k=1 k k k where ak are the complex amplitudes, fk the resonant frequencies and Qk the quality factors. Therefore, a fit of the spectrum with a large number of Lorentzian line shapes could be used to retrieve the parameters. However, it is difficult in such situation to obtain convergence with conventional non-linear fitting algorithm (such as the Levenberg-Marquardt algorithm), because this kind of algorithm needs a good starting point to converge. Moreover, the correct number M of resonant peaks in a given frequency band is unknown a priori. Lebedev(2002) proposed an approach that does not try to fit the spectrum with a sum of Lorentzian in the frequency domain but works mainly in the time domain, using the inverse Fourier transform of the spectrum to recover the resonant frequencies. While this method produces good estimates of the parameters, it was observed to lack accuracy in fitting the spectrum by Bernard et al.(2013). In this last study the estimates from the time domain method were used as the initial point of a non-linear fitting algorithm in the frequency domain, which slightly modified the estimated parameters to improve the fit. The combination of the two methods was observed to reduce the variability of the estimated parameters. The same combined approach was used here. We firstly describe the time domain part of the method, following the work of Lebedev and then the non-linear fitting in the frequency domain.

3.4.1 Estimation of the resonant frequencies in time domain

The inverse Fourier transform of the frequency response (5.1) is the impulse response of the system, which is a sum of damped exponentials involving the same parameters ak, fk and Qk. Methods to estimates these parameters from discrete samples of the impulse response have been developed in the signal processing literature. A method based on a linear predictive ﬁlter originally proposed by Kumaresan and Tufts(1982) has been used for RUS by Lebedev(2002). 3.4. Processing of the measured spectra 47

The N discrete samples of the measured spectrum in the range [fmin, fmax] are transformed into a time series y[n] via inverse discrete Fourier transform. From these samples, the matrix   y[L] y[L − 1] ··· y[1]   y[L + 1] y[L] ··· y[2]  A =  . . . .   . . .. .   . . .  y[N − 1] y[N − 2] ··· y[N − L] is constructed, with M ≤ L < N. In a noise free case, the N − L first samples would be sufficient to exactly predict the remaining samples via the prediction equation Ag = b, with b a column vector containing the N − L last samples [y[L + 1] y[L + 2] ··· y[N]]t and g the column vector of the linear prediction fil- t ter coefficients [g1 g2 ··· gL] . It can be shown (Kumaresan and Tufts, 1982) that the filter transfer function L X −k H(z) = 1 + gkz (3.9) k=1 2π(δ +if )τ has L zeros among which M are positioned on z = e k k with Qk = fk/2δk and τ = 1/(fmax − fmin). Thus, the resonance frequencies fk and loss factors Qk can be determined by solving the prediction equation and finding the zeros of H. The correct system order M (the number of resonant peaks) has to be determined. To this purpose, the autocorrelation matrix R = A∗ × A is constructed (with (.)∗ denoting the hermitian conjugate) and an eigenvalue decomposition of R is performed. In a noise free case, only M non-zero eigenvalues would be found (Kumaresan and Tufts, 1982) but, in the presence of noise, all the L eigenvalues are above zero. However, the L − M eigenvalues corresponding to random noise are weaker than the M signal values. Thus, by looking for abrupt change of slope in the eigenvalue spectrum of R, M can be determined (Lebedev, 2002). The coefficients of the linear predictive filter g could be obtained as g = Abˆ , where ˆ. denotes the pseudo-inverse of A. However, due to the presence of noise this system of equation is numerically instable. The solution is to built a “noise-free" approximation of A. Because the number of resonant peaks M has been determined it is possible to reconstruct an approximation R0 of R using the M largest eigenvalues λ of R and the corresponding eigenvectors V

M 0 X ∗ R = ViλiVi . (3.10) i=1 This approximation of R is noise-free in the sense that is has been constructed using only the eigenvalues and eigenvectors that correspond to resonant peaks present in the signal. Coeﬃcients of g are then determined from the system of equations 48 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids g = R0r, with r = A∗b (Kumaresan and Tufts, 1982). The zeros of the ﬁlter transfer function are localized, and the resonant frequencies and damping factors are obtained.

The last step estimates the complex amplitudes ak. They are estimated in the frequency domain. Indeed, the dependence of the Lorentzian to ak is linear (Eq. 5.1), and the ak can be obtain from a linear system of equations (Lebedev, 2002). The process requires us to compute the matrix A with a ﬁxed value of L before M has been determined. We used L = 128, this is much larger than the number of resonant frequencies present in the signal, and much smaller than the total number of samples. In the selection of the number M, there is generally a small ambiguity. Indeed, the break in the spectrum of the eigenvalues is not always very clear and 2 or 3 distinct values of M can be good candidates. In this case, one should select the highest candidate value, as selecting a value that is superior to the real number M will only introduce a little noise in R0, while selecting a smaller number could discard an actual resonant peak. The method presented here was originally described in the context of RUS by A.V. Lebedev. It was ﬁrst applied to a simulated frequency response (Lebedev, 2002) and subsequently to experimental signals (Lebedev et al., 2003). Since we applied the method in a similar way, these two last references contain relevant details and discussions about the processing and particularly about the determination of the number of resonances M.

3.4.2 Non-linear ﬁtting in frequency domain

In a second step, the parameters determined in time domain were used as initial parameters of a non-linear optimization algorithm (Levenberg-Marquardt), searching for the set of parameters minimizing the diﬀerence between Eq. (5.1) and the experimental frequency response in a least-square sense. The number of independent parameters increases rapidly with the number of peaks (two real and one complex parameter for each peak), making the convergence of the algorithm hard to obtain. Therefore, the processing was performed on small parts of the spectrum, each containing about 8-10 peaks (Fig. 3.4). The spectrum was cut in several overlapping sub-parts. Overlapping allows one to check for the good convergence (same frequencies should be obtained in the overlapping parts). Convergence was also checked visually by comparing the reconstructed and the experimental spectra. This additional step improves the repeatability of the parameters estimation and the agreement between experimental and reconstructed spectra compared to the time-domain method only (Bernard et al., 2013) (Fig. 3.4). It also makes the estimated parameters less sensitive to the choice of the model order M. If M is slightly over evaluated from the time domain processing, the least-square ﬁt runs 3.5. Elastic constants estimation 49

0.1

0.05 Amplitude(a.u.)

0 140 160 180 200 220 240 Frequency (kHz)

Figure 3.4: Portion of a measured spectrum (plain line), spectrum reconstructed after the linear prediction filter method (dashed line), and spectrum reconstructed after the non-linear optimization method (dot line, almost indistinguishable from the measured spectrum). Resonant frequencies obtained from linear prediction (•) and non-linear optimization (+). Measurement on a epoxy/glass composite sample. with more degrees-of-freedom than necessary. In that case, the fitting algorithm is able to discard some components, giving them a very low amplitude, a very low Q, or a very high or low frequency so that they do not add any significant contribution to the model.

3.5 Elastic constants estimation

In RUS, the elastic constant are generally obtained through an optimization process, in which the stiffness constants are adjusted until the best match between measured and predicted frequencies has been obtained. For that purpose, a cost function of the type 2 meas. pred. ! X f − fn (Cij) F = n (3.11) f meas. n n is defined and minimized. This process requires that each measured frequency is correctly paired with its corresponding predicted frequency. The pairing is not straightforward because some predicted modes can be missed during the measurement (this is particularly true for highly attenuating materials, such as bone (Bernard et al., 2013) and because vibration modes predicted for the initial guess of the stiffness constants can appear in a different order than in the measurement. We here describe a probabilistic formulation of the estimation of the elastic constants, in a Bayesian framework, which provides a quantitative criterion for the 50 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids automated pairing of the predicted and measured resonant modes. This Bayesian formulation of the problem requires the definition and use of prior information on the elastic constants.

3.5.1 Bayesian formulation of the RUS inverse problem

We denote m a random vector in the m dimensional space of the parameters (the stiﬀness constants) and pM (m) the prior distribution of the parameters. The observed data (the measured frequencies) are also represented by a probability distribution pD(d), where d is a vector of the p dimensional data space. The functional relation between the data and the parameters is noted d = g(m) and will be supposed to be exact (all the errors are included in the distribution of the frequencies). The posterior distribution of the parameters for given data d (the solution of the inverse problem) is given by the Bayes’ rule. Following Tarantola(2005), it is expressed by pD(g(m))pM (m) σ(m) = R 0 0 0 . (3.12) pD(g(m ))pM (m )dm Let us assume that the probability distributions on the parameters as on the data are multivariate normal distributions. The distribution over the data space is characterized by a mean dobs and a covariance matrix Cd and is given by: 1 p (g(m)) ∝ exp − (g(m) − d )t C −1 (g(m) − d ) . (3.13) D 2 obs d obs

We further assume that the measured frequencies are mutually independent, i.e. Cd i is diagonal with entries Cd. The prior distribution on the parameters is characterized by a mean mprior and a covariance matrix Cm and is given by: 1 p (m) ∝ exp − (m − m )tC −1(m − m ) . (3.14) M 2 prior m prior Under the multivariate normal assumption, Eq. (3.12) can be written

σ(m) ∝ exp(−S(m)), (3.15)

Where S is the misﬁt function:

t −1 t −1 2S(m) = (g(m) − dobs) Cd (g(m) − dobs) + (m − mprior) Cm (m − mprior). (3.16) Therefore, finding the point of maximum a posteriori probability (MAP) is identical to finding the point that minimize the misfit function S. Under an assumption of small non-linearity, the MAP estimate can be obtained recursively using a quasi- Newton method (Tarantola, 2005):

t −1 −1 −1 t −1 −1 mn+1 = mn +(GnCd Gn +Cm ) (GnCd (dn −dobs)+Cm (mn −mprior)), (3.17) 3.5. Elastic constants estimation 51 where G is the matrix of the partial derivatives of the frequencies with respect to each elastic constant evaluated at step n from Eq. (3.3). Parameters were adjusted using Eq. (4.15) until there was no more significant changes in the value of the misfit function (10−5 relative change). Once the final point was obtained, the posterior covariance matrix of the parameters C˜ m was calculated (Tarantola, 2005):

t −1 −1 −1 C˜ m ≈ (G Cd G + Cm ) . (3.18)

The diagonal entries of this matrix are the posterior variances of each parameter while the off-diagonal entries represent correlations between parameters. At this point, we have a solution to the RUS inverse problem that allows one to properly include prior information on the stiffness constants and uncertainties on the measured frequencies. This solution is more general than the least-square solution usually defined in RUS, Eq. (3.11), which can be obtained here as a special case (assuming infinite prior dispersion of the parameters and diagonal covariance matrix Cd with variances proportional to the squared frequencies). However, it still requires the correct pairing of the measured and predicted frequencies. The objective of the next section is to define a criterion for an automated pairing of the frequencies.

3.5.2 Automated pairing of the resonant modes

i Suppose that we have a set of p measured frequencies dobs (i = 1, ..., p) and a set of q predicted frequencies gj(m) (j = 1, ..., q, q ≥ p). Stating that we do not know the correct pairing of the modes is equivalent to stating that we do not know which j i component g (m) of the vector g(m) to pair with the i-th measured frequency dobs to calculate the density pD(g(m)) in Eq. (3.13). To ﬁnd the correct pairing, we will use the approach of Bayesian model selection and hence make the assumption that the correct pairing of the frequencies is the one which maximize the evidence R 0 0 0 (Wasserman, 2000), i.e. maximize the integral pD(g(m ))pM (m )dm (denomina- tor in (3.12)). Therefore, one solution could be to calculate the evidence for each possible pairing combination of frequencies. However for about p = 20 measured frequencies that can reasonably be supposed to correspond to one of the q = 30−40 ﬁrst predicted frequencies, there is an extremely large number of combinations and it is not possible to evaluate all of them in a reasonable amount of time. Then, we consider each frequency independently and compute

R pi (gj(m))p (m)dm P j = D M (3.19) i P R i j0 j0 pD(g (m))pM (m)dm 52 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids for each possible combination of i and j (it gives a reasonable number p × q of combinations), where

1 (gj(m) − di )2 pi (gj(m)) ∝ exp − obs . (3.20) D i 2 2 (Cd)

j We then look for the maximal individual normalized evidence maxi,j(Pi ) and as- sociate the corresponding measured and predicted frequencies. For this couple of frequencies we compute the MAP estimate from (4.15) and the posterior covariance matrix from Eq. (3.18). This posterior distribution becomes the new prior distribution, for which the evidences Eq. (3.19) are recomputed. A second pair of frequencies is associated. The process is repeated for p steps, until each measured frequency has been associated to one of the predicted frequencies (Fig. 3.5). The solution of the inverse problem is the MAP estimate obtained when all the measured frequencies have been taken into account. Uncertainty on this solution is obtained from the ﬁnal posterior covariance matrix.

3.5.3 Implementation

Integrals in Eq. (3.19) were calculated using Monte-Carlo simulation. We generated a number N of samples m1, ..., mN of the prior distribution and estimate the integrals as 1 X P j ≈ pi (gj(m )). (3.21) i N D n n j We found that for N = 500 the variance on the estimated Pi was sufficiently small for the whole process to be repeatable. The total number of computations of the resonant frequencies (evaluations of g(m)) is then equal to 500 × p. Samples were generated in a unit normal distribution and transformed into samples of the desired normal distribution pM (m) using the Cholesky decomposition of the covariance matrix (Kroese et al., 2011). We assumed that this distribution was multivariate normal because it allows iterative computing of the MAP estimate. However, this is not strictly consistent with physical boundaries on the stiffness constants. Indeed, any multivariate normal distribution attributes a non-zero probability to elastic tensors that are not positive-definite. Therefore, when generating samples of this distribution we tested each sample and rejected all the samples for which the elastic tensor was not positive-definite. There is an infinite number of theoretical vibration modes for a solid of finite shape. However, the Rayleigh-Ritz approximation method that is used to compute the frequencies (section 3.2) only considers the ∼ 103 first modes (852 for a RP sample with a polynomial order N = 10). Moreover, the experimental spectra were measured in a limited frequency band containing the ∼ 30 first modes. Then, 3.5. Elastic constants estimation 53

Figure 3.5: Diagram of the algorithm for the automated pairing of the resonant frequencies and resolution of the inverse problem. Starts at step n = 0 and stops when n = p. At the end of the last step each measured frequencies is paired with one of the predicted frequencies and the solution of the inverse problem is the posterior distribution. See text of section 3.5 for details. 54 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids the number q of predicted modes that are considered in the inverse problem can be reasonably ﬁxed to about 50 without risking to not consider a mode that was actually observed.

The distribution pD(d) should reflect the knowledge of the experimental data. Because we assumed that the frequencies were mutually independent, we only have to specify the mean dobs and the diagonal terms of the covariance matrix. The mean values of the frequencies measured over several measurement (section 3.4) were naturally chosen as the central values of the distributions. Variance terms must take into account measurement uncertainties and potential model errors (such as errors due to imperfect sample geometry), which sum under the Gaussian assumption (Tarantola, 2005). Uncertainties on the measurement are given by the standard deviation over the measured frequencies, but modeling errors are more difficult to evaluate a priori. A solution is to start with a reasonable guess for model errors, and then to evaluate them a posteriori from the value of the misfit function. Based on previous experiment in RUS measurements as on literature review, we started by postulating variances equal to 0.5% of each frequency (measurement and model error combined). This value corresponds to usual misfit obtained in RUS. The uncertainty estimation was then refined at the end of the inverse problem, using the method which consists in adjusting the assumed error so that the final misfit value 2S is equal to the number of data points p (wich is approximately the expected value of a p degrees-of-freedom χ2 distribution) (Press et al., 1992).

3.6 Damping factors estimation

Viscoelasticity was modeled by adding an imaginary part to the stiﬀness tensor:

∗ 0 −1 Cij = Cij + iCij = Cij(1 + iQij ), (3.22)

∗ −1 where Cij represents the complex viscoelastic constants and Qij are the ratios of the −1 imaginary to the real part of the stiﬀness constants. If Qij 1, which is the case −1 −1 for the considered materials (Qij ∼ 0.02 − 0.04), the inverse damping factors Qij −1 are linearly related to the inverse of the peaks quality factors Qk by the following expression (Leisure et al., 2004; Ogi et al., 2003):

2 X ∂fk Q−1 = C Q−1. (3.23) k f ∂C ij ij k i,j ij

After the inverse problem for the stiﬀness constants Cij has been solved, this last −1 equation can be inverted to obtain the damping factors Qij . If more Qk have −1 been measured than there is unknown Qij , the system is overdetermined and has a unique least-square solution which is obtained using a conventional linear least- square routine. 3.7. Results 55

Table 3.2: Maximum a posteriori probability estimates of the isotropic stiﬀness constants and engineering moduli for the PMMA samples (in GPa, except Poisson’s ra-

tio). Estimated standard deviations (SD) from the inverse problem are given (σC11

and σC44 ), as experimental mean and SD. The number of measured frequencies used in the inverse problem and the Root Mean Square error (RMSE) between measured and predicted frequencies are also given.

Sample C11 σC11 C44 σC44 E ν Nb of freq. RMSE (%) Cyl. 1 8.77 0.11 2.211 0.004 5.89 0.331 21 0.29 Cyl. 2 8.84 0.10 2.209 0.003 5.88 0.333 17 0.22 Cyl. 3 8.73 0.14 2.218 0.005 5.90 0.330 18 0.32 Cyl. 4 8.68 0.10 2.206 0.003 5.87 0.330 24 0.27 Mean 8.76 2.211 5.89 0.331 - - SD 0.07 0.005 0.01 0.001 - - RP 1 8.66 0.16 2.215 0.006 5.88 0.317 18 0.31 RP 2 8.56 0.29 2.196 0.007 5.83 0.328 14 0.36 RP 3 8.59 0.26 2.233 0.006 5.91 0.324 13 0.33 RP 4 8.33 0.16 2.234 0.005 5.87 0.314 18 0.26 Mean 8.54 2.219 5.87 0.320 - - SD 0.14 0.018 0.03 0.006 - -

3.7 Results

All the six spectra measured for each sample (see section 3.3) were processed as described in section 3.4. Mean and standard deviation of the resonant frequencies were calculated over the diﬀerent measurements, which allow one to estimates the precision of the measurement. It should be noted that all the resonant frequencies are not retrieved in each of the six spectra. We observed that almost all the standard deviations were below 0.5%, and where about 0.2% in mean for PMMA samples and 0.3% for composites samples. The spectra measured for a PMMA cylindrical sample are plotted on Fig. 3.6, with the resonant frequencies extracted from signal processing (17 in total). The numbers of resonant frequencies retrieved from the measured spectrum and included in the inverse problem for each sample are given in Table 3.2 for PMMA and Table 3.5 for the composite. The algorithm for the automated pairing of the measured resonant frequencies with the predicted frequencies was applied as explained in section 3.5. For PMMA, the prior distribution was constructed using the results of Minonzio et al.(2011),

C11 = 8.5 and C44 = 2.2 GPa, as central values of the distribution and a covariance matrix constructed assuming standard deviations of 33% of the mean values for each constant and no correlations between the constants. For the composite material, we also used the results of Minonzio et al.(2011) as central values for the diagonal 56 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids

rbblt auso h tffescntnsaerpeetdwt etclpanlnsfrfeunista aebe ardwt measured a with paired been the have observed. Hence, that not frequencies were polarization. for that transducers lines frequencies the plain for vertical to lines with respect dot represented with vertical are and sample constants frequency the stiffness of the orientation of values variable probability the ( to frequencies due resonant spectrum, observed to spectrum from differ 3.6: Figure Amplitude (a.u.) 0.02 0.04 0.06 0.08 0.12 0.1 0 120 esrdsetafraclnrclPM ape(egt=56) h eaieapiue ftedffrn eoatmodes resonant different the of amplitudes relative The 5.67). = (height sample PMMA cylindrical a for spectra Measured 140 + 160 r o h aei aho h i pcr.Tefeuniscluae o h aiu posterior maximum the for calculated frequencies The spectra. six the of each in same the not are ) 180 200 Frequency (kHz) Frequency 220 240 260 280 300 320 3.7. Results 57

Table 3.3: Velocities (m/s) of longitudinal and shear waves for PMMA, deduced from the isotropic stiﬀness constants compared to literature results. Uncertainties for our results are standard deviations over four measured samples. When available, we indicate uncertainties given in the publications.

Study Longit. veloc. Shear veloc. Present (Cylinders) 2720 ± 11 1366 ± 2 Present (RP) 2690 ± 22 1372 ± 6 Minonzio et al.(2011) 2685 ± 20 1365 ± 10 Hartmann and Jarzynski(1972) 2690 1340

Table 3.4: Damping ratios Q−1 (×10−2) of the PMMA samples. Estimated standard

deviations (SD) from the least-square inversion of Eq. (5.23) are given (σ −1 and σ −1 ), Q11 Q44 as experimental mean and SD.

−1 −1 −1 −1 Sample Q σ −1 Q σ −1 Q Qν 11 Q11 44 Q44 E Cyl. 1 1.73 0.33 2.40 0.05 2.29 -0.45 Cyl. 2 2.18 1.33 2.40 0.09 2.36 -0.15 Cyl. 3 2.05 0.90 2.48 0.05 2.41 -0.29 Cyl. 4 1.65 0.41 2.50 0.06 2.35 -0.59 Mean 1.90 2.45 2.35 -0.37 SD 0.25 0.05 0.05 0.19 RP 1 1.54 0.40 2.51 0.04 2.34 -0.67 RP 2 1.70 0.49 2.54 0.04 2.33 -0.58 RP 3 1.36 0.36 2.48 0.04 2.28 -0.83 RP 4 2.60 0.55 2.45 0.04 2.48 0.13 Mean 1.80 2.50 2.36 -0.49 SD 0.55 0.04 0.09 0.42

stiﬀness constants: C11 = 13.5, C33 = 22.5, C44 = 4.1, and C66 = 3.8. No result for the last independent constant C13 was available; we choose C13 = C11 − 2 × C44 = 5.9. The covariance matrix was constructed with standard deviations of 15% of the mean values for each constant and no correlations between the constants. The algorithm paired each measured frequency to a predicted frequency and gave the stiﬀness constants in the form of a MAP estimate and a posterior covariance matrix.

