Outline of the thesis

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This first chapter introduced the research context and motivations of the present work. State-of-the art methods for the assessment of cortical bone elasticity were described, and the need for a new method was argued. Basic principles and history of RUS were briefly reviewed. Finally, the difficulties of the application of RUS to bone were presented.

Chapter2is a research article published under the tileAccurate measurement of cortical bone elasticity tensor with resonant ultrasound spectroscopy in The Journal of the Mechanical Behavior of Biomedical Materials in 2013, and referred to in this manuscript asBernard et al.(2013). In this work, the difficulties of the application to bone are tackled, mainly by adequately combining tools available in the RUS literature. This chapter is a demonstration of the feasibility of RUS for cortical bone elastic characterization.

Chapter 3 is a research article published under the title Resonant ultrasound spectroscopy for viscoelastic characterization of anisotropic attenuative solid mate-rials in the Journal of Acoustical Society of America in 2014, and referred to in this manuscript as Bernard et al. (2014). It consolidates the results of chapter 2 and provides more extensive technical details about the method. RUS is applied on parallelepipedic and cylindrical samples of two attenuative materials: PMMA, an isotropic polymer, and an anisotropic bone mimetic material. Results are compared with wave velocity methods and mechanical testing. Precision and accuracy of RUS in that context are quantitatively discussed. The imaginary parts of the stiffness co-efficients, reflecting viscoelasticity, were also inferred from the resonant peak width Additionally, a first Bayesian formulation of the inverse problem and an iterative algorithm solving the pairing problem are proposed.

Chapter4 generalize the Bayesian formulation of Chapter3, and introduce the use of Markov chain Monte Carlo methods for an automated and robust inversion.

The proposed methodology is validated on data from the literature for a weakly

1.5. Outline of the thesis 19

attenuative material (aluminium alloy) and on data from chapter 2 for cortical bone. This chapter is also a research paper, prepared for submission to the journal Mechanics of Materials.

Chapter 5 presents the application of the developments introduced in the pre-ceding chapters to a collection of 59 human tibial bone samples. The Bayesian approach is further extended to handles collection of specimens and the full stiffness tensor of the specimens is obtained under a transverse isotropy assumption, as well as some viscoelastic parameters. An expected strong dependence of bone stiffness to apparent mass density is observed, and some original results about the anisotropy of the viscoelastic properties are presented.

Chapter 2

RUS measurement of cortical bone elasticity:

a feasibility study

This chapter is a research article published under the tile Accurate measurement of cortical bone elasticity tensor with resonant ultrasound spectroscopyin The Journal of the Mechanical Behavior of Biomedical Materials in 2013, and referred to as Bernard et al. (2013) in this manuscript. The full text of the article is reproduced here with no addition and no modifications except in the form.

This Chapter presents the first results that were obtained on a cortical bone specimen and demonstrates the feasibility of RUS in that context. The technical points, notably computation of the resonant frequencies and signal processing, are only briefly described here, but will get a more extensive treatment in Chapter 3.

2.1 Introduction

Cortical bone elasticity is anisotropic at the millimeter length scale and shows im-portant inter-individual and intra-individual variations. This variability is largely determined by a variable volume fraction of Haversian pores (Granke et al., 2011) and to a lesser extent to variable tissue mineral content (Currey,1988b) and average orientation of mineralized fibrils (Deuerling et al.,2009;Zebaze et al.,2011). There is a strong demand of precise and practical measurement methods for document-ing cortical bone elasticity and understanddocument-ing structure-function relationships. A proper characterization of bone elasticity at the millimeter scale requires measure-ments of all the terms of the elastic tensor on a unique small volume of material to avoid any effect of specimen variability. A method based on the measurement of longitudinal and shear ultrasonic bulk waves velocities (BWV) propagating along various directions of a bone specimen was introduced in the 60’s by Lang (1969) as an alternative to static mechanical techniques , which are ineffective to measure several elastic constants on the same specimen. BWV measurements in the range 1-2 MHz are still the state-of-the-art method to retrieve longitudinal and shear elas-tic constants in the different anatomical directions of a specimen (Espinoza Orias et al.,2008; Granke et al.,2011). However the method has several drawbacks: (1) the specimen must have typical dimensions larger than a few millimeters (∼5mm).

This size limitation is related to the requirement that measured wave velocities be associated to bulk waves and not to bar waves, which propagate when the wavelength is close to the dimension of the specimen (Ashman et al., 1984). This limitation can not easily be overcome by scaling down the wavelength (use of transducer with

higher working frequency) because the wavelength must remain sufficiently large with respect to bone tissue heterogeneities (osteon, remodeling cavities, Haversian pores) in order to minimize the frequency dependence of the wave velocities and to measure bone effective properties. (2) The BWV method involves multiple measure-ments (one for each constant) with specimen repositioning and delicate transducers manipulation, in particular for shear wave measurements. (3) Depending on the degree of anisotropy of the specimen (transversely isotropic or orthotropic) one or several 45 degrees oblique cuts are necessary to retrieve all non-diagonal terms of the stiffness tensor. Because such cuts considerably increase the specimen preparation time and complexity, they were not realized in many of the studies using the BWV method. (4) The precision of the BWV method is determined by the consistency and the repeatability of the shape of the time-domain through-transmitted signal which is in fact highly sensitive to transducer positioning and quality of the acoustic coupling between the transducers and the specimen (Grimal et al.,2009).

In a resonant ultrasound spectroscopy (RUS) experiment, the frequencies of me-chanical resonances of a freely vibrating specimen are measured, and an iterative numerical procedure is used to adjust elastic constants until the calculated spec-trum corresponds to the measured frequencies (Maynard,1992;Migliori et al.,1993;

Migliori and Sarrao, 1997). In this way, all elastic constants are determined from a single measurement configuration, a clear advantage over BWV measurements.

Another advantage is that RUS takes into account all the complexity of wave prop-agation in a finite specimen instead of assuming propprop-agation of pure bulk waves.

Therefore, there is in principle no limitation in specimen size. It is noteworthy that the method has been applied to measure the elasticity of crystalline specimens smaller than 1 mm3 (Spoor et al.,1995).

While RUS has become the gold standard technique to characterize elasticity of small single crystals, its application to biological materials is, to our knowledge, limited to two studies. Kinney et al. (2004) were successful in the characterization of elastic anisotropy of human dentin. In contrast, Lee et al.(2002) showed that the usual implementation of the method, using a commercial device, failed to measure all the anisotropic elastic constants of bone. The failure in Lee’s study was attributed to the high viscoelastic damping of bone, which causes resonant peaks to overlap and prevent the measurement of the resonant frequencies. Indeed, a mechanical quality factor Q of an order of a few hundreds and up is advised in RUS (Migliori and Maynard,2005) since it implies sharp resonant peaks. This is usually the case for crystalline or metallic specimens. In the context of geophysics, RUS has been successfully applied to materials with a smaller Qof the order of150(Ulrich et al., 2002). Lebedev (2002) introduced a signal processing method for the resolution of overlapped peaks in order to apply RUS to materials with aQfactor of the order of 50. This method was applied to simulated (Lebedev,2002) and experimental signals

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