Anisotropic elastic properties of human cortical bone tissue inferred from inverse homogenization and resonant ultrasound spectroscopy

Abstract

Bone extravascular matrix (EVM) elasticity at several tens micrometer scale plays a key role in the mechanical behavior of bone at different length scales with implications on bone biology through mechanotransduction.The elastic properties of cortical bone EVM have been evaluated by several experimental methods, including nanoindentation, scanning acoustic microscopy (SAM) and mechanical testing on µm sized bone specimens.Nevertheless, these methods hardly give access to elastic anisotropy. In this work, we propose a novel inverse homogenization method to evaluate the anisotropic elastic properties of cortical bone EVM based on the transverse isotropic elastic tensor of millimeter-sized bone specimens measured by using resonant ultrasound spectroscopy and Fast Fourier Transform homogenization method. With the inverse homogenization method, the anisotropic EVM stiffness constants were evaluated on 50 human femoral cortical bone specimens from an elderly group. To our knowledge, this is the first time that the whole set of the EVM stiffness tensor is evaluated on the same specimen and on a large number of samples. Further comparison with the results from SAM and the degree of mineralization of bone (DMB) showed the potential of this method. Empirical laws between DMB and EVM anisotropic stiffness constants were also provided for the first time. With the anisotropic elasticity evaluated by the proposed method, more accurate models can be developed to better understand bone mechanics and biology, such as mechanotransduction.

Maximum effect of the heterogeneity of tissue mineralization on the effective cortical bone elastic properties

“…In the elderly population without specific bone pathology, the coefficient of variation of the average mineral content was found to be 2% (Cai et al 2019b). Using inverse homogenization based on mesoscale anisotropic stiffness data for the same samples, Cai et al (2020) calculated a coefficient of variation of matrix elastic coefficients between 3 and 7% depending on the elastic constant, which was correlated to the mineral content. (Note that above reported percentages are for coefficients of variations, and that the actual ranges of variation are somewhat larger).…”

“…with E the macroscopic strain and Γ (0) denoting the strain Green operator corresponding to a reference homogeneous medium with elasticity C (0) . This numerical method, which is widely used in engineering material mechanics, has been recently used for cortical and trabecular bones (Colabella et al 2017;Gagliardi et al 2018;Cai et al 2019aCai et al , 2020. It can be mentioned that it allows to perform calculations directly on the microstructure digital image.…”

A 3D trabecular bone homogenization technique

“…Fitting Agic et al data it was possible to define two fourth order polynomial equations, Eqs. (4) and (5), which enabled us to describe the behavior of v 2 and v 3 analytically, considering  app .…”

“…In the literature, numerous examples of homogenization techniques application are found, with different approaches to define bone mechanical properties. The majority of the works use micro-CT images as a source for representing bone structure, [10], [16]- [19], but other authors use different methods, such as scanning acoustic microscopy (SAM) [5]. The method to define the mechanical properties of the homogeneous domain usually relies on the definition of a tensor that enables the researchers to relate the bone microstructure with bone mechanical properties.…”

“…The method to define the mechanical properties of the homogeneous domain usually relies on the definition of a tensor that enables the researchers to relate the bone microstructure with bone mechanical properties. Imaging-based approaches, such as gray-level structure tensor (GST) or Mean Intercept Length (MIL) method, are the most often used to acquire these tensors [12], [16]- [18], although there are other approaches, such as using Fast Fourier Transform (FFT) homogenization methods [5]. These homogenization techniques make it possible to determine equivalent mechanical properties of a heterogeneous material by substituting the volume of the heterogeneous material with an equivalent volume of homogeneous material so as to predict the behavior of the macro-scale using the mechanical properties coming from the micro-scale.…”