MAP estimates of the longitudinal (C11) and shear (C44) stiﬀness constants for the 8 PMMA samples are presented in Table 3.2, with there estimated standard de-

viations (SD) (σC11 and σC44 ), and the Root-mean-square errors (RMSE) between measured and calculated frequencies. Young modulus E and Poisson’s ratio ν were calculated from the stiﬀness constants. To evaluate repeatability, mean and SD of

C11, C44, E, and ν over the four samples of each shape are also given. Repeatability 58 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids

Table 3.5: Stiﬀness constants (in GPa) of the composite samples measured using RUS and using time-of-ﬂight (TOF). Uncertainties are standard deviations obtained from the posterior covariance matrix. RUS analysis was performed for a transversely

isotropic symmetry, which assumes C11 = C22, C44 = C55 and C12 = C11 − 2 × C66. TOF were measured in the two equivalent directions orthogonal to the composite ﬁber

axis, hence two values were obtained for C11 and C44. The number of measured frequencies are respectively 18 and 15 and the RMSE are respectively 0.42% and 0.45%

C11 C33 C13 C44 C66 Comp.1 RUS 15.0 ± 0.8 23.1 ± 0.9 8.4 ± 0.9 4.30 ± 0.02 3.69 ± 0.03 TOF 14.5 15.1 21.8 - 4.40 3.98 3.65 Comp.2 RUS 14.5 ± 0.6 22.5 ± 0.9 7.3 ± 0.8 4.41 ± 0.03 3.75 ± 0.03 TOF 14.9 15.2 23.4 - 4.43 4.26 3.68

Table 3.6: Young moduli (in GPa) and Poisson’s ratios of the composite samples measured in RUS compared to the values indicated by the manufacturer. Only one Pois- son’s ratio is given by the manufacturer, without any indication of which anisotropic Poisson’s ratio it corresponds to. Therefore, its value should not be considered mean- ingful.

Et Ea νt = ν12 ν13 νa = ν31 Comp.1 10.2 ± 0.1 16.8 ± 0.2 0.381 ± 0.020 0.226 ± 0.011 0.373 ± 0.017 Comp.2 10.4 ± 0.1 17.5 ± 0.2 0.376 ± 0.020 0.203 ± 0.012 0.341 ± 0.021 Ref 10 16 - 16.71 0.26

is better than 1% for C44 and 1 − 2% for C11 for both shapes. Repeatability on the Young modulus is better than 1% and then signiﬁcantly better than on the longitudinal modulus C11. The Poisson’s ratio is also estimated with a good repeatability (1 − 2%). To compare our results to previously published ones, we calculated wave velocities form the measured elastic constants and mass density. Results are compared to values from the literature in Table 3.3. The largest discrepancies between −1 our results and literature results are about 2%. The inverse damping ratios Q11 −1 and Q44 obtained from the measured widths of the resonant peaks are given for the four PMMA samples in Table 3.4. The complex viscoelastic constants obtained were −1 converted in complex Young modulus E and Poisson’s ratio ν and the ratios QE −1 and Qν are also indicated. Repeatability on the damping ratios is good (about 2%) for shear and Young moduli of PMMA but poor for C11 and ν. Despite the −1 somewhat large variations on Qν , we observe that this ratio is negative and about one order of magnitude smaller than the other ratios.

For the composite samples, stiﬀness constants Cij and their uncertainties obtained in RUS are given in Table 3.5 together with results from Time-of-Flight measurements and RMS errors between predicted and measured frequencies. The 3.8. Discussion 59

Table 3.7: Transverse isotropic Q−1 of the composite samples (×10−2)

Q−1 Q−1 Q−1 Q−1 Q−1 Ea Et ν31 Ga=C44 Gt=C66 Comp.1 2.98 ± 0.58 3.55 ± 0.35 -4.91 ± 3.09 3.55 ± 0.20 4.08 ± 0.52 Comp.2 2.44 ± 0.09 3.75 ± 0.09 -1.96 ± 0.68 3.85 ± 0.06 4.36 ± 0.10 measured elastic constants are in agreement between samples and between methods. The stiffness constants were converted in axial (Ea) and transverse (Et) Young moduli and Poisson’s ratios by inverting the elastic tensor. Results are presented in Table 3.6 and compared to reference values indicated by the manufacturer of the material, measured with quasi-static mechanical tests 2. Good agreement on the Young moduli can be observed. Uncertainties for the longitudinal stiffness constants of the composite (4 − 5%) are higher than for isotropic PMMA (1 − 2%) but uncertainties are still about 1% for shear and Young moduli and about 5% for the anisotropic Poisson’s ratios. Damping ratios where obtained on the stiffness constants from inversion of Eq.(5.23) and converted into ratios on the anisotropic engineering moduli. Results are given in Table 3.7. We observe that, as for PMMA, damping is stronger in shear (Q ≈ 22 − 28) than in compression (Q ≈ 26 − 40). As for PMMA, we

observe a negative ratio for the axial Poisson’s ratio ν31. Anisotropy of damping also appears. For Young moduli, damping is weaker in the axial direction of the fibers. Shear deformation in the plane orthogonal to the fibers axis, governed by the constant C66, suffer from more damping than other shear deformations.

3.8 Discussion

From the measurement setup using shear transducers, combined with signal processing, between 13 and 24 of the lower predicted resonant frequencies of the samples were retrieved despite damping. This was enough information to perform the inverse identification of the elastic properties with good precision. Remaining frequencies were not observed due to a combination of low excitation and masking by strong resonant modes. The number of retrieved frequencies is different for each sample because it strongly depends on the dimensions of the samples, which influences both excitability and distribution of the modes. Several experimental solutions to the pairing problem have been proposed in the literature, among which the most popular is the measurement of the modal shapes associated with each resonant frequency on a face of the sample by the mean of a laser interferometer (Ogi et al., 2002). Because the modal shapes are weakly

2Material properties indicated by the manufacturer are based on standards ASTM D-638, D-695, and D-1621. http://www.sawbones.com/products/bio/composite.aspx (date last viewed 04/10/13). 60 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids sensitive to the elasticity, they allow to pair the frequencies even if the initial guess is far from the actual stiffness constants. This method is powerful but significantly increases the complexity of the measurement setup. Moreover, it could not be suited to other specimen shapes than the parallelepiped shape or to very small samples (Ogi et al., 2002). Another solution, used in combination with laser interferometer measurement by Landa et al.(2009), is to pair the frequencies progressively, while the elastic constants are adjusted. This way, information contained in the most easy to pair frequencies is used to update the initial guess on the elastic constants so that more frequencies can be paired. This progressive approach to the inverse problem was successfully used in the application of RUS to a cortical bone specimen by Bernard et al.(2013), without measuring the modal shapes. However, resolution of the inverse problem was long and required many intervention of the user. In this work, we used a different approach relying on a Bayesian formulation of the inverse problem, which uses prior information of the elastic properties. The distribution representing the prior information on the material stiffness can be constructed in several ways. For example, it can be obtained from literature review of the elastic properties of the material, from the predictions of a micro mechanical model of the material or from previous sound velocity measurements. In this work, we used published values of the elastic properties as the central values of the distributions. Selected values of 33% for the SD on the a priori values for PMMA does not actually reflect our knowledge of the material, but were chosen to gives a very broad, almost flat distribution, in order to show that the algorithm described in section 3.5 does not require precise a priori information to perform the automated resolution of the inverse problem. Similarly, for the selected prior covariance matrix for the composite material gives a broad distribution. It is difficult to discuss the sensitivity of the analysis to the center of the a priori distribution since it may be very different for each particular case. However, in our experience with the data exposed here, the algorithm still converges properly for distributions off-centered of about 20% of the correct values, which we think is convenient for many applications, including bone characterization. For cases where such a priori knowledge of the material elasticity is not available, preliminary mechanical tests or sound velocity measurements may be required. Although no direct validation of the pairing of the frequencies was done, using for example laser interferometer measurements, two considerations indicate the cor- rectness of the pairing: 1) the stiffness constants obtained for several samples of the same material are in good agreement and 2) the agreement between predicted and measured frequencies for the obtained pairing is good, comparable to what is usually obtained in RUS Ulrich et al.(2002) considered a fit as good for RMS errors below 1% and Migliori and Maynard(2005) advised RMS error below 0.8%) and only slightly larger than the estimated uncertainty on the measured frequencies. This 3.8. Discussion 61 would not have been the case for a wrong pairing of the frequencies. From the estimated Qij it was possible to predict the width of the non-observed resonant peaks. Their Q factor were in all cases comparable to those of the observed modes and non- observation could therefore not be attributed to higher damping. We attempted to evaluate the excitability of each resonant mode by shear motion by computing, with the Rayleigh-Ritz method, the norm of the corner displacement vector projected into the plane of the transducers. However, we observed that this amplitude did not accurately predict the actual observation of the resonant modes. This may be explained by the fact that, with considerable overlapping, the detectability of a mode does not only depend on its own amplitude but also on the amplitudes, phases and closeness of adjacent modes.

Repeatability obtained on the isotropic elastic constants is in agreement with what can be expected from RUS measurements on low damping materials (Migliori et al., 1993) and therefore demonstrates the ability of RUS to measure elastic properties of isotropic attenuating materials. The fact that the engineering moduli are better estimated from RUS than the Cij constants was already observed in the application of RUS to a bone sample (Bernard et al., 2013). Higher uncertainties for the transversely isotropic material are due to the increased number of degrees-of- freedom in the inverse problem for a comparable number of included frequencies and to a slightly worse agreement between model and prediction, reflected in the RMS error. This worse agreement could be due to a small misalignment of the fibers with the axis of the samples. For PMMA, repeatability is better for the cylindrical samples (both in estimated and effective SD). This is due to the fact that there were more measured frequencies for the cylinders, and then more information was used in the inverse problem. Additionally, cylindrical samples are less subject to geometrical errors than RP since there is only one pair of parallel faces to cut. Hence, there are less potential errors in the final estimated elastic constants. Finally, we observe that the SD given for each sample by the probabilistic resolution of the inverse problem are in good qualitative agreement with the effective SD calculated over four measurements, which indicates that the estimated uncertainty could be a useful indication of the effective uncertainty for cases were only one sample of the material is available.

For both materials, good agreement was obtained between results from independent measurement methods. For PMMA, published ultrasonic velocities (Hartmann and Jarzynski, 1972; Minonzio et al., 2011) compared well to our results. For the composite material, ultrasonic velocities measurements also gave results close to RUS. Although the elastic properties from RUS are obtained at ultrasonic frequencies, they are in good agreement with the quasi-static elastic moduli provided by the manufacturer. As the attenuative behavior of the material entails dispersion, 62 Chapter 3. Resonant ultrasound spectroscopy for attenuative solids i.e. variation of the elastic properties with frequency, one could have expected larger discrepancies between elastic properties obtained at diﬀerent frequencies.

For the inverse estimation of the elastic constants we compared the natural frequencies of the undamped mechanical system, computed from Rayleigh-Ritz method, to the resonant frequencies of the forced damped system (Eq. 5.1). Equality between these frequencies holds only in the case of proportional damping (Rayleigh damping), i.e. if the damping matrix of the vibrational system is a linear combination of the stiffness and mass matrix K and M. In other cases, the resonant modes are coupled and the resonant frequencies are perturbed. Although proportional damping is generally not true, this hypothesis is common because the influence of the non-diagonal damping terms is negligible at first order for small damping (Ad- hikari and Woodhouse, 2001). The level of damping observed here (Q ≈ 30) is such that the first order approximation is still reasonable and we therefore did not consider any perturbations of the resonant frequencies due to damping. Moreover, this is also justified a posteriori: the elastic constants estimated in this way are in good agreement with values obtained from other methods.

Eq. (5.23), that was inverted to obtain the Q−1, assumes that viscoelasticity is the only source of vibration damping, which is not strictly true. As the samples vi- brate in air and in slight contact with the transducers, couplings exist that dissipate energy. Migliori and Maynard(2005) estimate the Q factor due to the measurement system to be of order of 106. Zhang et al.(1998) predicted and measured a Q factor of an order of 104 due to the coupling with air at 1 atm. In this paper we deal with Q factors lower than 102, and viscoelastic dissipation in the samples is then at least 100 times larger than the other sources of dissipation, which can thus be neglected. From the results on PMMA, we observe again that RUS characterizes more precisely the Young modulus than the longitudinal constant C11. The negative imaginary part of the Poisson’s ratio is a consequence of larger damping in shear than in tension/compression, a usual property of polymeric solids (Lakes, 2009). As Poisson’s ratio expresses the ratio of transverse to longitudinal strain, a larger damping for shear motion induces a phase lag on transverse strain that results in a negative imaginary part for the Poisson’s ratio. Hosten and Castaings(2008) re- −1 −2 ported Qν = −0.4 × 10 for PMMA, in good agreement with the present results. Anisotropic damping in the composite material was to be expected, since the damping in the aligned ﬁbers (glass) is weaker than in the matrix (epoxy). This insight on anisotropy of damping is the main advantage of RUS over other viscoelasticity measurement methods, which are usually limited to only one complex modulus. On the other hand, the small frequency band investigated in RUS is determined by the size of the sample and is not tunable. Therefore, RUS is unable to investigate the frequency dependence of the complex viscoelastic moduli. 3.9. Conclusion 63

The adaptations of the RUS method proposed in this work are certainly not the only possibility to tackle the difficulties due to attenuation and we consider that there is room for further improvements, both regarding the precision of the results and the ease of implementation and use. Concerning the measurement setup, several predicted resonant frequencies could not be observed, despite the use of shear transducers with several orientation of the sample. Additional measurements using compressional transducers, or a combination of shear and compression, could allow observation of some other predicted resonant modes. Additional care in the sample preparation should lead to lower residual error between predicted and measured frequencies and hence to improved repeatability in the stiffness constants. For the estimation of the stiffness constants we assumed multivariate normal distributions because it allows us to find the MAP estimate easily. However, it leads to inconsis- tency with the physical boundaries on the elastic properties and may not be adapted to some practical cases, where the prior knowledge cannot be represented by a normal distribution. In such cases, the normal distribution hypothesis may be dropped and the posterior distribution obtained with a Markov Chain Monte Carlo method (Tarantola, 2005).

3.9 Conclusion

This paper proposed some adaptations of the RUS method for the viscoelastic characterization of highly attenuative materials. Thanks to adequate measurement setup and signal processing of the RUS spectrum, enough resonant frequencies to identify the viscoelastic properties were retrieved, despite overlapping of the resonant peaks. Formulation of the inverse problem in a Bayesian framework allowed auto- matic pairing of the predicted and measured resonant frequencies and solving the inverse problem. Finding the correct pairing is usually a tedious task in RUS, and we think that the automatization of this step is important to promote the use of RUS as a routine method for material viscoelastic characterization. The method was applied to samples of isotropic and transversely isotropic materials. In both case, good agreement was obtained between results on diﬀerent samples and with results from independent measurement methods. This demonstrates both repeatability and accuracy of the RUS method for the viscoelastic characterization of isotropic and anisotropic attenuating materials (Q ≥ 20 ) on the form of small rectangular parallelepipeds or cylinders.

Chapter 4 Bayesian mode identiﬁcation and stiﬀness estimation using Markov Chain Monte Carlo

Forewords

This Chapter is a research paper prepared for submission to the journal Mechanics of materials but not yet submitted at the time of writing this thesis. The Bayesian formulation of the RUS inverse problem introduced in the preceding Chapter was only partial, in the sense that the elastic coefficients were considered as random variables, but given a fixed pairing of the predicted and measured frequencies. This pairing was selected iteratively using a criterion of Bayesian model selection: the maximum of evidence. However, a full Bayesian approach should not only consider the ’best’ pairing, but all the pairings that are probable given the data. In this Chapter, the pairing will be considered as a random variable whose posterior distribution is explored jointly to the posterior distribution of the elastic parameters. Moreover, a random exploration of the distribution will be used, using Markov Chain Monte Carlo, instead of iterative schemes. A random method was required due to the complexity of the posterior distribution, but also presents the advantage of being less sensitive to the user-specified starting point than iterative algorithm. The formalism to be presented was initially developed with the application to bone or other attenuative materials in mind, but turned out to be more general. The approach is actually useful to solve the pairing problem for frequencies measured on low damping materials as well, as it will be illustrated using previously published data sets. In that case, there is no or just a few resonant frequencies that are not observed in the measurement, but the pairing is still a problem if no good initial guess is available for the stiffness coefficients.

4.1 Introduction

Resonant ultrasound spectroscopy (RUS) is an experimental technique for measuring the stiffness of anisotropic solid materials (Maynard, 1992; Migliori et al., 1993; Migliori and Sarrao, 1997; Migliori and Maynard, 2005). In RUS, the free vibration resonant frequencies of a material specimen of simple shape (e.g. cylinder, sphere, parallelepiped) are measured and the stiffness coefficients adjusted until model-predicted frequencies match the experimental results. This inverse approach gives accurate elastic parameters because the resonant frequencies, which are en- 66 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo tirely determined by elasticity, specimen dimensions and material mass density, can be measured with high accuracy.

An issue of that inverse approach is that each measured resonant frequency taken individually does not carry information about which of the many predicted vibration mode it corresponds to (Ohno, 1976; Stekel et al., 1992; Migliori and Sarrao, 1997; Ogi et al., 2002). As the inversion rely on the minimization of the differences between pairs of predicted and measured frequencies, ambiguities in the identification of the frequencies leads to ambiguities in the stiffness estimation. Identification, or pairing, is particularly difficult if the initial guess of the stiffness coefficients is far from the actual values and if some predicted resonant modes are not observed experimentally. It is usually argued that without knowledge of the pairing, RUS cannot lead to accurate estimates of the stiffness coefficients. One purpose of this work is to show, through a Bayesian formulation of the inverse problem, that knowledge of the exact pairing is not necessary to get precise and accurate stiffness estimates.

Several experimental solutions have been proposed to find the frequency pairing, the most popular being that of Ogi et al.(2002). They proposed to image the vibration patterns of the resonant modes on a face of the specimen using a laser interferometer. These unique patterns are predicted by the model and have a low sensitivity to the stiffness, so that the resonant modes can be identified. This method has been used by several groups (Ogi et al., 2003; Ledbetter et al., 2004; Reese et al., 2008; Landa et al., 2009; Farzbod and Hurley, 2012; Sedlák et al., 2014) but suffers from a number of limitations. (1) It requires specimens with a plane face large enough to be correctly sampled by the laser dot. In consequence, it is not adapted to very small (sub-millimeter) specimens nor to complex shapes. (2) The scanned face must be perfectly plane and highly reflective, which is not always the case, particularly for biological materials such as dentin (Kinney et al., 2004) or bone (Lee et al., 2002; Bernard et al., 2013). (3) The use of a Laser interferometer considerably increases the cost and difficulty of implementation of RUS, preventing its use as a routine method for material characterization in some contexts, e.g. industrial or biomedical. The growing interest in applications of RUS where the above limitations are problematic motivates the development of an alternative method.

Without measuring the modal shapes, the correct identiﬁcation can be obtained through trial-and-errors methods (Migliori et al., 1993), or progressive approaches identifying modes one after another (Landa et al., 2009; Bernard et al., 2013, 2014). Although each individual frequency does not bring information on its corresponding vibration mode, the distribution of the frequencies does. Indeed, the correct pairing is expected to lead to the best match between the experiment and the model. These methods however are time-consuming and require the user to manually identify the frequencies. 4.2. Method 67

In this work we propose a Bayesian formulation of the inverse problem that introduces probability distributions for the possible pairings of predicted and measured frequencies and for the stiffness coefficients. Estimating the joint posterior distribution given a set of measured frequencies then solve the combined problem of finding the correct pairing and estimating stiffness. This is done through Markov Chain Monte Carlo (MCMC) sampling of the posterior distribution. It requires no input of the user, except for the specification of a priori probability densities. We show on two examples that the method leads to results that are almost as accurate and precise as with the exact known pairing.

4.2 Method

4.2.1 Forward problem

The Rayleigh-Ritz method is efficient to predict the resonant frequencies of a solid of simple shape given its stiffness, geometry and mass density. This method has been extensively used in RUS (Ohno, 1976; Maynard, 1992; Migliori et al., 1993; Migliori and Sarrao, 1997; Migliori and Maynard, 2005; Ogi et al., 2002). In a recent work (Bernard et al., 2014) we summarized the method with an emphasis on the efficiency of the computation, which is crucial for Monte Carlo methods where the forward problem is solved thousands of times. We used the same forward solver in the present work and therefore only describe the principle of the method. The free-vibration natural frequencies of a solid body are the stationary points of the Lagrangian L Z 1 2 2 L = ρω ui − Cijklεijεkl dV, (4.1) 2 V

where Cijkl are the stiffness coefficients, V and ρ are respectively the volume and mass density, ui is a component of the displacement field, and εij is a component of the strain tensor. The Rayleigh-Ritz method solves the stationary equation ∂L = 0 as an eigenvalue problem ω2Mα = Kα (4.2)

by expanding the displacement field as a finite sum of known basis functions. In (4.2), M and K are respectively the mass and stiffness matrices of the vibration problem and α is a vector containing the coefficients of the expansion. For simple shapes and well-chosen basis functions, components of the matrices have simple analytical expressions and Eq. (4.2) is numerically solved. The solution eigenvectors α contain the displacement patterns of the resonant modes. Note that the forward relation between the resonant frequencies and the stiffness coefficients is non-linear and has no analytical expression. 68 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo

A RUS measurement usually focuses on a limited frequency band containing a few tenth of resonant frequencies. Hence, it is not necessary to consider all the solutions of the approximation (4.2) in the inverse problem, but only a number K slightly superior to the number N of frequencies measured in the investigated band (K > N to account for the possibility of non-observed frequencies).

