Assessment of anisotropic elasticity of small bone samples with resonant ultrasound spectroscopy: Attenuation does not prevent the measurements

Abstract

Resonant Ultrasound Spectroscopy (RUS) is the gold standard method for anisotropic elasticity assessment of millimeter-size samples. However, the usual implementation of RUS fails to measure bone because of its high mechanical damp- ing, which prevents the resonance frequencies measurement. The aim of this study is to demonstrate that RUS method can be adapted to assess the elasticity of highly attenuating anisotropic materials such as cortical bone. A procedure for sample dimen- sions optimization is defined in order to reduce the number of frequencies to be measured for a precise elasticity assessment. We applied a dedicated signal processing method able to retrieve resonance frequencies even in the case of peaks overlapping. For a sample of transversely isotropic, attenuating, bone-mimetic material, we were able to measure 13 resonance frequencies among the 20 first predicted by theory in the frequency band (0-240 kHz). Using a dedicated iterative optimization method we estimated the full set of independent elastic coefficients from the measured frequencies. The determined coefficients were validated by comparison with independent measurements. This work opens the way for precise elasticity assessment of small bone sample by RUS. I. INTRODUCTION Resonant Ultrasound Spectroscopy (RUS) is an elegant method to characterize from a single experiment all the terms of the elasticity tensor of anisotropic materials (1). RUS is well-established to measure samples of low damping materials, for which resonance peaks are well separated. It has particularly been developed for materials for which only small samples can be prepared. RUS involves (1) measuring the resonance frequencies of a freely vibrating sample; (2) independently computing these frequencies with an initial guess of the elastic coefficients; and (3) adjusting the elastic coefficients in a iterative process until computed and measured frequencies match. Compact bone is an orthotropic material which constitutes approximately 80% of the skeletal mass. It is essentially found in the cortex of bones and its mechanical properties determine the resistance of bones. Compact bone properties vary among individual, across the skeleton, and within a single bone depending on the anatomical location. A proper characterization of bone properties (measurement of all the terms of the elasticity tensor) must be conducted on the same volume of material to prevent the effects of samples variability. State-of-the-art methods to measure bone anisotropic elasticity measure the time-of-flight of longitudinal and shear waves in