Estimation of the effective bone-elasticity tensor based on μ CT imaging by a stochastic model. A multi-method validation

Abstract

Abstract In previous work, we proposed a stochastic model to describe the elasticity of bone matrix (so-called ultrastructure, US) based on basic statistical information on the tissue mineral density ( TMD ). This information was obtained by analyzing high-resolution images of a human femoral neck realized by means of synchrotron radiation micro-computed tomography ( SR − μ CT ). In this paper, we extend this study by focusing at the upper scale where cortical bone is described as a two-phase mixture made up of water-filled Haversian pores (HP) embedded in the surrounding solid US. The goal of this paper is to develop a stochastic model of cortical bone elasticity accounting for the effect of uncertainty affecting both phases, the US via the TMD and the HP. Experimental information was assumed to be given in terms of mean values and dispersions of the average TMD (denoted TMD ¯ ) and HP at the millimeter scale. To this aim, SR − μ CT images were used to extract several representative volume elements (RVEs) spanning the whole cortical tissue which, in turn, were analyzed to obtain the required statistical information on TMD ¯ and HP. This information has been used for constructing a stochastic multiscale model of cortical bone based on the Maximum Entropy (denoted MaxEnt) principle. This stochastic multiscale model is used in the estimation of the effective elastic properties of cortical bone (CB-S μ M) based on continuum micromechanics ( μM). In parallel, a deterministic nominal multiscale model of cortical bone (CB-NμM) was developed by using as input data the mean values of TMD ¯ and HP. The elastic moduli estimated with CB-NμM has been compared with the average value of the same counterpart obtained with the CB-SμM model. The two estimates differ for less than 1 % , proving the robustness of the CB-NμM model to uncertainties. Moreover, the accuracy of the SμM stochastic model has been tested. Estimations of the cortical bone elasticity of each RVE with the μ M (cylindrical HP and homogeneous US), has been compared with other estimation obtained through others homogenization techniques: finite elements (FEM) and fast Fourier transform (FFT), both able to account for the real morphology of Havesian porosity and the heterogeneity of the ultrastructure. Results show that (1) the transverse isotropic (TI) elastic μM model correctly approximates the FEM and FFT estimates (∼98% TI); (2) the μM model accurately estimates the axial modulus ( Y 3 ) in longitudinal direction and the lateral contraction ratio ν 31 .