4.2.2 Bayesian formulation of the inverse problem

The Bayesian formulation of the inverse problem includes the estimation of the stiffness coefficients from measured resonant frequencies, as well as the estimation of the correct pairing of the frequencies. We only briefly review the required concepts of Bayesian inference. Extensive descriptions can be found elsewhere (Tarantola, 2005; MacKay, 2003). The purpose of Bayesian inference is to evaluate the posterior probability density function (pdf) p(m|y) of a d-dimensional vector of parameters m (e.g. stiffness coefficients) of a probabilistic model, given some measured data y and some prior information on the parameters p(m). In our problem, y is a vector containing N measured resonant frequencies. Bayes’ theorem expresses the posterior pdf as a normalized product of the prior density and the likelihood p(y|m) (the probability of having obtained the data given the parameters)

p(m)p(y|m) p(m|y) = . (4.3) p(y)

The normalizing term is the marginal likelihood of the data and does not depend on the parameters. The solution of the inference problem can then be obtained from the simpler equation p(m|y) ∝ p(m)p(y|m). (4.4)

The likelihood p(y|m) describes both the physics of the problem – the assumed relation between the parameters and the measured data y = g(m) – and the statis- tics of measurement and modeling errors. It is usually not possible to separate the sources of errors and their distribution is therefore difficult to specify. A common and convenient practice is to assume a multivariate normal distribution of the errors (Tarantola, 2005). We further assume that the errors are uncorrelated and that the variance of each particular frequency is proportional to the squared frequency, with a unique proportionality factor σ2. As previously stated, we do not know which of the over-numerous predicted frequencies in the investigated band correspond to the measured frequencies. A correspondence between the predicted and measured frequencies is called a pairing and is defined as follows: 1) N predicted frequencies among K are selected, and are considered as present in the measurement and 2) each of these selected frequencies 4.2. Method 69 is uniquely identified to one of the N measured frequencies. Formally, the possible pairings are represented by a vector a of K integers. The n-th component of a is equal to 0 if the corresponding predicted frequency is not paired to a measured frequency (frequency not observed during the experiment) and equal to k if the corresponding frequency is paired to the k-th measured frequency (see Fig. 4.1). With this notation, the likelihood is

" N 2# 1 X gn(m, a) − yn p(y|m, σ2, a) ∝ (σ2)−N/2 exp − , (4.5) 2σ2 y n=1 n

where gn is the predicted frequency paired to the nth measured frequency yn for the pairing a. If we want to estimate the pairing from the data and if we ignore the magnitude of the errors, the inference problem is threefold. We want to obtain the joint posterior pdf of m, a, and σ2,

p(m, σ2, a|y) ∝ p(m)p(σ2)p(a)p(y|m, σ2, a), (4.6)

where we also introduce prior information on the pairing and the error term. As often in Bayesian inference, the posterior pdf is too complex for being studied analytically. A convenient strategy is then to sample the pdf using Monte Carlo sim- ulations. Then, from a large sample of the pdf, any quantity of interest (e.g. mean and covariances) can be calculated. In the present work we use Gibbs sampling. Ideally, the prior distributions should reﬂect our exact state of knowledge on the parameters before the data have been collected. However, because that knowledge can be diﬃcult to translate perfectly into probability distributions and because some families of distribution are more mathematically convenient for the sampling of the posterior distribution, the prior distributions are constructed as a compromise between convenience and introduced information. We describe the prior distributions p(m), p(a), and p(σ2) in the next sections along the description of the sampling strategy.

4.2.3 Gibbs sampling

MCMC methods provide a way to sample complicated and multidimensional probability distributions such as the pdf of Eq. (4.6). The idea of MCMC methods is to generate dependent samples during a random exploration of the posterior pdf. At a given state, a move is proposed based only on the current values of the variables (Markov property), and accepted or rejected based on probabilistic rules. Diﬀer- ent set of rules exist to generate and accept moves, such as the Metropolis and Metropolis-Hasting algorithms (Metropolis et al., 1953; Hastings, 1970; Gilks et al., 1995), and Gibbs sampling (Geman and Geman, 1984; Casella and George, 1992; 70 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo

predicted freq.

y1 y2 y3 y4 y5 y6 measured freq. a = [1, 2, 4, 3, 0, 5, 0, 6, 0]

Figure 4.1: Example of a possible pairing for a simple ﬁctitious case with N = 6 measured and K = 9 predicted frequencies. The illustrated pairing a shows missing of some predicted mode during measurement and diﬀerent orders of appearance of the vibration modes in the experiment and the model.

Gilks et al., 1995). All ensure that the resulting Markov chain asymptotically samples the target distribution. In Gibbs sampling only one variable or sub-group of variables is moved at a time. The new state of the updated variable is generated from the conditional distribution of this variable given the data and all other variables at their current 2 values. Our algorithm is initialized at (m0, σ0) and then samples alternatively from each conditional distribution:

2 step 1: a1 ∼ p(a|y, m0, σ0), 2 2 σ1 ∼ p(σ |y, m0, a1), 2 m1 ∼ p(m|y, σ1, a1), (4.7) 2 step 2: a2 ∼ p(a|y, m1, σ1), ...

2 After an appropriate number of iterations, the (an, mn, σn) are samples of the joint pdf Eq. (4.6). One advantage is that there is no need to choose and fine tune a rule to generate new states, on which the efficiency of many MCMC algorithms is dependent (Gilks et al., 1995). Particularly, we want to sample jointly continuous (m, σ2) and discrete (a) variables, and it might be difficult to find efficient jumping rules. However, there are convenient ways to sample from the conditional distributions, which make Gibbs sampling a sensible choice here. The three following sections ex- pose how we sample from the conditional distributions p(a|m, σ2, y), p(m|σ2, a, y), and p(σ2|m, a, y) 4.2. Method 71

4.2.3.1 Sampling the conditional distribution of a

The conditional distribution of a is proportional to the product of the likelihood (with known m and σ2) and the prior on a: p(a|m, σ2, y) ∝ p(a)p(y|m, σ2, a). (4.8) Due to the complicated dependence of the likelihood on a, there is no way to sample this distribution directly. A solution is to sample it using a MCMC method embedded inside Gibbs sampling (i.e. called at each iteration of the Gibbs sampler). Because a is a discrete vector, finding the best a or exploring its distribution is a combinatorial problem which is in some aspects similar to classical combinatorial problems such as the traveling salesman problem (Cerný, 1985) (TSP; find the shortest route that visits each city in a given list exactly once and returns to the origin city). An efficient way to solve this kind of problem is simulated annealing (Kirkpatrick et al., 1983; Cerný, 1985), briefly explained here. From an initial state (a route in the TSP, a pairing in our problem), a new state is generated and randomly accepted or rejected based on the ratio of the probabilities of the proposed and current states and on an additional temperature parameter T . For high temperature, almost all states can be reached, even those of low probability, while for low temperature only the moves leading to states of higher probability are accepted. Temperature is then slowly reduced from high to low values, mimicking the cooling of materials and ensuring convergence to the global optimum of the problem. Here, we are not interested only in the best solution but in the distribution of the solutions. We will therefore stop the temperature decrease at T = 1, the value for which the chain samples the target distribution, much before “freezing” of the chain in the most probable state.

From a given state aj, a new state aj+1 is generated by inverting two randomly selected successive components of aj. For that purpose a random integer n is uni- formly generated in [1,M] and the index i of the component n in a is determined. Then this component is moved to the right or left with probability 1/2. For a move to the right we have   aj+1[i + 1] = aj[i] = n,  aj+1[i] = aj[i + 1], , (4.9)   aj+1[l] = aj[l]; l 6= i, i + 1. and similarly for a move to the left. If i = 1 or i = K, the moves to the left and right respectively are not possible; the only remaining possible move is then selected with probability 1. The chain moves to the new state with probability ( ) p(a → a )p(a )p(y|m, σ2, a )1/T P = min 1, j+1 j j+1 j+1 . (4.10) 2 1/T p(aj → aj+1)p(aj)p(y|m, σ , aj) 72 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo

The ratio of the proposal probabilities p(• → •) is equal to 1 except for moves involving a component at index 1 or K, in which cases it can be 1/2 or 2. In all other cases, only the ratio of the prior probabilities p(aj+1)/p(aj) and the ratio of the likelihoods need to be calculated. We start the chain at T = 1000 and then decrease T using a scheme similar to that of Kirkpatrick et al.(1983): after 10 × N accepted moves or 100 × N attempted moves (first reached) T is multiplied by 0.9. This is iterated until T = 1, the temperature for which the chain samples the conditional distribution of a. The same stopping criterion is used at T = 1 and the last sampled pairing is then selected as the sample from the conditional posterior of a, see Eq. (4.8). Since it is done for fixed elasticity, running the simulated annealing algorithm requires no evaluation of the forward problem, but only computation of the squared difference in Eq. (5.6), a much less costly operation. Therefore, running a simulated annealing chain long of ∼ 104 samples at each step of the Gibbs sampling only accounts for a small portion of the total computation time. The formal representation of the pairings considers a fixed number K of predicted frequencies, that is superior to the number of measured frequencies N to account for the possibility of non-observed frequencies. However, the number of frequencies that have been actually missed during the measurement is not always K − N (the total number of zeros in a). The number of missed frequencies is equal to the number of zeros before the last non-null component of a. Indeed, the zeros after this component represents resonant modes whose frequency is higher than the highest measured frequency, so that are out of the investigated frequency band. For example, on the simple case depicted on Fig. 4.1 there is three zeros in the vector a, but the last one does not count as a missed frequency because it correspond to a frequency predicted above the last measured frequency. Another approach could have been to consider a fixed frequency band instead of a fixed number of frequencies. However, this would have made the length of the paring vector variable, which would not have been convenient for the implementation. Our prior distribution on a depends on the (positive) number k of missed resonant frequencies, i.e. on the number of 0 in a before its last non-null component. This prior is a truncated Poisson’s distribution  λk  p(k) ∝ exp(−λ) if k ≤ k , k! max (4.11)   p(k) = 0 if k > kmax, where kmax = K − M and λ is the expected number of missed frequencies. The Poisson’s distribution describes the number of occurrences of a random event in a fixed interval of time if this event occurs at random, independently of the time since the last occurrence. In our case, this mean that we consider the probability of 4.2. Method 73 missing the frequency equal for all modes and independent of whether the preceding has been missed or not. For a given k, all the possible pairings are considered equiprobable a priori, at the exception of pairings that identify the first predicted frequency as missed, which are not considered, i.e p(a) = 0 if a(1) = 0 and p(a) ∝ cte otherwise. Indeed, in our experience with RUS measurement, the first resonant mode is usually easy to observe, hence we excluded the possibility of having missed it in our treatment of the inverse problem. Without this assumption, it would be possible to find pairings of all the measured frequencies with high order resonant modes still producing a good fit, since the density of predicted modes increase with frequency. The Poisson’s distribution of Eq. (5.13) can be rather sharply peaked around λ, which may not be adequate if only vague information is known about k. In that case, a gamma prior probability can be attributed to λ instead of a determined value

βα p(λ) = λα−1 exp(−βλ). (4.12) Γ(α)

Given the pairing a (and then given k), an additional Gibbs step must be done to update λ according its conditional distribution

λk/k! p(λ|k) ∝ p(λ) . (4.13) Pkmax l l=0 λ /l!

4.2.3.2 Sampling the conditional distribution of m

The conditional distribution of m is proportional to the product of the likelihood (with known a and σ2) and the prior on m

p(m|σ2, a, y) ∝ p(m)p(y|m, σ2, a). (4.14)

For fixed a, g(m, a) is still a non-linear function and the conditional probability has no simple exact analytical expression. However, non-linearities are moderate, particularly around the point of maximal conditional probability for m. A sensible strategy for this kind of situations, advocated by Tarantola(2005), is to find the point of maximal probability using an iterative gradient-based method and then to approximate the distribution around this point using the partial derivatives of the function. This is indeed what it is usually done in RUS, where the stiffness coefficients are found by minimizing the differences between experimental and predicted frequencies in a least-square sense using the Levenberg-Marquardt algorithm, and the uncertainties are obtained from an expansion of the quadratic cost function around the solution (Migliori and Sarrao, 1997). A method to find the point maximizing the probability in Eq. (4.14) and to approximate the conditional around this point for a multivariate normal prior on the elastic parameters m was exposed in a previous work (Bernard et al., 2014) 74 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo based on Tarantola. It consists of a quasi-Newton iterative algorithm where m is updated using

t −1 −1 −1 t −1 −1 mn+1 = mn +(GnCy Gn +Cm ) (GnCy (g(m)−y)+Cm (mn −mprior)), (4.15) where Gn is the matrix of partial derivatives of the resonant frequencies with respect to each elastic parameter at step n, Cy is the (diagonal) covariance matrix of the data and mprior and Cm are respectively the mean and covariance of the prior distribution of m. The conditional (4.14) is then approximated by a multivariate normal centered on mfinal with covariance t −1 −1 −1 C˜ m ≈ (G Cy G + Cm ) . (4.16) A sample of the approximate conditional can then be generated using the Cholesky decomposition of C˜ m (Kroese et al., 2011). The quasi-Newton method is stopped when the relative change in m or in the misfit become smaller than a specified value, set to 10−2. Because we generate a sample in a normal approximation of the conditional, a more precise estimation of the center of the distribution (relative to its width) is useless. It usually requires 1 to 8 computations of the forward problem. These few forward problem solving steps at each iteration of the Gibbs sampler account for the main part of the total computation time. Due to the approximation, the random sample drawn that way is not a sample of the full conditional (4.14), which causes the stationary distribution of the Gibbs sampler to be different from the target distribution (4.6). To ensure that we sample the target distribution, the approximation is corrected by introducing a Metropolis-Hasting rejection step for the samples m, considering the approximation as a proposal density (Gilks et al., 1995; Gelman et al., 2013). If we denote ˜p(m) the proposal density, the probability of accepting the move from m to m∗ is ˜p(m)p(m∗)p(y|m∗, σ2, a) P (m∗, m) = min 1, . (4.17) ˜p(m∗)p(m)p(y|m, σ2, a) Strictly speaking, we no longer use Gibbs sampling but a Metropolis-Hasting-within- Gibbs algorithm (Gilks et al., 1995). However, since the normal approximation of the full conditional is actually very good, the rejection rate is low and we continue to refer to Gibbs sampling for simplicity. The described approach requires a multivariate normal prior on the d- dimensional vector of the elastic parameters m, where d depends on the elastic symmetry of the material (e.g. for isotropic material d = 2). However, a normal prior on the components of the elastic tensor tensor Cij is not consistent with the thermodynamical constraints on the tensor. We therefore did not use the Cij coefficients, but a set of transformed elastic parameters for which a normal prior is more consistent. The new elastic parameters are described in AppendixA for symmetries ranging from isotropy to orthotropy (d = 9). 4.2. Method 75

4.2.3.3 Sampling the conditional distribution of σ2

The conditional distribution of σ2 is proportional to the product of the likelihood (with known a and m) and the prior p(σ2):

1 NS2 p(σ2|m, a, y) ∝ p(σ2) exp − , (4.18) σN 2σ2 with the sum of squares

N 2 1 X gn(m, a) − yn S2 = . (4.19) N y n=1 n

A convenient family of prior distribution p(σ2) is the scaled-inverse-chi-square distribution, which is conjugate to the likelihood. It means that the conditional Eq. (4.18) is scaled inverse-chi-square as well (Gelman et al., 2013). Formally if,

2 2 2 2 p(σ ) = Inv-χ (σ |ν0, σ0), (4.20)

where we use the notation from Gelman et al.(2013), then the conditional writes

2 2 2 2 2 ν0σ0 + NS p(σ |m, a, y) = Inv-χ σ ν0 + N, . (4.21) ν0 + N This choice is particularly convenient since it is easy to sample directly from the 2 2 2 Inv-χ distribution. For ν0 → 0 the prior distribution is p(σ ) = 1/σ and becomes nonimformative, as the conditional (4.21) then depends only on N and S. In practice, we found useful to truncate the prior on σ2 to an upper bound. It is useful in the early states of the chain, when no pairings producing good fit have yet been reached. The sampled values of σ2 then tend to be large, which makes the likelihood flat and the chain to stay in states of poor fit. Truncating the prior on σ2 breaks this vicious cycle and prevents the chain from staying for a long time in states of low probability before reaching by chance a state of good fit. The upper bound is set larger than the expected misfit in a successful RUS experiment, so that truncation only improves convergence with no effect on the stationary distribution.

4.2.3.4 Initialization and convergence

To diagnose convergence, L parallel Gibbs chains starting from dispersed points were run. Since the algorithm starts by sampling the distribution of the pairings (5.12), 2 we have to choose or generate L initial values m0 and σ0. The initial variance was set 2 2 to σ0 = 0.01 for each chain. It corresponds to an assumption of relative standard deviation of 1% on the resonant frequencies, which is of the order of the misfit usually obtained in RUS (Migliori and Maynard, 2005). A simple way to generate initial stiffness coefficients is to sample the prior distribution, but in the case of a 76 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo broad prior, this starts the chains very far from the high probability zone and leads to slow convergence. To overcome this problem, we generated L initial values m0 using importance re-sampling (Gelman et al., 2013). M L samples were generated in the prior distribution, and the distribution of the pairing explored for each sample using the simulated annealing algorithm (section 4.2.3.1). The probabilities of all the distinct pairings sampled at T = 1 were summed, giving an approximation of the marginal probability of each proposed m0

2 X 2 p(m0|y, σ0) ∝ p(a, m0|y, σ0). (4.22) a

The marginal probabilities (4.22) were used as weights to randomly select L values among the M proposed, each having a chance proportional to its weight to be selected. After letting the chains run for a while, we assessed convergence using the criterion from Gelman et al.(2013). For each scalar parameter, a ratio R of the between-chain to within-chain variance was computed. A ratio close to one indicates that all the chains are sampling the same area of the posterior pdf. Although it is not our purpose to discuss the diﬃcult problem of assessing convergence of MCMC, we note that it is only an indication of possible convergence, but in no way a proof. We considered that convergence was reached when R < 1.1 for all scalar parameters (stiﬀness and variance). All the computations in this paper were performed on a desktop computer with 8 computing cores (2 quad-core Intel Xeon E5620 @2.4GHz processors), with a chain running independently on each core. The algorithm was coded using Matlab R . To assess convergence, it was necessary to regularly pause the chains and pool them to compute the ratios R. This was done after each segment of 100 iterations. The L = 8 starting points were selected from M = 1000 draws from the prior.

4.3 Application 1 - Data from Ogi et al., 2002

For the first illustration of the proposed method we used the resonant frequencies published by Ogi et al.(2002) for a specimen of polycrystalline aluminum alloy (isotropic symmetry). The paper contains all the necessary information about the specimen: dimensions, mass, and the values of the 42 first measured resonant frequencies. Additionally, the exact mode identification obtained by laser interferometry is given, in the form of a group symmetry label identifying uniquely each measured resonant mode (Mochizuki, 1987). The group symmetry labels are an output of the forward model, and can be unambiguously attributed to the experimental frequencies in laser-based RUS from a comparison of the predicted and experimental modal shapes. This application is an interesting test for our method, 4.3. Application 1 - Data from Ogi et al., 2002 77 since this dataset was initially presented to demonstrate the ability of laser-based RUS to solve the inverse problem in a case where, according to the authors, it would have been very difficult with the measurement of the frequencies alone. We report here that the method presented above can be used to estimate accurate stiffness coefficients without using the laser data and without requiring an informative prior on the stiffness.

4.3.1 Prior distributions

Ogi et al.(2002) emphasize on the ability of laser-based RUS to find the correct pairing of frequencies even in difficult situation by starting with very unrealistic guess of the stiffness constants: C11 = 300 and C44 = 50 Gpa (“true” values are close to 109 and 27 Gpa, respectively). We therefore centered our prior distribution

on these values by ﬁxing mprior = [0 ; 0], with (see AppendixA)

4 ! C11 − 3 C44 m1 = log 4 , (4.23a) 300 − 3 50

C m = log 44 . (4.23b) 2 50 We then constructed a weakly informative prior by choosing the covariance matrix to be 0.1 times the 2-by-2 identity matrix I2, which gives a broad distribution in the stiffness space (Fig. 4.2). For the pairing, we considered the M = 50 first predicted frequencies, letting room for up to kmax = 8 potentially non-observed frequencies. The expectation λ was set to 1. It gave equally high prior probability (≈ 0.37) of having missed k = 0 and k = 1 resonant frequencies and a decreasing probability for larger k, thus reflecting our prior knowledge that the frequencies are easily measured on a very low damping material. We used the noninformative prior p(σ2) = 1/σ2 for the variance. We set an upper boundary of σ2 < 0.032, corresponding to a misfit of 3% between predicted and measured frequencies, much larger that what can be expected from a successful RUS experiment.

4.3.2 Results and discussion

The 8 starting points generated from the procedure described in Section 4.2.3.4 are located by crosses in Fig. 4.2. We see that the 8 starting values of C44 are close. This is due to the fact that the first predicted mode is usually an almost pure shear mode, depending mainly on C44. Since this mode is always paired with the first measured frequency, it has the effect of almost fixing the shear coefficient C44 close to the correct value. The coefficient C11 remains largely undetermined at this step. 78 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo

200

150

(GPa) 100 44 C

0 0 200 400 600 800 1000 C (GPa) 11

Figure 4.2: Application 1: prior probability in the space of the stiﬀness coeﬃcients

corresponding to the bivariate normal prior distribution on m, centered on m1 and m2 with covariance equal to 0.1 × I2. The contour lines enclose respectively 10, 68, 95, 99, and 99.99 % of the distribution. The crosses locate the randomly generated starting points of the 8 Gibbs chains.

450 400 112 350 110 300 108 (GPa) 250 106 11 40 50 60 70 80 90 100 C 200 150 100 20 40 60 80 100 Gibbs sampler iteration

Figure 4.3: Application 1: sampled values of C11 for the 100 ﬁrst iterations of the 8 Gibbs chain. After 40 iterations, all the chains have converged to the same location. 4.3. Application 1 - Data from Ogi et al., 2002 79

C (GPa) C (GPa) −3 11 44 σ (x10 )

104 106 108 110 112 114 26.55 26.6 26.75 26.85 1 2 3 4 5

Figure 4.4: Application 1: Histograms of the stiﬀness coeﬃcients and error term σ obtained from Gibbs sampling using the frequency data of the alluminium alloy specimen from Ogi et al.(2002).

Convergence of the 8 parallel chains was observed at the first convergence test, after 100 iterations of the Gibbs sampler, but we let them run for 900 more iterations (∼ 4 min of computation time). The convergence was indeed reached after about 40 iterations, as it can be seen on Fig. 4.3. After discarding the 100 first samples, chains were pooled and the sampled m transformed back to stiffness coefficients to evaluate the posterior distribution. Any other elastic modulus (e.g. Young’s modulus or Poisson’s ratio) could also have been calculated.

The histograms for C11, C44, and σ are plotted on Fig. 4.4. It can be observed that the marginal posterior distributions of the stiffness constants have a normal shape while the marginal for σ is asymmetric. The posterior means and standard deviations for C11, C44 were respectively taken as the estimate of the stiffness constant and uncertainties. For σ, the point of maximum marginal posterior probability was estimated from the histogram. The results are given in Table 4.1, and compared to the estimates obtained assuming the exact pairing. These estimates were obtained from the quasi-Newton method described in section 4.2.3.2. In that case the estimated standard deviation is the relative root-means-square error between measured and predicted frequencies at the end of the fit. The result obtained with uncertain and exact pairing are almost identical and the uncertainties are only slightly larger in the first case, which shows that the approach proposed here produces results that are almost as accurate and precise as the results obtained using the exact pairing deduced from laser measurements. We also recall the values obtained by Ogi et al. using more than 80 frequencies with exact pairing (only the 42 first frequencies were given in the paper). The values are slightly different but agree with the uncertainty intervals. The 42 published frequencies correspond to the actual 42 first resonant modes of the specimen (no missed frequencies). Our algorithm correctly and unambiguously identified that no frequency was missed, i.e. we obtained p(k = 0) = 1. Moreover, 80 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo

Table 4.1: Application 1: parameters estimated from the proposed Gibbs sampling method for the data from Ogi et al.(2002), compared to estimations knowing the exact pairing of the frequencies.

Gibbs Exact pairing Ogi et al.(2002) Nb freq. 42 42 > 80

C11 (GPa) 108.8 ± 0.9 109.0 ± 0.7 109.26 C44 (GPa) 26.68 ± 0.03 26.68 ± 0.02 26.72 Variance (%) 0.24 0.23 0.2 since the exact pairing is known we can compare the pairings obtained through Gibbs sampling to the correct pairing. In Table 4.2, the sampled group symmetry labels for each mode are listed against the exact label obtained from Laser interferometry. We can see that 18 out of 42 modes are uniquely paired and that a total of 32 modes are confidently identified (> 95% probability) with the correct predicted mode. The other modes are identified with larger uncertainties and with up to 4 possibilities of pairing. However, the most probable mode label is always the correct one, except for two pairs of modes that were inverted (modes 38 − 39 and 41 − 42). All cases of uncertain or inexact identifications correspond to resonant frequencies that are very close to each others, compared to the estimated standard deviation of the experimental frequencies (0.24%). It is then not surprising that these frequencies cannot be confidently paired to a unique predicted frequency. But we argue that it is precisely in those cases that the exact pairing matters the less. Indeed, inverting two or more frequencies that are separated by less than the standard deviation can only have a negligible influence on the estimated stiffness. The correct pairing was obtained by the method (except for some very close modes) using only the frequencies, which shows that the information on the vibration modes is indeed contained in the distribution of the frequencies. Then, getting the modal shapes experimentally only provides redundant information here.

Table 4.2: Application 1: experimental resonant frequencies and group symmetry label from Laser-based RUS measurement (Ogi et al., 2002) and group symmetry labels proposed during Gibbs sampling with their percentage of occurrence. Labels indicate belongings to one of the 8 distinct symmetry groups and the ordering in each group, following the notation introduced by Mochizuki(1987)

fexp (kHz) Mode Modes Gibbs (% of occurrence)

116.716 Au-1 Au-1 (100) - - - 143.783 Au-2 Au-2 (100) - - - 158.081 B1u-1 B1u-1 (100) - - - Continued on next page 4.3. Application 1 - Data from Ogi et al., 2002 81

Table 4.2 : Continued from previous page

fexp (kHz) Mode Modes Gibbs (% of occurrence)

166.500 B2u-1 B2u-1 (100) - - - 169.523 B1g-1 B1g-1 (100) - - - 177.846 B2g-1 B2g-1 (100) - - - 183.875 B3u-1 B3u-1 (99.2) B3g-1 (0.8) - - 186.047 B3g-1 B3g-1 (99.2) B3u-1 (0.8) - - 190.341 Ag-1 Ag-1 (100) - - - 197.386 B1u-2 B1u-2 (100) - - - 201.136 Ag-2 Ag-2 (100) - - - 207.386 B3g-2 B3g-2 (100) - - - 209.836 Ag-3 Ag-3 (100) - - - 214.753 B2g-2 B2g-2 (100) - - - 223.548 B3u-2 B3u-2 (100) - - - 231.266 B2u-2 B2u-2 (100) - - - 233.538 B3g-3 B3g-3 (99.1) B1g-2 (0.9) - - 234.717 B1g-2 B1g-2 (99.1) B3g-3 (0.9) - - 250.980 Ag-4 Ag-4 (55.0) Au-3 (44.7) B2g-3 (0.3) - 251.256 Au-3 Au-3 (54.6) Ag-4 (44.7) B2g-3 (0.7) - 252.742 B2g-3 B2g-3 (99.0) Au-3 (0.7) Ag-4 (0.3) - 256.122 B1u-3 B1u-3 (98.4) B3u-3 (1.5) Ag-5 (0.1) B2u-3 (0.0) 257.595 B3u-3 B3u-3 (72.6) Ag-5 (19.6) B2u-3 (6.3) B1u-3 (1.5) 258.118 Ag-5 Ag-5 (51.0) B2u-3 (25.9) B3u-3 (23.0) B1u-3 (0.1) 259.035 B2u-3 B2u-3 (67.9) Ag-5 (29.3) B3u-3 (2.8) - 268.540 B1g-3 B1g-3 (100) - - - 277.113 B2u-4 B2u-4 (99.9) B3u-4 (0.1) - - 278.762 B3u-4 B3u-4 (99.9) B2u-4 (0.1) - - 282.311 B1u-4 B1u-4 (100) - - - 293.686 B3u-5 B3u-5 (100) - - - 304.159 B2u-5 B2u-5 (58.9) B1u-5 (41.1) - - 304.464 B1u-5 B1u-5 (58.9) B2u-5 (41.1) - - 310.109 B1u-6 B1u-6 (100) - - - 316.197 B1g-4 B1g-4 (97.7) B2g-4 (2.3) - - 317.392 B2g-4 B2g-4 (97.7) B1g-4 (2.3) - - 326.462 Au-4 Au-4 (100) B3g-4 (0.0) - - 329.034 B3g-4 B3g-4 (100) Au-4 (0.0) - - 332.441 Ag-6 B2u-6 (70.9) Ag-6 (29.1) - - 333.364 B2u-6 Ag-6 (70.9) B2u-6 (29.1) - - 336.650 B1g-5 B1g-5 (69.6) Ag-7 (18.8) B2g-5 (11.6) - Continued on next page 82 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo

Table 4.2 : Continued from previous page

fexp (kHz) Mode Modes Gibbs (% of occurrence)

337.359 B2g-5 Ag-7 (38.1) B2g-5 (36.7) B1g-5 (25.1) - 338.276 Ag-7 B2g-5 (51.7) Ag-7 (43.1) B1g-5 (5.2) -

4.3.3 Additional results

We solved the inverse problem for the same data set but for a more general assumption of orthotropic elastic symmetry. The prior distribution was centered on

C11 = 100, C22 = 150, C33 = 200, C12 = 70, C13 = 60, C23 = 50, C44 = 20, C55 = 30, C66 = 40, with a covariance matrix again equal to 0.1 times the identity matrix. After about 1000 iterations the 8 chains were eventually sampling the same peak of the pdf and were run for 20000 additional samples. The sampled values of the orthotropic elastic coefficients, presented in Table 4.3, are very similar to the isotropic ones and to those obtained with the exact pairing. It shows that the proposed algorithm is able to accurately and automatically estimates 9 orthotropic stiffness constants, without requiring to measure the modal shape of the modes nor to specify an informative prior. Convergence could have been greatly improved with the use of an informative prior, which may actually be available in many practical applications. We also applied the method to two additional data sets, published in another paper proposing an experimental method for the measurement of the modal shapes of the modes (Stekel et al., 1992). The data consists in 15 resonant frequencies measured on an isotropic Ni80P20 alloy specimen and 31 frequencies measured on a cubic YbAl specimen. In both case the algorithm performed very well and accurately estimated the stiffness constants, starting from a broad prior distribution (results not shown). The first of these two data sets is particularly interesting since three predicted resonant frequencies were not measured. The algorithm unambiguously identified k = 3 missing frequencies: p(k = 3) = 1. Finally, we tested the robustness of the method to the expected number of missed frequencies λ. We attributed to λ a noninformative prior, in the form of a gamma prior – Eq. (5.19) – with α = 1/2 and β = 0 (Yang and Berger, 1997). For all the applications described above, the algorithm converged to the same results, although convergence was somewhat longer. It shows that the method is weakly sensitive to λ, at least for these particular data sets. Hence, setting λ = 1 or another low value should be adequate for applications to low damping materials. 4.4. Application 2 - Data from Bernard al., 2013 83

Table 4.3: Application 1: stiﬀness constants (in GPa) estimated from the data from Ogi et al.(2002) for a more general orthotropic elasticity assumption (see section 4.3.3).

Gibbs Exact pairing Ogi et al.(2002)

C11 108.9 ± 0.8 109.3 ± 0.6 109.26 C22 108.7 ± 0.6 109.0 ± 0.5 C33 109.3 ± 0.6 109.5 ± 0.4 C12 55.5 ± 0.7 55.9 ± 0.5 55.82 C13 55.5 ± 0.7 55.8 ± 0.5 C23 56.3 ± 0.6 56.6 ± 0.4 C44 26.95 ± 0.06 26.96 ± 0.05 26.72 C55 26.52 ± 0.06 26.52 ± 0.05 C66 26.85 ± 0.05 26.84 ± 0.04

4.4 Application 2 - Data from Bernard al., 2013

The second application deals with the recently developed application of RUS to highly attenuating materials (Bernard et al., 2014) and in particular to cortical bone (Bernard et al., 2013). This application leads to an even more difficult identification problem. Indeed, due to the high amount of vibration damping, the spectrum of the specimen is not composed of sharp resonant peaks, but of broad and overlapping peaks. The frequencies must then be estimated through signal processing (Lebedev, 2002). Due to overlapping, it is likely that some frequencies are missed during the measurement and there is no simple way to tell which modes are more likely to be missed. We do not discuss here signal processing and we consider the resonant frequencies as given input data. We applied the method described above to the 20 resonant frequencies measured by Bernard et al.(2013) for a specimen of human femoral cortical bone. In that case, the exact pairing of the measured frequencies was not known, but an independent measurement of the stiffness coefficients was available to evaluate the results.

4.4.1 Prior distributions

The measured specimen was taken in a population of 21 femoral bone specimens whose diagonal elastic coefficients (Cii, i = 1, ..., 6) were previously measured using ultrasonic bulk waves velocities (Granke et al., 2011). It was then possible to construct a prior distribution using the previous results obtained for the whole population. For the transformed parameters m corresponding to the diagonal coefficients (see AppendixA) the center of the prior distribution was fixed using the mean val-

ues of the measured coeﬃcients, which are C11 = 19.3, C22 = 19.8, C33 = 29.2, C44 = 5.8, C55 = 5.6, and C66 = 4.3 (in GPa). Since the three oﬀ-diagonal 84 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo

50 20 9

45 8 15 40 7

35 6 10 (GPa) (GPa) (GPa) 23 33 30 66 5 C C C

25 5 4

20 3 0 15 2 10 15 20 25 30 35 0 5 10 15 20 2 4 6 8 10 12 C (GPa) C (GPa) C (GPa) 11 13 55

Figure 4.5: Application 2: prior probability in the space of the stiﬀness coeﬃcients corresponding to the multivariate normal prior distribution on m. The contour lines embed respectively 10, 68, 95, 99, and 99.99 % of the distribution. The cross locate the randomly generated starting points of the 8 Gibbs chains.

coefficients C12, C13, and C13 were not measured, their prior were centered on Cij = C11 − 2 × C66 = 10.7 GPa. Variances and covariances of the parameters corresponding to the diagonal coefficients were estimated from the population results. No covariances were assumed between the off-diagonal and the diagonal coefficients, nor between the three off-diagonal coefficients. Their variances were fixed to 0.05. This gives the following prior covariance matrix for m:

C11 C22 C33 η12 η13 η23 C44 C55 C66   C11 1.35 1.24 1.19 0 0 0 1.41 1.71 1.72 C  1.24 1.40 1.19 0 0 0 1.48 1.63 1.79  22     C33  1.19 1.19 1.27 0 0 0 1.42 1.66 1.64    η  0 0 0 5 0 0 0 0 0  12     0.01 × η13  0 0 0 0 5 0 0 0 0 . (4.24)   η23  0 0 0 0 0 5 0 0 0    C  1.41 1.48 1.42 0 0 0 1.84 1.93 2.04  44     C55  1.71 1.63 1.66 0 0 0 1.93 2.48 2.40  C66 1.72 1.79 1.64 0 0 0 2.04 2.40 2.58

Three 2D sections of the prior distribution are plotted on Fig. 4.5. The assumed a priori correlation between C11 and C33 on one hand, and C55 and C66 on the other hand, can be observed. For σ2, the same prior as in application 1 was used. The expected number of missed frequencies λ was not ﬁxed, but an noninformative prior was attributed to it (section 4.3.3). 4.5. Conclusion 85

4.4.2 Results and discussion

The 8 randomly generated starting points are shown on Fig. 4.5. In that case, all coefficients show some scattering. This is due to anisotropy. Indeed, although the first predicted mode is still an almost pure shear mode, it depends on a combination of the three shear coefficients. Convergence of the Gibbs chains was indicated after about 500 iterations, but the chains were stopped after 5000 samples. The values of the elastic parameters sampled after convergence were transformed back to the coefficients Cij. Histograms of the 9 stiffness coefficients and the standard deviation σ are represented on Fig. 4.6. Again, the marginal posterior distributions for the elastic coefficient look normal, while the distribution of σ is asymmetric. The mean values of the Cij are given in Table 4.4. They are compared to the results obtained for a fixed pairing, that was obtained manually (Bernard et al., 2013) and to the independent results obtained from wave velocities measurements. Good agreement is observed between the three results. Particularly, no significant difference is observed between results from RUS with fixed pairing and from Gibbs sampling. The number of missed frequencies was unambiguously identified to k = 10 [p(k = 10) = 0.988], the same number that was obtained from manual pairing (Bernard et al., 2013), despite the non-informative prior on λ. In this application to a highly attenuating material, an informative prior on the stiffness is necessary for the identification of the pairing and then the estimation of the stiffness. Due to the low number of measured frequencies (= 20) relative to the number of elastic parameters (= 9), the pairing cannot be entirely determined from the distribution of the frequencies, contrary to the previous application. This is a limitation of the proposed approach, however compensated by the potential availability of prior information in many applications. The prior distribution of the elastic parameters was constructed from available results for a population of similar samples. In some practical applications, this might be done through review of reported elasticity for similar materials. In some context, like composites materials, a prior distribution might also be obtained from theoretical predictions of elasticity. If no prior is available, it might be necessary to perform preliminary measurements, for instance of ultrasonic bulk wave velocities in some material directions.

4.5 Conclusion

We presented a method based on Bayesian inference and MCMC sampling to automatically solve the inverse problem in RUS, including the identification of the 86 Chapter 4. Bayesian inversion using Markov Chain Monte Carlo ( 2013 ). al. et Bernard from specimen bone the of data 4.6: Figure 12 19 20 plcto :hsorm ftesins offiinsaderrterm error and coefficients stiffness the of histograms 2: Application 14 C C 11 21 23 22 16 23 6 20 6.2 C C 6.4 44 22 22 6.6 24 28 5.8 6 30 C C 6.2 55 33 32 6.4 34 6.6 σ 4.4 9 bandfo ib apiguigtefrequency the using sampling Gibbs from obtained 10 4.6 11 C C 12 66 12 4.8 13 5 0 12 0.01 C σ 13 14 16 0.02 4.5. Conclusion 87

Table 4.4: Application 2: stiﬀness constants (in GPa) estimated using Gibbs sampling and using a ﬁxed pairing, and results from bulk wave velocity measurements. Data for a human femoral cortical bone specimen (Bernard et al., 2013).

Gibbs Fixed pairing wave velocity

C11 20.9 ± 0.6 20.3 ± 0.6 21.6 C22 21.3 ± 0.7 20.2 ± 0.6 21.4 C33 31.7 ± 0.9 31.7 ± 0.8 31.3 C12 11.6 ± 0.6 10.7 ± 0.6 - C13 13.8 ± 0.6 13.4 ± 0.7 - C23 14.1 ± 0.7 13.4 ± 0.7 - C44 6.38 ± 0.06 6.38 ± 0.02 6.5 C55 6.31 ± 0.07 6.32 ± 0.03 6.5 C66 4.79 ± 0.04 4.80 ± 0.02 4.8 Variance (%) 0.43 0.30 - resonant frequencies, without requiring the measurement of the modal shapes associated to each resonant frequency. In the context of RUS applied to weakly attenuative materials (application 1), where many resonant frequencies are easily measured, we showed that the method allows estimating the stiffness constants with high accuracy and precision, without requiring introduction of a prior information. For the more difficult application of RUS to attenuating materials (application 2), less resonant frequencies can be measured. The joint estimation of the pairing and stiffness constants can then be ambiguous. However, the Bayesian formalism allows to including the available prior information to resolve the ambiguities. The method was illustrated for anisotropy up to the orthotropic symmetry (9 stiffness constants). There is however no restriction on the number of parameters that can be estimated, and the method could be generalized to lower elastic symmetry and/or to the joint estimation of the Euler’s angles in the case of misoriented specimens or to the measurement of piezoelectric properties of materials.

Chapter 5 Application to a collection of human tibial cortical bone specimens

5.1 Introduction

This chapter presents the application of the RUS method adapted for highly attenuative materials, as described in the previous chapters, to a large collection of human cortical bone specimens. This work is a part of a cooperative study involving several research laboratories in Europe, including the three members of the European laboratory for ultrasonic assessment of bone (ULAB, CNRS) : the Laboratoire d’Imagerie Biomédicale in Paris (UPMC, CNRS), the Julius Wolﬀ institute in Berlin (Charité Universitätsmedizin) and the medical physics research group in Kiel university. State-of-the-art and recently developed ultrasonic methods for bone assessment, as well as complementary approaches, were applied to a collection of human bone specimens. Among the used methodologies, some are listed here:

• Scanning acoustic microscopy (SAM) provides a map of tissue elasticity at the micro-scale. Both 50 MHz and 200 MHz SAM were used, with resolutions of about 30 and 8 µm respectively (Raum, 2011).

• Bi-directional axial transmission (BDAT) uses guided waves propagating over a few centimeters in long bones such as the tibia and radius to infer bone thickness and elasticity (Foiret et al., 2014). This method is in development for in vivo applications, but was used ex vivo in this study.

• X-ray computed tomography (CT) provides the geometry of bone and bone mineral density.

• Resonant ultrasound spectroscopy to characterize the full anisotropic stiﬀness tensor at the millimeter scale, as well as the anisotropic damping.

The goals of this study were: 1) to validate the in vivo BDAT method by rigor- ous site-matched comparison with RUS, and 2) to increase the fundamental understanding of relationships between bone structure (CT, SAM), mechanical properties (RUS) and tissue elasticity (SAM). Accurate measurement of bone anisotropic elasticity with RUS is crucial regarding these two objectives. The results of the RUS measurements are described in the present Chapter. Moreover, a particular method to construct the prior distribution needed in the 90 Chapter 5. RUS on a collection of human tibial bone specimens

20 mm PMMA 5 mm

Medial face

C+2/3 C+1/3 C C-1/3

Figure 5.1: Scheme and photograph of the specimens location. A cross-section was harvested from the mid-diaphysis and two to four specimens extracted from the medial face at registered locations. See the text for details.

Bayesian analysis introduced in Chapter4 is presented. This method uses the actual distribution of the elastic parameters in the specimen collection as a prior for the inversion of each individual specimen.

5.2 Materials and methods

5.2.1 Specimens

A 20-mm thick cross-sections was cut perpendicular to the bone axis in 19 femurs from human cadavers at mid-diaphysis (Fig. 5.1). The cross sections were partially embedded in PMMA for ease of manipulation (on about 5 mm of the thickness) and stored at −20◦C after cutting. Among the 19 donors, 13 were females (ages [69-94] years, mean±SD = 82.7±8.4 years) and 6 were males (ages [70-94] years , 82.2±10.1 years). Up to four rectangular parallelepiped specimens were extracted from the medial face of each cross-section on a part that was not contacting the PMMA layer. The cutting protocol used in that study is described and illustrated in AppendixB, and summarized here. Cutting was done using a precision linear saw (Isomet 4000, Buehler GmbH, Düsseldorf, Germany). A face of each parallelepiped was carefully aligned with the medial surface and another face aligned with the bone axis. At least two specimens were prepared for each cross section: one in the center of the medial face (centered on the C line, see Fig. 5.1) and one in the extremity, centered on the C+2/3 line. When the curvature of the medial face was low, one or two additional specimens could be extracted together with the central specimen: one immediately adjacent to it, limited by the C+1/3 line and another one centered on the C−1/3 line. These lines mark the positions where other measurement techniques were applied, allowing site matched comparison of the results (not reported in this 5.2. Materials and methods 91 thesis). In total, 59 specimens were harvested and marked so as to identify the three anatomical directions. The typical specimen size was 2x3x4 mm3 in radial, circumferential and axial direction, respectively. All the specimens were stored at ambient temperature in Phosphate buﬀered saline solution and measured in a fully hydrated state no more than 24 hours after cutting. Apparent mass density of each specimen was deduced from the measured mass (±0.1 mg) and dimensions (±0.02 mm).

5.2.2 Measurement setup and signal processing

The setup and signal processing methods dedicated to the RUS measurement of attenuative materials such as cortical bone, extensively described in Bernard et al. (2013, Chapter2) and Bernard et al.(2014, Chapter3) were used. The specimens were placed with slight contact between a pair of shear ultrasonic contact transducers (V154RM, Panametrics, Waltham, MA), see Fig. 5.2, one acting as an emitter and the second one as a receiver. The frequency response was recorded using a vectorial network analyzer (Bode 100, Omicron Electronics GmbH, Klaus, Austria) and a broadband charge amplifier (HQA-15 M-10 T, Femto Messtechnik GmbH, Berlin, Germany). The frequency band of analysis was tuned for each specimen so as to contain the ∼20-30 first resonant frequencies (typically 150 to 500 kHZ). Six successive measurements on each specimens were done, with intermediate rotation of the specimens by a small angle between each measurement to vary the relative amplitudes of the excited resonant modes and then to maximize the number of detectable resonant frequencies (Fig. 5.3). The spectra were fitted to a sum of M Lorentzian lineshapes, each describing the behavior of a one degree-of-freedom mechanical resonator

M X ak L(f) = , (5.1) (f 2 − f 2) + i(f f/Q ) k=1 k k k

Figure 5.2: A bone specimen inserted between the two transducers of the RUS setup. One tranducer acts as a emitter and the other as a reciever to record the vibration spectrum of the specimen. 92 Chapter 5. RUS on a collection of human tibial bone specimens

0.15

0.1

Amplitude (a.u.) 0.05

0 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Frequency (MHz)

Figure 5.3: Typical set of spectra measured for a cortical bone specimen. The relatives amplitudes of the resonant modes vary as the specimen is rotated, allowing to detect more resonant frequencies than from a single spectrum. Due to overlaping, the local maxima of the spectra do not always correpond to resonant frequencies. Signal processing must be used to accuratly and reproductibly estimate the frequencies.

with ak the complex amplitudes, fk the resonant frequencies and Qk the quality factors. To estimate these parameters as well as the number of peaks M, we used a time domain estimation method based on a linear predictive ﬁlter (Lebedev, 2002) followed by a frequency domain non-linear optimization.

5.2.3 Estimation of the elastic properties

In RUS, the elastic properties are estimated from a comparison of the measured resonant frequencies to model-predicted frequencies. The geometry, mass density and stiffness of the specimen entirely determine the frequencies, so that the stiffness can be adjusted until the best match between the model and the experiment has been reached. This is usually done thought the minimization of a quadratic objective function exp cal 2 X fn − f (Cij) F (C ) = n . (5.2) ij f exp n n An issue of that approach is that it requires to correctly pair the measured frequencies to their predicted counterpart, while the measurement does not contain the information on which of the many predicted resonant mode have been measured. If the initial guess is far form the actual elasticity and if the predicted resonant modes are not all observed experimentally, as it is the case for high damping materials, the correct pairing is difficult to obtain. In Chapter4, a Bayesian formulation of the inverse problem was proposed, that automatically solves the joint problem of finding the correct pairing of frequencies and estimating the elastic properties. This approach was used here, with an exten- 5.2. Materials and methods 93 sion allowing one to consider a collection of similar specimens instead of processing each specimen independently. A transversely isotropic symmetry, which requires 5 elastic parameters, was assumed for the elastic properties of tibial cortical bone. In the coordinates system defined by the orthogonal faces of the specimen, which were aligned with the anatomical directions, the elasticity matrix takes the form   C11 C12 C13 0 0 0   C12 C11 C13 0 0 0    C13 C13 C33 0 0 0  Cij =   , (5.3)  0 0 0 C 0 0   44     0 0 0 0 C44 0  0 0 0 0 0 C66

with the relation C12 = C11 − 2C44, and where the axis 3 is in the axial direction of bone and axis 1 and 2 orthogonal to the bone axis. The inverted elasticity matrix, containing the engineering moduli, is   1/E1 ν12/E1 ν13/E1 0 0 0   ν21/E1 1/E1 ν13/E1 0 0 0    −1 ν31/E3 ν31/E3 1/E3 0 0 0  C =   , (5.4) ij  0 0 0 1/C 0 0   44     0 0 0 0 1/C44 0  0 0 0 0 0 1/C66

with ν12 = ν21 and ν13/E1 = ν31/E3. This symmetry assumes that elasticity is isotropic in the plane orthogonal to the bone axis. It has been reported to be an accurate assumption for the elastic properties of femoral cortical bone at the mid- diaphysis (Espinoza Orias et al., 2008; Granke et al., 2011). The elastic coeﬃcients

Cij are not used directly in the Bayesian estimation, but new parameters m more consistent with an assumption of a normal distribution are used (see AppendixA).

Bayesian formulation of the inverse problem The pairing of the frequencies is formally represented by a vector a of integers. The length of this vector is equal to the number of predicted modes in the investigated frequency band, and contains a 0 if a particular mode is not observed and an integer is the mode is observed (see Fig. 4.1 in Chapter4). The joint posterior probability distribution of the elastic properties m, measurement noise σ2 and pairing of predicted and measured resonant modes a (the solution of the inverse problem) is then expressed as the product of prior distributions and the likelihood of the measured data y through Bayes theorem

p(m, σ2, a|y) ∝ p(m)p(σ2)p(a)p(y|m, σ2, a), (5.5) 94 Chapter 5. RUS on a collection of human tibial bone specimens with the likelihood

" N 2# 1 1 X gn(m, a) − yn p(y|m, σ2, a) ∝ exp − , (5.6) σN 2 σy n=1 n where g(m, a) includes the forward relation between the elastic parameters and the frequencies and a indicates which output of the model is to be compared with the nth measured frequency. The posterior distribution (5.5) can not be studied analytically and must be sampled using Markov Chain Monte Carlo methods (MCMC). Gibbs sampling, a particular MCMC, is particularly adapted to that problem. In Gibbs sampling, the conditional distribution of one variable given the other variables in their current state are successively sampled

2 step 1: a1 ∼ p(a|y, m0, σ0) 2 2 σ1 ∼ p(σ |y, m0, a1) 2 m1 ∼ p(m|y, σ1, a1) (5.7) 2 step 2: a2 ∼ p(a|y, m1, σ1) ... which asymptotically produces samples of the joint posterior distribution (5.5). From a large number of samples from the posterior distribution, any characteris- tic of the distribution can be computed (e.g. mean, maximal posterior estimate, confidence interval). This Bayesian formulation requires the specification of prior probability densities that might have a significant impact on the results but are necessary to constrain the problem. Notably, a multivariate normal prior distribution must be given to the elastic parameters and a prior on a depending on the number of predicted frequencies that have been missed during the measurement must be specified. These distributions could have been fixed once for all before starting the analysis, for example using published data on bone elasticity. This however might have biased the results. Perhaps the best prior distribution that could be used to analyze the collection of bone specimens at hand is the actual distribution of elastic properties in that particular population. That way, information from the whole population will be used to help the estimation of the parameters for each individual specimen, instead of a priori information from another source. The population distribution is however unknown before the estimation of elasticity for each specimen has been solved. Bayesian theory provides a tool to estimate jointly the population distribution and the parameters of each specimen. This tool is called hierarchical modeling and is described below. 5.2. Materials and methods 95

Hierarchical Bayes modeling Let mj and yj denote the parameters and data for the j-th specimen of the collection, while m and y are parameters and data for the whole collection (to simplify notations, the dependence on the pairing vector a and experimental error σ2 is implicit in the following equations). Hierarchical modeling assumes that the parameters mj are themselves samples of a distribution characterized by the hyperparameters θ. The joint posterior probability distribution of θ and m writes p(θ, m|y) ∝ p(θ)p(m|θ)p(y|m, θ). (5.8) Common assumptions in hierarchical modeling (Gelman et al., 2013) consider that the hyperparameters inﬂuence the likelihood of the data only through the parameters m, i.e. p(y|m, θ) = p(y|m), (5.9) and that the j-th data set depends only on the j-th parameter set mj (i.e. the j measurements are independent from each others) Y p(y|m) = p(yj|mj). (5.10) j Moreover, Eq. (5.9) implies that the conditional posterior of the hyperparameters is independent of the data p(θ|m, y) = p(θ|m). (5.11) The structure of the model is illustrated on ﬁgure 5.4. With the above assumptions, the joint posterior distribution (5.8) can be easily sampled using Gibbs sampling. Indeed, given a value of the hyperparameters θ, each set of parameters can be sampled from its conditional distribution. Then, given all the set of parameters, the hyperparameters can be sampled:

j j j step 1: m1 ∼ p(m |y , θ0) for j = 1, . . . , n

θ1 ∼ p(θ|m1) (5.12) j j j step 2: m2 ∼ p(m |y , θ1) for j = 1, . . . , n ...

Three hyperparameters θ = (λ, µ, Σ) were introduced. First, the number of

missed frequencies in each experiment ki is supposed to follow a truncated Poisson distribution of expectation λ  λki  p(ki) ∝ exp(−λ) if ki ≤ kmax, ki! (5.13)   p(ki) = 0 if ki > kmax, where kmax is the maximal number of missed frequencies permitted by the deﬁnition of the pairing vector a. The Poisson’s distribution is adequate to describe the 96 Chapter 5. RUS on a collection of human tibial bone specimens

θ∼p(θ)

1 2 n m ∼p(m∣θ) m ∼p(m∣θ) ... m ∼p(m∣θ)

y1∼p(y∣m1) y2∼p(y∣m2) ... yn∼p(y∣mn)

Figure 5.4: Hierarchical modeling assumes that the parameter set mj of each specimen in the population is a sample of a common population distribution p(m|θ), which depends on the hyperparameters θ. Each data set yj follows the distribution p(y|mj), indepently of all the other parameters. number of occurrence of a random event. Then, the elastic parameters m in the population are suppose to follow a multivariate normal distribution of mean µ and covariance matrix Σ. A multivariate normal distribution was used because the sampling strategy described in Chapter4 requires a normal prior on m and because this gives simple expressions for sampling the hyperparameters given the elastic parameters for each specimen. The prior on σ2 was not included in the hierarchical approach but was ﬁxed once for all. The conjugate prior for the mean µ and covariance matrix Σ of a normal distribution is a normal-inverse-Whishart distribution (NIW; inverse-Whishart distribution for the covariance matrix, and normal distribution for the means given the covariance matrix; Gelman et al., 2013):

( −1 Σ ∼ Inv-Wishartν (Λ ) 0 0 (5.14) µ|Σ ∼ N(µ0, Σ/κ0), where ν0 and Λ0 describe the degrees of freedom and scale matrix for the IW distribution on Σ, µ0 is the prior mean, and κ0 is equivalent to a number of prior measurements. Given m observed sets of elastic parameters m, the posterior distribution p(µ, Σ|m) for the collection mean and covariance matrix is then a NIW distribution (5.14) but with parameters

 κ0 m µ = µ + m  m κ + m 0 κ + m  0 0   κm = κ0 + m (5.15)  ν = ν + m  m 0  κ m  Λ = Λ + S + 0 (m − µ )(m − µ )t, m 0 m 0 0 5.2. Materials and methods 97

instead of ν0; Λ0, µ0 and κ0. S is the sum of square matrix about the sample mean m X S = (mi − m)(mi − m)t. (5.16) i=1 with n 1 X m = mi. (5.17) n i=1 For the mean µm, we used a noninformative prior by setting κ0 → 0. In that case, µ0 ∝ cst and the mean of the population is equal to the sample mean m. Noninformative IW priors for covariance matrices are notoriously diﬃcult to specify.

Nevertheless, this prior can be interpreted as equivalent to ν0 prior observations with covariance Λ0. Therefore, we speciﬁed these parameters based on previous experimental observation of bone elasticity. The diagonal elastic coeﬃcients Cii of 21 mid-diaphysis femoral bone specimens measured by Granke et al.(2011) were used

to construct Λ0. Transversely isotropic elastic coeﬃcients were inferred (averaging the values of C11 and C22, C44 and C55, and assuming C13 = C12 − 2C66 + ε with ε ∼ N(0, 1)). They were then converted to parameters m (see AppendixA) and their empirical covariance matrix Σemp. computed. We obtained   0.0110 0.0107 −0.0027 0.0138 0.0154    0.0127 −0.0030 0.0154 0.0164    Σemp. =  0.0208 −0.0038 −0.0080 . (5.18)    Sym. 0.0203 0.0220    0.0258

Then, Λ0 was set to ν0 × Σemp., so that the mode of the prior distribution p(Σ) was equal to Σemp.. The degree of freedom number ν0 was set to d + 1 = 6. The prior on the covariance matrix of the collection distribution is then an informative prior, constructed from previous data. However, the prior is expected to be weak, as its relative weight in the posterior covariance matrix Λm, expressed in Eq. (5.15), will be 6/(59 + 6) ≈ 1/11 (since the sum of square matrix S scales with the number of specimens). For λ, the expected number of missed frequencies, a gamma prior is adequate βα p(λ) = λα−1 exp(−βλ). (5.19) Γ(α)

Given the number of missed frequencies for each of the m experiment: ki, i = 1, ..., m, the posterior distribution of λ, p(λ|k), is m Y p(λ|k) ∝ p(λ) p(ki|λ) i=1 (5.20) m k Y λ i /ki! ∝ p(λ) , Pkmax kl i=1 l=0 λ /kl! 98 Chapter 5. RUS on a collection of human tibial bone specimens where the numerator term is due to the truncation of the Poisson’s distribution. P The posterior (5.20) is close to the Gamma distribution Gamma(α + ki, β + n) (it would be exactly proportional to it without the truncation on k), therefore to generate samples of the posterior, a sample is generated in the corresponding gamma approximation and a rejection step is applied. In this work we used a non informative prior for λ by setting α = 1/2 and β = 0 (Yang and Berger, 1997). The noise variance σ2 was not treated as a hierarchical parameter but its prior was ﬁxed once for all. The conjugate prior for this parameters is

2 2 2 2 p(σ ) = Inv-χ (σ |η0, σ0). (5.21)

2 2 Parameters were set to η0 = 2 and σ0 = 0.05 , reflecting the fact that the expected error on the measured frequencies is about 0.5%. This prior is however very weak, 2 as an Inv-χ with ν0 = 2 degrees of freedom has an infinite variance and no defined mean.

Convergence and post processing Convergence of the Markov Chain to the target distribution was monitored as in Chapter4, following recommendations of Gelman et al.(2013). 8 chains starting from randomly generated starting points were run in parallel, and the ratios of within-chain variance to between-chain variance where calculated for each scalar parameter of the problem. Ratios of variances lower than 1.1 indicate that all the chains have converged to the same posterior distribution. After convergence, samples from the 8 chains were pooled. The maxima of the marginal posterior distribution of the elastic parameters were chosen as the estimates. They were obtained from 1D kernel density estimation (Kroese et al., 2011), which provides a smoothed estimate of the marginal distribution (Fig. 5.6 and 5.7). Conﬁdence intervals containing 68% of the marginal distribution were also computed. The boundary of these intervals were placed such that 16% of the samples lie above the upper boundary, and 16% lie below the lower boundary. For a normal distribution, the half-width of this 68% interval would be equal to the standard deviation.

5.2.4 Viscoelasticity

Viscoelasticity was modeled by adding an imaginary part to the stiﬀness tensor, as in Chapter3: ∗ 0 −1 Cij = Cij + iCij = Cij(1 + iQij ), (5.22)

−1 where Qij are the ratios of the imaginary to the real part of the stiﬀness constants. −1 −1 If the inverse damping ratios Qij are small (Qij 1) they can be linearly related 5.2. Materials and methods 99

−1 to the inverse of the peaks quality factors Qk (Leisure et al., 2004; Ogi et al., 2003):

X 2Cij ∂fk Q−1 = Q−1. (5.23) k f ∂C ij i,j k ij

Where the coeﬃcients of the linear system are the normalized relative sensitivities of the resonant mode k to the elastic coeﬃcients, i.e.

X 2Cij ∂fk = 1. (5.24) f ∂C i,j k ij

When the elastic constants Cij have been estimated, Eq. (5.23) can be inverted to −1 −1 yield the damping ratios Qij from the measured quality factors Qk , as done in Chapter3.

However, this analysis presents two diﬃculties: 1) the peak quality factors Qk are diﬃcult to estimate precisely particularly when overlapping is strong. More- over, as the relative sensitivities of some frequencies to some elastic modulus are

low, the inversion is badly conditioned and sensitive to errors in the measured Qk. 2) Eq. (5.23) is dependent the pairing of the frequencies, which in our approach is not ﬁxed. A good approach would then be to solve Eq. (5.23) for each pairing and set of elastic parameters sampled during the exploration of the posterior probability distribution, which would lead to a distribution of viscoelastic parameters. Perhaps an even better solution would be to formulate the inversion of Eq. (5.23) −1 in a Bayesian framework, with the introduction of a prior distribution for the Qij . None of these two approaches was pursued in these work, but we went for simpliﬁed analysis based on the selection of the most reliable measurements of the quality

factors Qk.

Analysis based on the ﬁrst resonant mode

The ﬁrst vibration mode of rectangular specimen with small aspect ratios is usually a pure shear mode (Migliori and Sarrao, 1997), because shear waves have a smaller velocity than longitudinal waves. Moreover, many of the specimens in that study are longer in the third direction (axial direction of bone), so that the ﬁrst resonant

mode depends mainly on the axial shear modulus C44. Hence, the mechanical quality factor of the first resonant mode Qfirst can be directly related to the damping ratio Q44. Eq. (5.23) reduces to Q44 ≈ Qfirst. (5.25) An histogram of the relative sensitivity of the first peak to the axial shear coefficient

C44, computed from the specimens dimensions and estimated stiﬀness, is shown on ﬁgure 5.5. The sensitivity is larger than 0.8 for the majority of the population, which justify the approximation (5.25). 100 Chapter 5. RUS on a collection of human tibial bone specimens

5 Number of specimens

0 0.2 0.4 0.6 0.8 1 Relative sensitivity to C 44

Figure 5.5: relative sensitvity of the ﬁrst resonant mode to the axial shear stiﬀness

coeﬃcient C44. The sensitivity is larger that 0.8 for the majority of the specimens, which justify the approximation Q44 ≈ Qfirst.

The main advantage of that simple analysis is that a damping ratio is directly estimated from a single measured peak width. As the ﬁrst resonant peak is usually strong and isolated from the following peaks (see Figure 5.3), its width is measured easily and with good precision.

Anisotropic analysis of some selected specimens

Some specimens were selected for an anisotropic analysis of the viscoleaticity based on the inversion of Eq. (5.23). The criterion for selection was that several resonant peaks were clearly marked and separated from the other, and that those peaks could be unambiguously associated to their predicted counterpart. The specimen corresponding to the spectra in Fig. 5.3 is a good example: the four ﬁrst peaks are not close to each other, their peaks are very well deﬁned and their width can hence be measured with good precision. For these specimens, the damping ratios were inverted using the analogous of Eq. (5.23) formulated for the Young’s moduli, shear moduli, and Poisson’s ratio. Moreover, as RUS is intrinsically less sensitive to the Poisson’s ratios than to other engineering moduli (Chapter3) and because the imaginary part of the Poisson’s ratios is expected to be small, their imaginary part was neglected in the inversion. This reduces the number of parameters to three, as the relation between the transverse shear modulus C66, Young’s modulus E1 and Poisson’s ratio ν12

E1 C66 = (5.26) 2(1 + ν12)

entails the equality of the damping ratios Q66 and QE1 when ν12 is real. Then 2E ∂f 2E ∂f 2C ∂f Q−1 = 3 k Q−1 + 1 k Q−1 + 44 k Q−1 (5.27) k E3 E1 44 fk ∂E3 fk ∂E1 fk ∂C44 5.3. Results 101

C C C C 22 13 44 66

C 11

15 16 17 18 19 C 11

C 22

30 32 34 C 22

C 13

7 8 9 10 11 C 13

C 44

6.4 6.5 6.6 6.7 C 44

4.5 4.6 4.7 C 66

Figure 5.6: 1D marginal posterior distribution of the elastic coeﬃcients for a given specimen (diagonal), obtained using kernel density estimation, and 2D scattering plots showing the posterior correlations. The maxima of the marginal distribution are indicated by crosses and the 68% conﬁdence intervale represented by horizontal lines.

Reduction to a three-parameters model drastically improves the conditioning of the inversion, even for a low number of measured peak width.

5.3 Results

Between 9 and 21 frequencies were obtained form the measured spectra for each specimen, with an average of 16 frequencies per specimen. The first resonant frequency of each specimen was in the range 100−400 kHz and the highest frequency in the range 200 kHz−1 MHz. Typical posterior marginal distributions and 2D scatter plots obtained for a specimen from the sampled distribution are plotted on figure 5.6. This representation allows to visualize both the variance and covariances of the coefficients. The majority of the posterior distributions were clearly uni-modal, as on figure 5.6, but some were more complex and multimodal. The posterior distribution presented on figure 5.7 was, subjectively, the worst looking distribution. However, in all cases the positions of the maximum a posteriori estimates were clearly identified. Such representation are also available for the other elastic parameters (Young’s modulus, Poisson’s ratios). 102 Chapter 5. RUS on a collection of human tibial bone specimens

C C C C 22 13 44 66

C 11

15 20 25 C 11

C 22

25 30 35 C 22

C 13

5 10 15 C 13

C 44

5.6 5.8 6 6.2 6.4 6.6 C 44

3.6 3.8 4 4.2 4.4 4.6 C 66

Figure 5.7: Same as Fig. 5.6 above, but for another specimen. In that particular case the distribution has two modes.

The median values of the relative half-width of the 68% conﬁdence interval were

7.3, 4.4, 10.9, 1.3, and 2.0 % for C11, C33, C13, C44, and C66, respectively and 2.3, 2.8, 10.1, 7.6, 8.4 % for E1, E3, ν12, ν13, and ν31, respectively. For 4 specimens, the sampled values of σ were very high, about 0.05, corresponding to an error of 5% between measured and predicted frequencies. This is much higher that what should be obtained in a successful experiment. These specimens were excluded of all the subsequent analysis. For the other specimens, the error is about 1%. The median value of the MAP estimate of σ was 0.83%. It reflects the relative agreement between the predicted and measured frequencies. The estimated stiffness coefficients and Young’s moduli, as well as the Poisson’s ratios, are plotted as a function of bone density in figure 5.8. The anisotropy ratios are plotted on figure 5.9. The range of apparent mass density was 1.58 - 2.01 g/cm3. Additionally, the mean values of the Poisson’s ratios are compared to results of the literature in Table 5.2. The damping ratio of the axial shear coefficient is plotted on figure 5.10 as a function of density, while results of the anisotropic viscoelastic analysis for seven selected specimens are presented in Fig. 5.11. From 7 to 9 peak widths, the damping ratios for the shear and Young’s moduli were estimated. The predicted peak width agreed with measurement within 5% in average (Fig. 5.12), which is comparable to the precision of the measurement. Despite the somewhat large uncertainties 5.3. Results 103

C C 11 66 30 7 C C 33 44 6 25

5 20

4 15 Stiffness (GPa) Stiffness Stiffness (GPa) 3 10 2 5 1 1.6 1.7 1.8 1.9 2 1.6 1.7 1.8 1.9 2 Bone density Bone density E ν 25 1 0.65 12 E ν 3 0.6 13 ν 31 20 0.55 0.5 0.45 15 0.4 0.35 10 Poisson’s ratio 0.3 Young’s modulus (GPa) modulus Young’s 0.25 5 0.2

1.6 1.7 1.8 1.9 2 1.6 1.7 1.8 1.9 2 Bone density Bone density

Figure 5.8: Longitudinal stiﬀness (C11 and C33; top left), shear stiﬀness (C44 and C6; top rigth), Young’s moduli (E1 and E3; bottom left), and Poisson’s ratios (ν12, ν13 and ν31; bottom rigth) as function of apparent bone mass density.

C /C 3 44 66 C /C 33 11 2.8 E /E 3 1 2.6

2.4

2.2

2 Anisotropy ratio 1.8

1.6

1.4

1.6 1.7 1.8 1.9 2 Bone density

Figure 5.9: Anisotropy ratios as functions of apparent bone density. 104 Chapter 5. RUS on a collection of human tibial bone specimens

35 R = 0.84, p < 10−5

30 44 Q ≈ 25 firstpeak Q

15 1.6 1.7 1.8 1.9 2 Bone density

Figure 5.10: Damping factor of the ﬁrst resonant peak, approximetly equal to the

axial shear damping ratio Q44, as a function of bone mass density. on the results, anisotropy of the damping ratios can be observed: Q−1 < Q−1 < E3 44 Q−1 = Q−1, i.e. the damping is weaker in axial direction of bone, both for Young’s E1 66 and shear moduli. Adding the Poisson’s ratios in the analysis did not signiﬁcantly improve the agreement, as represented on Fig. 5.12, which shows that their eﬀect on the peak widths was indeed negligible.

5.4 Statistical analysis

Linear correlations of the elastic coefficients, engineering moduli, anisotropy ratios and axial shear damping ratio with the apparent mass density were investigated. The results (correlation coefficient, significance, slope, intercept) are reported in Table 5.1. All the parameters are significantly correlated to the apparent density (all p < 0.05). The elastic coefficients and engineering moduli increase with density, while anisotropy and Poisson’s ratio ν31 and ν13 decrease with density. To investigate the influence of the age (69 − 94 years) and sex of the donors, the medians of the elastic parameters and mass density were calculated for each donor over the two to four specimens corresponding to different locations. This reduces the data set to one value of each parameter per donor. These median values showed no linear correlation (Pearson’s) with age (all p > 0.25) nor rank correlation (Spearman’s; all p > 0.1). Differences between sexes were investigated using a non-parametric Wilcoxon rank-sum test. No differences were found (all p > 0.05). A Kruskal-Wallis test (non-parametric equivalent to classical one-way analysis of variance) was done for each elastic parameter and for the density, with the specimens grouped according to their anatomical position. A significative difference (at a 0.05 5.4. Statistical analysis 105

0.055 1/Q E3 1/Q =1/Q 0.05 E1 66 1/Q 0.045 44

0.04

0.035 −1

Q 0.03

0.025

0.02

0.015

0.01 1.9 1.95 2 Bone density

Figure 5.11: Results of the three parameters anisotropic viscoelasticity analysis for seven selected specimens. Despite some large uncertainties, anisotropy is clear. The −1 damping is stronger (Q is large) for the transverse parameters E1 and C66.

40 Predicted (w/ Poisson) 40 Predicted (w/o Poisson) 35 Measured 35

30 30

25 25

20 20

15 15

Q factor of the mode 10 Q factor of the mode 10

5 5

0 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 Resonant mode # Resonant mode #

Figure 5.12: Predicted quality factors of the peaks, after fitting the viscoelastic parameters, compared to the measurents, for two specimens. Error bars on the measurement assume 5% uncertainty. The variations of the peak widths are well captured by the model, whether the Poisson’s ratios are included in the analysis or not. 106 Chapter 5. RUS on a collection of human tibial bone specimens confidence level) in the median value was found for only one parameter: the stiffness coefficient C13. That stiffness coefficient was larger for the position C-1/3 than the for the position C+2/3.

5.5 Discussion

RUS technique and hierarchical modeling

In the previous chapters, the applicability of RUS to highly attenuative materials was established for a single cortical bone specimen (Bernard et al., 2013, Chapter2) and a few specimens of plastic materials with damping and stiffness comparable to that of bone (Bernard et al., 2014, Chapter2). The present study is the first one to report the measurement of bone anisotropic elasticity using RUS on a large collection of specimens. Moreover, to the best knowledge of the author, this work involves the largest specimen collection ever characterized using RUS in a single study, whatever the research context (material sciences, geophysics, biomechanics). The application on a large collection was made easier by the automated resolution of the inverse problem, based on a Bayesian framework. Resolution of the inverse problem usually requires a tedious manual pairing of the resonant frequencies by the user, or the use of a complex experimental setup to measure the shapes of the resonant modes in addition to their frequencies (Ogi et al., 2002). Both methodologies limit the number of specimens measurable in a given amount of time. In contrast, the method described here does not require the user to play a role during the inversion, nor to modify the original simple RUS setup. The Bayesian inversion scheme developed in Chapter4 requires a normal prior distribution on the elastic parameters. Here, the approach was extended to a collection of specimens through hierarchical modeling. All the specimen are processed together, and the distribution of parameters in the collection is used as a prior for each specimen. To infer the properties of the population distribution, a nonifor- mative prior has been used for the means, and a weekly informative prior for the covariance matrix, so that the final estimates are not biased by information independent from the measurements. The final estimates appears to be weakly sensitive to a change in the prior distribution of the covariance matrix. Indeed, both changing the number of degree of freedom ν0, and scaling up the matrix Λ0 by a factor 2 or 3, did not affect the results significantly (i.e. produced a deviation larger than the confidence interval). The hierarchical approach proves to be necessary here, because the available experimental elastic parameters from which a prior distribution could have been deduced were different from the parameters of the specimen collection at hand.

Indeed, the mean values of the elastic coeﬃcients in Granke et al.(2011) were C11 = 5.5. Discussion 107

19.5, C33 = 29.2, C44 = 5.7, and C66 = 4.2 GPa, while there are C11 = 14.0, C33 = 26.2, C44 = 5.3, and C66 = 3.4 GPa for the present population. Using the empirical distribution would than have biased the results towards higher stiffness, particularly for the less stiff specimens. A simpler alternative to hierarchical modeling could have been to process first a few representative specimens with manual pairing of the frequencies (Chapter2) to deduce a prior distribution, process all the specimens with that prior, and then possibly update the prior and re-process the whole population. Such a methodology is known as empirical Bayes, and is only an approximation of the full hierarchical approach.

The hierarchical modeling suppose that all the specimens are samples of a single distribution, and that this distribution is normal. Both hypothesis are arguable. First, the specimens were harvested on close but different locations, on donors of both sexes and with different ages. None of this parameters was found to cause significative differences on stiffness in that study (except for only one stiffness coef-

ﬁcient: C13). The assumption of a single distribution is then justiﬁed by the results. This might not be the same in, for example, studies comparing two or more popula- tions (e.g. healthy/unhealthy). Secondly, the normal hypothesis might not be fully adapted. Indeed, the empirical distributions of the elastic parameters were found to deviate from normal distributions, due the presence of a few soft specimens whose mass densities were lower than 1750 g/cm3 (Fig. 5.8). Hence it might have been more adapted to drop the normal hypothesis, and the associated mathematical convenience, to use an alternative distribution (e.g. a long-tailed distribution such as the Student’s distribution). This was not pursued in this work.

The precision of the results, quantified by the half-width of the 68% confidence intervals deduced from the posterior probability distribution, is 1-2% for the shear coefficients and 2-3% for the Young’s moduli (median values). The other coefficients are determined with a lower precision: 4% for C11, 7% for C33, 11% for C13 and 8-10% for the Poisson’s ratios. This ordering of the precision, depending on the nature of the elastic parameter, is consistent with results reported in the previous chapters and the RUS literature in general (e.g. Migliori and Sarrao, 1997; Landa et al., 2009). The precision is however slightly worst in the present study than in Chapters2 and3. This is in part due to the inversion method, which does not assume the pairing to be known here. Therefore, the posterior distribution of the elastic parameters is a compound of several distributions for several probable pairings, which gives more or less complicated posterior distribution depending on the specimen.

Another reason of the degraded precision is the slightly worst agreement between predicted and measured frequencies, reﬂected in the median value of the maximum a posteriori estimates of σ (0.83%), which is larger than the RMS errors reported 108 Chapter 5. RUS on a collection of human tibial bone specimens in Chapter2 for a bone specimen (0.3 %) or in Chapter3 for two bone mimetic material specimens (0.42 and 0.45 %). This has several potential causes:

• The geometry of the specimens might diﬀer from a perfect rectangular parallelepiped. Several specimens were harvested from the same bone cross-section, which required several repositioning of the section during cutting. Although alignment was carefully checked, some geometrical imperfection inevitably oc- curred. Deviations from perpendicularity of about 1◦ or larger can shift the frequencies of a few tenth of percents (Migliori and Sarrao, 1997).

• Also because of cutting, the bone axial direction might be tilted with respect to the edges of the specimens. A tilt angle of a few degrees can shift the frequencies of a few tenth of percents (Migliori and Sarrao, 1997).

• Due to the large quantity of data, the signal processing – which is still a user- dependent step – might not have been done as carefully than is was previously done for a few specimens.

• The transversely isotropic symmetry might not be an accurate model for all the specimens. Although Espinoza Orias et al.(2008) and Granke et al.(2011) found weak or nonexistent statistical diﬀerences in elastic properties in the plan orthogonal to the bone axis for femur mid-diaphysis, some anisotropy might still occur for individual specimens, for example due to a gradient in density between the inner an outer parts of the cortical shell.

• The forward model assumes homogeneity of the material, while bone is fun- damentally inhomogeneous. RUS has been shown to be weakly sensitive to small linear gradients of elastic properties (Seiner et al., 2012), but this might not be an accurate model for bone. Ulrich et al.(2002) studied the eﬀect of a localized mass perturbation (that could be a large pore in bone), and found that a frequency shift up to 0.2% could be observed.

All the arguments above might also explain why no satisfactory agreement between predicted and measured frequencies could be obtained for four of the specimens. Viscoelasticity entails a variation of the real and imaginary part of the elastic parameters with frequency (Lakes, 2009). Hence, because the frequency of the first peak spans a large range (100-400 kHz), it was questionable whether the width of the first resonant peak was dependent on its central frequency. No linear correlation was found between these two quantities (p=0.99), which indicates that the effect of frequency on the viscoelastic parameter is smaller than the effects of bone structure and composition. Moreover, Qfirst was not correlated to the relative sensitivity of the first peak to the axial shear stiffness C44 (p=0.64). Therefore, the approximation made to estimate the axial shear viscoelaticity from the first peak alone did not 5.5. Discussion 109

Table 5.1: Linear correlation of the elastic coeﬃcients and engineering moduli (GPa,

except Poisson’s ratios), viscoelastic parameter Q44 (no dimension) and anisotropy ratios (no dimension) to the apparent mass density (g/cm−3).

Parameter Corr. coef. r2 p-value Slope Intercept Range

−5 C11 0.80 < 10 28.9 -39.4 5.3 - 20.3 GPa −5 C33 0.85 < 10 30.6 -30.8 16.8 - 31.1 GPa −5 C13 0.53 < 10 7.0 -4.8 5.1 - 11.0 GPa −5 C44 0.89 < 10 9.4 -12.2 2.4 - 7.2 GPa −5 C66 0.87 < 10 8.0 -11.3 1.2 - 5.0 GPa −5 E1 0.86 < 10 22.7 -32.3 3.1 - 14.3 GPa −5 E3 0.80 < 10 33.5 -42.7 7.9 - 25.4 GPa ν12 0.11 < 0.05 0.13 0.14 0.21 - 0.49 −4 ν13 0.25 < 10 -0.11 0.40 0.14 - 0.30 −5 ν31 0.72 < 10 -0.61 1.54 0.29 - 0.68 −5 C33/C11 0.60 < 10 -2.5 6.58 1.35 - 3.15 −5 C44/C66 0.77 < 10 -1.3 4.01 1.38 - 2.17 −5 E3/E1 0.54 < 10 -2.0 5.85 1.67 - 3.31 −5 Q44 0.69 < 10 31.8 -31.6 17.3 - 31.1 introduce a bias. This is a very straightforward analysis, not even requiring to estimate the elasticity. The anisotropic viscoelastic analysis revealed a signiﬁcant anisotropy in both shear and Young’s moduli. The agreement between measured and predicted peak widths was about 5%, which is comparable to the precision of the measurement, and comparable to previous reported results for similar analysis on low damping materials (Seiner et al., 2013). We observed that the contribution of the imaginary part of the Poisson’s ratios was negligible given that precision, which means that a simple three parameter model can be used, requiring only a few measured peak widths to lead reliable results.

Tibial cortical bone viscoelasticity

All the elastic coefficients, engineering moduli, and anisotropy ratios show a statistically significant dependence to the apparent mass density (Table 5.1, Fig. 5.8 2 2 and Fig. 5.9). This dependence is strong (r > 0.5), except for ν12 (r = 0.11) 2 and ν13 (r = 0.25) which are almost constant functions of density. The elastic coefficients Cij, the Young’s moduli and the Poisson’s ratio ν12 increase with mass density, while the two other Poisson’s ratios and the three anisotropy ratios decrease with mass density. Such a strong dependence of bone stiffness to the apparent mass 110 Chapter 5. RUS on a collection of human tibial bone specimens density was expected. Mass density is indeed known to be a good predictor of bone millimeter-scale stiffness, both for trabecular (Carter and Hayes, 1977) and cortical tissues (Schaffler and Burr, 1988; Keller, 1994). Linear or power law relations between density and stiffness are used in studies of bone mechanics at organ scale to set the stiffness in the numerical models from density measurement (Schileo et al., 2007, 2008; Austman et al., 2008). Measuring femoral bone elasticity using ultrasound waves, Espinoza Orias et al.(2008) also observed correlations between apparent mass density and longitudinal stiffness, with similar strength, but no correlation with shear stiffness. Keller(1994) found a strong linear correlation ( r2 = 0.69) between the Young’s modulus (mechanical testing) and dry mass density in human femoral bone.

The observed decrease in longitudinal stiffness anisotropy ratio C33/C11 with increasing density is also consistent with the findings of Espinoza Orias et al.(2008), who reported a slope of -2.95 (r2 = 0.61), close to the present result (Table 5.1). Moreover, micro-mechanical models relating bone elasticity to porosity and microscale elastic properties predict an increase of the anisotropy with increasing porosity (Parnell and Grimal, 2009; Parnell et al., 2011). As density and porosity are in first approximation inversely related, the observed variation in anisotropy is consistent with the models. However, the longitudinal stiffness anisotropy ratios C33/C11 reported here are larger than previously measured or predicted values for femoral bone: Espinoza Orias et al.(2008) found C33/C11 in the range 1.25 - 1.75, and Parnell et al.(2011) predicted 1.3 - 1.5 for the physiological range of porosity. We observe a larger range of 1.35 - 3.15. This might be due to physiological differences between tibia and femur, but could also be partially explained by the large range of mass densities in our specimen collection. To our knowledge, experimental observations of the decrease in the two other anisotropy ratios – C44/C66 and E3/E1 – are not reported in the literature. Overall, the increase in anisotropy with decreasing density indicates that the transverse elastic properties (C11, E1, and C66) are more sensitive than the axial properties (C33, E3, and C44) to bone material losses. This might be relevant for bone quality assessment using quantitative ultrasound methods. Physiologically, it is consistent with the function of the tibia, which withstand mainly axial stresses.

Two of the Poisson’s ratios – ν12 and ν13 – were found to vary weakly with apparent density (Table 5.1 and Fig. 5.8). This is consistent with the model of Parnell et al.(2011), which predicts small variations in Poisson’s ratios as a function of porosity. The mean values are ν12 = 0.397 and ν13 = 0.191, in good agreement with values previously measured on femur and tibia using ultrasound wave velocities, reported in Table 5.2. The third Poisson’s ratio ν31 is related to ν13 trough ν ν 31 = 13 . (5.28) E3 E1 5.5. Discussion 111

Table 5.2: Mean Poisson’s ratios measured in this study compared to values of the literature, measured using ultrasound wave velocities. Standard deviations are indicated in parenthesis, if available. In all studies, direction 3 is the axial direction of bone. Our study assumes a transversely isotropic symmetry, which impose equality of pairs of Poisson’s ratios.

This study Ashman et al.(1984) Rho(1996)

Bone Tibia Femur Tibia Number of specimens 55 60 96

ν12 ν12=ν21 0.376 0.420 (0.075) ν21 =0.397 (0.042) 0.422 0.435 (0.057) ν13 ν13=ν23 0.222 0.237 (0.041) ν23 =0.191 (0.024) 0.235 0.231 (0.035) ν31 ν31=ν32 0.371 0.417 (0.048) ν32 =0.390 (0.077) 0.350 0.390 (0.021)

Therefore, the observed variations of ν31 with density reflects the variation of the anisotropy ratio E3/E1 (Fig. 5.8). Bone viscoelasticity is difficult to characterize using ultrasound methods based on wave propagation, because these methods can not usually separate two cumulative effects of energy losses: intrinsic viscoelastic losses and scattering effects (Lakes et al., 1986; Haïat, 2011). In contrast, RUS is free from any scattering effect, and is only sensitive to the intrinsic viscoelastic losses (neglecting the energy leakage due to contact with air and with the transducers, as discussed in Chapter3). This advantage makes RUS a valuable tool to study the phenomena responsible for energy dissipation in bone. To the best of our knowledge, the dependence of bone axial shear damping to mass density was not reported before. However, a negative correlation of broadband ultrasonic attenuation with bone apparent density and mineral density was observed in bovine cortical bone (Sasso et al., 2008), which is consistent with our result. Anisotropic viscoelasticity for the Young’s modulus has been previously observed in bovine cortical bone using mechanical tests (Iyo et al., 2004). Here, anisotropy is observed in both shear and Young’s viscoelastic moduli. It is not clear which physical phenomena are responsible for the viscoelastic behavior of bone (Lakes and Katz, 1979), particularly at ultrasonic frequency. Among several potential causes we can mention fluid motions inside the pores, thermomechanical effects and motions at interfaces (e.g. motion of at the cement lines between osteons or motion of the collagen lamellae inside osteons). Due to the structural anisotropy of cortical bone, all these phenomena are expected to entail an anisotropic viscoelastic behavior. 112 Chapter 5. RUS on a collection of human tibial bone specimens

No effects of age or sex were found on bone apparent density, elasticity, anisotropy and damping. However, the number of donors (6 males, 13 females) might be too low for any effect of the sex to be observable, and the age range (69 - 94 years) was quite narrow. The influence of these two parameters is worth being investigated further using RUS. As a mechanism of energy dissipation, viscoelaticity is intimately related to crack initiation and propagation in bone (Schapery, 1975; Ritchie et al., 2009), and bone strength and toughness have been found to be rate-dependent (Carter and Hayes, 1977; Currey, 1988a). Crack-initiation and crack-growth toughness in human cortical bone are also age-dependent (Nalla et al., 2006; Zimmermann et al., 2011). Using nanoindentation, Wu et al.(2012) observed a difference in creep time in cortical bone for human females and males.

5.6 Conclusion

RUS was for the first time applied on a large collection of cortical bone specimens. The full stiffness tensor was obtained under a transversely isotropic symmetry assumption for 55 out the 59 specimens, with a precision ranging from about 1% (axial shear stiffness) to about 10% (diagonal stiffness coefficient). This demonstrate that RUS is a valuable tool for a routine assessment of bone anisotropic elasticity on millimeter-sized ex vivo specimens. Additionally, the shear viscoelasticity in the axial direction of bone was characterized for all specimens from the first resonant peak, and three viscoelastic parameters could be obtained for some specimens. We observed that bone elasticity is strongly dependent on bone apparent mass density, which was expected. The axial shear viscoelastic parameter also correlates strongly with mass density. Anisotropy of the viscoelastic behavior was observed: the Q ratios of the real to the imaginary part of the moduli are larger in the axial direction of bone, both for Young’s and shear moduli. This might help to better understand the phenomena responsible of bone viscoelasticity, which is intimately related to bone fracture mechanisms. In the view of the promising results on the anisotropic damping, it might be worthwhile to reformulate the estimation of the damping parameters in Bayesian terms. The introduction of prior information might allow a reliable inversion for more than a few selected specimens, and then permit to investigate the variation of damping anisotropy with bone apparent density. About the precision of the elastic coefficients estimation, the main limiting factors are the imperfect geometry of specimens and the limited number of measured frequencies. The first difficulty could be overcome by a more complex cutting protocol, or by measuring and tacking into account the imperfections of the specimens in the forward modeling (e.g. using finite element modeling). The second difficulty is intimately related to the strong damping of ultrasound in bone. Better signal processing methods could perhaps 5.6. Conclusion 113 be developed to find more resonant frequencies. Particularly, the spectrum were processed independently of any information on the material properties and specimen shape, while it is clear that this information could help the estimation of the peak parameters.

Chapter Summary and conclusion

Deep understanding of the structure-function relationships of cortical bone in the context of bone quality assessment is still missing. Cortical bone millimeter-scale elasticity is an important aspect of bone quality but is diﬃcult to measure, even ex vivo. Currently available methods – mechanical testing and ultrasound waves velocity measurements – have limitations arising mainly from the anisotropy of the tissue, its heterogeneity, and the small size of the specimens. In this work, resonant ultrasound spectroscopy (RUS), a method particularly suitable for anisotropic elasticity assessment on small specimens, has been adapted to highly attenuative materials such as cortical bone. Additionally, RUS also provided insight into the viscoelastic behavior of bone.

RUS estimates the elastic properties from the free vibration resonant frequencies of a single millimeter-sized specimen, and the viscoelastic damping from the width of the resonant peaks. Its application to cortical bone was considered particularly challenging. Indeed, the high damping of the vibrations modes causes overlapping of the resonant peaks, which prevents a direct measurement of the resonant frequencies. Moreover, the observed resonant modes are not easily identiﬁed to their corresponding predicted modes, thereby causing ambiguities in the inverse estimation of the mechanical parameters. To overcome these diﬃculties, adaptations of all the steps of the RUS method – measurement, signal processing and inverse estimation of the material properties – have been introduced:

• Shear transducers are used instead of longitudinal transducers (Chapter3) because 1) they produce a stronger signal and 2) the relative amplitudes of the resonant modes can be varied by rotating the sample with respect to the polarization axis of the transducers, allowing to detect more resonant modes.

• Two signal processing methods have been introduced to retrieve the resonant frequencies and peaks width in the overlapped spectra (Chapters2 and3). The first signal processing approach is based on the construction of a linear filter predicting the mechanical response of the specimen in the time domain. An additional curve-fitting step in the frequency domain was combined to the first method to improve accuracy and repeatability.

• An original Bayesian formulation of the inverse problem has been proposed (Chapters4). It gives an automated solution to the joint problems of mode 116 Summary and conclusion

identification and stiffness estimation, without requiring measurement of the modal shapes or manual mode pairing. A sampling method based on Markov Chain Monte Carlo was used to compute the solution. The proposed adaptations of the measurement setup and signal processing were carefully tested and validated by applications to a cortical bone specimen and to PMMA and bone-mimicking composite material specimens. The accuracy and repeatability were quantified and compared to the concurrent methods. RUS measurements in bone were found in good agreement with wave velocity measurements. A comparable (longitudinal stiffness) or better (shear stiffness) precision was obtained from RUS. Results on the bone-mimicking anisotropic composite showed good agreement with results from both wave velocity measurements and mechanical testing. On PMMA, an isotropic material, precision of the elastic characterization was shown to be better than 1% for the shear and Young’s moduli. The viscoelastic characterization was precise enough the reveal a slightly negative imaginary part of the Poisson’s ratio for PMMA, and to detect anisotropic damping for the composite material. The Bayesian formulation of the inverse problem was validated using a data set from the literature, in which the exact mode identification is reported. A comparable accuracy and precision in stiffness estimation could be obtained from a completely automated identification of the resonant frequencies, without requiring the measurement of the modal shapes. In the application to bone, results similar to those of a manual pairing were obtained. However, in that case, the introduction of prior information on the expected elasticity parameters was necessary to regularize the problem, due to a low ratio of the number of measured frequencies to the number of elastic parameters. The Bayesian formulation was complemented in Chapter5 by a method for constructing such a prior from the actual distribution of elastic properties in a collection of specimens. Hence, the prior information required to regularize the inverse problem for each specimen of the collection was obtained from the whole collection instead of being set independently from the measurements. Thanks to automation, the method could be applied to a large number of bone specimens (n = 59), providing a unique data set on human tibial cortical bone anisotropic viscoelasticity. To the best knowledge of the author, it is also the largest specimen collection ever investigated using RUS in a single study, whatever the research context (materials sciences, geophysics, biomechanics). The full stiffness tensor, under a transversely isotropic symmetry assumption, and the axial shear damping ratio were obtained for 55 out of 59 specimens. For some specimens, anisotropy of the damping ratios was also observed. This might help to better understand the physical phenomena of mechanical energy dissipation in bone. 117

Throughout this work, the simplest form of the RUS measurement setup – two ultrasonic transducers in contact with the specimen – was used. Particularly, to solve the challenging problem of identification of the resonant modes, emphasis was placed on data processing rather than on addition to the experimental setup (e.g. laser interferometer). Simplicity is an advantage of RUS over concurrent methods which require expensive and complex apparatus, difficult specimen preparation, and/or measurements in several directions. A simple and low-cost setup could help to promote the use of RUS as a routine method for bone characterization. The developed tools might also prove useful for the application of RUS in other contexts (e.g. industrial). There is of course room for improvements. Particularly, the crucial step of frequency retrieving in the overlapped spectra appears to rely on the experience of the user. The choice has been made throughout this work to consider separately the problems of spectral analysis and stiffness estimation, as they were naturally separated in the pre-existing RUS literature. The pertinence of this choice for the application to bone is however now questionable. Signal processing could indeed benefit from information related to the viscoelasticity of the material, for example to set the number of expected resonant peaks in a frequency band or to constrain the peak widths. Improved spectral analysis would in turn improve the mechanical parameters estimation. The Bayesian framework might be adapted to formulate the problem that way. Extension of the proposed methodology in the application of RUS to complex specimen shapes, using finite element analysis, is envisaged. Some technical chal- lenges will have to be overcome, but the result could be rewarding. Indeed, it is challenging to prepare regular specimens from bones of small animals (e.g. mice, rat, rabbits), but such studies are interesting because of the control that the exper- imenter has on many parameters, e.g. genetics.

In conclusion, RUS has been shown to be an advantageous method for the assessment of anisotropic viscoelasticity of cortical bone – or of any other material whose viscoelastic properties are similar – from millimeter-sized specimens. Some additional eﬀorts could even enlarge the spectrum of potential applications. Hopefully, RUS will contribute to answer some intriguing and important questions related to bone mechanics.

Appendix A Appendix: Transformed elastic parameters

A.1 Isotropic symmetry

The isotropic stiﬀness tensor is, in the two index notation:   C11 C12 C12 0 0 0   C12 C11 C12 0 0 0    C12 C12 C11 0 0 0  Cij =   , (A.1)  0 0 0 C 0 0   44     0 0 0 0 C44 0  0 0 0 0 0 C44

with the relation C12 = C11 − 2C44. It involves two parameters. Thermodynamics constraints requiring the tensor to be positive are 4 C > C > 0. (A.2) 11 3 44

The use of C11 and C44 as random variables in the Bayesian formulation implies two diﬃculties in the deﬁnition of a prior distribution: 1) their positiveness is not consistent with a normal distribution and 2) they must satisfy C11 > (4/3)C44, and should then be statistically dependent variables. Constraints (A.2) transform to: 4 K = C − C > 0 ; G = C > 0 , (A.3) 11 3 44 44 where K and G are respectively the bulk and shear modulus, two statistically independent random variables. Their logarithms

m1 = log(K/K0), (A.4a)

m2 = log(G/G0), (A.4b) with arbitrary constants K0 and G0 are consistent with a normal distribution on m = [m1; m2].

A.2 Cubic symmetry

The cubic elastic tensor is (A.1), without the relation between C12 and the two other parameters. Positive-deﬁniteness is obtained with positiveness of the bulk modulus

K and of the two shear modulus G1 and G2, deﬁned as (Walpole, 1986):

K = (C11 + 2C12)/3, (A.5a) 120 Appendix: Transformed elastic parameters

G1 = (C11 − C12)/2, (A.5b)

G2 = C44. (A.5c)

The logarithms m1, m2, and m3 of K, G1 and G2 are statistically independent variables consistent with a normal distribution.

A.3 Transversely Isotropic symmetry

The transversely isotropic stiffness tensor is:   C11 C12 C13 0 0 0   C12 C11 C13 0 0 0    C13 C13 C33 0 0 0  Cij =   , (A.6)  0 0 0 C 0 0   44     0 0 0 0 C44 0  0 0 0 0 0 C66 with the relation C12 = C11 − 2C44, and therefore involves five parameters. A necessary and sufficient set of conditions for positive-definiteness is (Auld, 1990)

2 C11 > C66 > 0; C44 > 0; C33(C11 − C66) > C13 . (A.7)

Denoting κ = C11 − C66, it transforms to

{κ > 0 ; C66 > 0; C44 > 0; C33 > 0} , and (A.8a) C − 1 < x = √ 13 < 1. (A.8b) C33κ This last variable x can be transformed to a positive variable using η = (1 + x)/(1 − x), and we then have ﬁve independent positive variables {κ; C33; η; C44; C66} that can be mapped onto variables mi (i = 1, ..., 5) deﬁned from −∞ to +∞ by taking their logarithm.

A.4 Orthotropic symmetry

The orthotropic stiffness tensor is:   C11 C12 C13 0 0 0   C12 C22 C23 0 0 0    C13 C23 C33 0 0 0  Cij =   , (A.9)  0 0 0 C 0 0   44     0 0 0 0 C55 0  0 0 0 0 0 C66 A.4. Orthotropic symmetry 121 with no relation between the coefficients. It involves nine parameters. The conditions for positive-definiteness are that all the principal minors (determinants of the k × k sub-matrices for k = 1, ..., 6) of (A.9) are positive (Auld, 1990). Al- though it may exists a set of nine independent real-valued variables that satisfies this conditions, we did not find a simple transformation leading to such parametrization. Instead, we use the logarithms of the positives parameters

{C11; C22; C33; η12; η13; η23; C44; C55; C66;} with p 1 + Cij/ CiiCjj ηij = p . (A.10) 1 − Cij/ CiiCjj

This parametrization ensures that all the principal minors of order 1 and 2 are positive but does not guaranty   C11 C12 C13   det C12 C22 C23 > 0. (A.11) C13 C23 C33

Hence the prior distribution on m must be a truncated normal, attributing zero probability for non positive-deﬁnite tensors. It was obtained by introducing in Gibbs sampling a rejection of the non positive-deﬁnite samples.

Appendix B Appendix: Cutting protocol for human tibia specimens

This section describes and illustrates the cutting protocol used for the preparation of the tibial cortical bone specimens measured in Chapter5. This protocol was designed to allow for a fast specimen harvesting while still maintaining a good specimen geometry, which is crucial for RUS (particularly perpendicularity of the faces of the specimen). The tibia specimens were harvested from the donors in Germany, and arrived in our lab in the form of 20 mm-thick cross sections, partially embedded in PMMA. The objective was then to prepare four specimens form the medial face of each cross section, at specific sites registered by the lines on Fig. B.1 (C: center of the medial face, C-1/3: one third off the center to the right, and so on). Due to the curvature of the medial face, the C and C+2/3 required different cutting directions. It was therefore necessary to split the cross section in two parts before cutting the specimens. This splitting was done along the C+1/3 line. Un- fortunately it implied that one of the specimen was not exactly centered on its line. Moreover, when the curvature the medial face was strong, the C+1/3 or C-1/3 specimens could not always be harvested.

B.1 Material

The saw was an Isomet 4000 linear saw (Buehler GmbH, Düsseldorf, Germany), used with a 0.3 mm thick diamond coated Isomet wafering blade (Fig. B.2). All cuttings

20 mm PMMA 5 mm

Medial face

C+2/3 C+1/3 C C-1/3

Figure B.1: Scheme and photograph of the specimens location. A cross-section was harvested from the mid-diaphysis and two to four specimens extracted from the medial face at registered locations. See the text for details. 124 Appendix: Cutting protocol for human tibia specimens

Figure B.2: Left: Buelhler Isomet 4000 linear saw. Right: Diamond coated Isomet wafering blade and ﬁxing material. were done at 3800 rpm and 7 mm/min feed rate. A drill press was used to drill a hole in the PMMA layer in order to attach the specimen. A custom made sample holder and a rectangular aluminum bar were used to ensure 90◦ cutting angles. The bones were already prepared as 20 mm thick cross sections, partially embedded in PMMA for ease of manipulation. A real size print-out of the specimen, with the C, C-1/3, C+1/3, and C+2/3 lines drawn on it, is required for site-matched specimen preparation. We used images from 50 MHz scanning acoustic microscopy.

B.2 Specimen preparation

• Unfreeze the tibia cross-section in phosphate buﬀered saline solution with calcium chloride an magnesium chloride (PBS ; Sigma-Aldrich) for at least 15 minutes.

• Drill a hole in the center of the PMMA support disk through the bone marrow of the sample and ﬁx it on aluminium bar, putting a washer between both materials (Fig. B.3). To avoid that the bone dries out, store it in PBS.

• Deﬁne the cut position of the C+2/3 specimen on the print-out. For that, place the tangent on C+2/3 position and draw a line perpendicular to the tangent/bone surface (Fig. B.4). Cut out the print-out and place it on the wet bone sample.

• Place the specimen with print-out into the saw and use a piece of sand paper to avoid movements of aluminium bar when tightening the screws (Fig. B.5, a). Align the C+1/3 line with the blade and turn the PMMA disk carefully/slightly in order to make it parallel (Fig. B.5, b). Tighten the screw in the center of the PMMA disk again to make sure that the sample does not move during the whole cutting procedure. Cut the bone sample and B.2. Specimen preparation 125

Figure B.3: Left: A hole was drill in PMMA in the center of the cross-section. Right: The section in then ﬁxed on a rectangular aluminium bar, that will be used to obtain 90◦ angles.

Figure B.4: Left: Cutting lines for the C+2/3 specimen. Right: Real size print-out positioned on the section. 126 Appendix: Cutting protocol for human tibia specimens

b a

c d

Figure B.5: Separation of the C+2/3 part form the rest of the cross-section.

PMMA disk along the C+1/3 line. Accordingly, divide the print-out along the C+1/3 line as well (Fig. B.5, c) and store the separated outer bone piece (C+2/3 part, Fig. B.5, d) in PBS.

B.3 Transversal cuts

• Place the separated print-out on the bone, which is still ﬁxed in the saw. Position the blade along the C-line (Fig. B.6) and choose the transversal (T) dimension of the bone cube. Dimension in mm = T, Thickness of blade = B (here, B = 0.3mm).

• Deplace the blade of T/2 + B/2 mm to the right and cut until you reach your prefered axial length.

• Position the blade again along the C-line. Deplace the blade of T/2 + B/2 mm to the left and cut until you reach your prefered axial length.

• Repeat the same procedure for the C-1/3 line.

B.4 Radial cuts

• Turn the aluminium bar of 90◦ and screw it into the sample holder again (Fig. B.7, left). The ﬁrst radial cut should remove the round outer surface of B.5. Axial cuts 127

Figure B.6: Transversal cut of the C specimen.

the bone. Cut the PMMA disk at the same time to improve cutting access for later axial cuts .

• The second radial cut (Fig. B.7, right) should completely remove the inner trabecular part of bone in order to create proper cortical bone cubes. After that, demount the sample.

B.5 Axial cuts

• 1st cut: Turn the position of the aluminium bar of 90◦ (horizontal direction) and carry out the ﬁrst axial-cut in order to remove a thin surface layer to have better parallelism (Fig. B.8).

• 2nd cut: use caliper to define the final axial lenght (A) of the specimens. Mark the cube with a waterproof pen according to a defined convention to identify anatomical direction before the last cut. Place sponges underneath the blade in order to avoid that the specimens are taken out by the water flow in the the saw when it falls down.

Figure B.7: Radial cuts. 128 Appendix: Cutting protocol for human tibia specimens

• Get the C, C-1/3 and C+1/3 specimens from inside the saw. The C+1/3 specimen is deﬁned by the left cut of the C specimen and the cut along the C+1/3 line made to split the cross section in two parts (Fig. B.1).

Figure B.8: Axial cuts.

B.6 C+2/3 specimen

• Drill a hole in the PMMA where the surface is plane. Fix it then on the aluminium bar using a washer to provide better support (Fig. B.9, left).

• Place the +2/3 part of the print-out on bone and ﬁx it to the sample holder (Fig. B.9, left).

• Align the perpendicular line at C+2/3 position with blade and cut out the C+2/3 cube following the steps explained above.

– First, two transversal cuts. – Then, two radial cuts. – Finally, two axial cuts (Fig. B.9, right). B.6. C+2/3 specimen 129

Figure B.9: Cutting of the C+2/3 specimen.

Bibliography

S. Adhikari and J. Woodhouse. Identiﬁcation of damping: part 1, viscous damping. J. Sound Vib., 243(1):43–61, 2001.

P. Ammann and R. Rizzoli. Bone strength and its determinants. Osteoporosis Int., 14(3):13–18, 2003.

R. B. Ashman, S. C. Cowin, W. C. Van Buskirk, and J. C. Rice. A continuous wave technique for the measurement of the elastic properties of cortical bone. J. Biomech., 17(5):349–361, 1984.

B. A. Auld. Acoustic ﬁelds and waves in solids, 2nd edition, volume 1. Krieger, Malabar, Florida, 1990.

R. L. Austman, J. S. Milner, D. W. Holdsworth, and C. E. Dunning. The eﬀect of the density–modulus relationship selected to apply material properties in a ﬁnite element model of long bone. J. Biomech., 41(15):3171 – 3176, 2008.

S. Bernard, Q. Grimal, S. Haupert, and P. Laugier. Assessment of anisotropic elasticity of small bone samples with resonant ultrasound spectroscopy: attenuation does not prevent the measurements. In Ultrasonics Symposium (IUS), IEEE, Sept. 2011.

S. Bernard, Q. Grimal, and P. Laugier. Accurate measurement of cortical bone elasticity tensor with resonant ultrasound spectroscopy. J. Mech. Behav. Biomed. Mater., 18(0):12–19, 2013.

S. Bernard, Q. Grimal, and P. Laugier. Resonant ultrasound spectroscopy for viscoelastic characterization of anisotropic attenuative solid materials. J. Acoust. Soc. Am., 135(5):2601–2613, 2014.

J.-P. Berteau, C. Baron, M. Pithioux, F. Launay, P. Chabrand, and P. Lasaygues. In vitro ultrasonic and mechanic characterization of the modulus of elasticity of children cortical bone. Ultrasonics, 54(5):1270–1276, 2014.

A. F. Bower. Applied mechanics of solids. CRC Press, Bocca Raton, Florida, 2009.

M. D. Brodt, C. B. Ellis, and M. J. Silva. Growing C57Bl/6 mice increase whole bone mechanical properties by increasing geometric and material properties. J. Bone Miner. Res., 14(12):2159–2166, 1999.

D. Carter and W. Hayes. The compressive behavior of bone as a two-phase porous structure. J. Bone Joint Surg., 59(7):954–962, 1977. 132 Bibliography

G. Casella and E. I. George. Explaining the gibbs sampler. Am. Stat., 46(3):167–174, 1992.

V. Cerný. Thermodynamical approach to the traveling salesman problem: An eﬃ- cient simulation algorithm. J. Optim. Theory Appl., 45(1):41–51, 1985.

N. L.-T. Chattah, A. Sharir, S. Weiner, and R. Shahar. Determining the elastic modulus of mouse cortical bone using electronic speckle pattern interferometry (ESPI) and micro computed tomography: A new approach for characterizing small-bone material properties. Bone, 45(1):84–90, 2009.

S. Cowin. Bone mechanics handbook. CRC Press, 2nd edition, 2001.

J. Currey. Strain rate and mineral content in fracture models of bone. J. Orthop. Res., 6(1):32–38, 1988a.

J. Currey. The eﬀect of porosity and mineral content on the Young’s modulus of elasticity of compact bone. J. Biomech., 21(2):131–139, 1988b.

T. Delaunay, B. Morvan, M. Guennou, H. Dammak, M. P. Thi, and G. Feuillard. Full tensorial characterization of PZN-12PT single crystal by resonant ultrasound spectroscopy. IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 55(2):476–488, February 2008.

H. H. Demarest. Cube resonance method to determine the elastic constants of solids. J. Acoust. Soc. Am., 49(3B):768–775, 1971.

J. M. Deuerling, W. Yue, A. A. E. Orias, and R. K. Roeder. Specimen-speciﬁc multi-scale model for the anisotropic elastic constants of human cortical bone. J. Biomech., 42(13):2061–2067, 2009.

G. N. Duda, M. Heller, J. Albinger, O. Schulz, E. Schneider, and L. Claes. Inﬂuence of muscle forces on femoral strain distribution. J. Biomech., 31(9):841 – 846, 1998.

A. A. Espinoza Orias, J. M. Deuerling, M. D. Landrigan, J. E. Renaud, and R. K. Roeder. Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur. J. Mech. Behav. Biomed. Mater., 2:255–263, 2008.

F. Farzbod and D. Hurley. Using eigenmodes to perform the inverse problem associated with resonant ultrasound spectroscopy. IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 59(11), November 2012.

J. Foiret, J.-G. Minonzio, C. Chappard, M. Talmant, and P. Laugier. Combined estimation of thickness and velocities using ultrasound guided waves: a pioneering Bibliography 133

study on in vitro cortical bone samples. IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 61(9):1478–1488, Sept 2014.

D. B. Fraser and R. C. LeCraw. Novel method of measuring elastic and anelastic properties of solids. Rev. Sci. Instrum., 35(9):1113–1115, 1964.

P. Fratzl. Biomimetic materials research: what can we really learn from nature’s structural materials? J. R. Soc. Interface., 4(15):637–642, 2007.

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin. Bayesian data analysis. CRC press, third edition, 2013.

S. Geman and D. Geman. Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell., 6:721– 741, Nov 1984.

W. Gilks, S. Richardson, and D. Spiegelhalter. Markov Chain Monte Carlo in Practice. Chapman & Hall, London, UK, 1995.

M. Granke. Multiscale investigation of bone quality using ultrasound. PhD thesis, Université Paris VII - Denis Diderot, 2012.

M. Granke, Q. Grimal, A. Saïed, P. Nauleau, F. Peyrin, and P. Laugier. Change in porosity is the major determinant of the variation of cortical bone elasticity at the millimeter scale in aged women. Bone, 49(5):1020–1026, 2011.

Q. Grimal, S. Haupert, D. Mitton, L. Vastel, and P. Laugier. Assessment of cortical bone elasticity and strength: Mechanical testing and ultrasound provide complementary data. Med. Eng. Phys., 31(9):1140–1147, 2009.

Q. Grimal, K. Raum, A. Gerisch, and P. Laugier. A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties. Biomech. Model. Mechanobiol., 10(6):925–937, 2011.

G. Haïat. Linear ultrasonic properties of cortical bone: in vitro studies. In P. laugier and G. Haïat, editors, Bone Quantitative Ultrasound. Springer, 2011.

B. Hartmann and J. Jarzynski. Ultrasonic hysteresis absorption in polymers. J. Appl. Phys., 43(11):4304–4312, 1972.

W. K. Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1):97–109, 1970.

P. Heyliger, A. Jilani, H. Ledbetter, R. G. Leisure, and C.-L. Wang. Elastic constants of isotropic cylinders using resonant ultrasound. J. Acoust. Soc. Am., 94(3):1482– 1487, 1993. 134 Bibliography

P. R. Heyliger and W. L. Johnson. Traction-free vibrations of ﬁnite trigonal elastic cylinders. J. Acoust. Soc. Am., 113(4):1812–1825, 2003.

G. Holzer, G. von Skrbensky, L. A. Holzer, and W. Pichl. Hip fractures and the contribution of cortical versus trabecular bone to femoral neck strength. J. Bone Miner. Res., 24(3):468–474, 2009.

B. Hosten and M. Castaings. Comments on the ultrasonic estimation of the viscoelastic properties of anisotropic materials. Composites Part A, 39(6):1054–1058, 2008.

D. G. Isaak and I. Ohno. Elastic constants of chrome-diopside: application of resonant ultrasound spectroscopy to monoclinic single-crystals. Phys. Chem. Miner., 30(7):430–439, 2003.

T. Iyo, Y. Maki, N. Sasaki, and M. Nakata. Anisotropic viscoelastic properties of cortical bone. J. Biomech., 37(9):1433–1437, 2004.

T. Jaglinski and R. S. Lakes. Resonant ultrasound spectroscopy of cylinders over the full range of poisson’s ratio. Rev. Sci. Instrum., 82(3):035105–035105–7, 2011.

J. L. Katz and H. S. Yoon. The structure and anisotropic mechanical properties of bone. IEEE Trans. Biomed. Eng., BME-31(12):878 –884, dec. 1984.

T. M. Keaveny, D. L. Kopperdahl, L. J. Melton, P. F. Hoﬀmann, S. Amin, B. L. Riggs, and S. Khosla. Age-dependence of femoral strength in white women and men. J. Bone Miner. Res., 25(5):994–1001, 2010.

T. S. Keller. Predicting the compressive mechanical behavior of bone. J. Biomech., 27(9):1159–1168, 1994.

G. Kim, A. L. Boskey, S. P. Baker, and M. C. van der Meulen. Improved prediction of rat cortical bone mechanical behavior using composite beam theory to integrate tissue level properties. J. Biomech., 45(16):2784–2790, 2012.

J. H. Kinney, J. R. Gladden, G. W. Marshall, S. J. Marshall, J. H. So, and J. D. May- nard. Resonant ultrasound spectroscopy measurements of the elastic constants of human dentin. J. Biomech., 37(4):437–441, 2004.

S. Kirkpatrick, C. D. Gelatt, Jr., and M. P. Vecchi. Optimization by simulated annealing. Science, 220(4598):671–680, 1983.

D. P. Kroese, T. Taimre, and Z. I. Botev. Handbook of Monte Carlo Methods. Wiley, Hoboken, New Jersey, 2011. Bibliography 135

R. Kumaresan and D. Tufts. Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise. IEEE Trans. Acoust., Speech, Signal Process., 30(6):833–840, dec 1982.

R. Lakes. Viscoelastic materials. Cambridge University Press, 2009.

R. Lakes, H. S. Yoon, and J. L. Katz. Ultrasonic wave propagation and attenuation in wet bone. J. Biomed. Eng., 8(2):143–148, 1986.

R. S. Lakes and J. L. Katz. Viscoelastic properties of wet cortical bone – II. Relax- ation mechanisms. J. Biomech., 12(9):679–687, 1979.

H. Lamb. On the vibrations of an elatic sphere. P. Lond. Math. Soc., 13:189–212, 1882.

M. Landa, P. Sedlák, H. Seiner, L. Heller, L. Bicanová, P. Šittner, and V. Novák. Modal resonant ultrasound spectroscopy for ferroelastics. Appl. Phys., A, 96: 557–567, 2009.

S. B. Lang. Elastic coeﬃcients of animal bone. Science, 165(3890):287–288, 1969.

P. Laugier. Instrumentation for in vivo ultrasonic characterization of bone strength. IEEE Trans. Ultrason., Ferroelectr., Freq. Control., 55(6):1179–1196, June 2008.

P. Laugier and Q. Grimal. Preface to the special section: ultrasonics of bone. Ultrasonics, 54(5):1123–1124, 2014.

P. Laugier and G. Haïat. Bone Quantitative Ultrasound. Springer, 2011.

A. V. Lebedev. Method of linear prediction in the ultrasonic spectroscopy of rock. Acoust. Phys., 48:339–346, 2002.

A. V. Lebedev, L. A. Ostrovskii, A. M. Sutin, I. A. Soustova, and P. A. Johnson. Resonant acoustic spectroscopy at low Q factors. Acoust. Phys., 49:81–87, 2003.

H. Ledbetter, C. Fortunko, and P. Heyliger. Orthotropic elastic constants of a boron-aluminum ﬁber-reinforced composite: An acoustic-resonance-spectroscopy study. J. Appl. Phys., 78(3):1542–1546, 1995.

H. Ledbetter, H. Ogi, and N. Nakamura. Elastic, anelastic, piezoelectric coeﬃcients of monocrystal lithium niobate. Mech. Mater., 36(10):941–947, 2004.

T. Lee, R. S. Lakes, and A. Lal. Resonant ultrasound spectroscopy for measurement of mechanical damping: Comparison with broadband viscoelastic spectroscopy. Rev. Sci. Instrum., 71(7):2855–2861, 2000. 136 Bibliography

T. Lee, R. S. Lakes, and A. Lal. Investigation of bovine bone by resonant ultrasound spectroscopy and transmission ultrasound. Biomech. Model. Mechanobiol., 1:165– 175, 2002.

R. Leisure, K. Foster, J. Hightower, and D. Agosta. Internal friction studies by resonant ultrasound spectroscopy. Mater. Sci. Eng., A, 370:34–40, 2004.

G. Li and J. R. Gladden. High temperature resonant ultrasound spectroscopy: A review. Int. J. Spectrosc., 2010.

G. Liu and J. Maynard. Measuring elastic constants of arbitrarily shaped samples using resonant ultrasound spectroscopy. J. Acous. Soc. Am., 131(3):2068–2078, 2012.

A. Love. A Treatise on the Mathematical Theory of Elasticity. Dover Pubilcations„ 1944.

D. J. C. MacKay. Information Theory, Inference, and Learning Algorithms. Cam- bridge University Press, 2003.

J. D. Maynard. The use of piezoelectric ﬁlm and ultrasound resonance to determine the complete elastic tensor in one measurement. J. Acoust. Soc. Am., 91(3): 1754–1762, 1992.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equation of State Calculations by Fast Computing Machines. J. Chem. Phys., 21:1087–1092, June 1953.

A. Migliori. Resonances and time-of-ﬂight: Ultrasonic probes of condensed matter go digital. Acoustics Today, 4(1):17–22, 2008.

A. Migliori and J. D. Maynard. Implementation of a modern resonant ultrasound spectroscopy system for the measurement of the elastic moduli of small solid specimens. Rev. Sci. Instrum., 76(12):121301, 2005.

A. Migliori and J. L. Sarrao. Resonant Ultrasound Spectroscopy: Applications to Physics, Materials Measurements, and Nondestructive Evaluation. Wiley, New York, 1997.

A. Migliori, W. M. Visscher, S. E. Brown, Z. Fisk, S.-W. Cheong, B. Alten, E. T. Ahrens, K. A. Kubat-Martin, J. D. Maynard, Y. Huang, D. R. Kirk, K. A. Gillis, H. K. Kim, and M. H. W. Chan. Elastic constants and speciﬁc-heat measurements

on single crystals of La2CuO4. Phys. Rev. B, 41:2098–2102, Feb 1990. Bibliography 137

A. Migliori, J. L. Sarrao, W. M. Visscher, T. M. Bell, M. Lei, Z. Fisk, and R. G. Leisure. Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids. Physica B, 183(1-2):1–24, 1993.

R. D. Mindlin. Simple modes of vibration of crystals. J. Appl. Phys., 27(12):1462– 1466, 1956.

J.-G. Minonzio, J. Foiret, M. Talmant, and P. Laugier. Impact of attenuation on guided mode wavenumber measurement in axial transmission on bone mimicking plates. J. Acoust. Soc. Am., 130(6):3574–3582, 2011.

E. Mochizuki. Application of group theory to free oscillations of an anisotropic rectangular parallelepiped. J. Phys. Earth, 35:159–170, 1987.

R. Nalla, J. Kruzic, J. Kinney, M. Balooch, J. A. III, and R. Ritchie. Role of microstructure in the aging-related deterioration of the toughness of human cortical bone. Mat. Sci. Eng. C, 26(8):1251–1260, 2006.

H. Ogi, K. Sato, T. Asada, and M. Hirao. Complete mode identiﬁcation for resonance ultrasound spectroscopy. J. Acoust. Soc. Am., 112(6):2553–2557, 2002.

H. Ogi, N. Nakamura, K. Sato, M. Hirao, and S. Uda. Elastic, anelastic, and piezoelectric coeﬃcients of langasite: resonance ultrasound spectroscopy with laser- doppler interferometry. IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 50(5): 553 –560, 2003.

C. Ohman, M. Baleani, C. Pani, F. Taddei, M. Alberghini, M. Viceconti, and M. Manfrini. Compressive behaviour of child and adult cortical bone. Bone, 49(4):769–776, 2011.

I. Ohno. Free vibration of a rectangular parallelepiped crystal and its application to determination of elastic constants of orthorhombic crystals. J. Phys. Earth, 24: 355–379, 1976.

I. Ohno. Rectangular parallellepiped resonance method for piezoelectric crystals and elastic constants of alpha-quartz. Phys. Chem. Miner., 17(5):371–378, 1990.

I. Ohno, S. Yamamoto, O. L. Anderson, and J. Noda. Determination of elastic constants of trigonal crystals by the rectangular parallelepiped resonance method. J. Phys. Chem. Solids, 47(12):1103–1108, 1986.

W. Parnell, M. Vu, Q. Grimal, and S. Naili. Analytical methods to determine the eﬀective mesoscopic and macroscopic elastic properties of cortical bone. Biomech. Model. Mechanobiol., 11(6):883–901, 2011. 138 Bibliography

W. J. Parnell and Q. Grimal. The inﬂuence of mesoscale porosity on cortical bone anisotropy. investigations via asymptotic homogenization. J. Roy. Soc. Int., 6 (30):97–109, 2009.

J. Plesek, R. Kolman, and M. Landa. Using ﬁnite element method for the determination of elastic moduli by resonant ultrasound spectroscopy. J. Acoust. Soc. Am., 116(1):282–287, 2004.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical recipes in C: the art of scientiﬁc computing, 2nd edition. Cambridge University Press, 1992.

K. Raum. Microscopic elastic properties. In P. laugier and G. Haïat, editors, Bone Quantitative Ultrasound. Springer, 2011.

K. Raum, Q. Grimal, P. Varga, R. Barkmann, C. Glüer, and P. Laugier. Ultrasound to assess bone quality. Curr. Osteoporos. Rep., 12(2):154–162, 2014.

S. J. Reese, K. Telschow, T. Lillo, and D. H. Hurley. On the establishment of a method for characterization of material microstructure through laser-based resonant ultrasound spectroscopy. IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 55(4):770–777, April 2008.

A. G. Reisinger, D. H. Pahr, and P. K. Zysset. Elastic anisotropy of bone lamellae as a function of ﬁbril orientation pattern. Biomech. Model. Mechanobiol., 10(1): 67–77, 2011a.

A. G. Reisinger, D. H. Pahr, and P. K. Zysset. Principal stiﬀness orientation and degree of anisotropy of human osteons based on nanoindentation in three distinct planes. J. Mech. Behav. Biomed. Mater., 4(8):2113 – 2127, 2011b.

M. C. Renken, S. Kim, K. Hollman, and C. Fortunko. Accuracy of resonant ultrasound spectroscopy: an experimental study. In Ultrasonics Symposium (IUS), IEEE, volume 1, pages 483–488, Oct. 1997.

J.-Y. Rho. An ultrasonic method for measuring the elastic properties of human tibial cortical and cancellous bone. Ultrasonics, 34(8):777–783, 1996.

J.-Y. Rho, L. Kuhn-Spearing, and P. Zioupos. Mechanical properties and the hierarchical structure of bone. Med. Eng. Phys., 20(2):92–102, 1998.

R. O. Ritchie, M. J. Buehler, and P. Hansma. Plasticity and toughness in bone. Phys Today, 62(6):41–47, 2009. Bibliography 139

D. Rohrbach, S. Lakshmanan, F. Peyrin, M. Langer, A. Gerisch, Q. Grimal, P. Laugier, and K. Raum. Spatial distribution of tissue level properties in a human femoral cortical bone. J. Biomech., 45(13):2264–2270, 2012.

D. Royer and E. Dieulesaint. Elastic waves in solids, volume 1. Springer, 2000.

D. J. Rudy, J. M. Deuerling, A. A. E. Orías, and R. K. Roeder. Anatomic variation in the elastic inhomogeneity and anisotropy of human femoral cortical bone tissue is consistent across multiple donors. J. Biomech., 44(9):1817–1820, 2011.

M. Sasso, G. Haïat, Y. Yamato, S. Naili, and M. Matsukawa. Dependence of ultrasonic attenuation on bone mass and microstructure in bovine cortical bone. J. Biomech., 41(2):347–355, 2008.

M. B. Schaffler and D. B. Burr. Stiffness of compact bone: Effects of porosity and density. J. Biomech., 21(1):13–16, 1988.

R. Schapery. A theory of crack initiation and growth in viscoelastic media. Int. J. Fract., 11(1):141–159, 1975.

E. Schileo, F. Taddei, A. Malandrino, L. Cristofolini, and M. Viceconti. Subject- speciﬁc ﬁnite element models can accurately predict strain levels in long bones. J. Biomech., 40(13):2982–2989, 2007.

E. Schileo, E. Dall‘Ara, F. Taddei, A. Malandrino, T. Schotkamp, M. Baleani, and M. Viceconti. An accurate estimation of bone density improves the accuracy of subject-speciﬁc ﬁnite element models. J. Biomech., 41(11):2483–2491, 2008.

E. Schreiber and O. L. Anderson. Properties and composition of lunar materials: Earth analogies. Science, 168(3939):1579–1580, 1970.

S. Schuit, M. van der Klift, A. Weel, C. de Laet, H. Burger, E. Seeman, A. Hofman, A. Uitterlinden, J. van Leeuwen, and H. Pols. Fracture incidence and association with bone mineral density in elderly men and women: the Rotterdam study. Bone, 34(1):195–202, 2004.

P. Sedlák, H. Seiner, J. Zídek, M. Janovská, and M. Landa. Determination of all 21 independent elastic coeﬃcients of generally anisotropic solids by resonant ultrasound spectroscopy: Benchmark examples. Exp. Mech., pages 1–13, 2014.

E. Seeman and P. D. Delmas. Bone quality - The material and structural basis of bone strength and fragility. N. Eng. J. Med., 354(21):2250–2261, 2006.

H. Seiner, P. Sedlak, L. Bodnarova, A. Kruisova, M. Landa, A. de Pablos, and M. Belmonte. Sensitivity of the resonant ultrasound spectroscopy to weak gradients of elastic properties. J. Acoust. Soc. Am., 131(5):3775–3785, 2012. 140 Bibliography

H. Seiner, P. Sedlák, M. Koller, M. Landa, C. Ramírez, M. Osendi, and M. Belmonte. Anisotropic elastic moduli and internal friction of graphene nanoplatelets/silicon nitride composites. Compos. Sci. Technol., 75:93–97, 2013.

R. Shahar, P. Zaslansky, M. Barak, A. Friesem, J. Currey, and S. Weiner. Anisotropic Poisson’s ratio and compression modulus of cortical bone determined by speckle interferometry. J. Biomech., 40(2):252–264, 2007.

H. Sievänen, S. Cheng, S. Ollikainen, and K. Uusi-Rasi. Ultrasound velocity and cortical bone characteristics in vivo. Osteoporosis Int., 12(5):399–405, 2001.

J. So and A. Leissa. Three-dimensional vibrations of thick circular and annular plates. J. Sound Vib., 209(1):15–41, 1998.

P. S. Spoor, J. D. Maynard, and A. R. Kortan. Elastic isotropy and anisotropy in quasicrystalline and cubic AlCuLi. Phys. Rev. Lett., 75:3462–3465, 1995.

A. Stekel, J. L. Sarrao, T. M. Bell, M. Lei, R. G. Leisure, W. M. Visscher, and A. Migliori. Method for identiﬁcation of the vibrational modes of a rectangular parallelepiped. J. Acoust. Soc. Am., 92(2):663–668, 1992.

A. Tarantola. Inverse Problem Theory and methods for model parameters estimation. SIAM, 2005.

T. Ulrich, K. R. McCall, and R. A. Guyer. Determination of elastic moduli of rock samples using resonant ultrasound spectroscopy. J. Acous. Soc. Am., 111(4): 1667–1674, 2002.

W. Van Buskirk, S. Cowin, and R. N. Ward. Ultrasonic measurement of orthotropic elastic constants of bovine femoral bone. J. Biomech. Eng., 103(2):67–72, 1981.

G. H. van Lenthe, R. Voide, S. K. Boyd, and R. Müller. Tissue modulus calculated from beam theory is biased by bone size and geometry: Implications for the use of three-point bending tests to determine bone tissue modulus. Bone, 43(4):717–723, 2008.

W. M. Visscher, A. Migliori, T. M. Bell, and R. A. Reinert. On the normal modes of free vibration of inhomogeneous and anisotropic elastic objects. J. Acoust. Soc. Am., 90(4):2154–2162, 1991.

S. Wallach, J. D. Feinblatt, J. H. Carstens, and L. V. Avioli. The bone "quality" problem. Calcif. Tissue Int., 51(3):169–172, 1992.

L. J. Walpole. The elastic shear moduli of a cubic crystal. J. Phys. D: Appl. Phys., 19(3):457, 1986. Bibliography 141

Y. C. Wang and R. S. Lakes. Resonant ultrasound spectroscopy in shear mode. Rev. Sci. Instrum., 74(3):1371–1373, 2003.

L. Wasserman. Bayesian model selection and model averaging. J. Math. Psychol., 44(1):92–107, 2000.

S. Weiner and H. D. Wagner. The material bone: structure-mechanical function relations. Annu. Rev. Mater. Sci., 28(1):271–298, 1998.

Z. Wu, T. C. Ovaert, and G. L. Niebur. Viscoelastic properties of human cortical bone tissue depend on gender and elastic modulus. J. Orthop. Res., 30(5):693–699, 2012.

R. Yang and J. Berger. A catalog of noninformative priors. ISDS Discussion paper, 97(42), 1997.

A. Yoneda. Intrinsic eigenvibration frequency in the resonant ultrasound spectroscopy: Evidence for a coupling vibration between a sample and transducers. Earth Planets Space, 54(7):763–770, 2002. ISSN 1343-8832.

H. S. Yoon and J. L. Katz. Ultrasonic wave propagation in human cortical bone, I. Theoretical considerations for hexagonal symmetry. J. Biomech., 9(6):407–412, 1976a.

H. S. Yoon and J. L. Katz. Ultrasonic wave propagation in human cortical bone, II. Measurements of elastic properties and microhardness. J. Biomech., 9(7):459–464, 1976b.

R. M. Zebaze, A. C. Jones, M. G. Pandy, M. A. Knackstedt, and E. Seeman. Diﬀer- ences in the degree of bone tissue mineralization account for little of the diﬀerences in tissue elastic properties. Bone, 48(6):1246–1251, 2011.

H. Zhang, R. S. Sorbello, C. Hucho, J. Herro, J. R. Feller, D. E. Beck, M. Levy, D. Isaak, J. D. Carnes, and O. Anderson. Radiation impedance of resonant ultrasound spectroscopy modes in fused silica. J. Acoust. Soc. Am., 103(5):2385–2394, 1998.

E. A. Zimmermann, E. Schaible, H. Bale, H. D. Barth, S. Y. Tang, P. Reichert, B. Busse, T. Alliston, J. W. Ager, and R. O. Ritchie. Age-related changes in the plasticity and toughness of human cortical bone at multiple length scales. PNAS, 108(35):14416–14421, 2011.