Effect of porosity on effective diagonal stiffness coefficients (cii) and elastic anisotropy of cortical bone at 1MHz: A finite-difference time domain study

Abstract

Finite-difference time domain (FDTD) numerical simulations coupled to real experimental data were used to investigate the propagation of 1 MHz pure bulk wave propagation through models of cortical bone microstructures. Bone microstructures were reconstructed from three-dimensional high resolution synchrotron radiation microcomputed tomography (SR-muCT) data sets. Because the bone matrix elastic properties were incompletely documented, several assumptions were made. Four built-in bone matrix models characterized by four different anisotropy ratios but the same Poisson's ratios were tested. Combining them with the reconstructed microstructures in the FDTD computations, effective stiffness coefficients were derived from simulated bulk-wave velocity measurements. For all the models, all the effective compression and shear bulk wave velocities were found to decrease when porosity increases. However, the trend was weaker in the axial direction compared to the transverse directions, contributing to the increase of the effective anisotropy. On the other hand, it was shown that the initial Poisson's ratio value may substantially affect the variations of the effective stiffness coefficients. The present study can be used to elaborate sophisticated macroscopic computational bone models incorporating realistic CT-based macroscopic bone structures and effective elastic properties derived from muCT-based FDTD simulations including the cortical porosity effect.

The influence of mesoscale porosity on cortical bone anisotropy. Investigations via asymptotic homogenization

“…Accordingly, the local mesoscale porosity in samples taken from cortical bone cross sections is that which is effectively assessed with standard techniques. It is defined as that due to the Haversian canals and resorption cavities and lies in the range of 5-15 per cent (Bousson et al 2001;Dong & Guo 2004;Baron et al 2007). However, note that the macroscopic porosity defined as an average of the mesoscopic porosity over local representative volume element (RVE) locations varies much less and is usually in the range of 7-9 per cent.…”

“…All these models consider elementary shapes for the pores and simple mesostructure. Recently, Baron et al (2007) and Grimal et al (2008) have developed numerical models to compute the effective properties of cortical bone with realistic mesoscale structures derived from images. However, these latter models are not appropriate to use when performing systematic parametric studies due to their high computational cost.…”

Bone as a Composite Material

“…The influence of porosity on material properties is directional anisotropy [108,109]. Young's and shear modulus in human femoral cortical bone samples correlate with porosity in the longitudinal, but not the transverse, direction [109].…”

“…Young's and shear modulus in human femoral cortical bone samples correlate with porosity in the longitudinal, but not the transverse, direction [109]. In addition, increasing porosity correlates with decreased anisotropy [108,109]. In general, pore spaces in bone preferentially align with the long axis of the bone.…”

A numerical study of ultrasonic response of random cortical bone plates

“…It is has been shown that the porosity in the radial direction (which is defined in the bone's cross-section plan) is heterogeneous [13,14]. As the macroscopic mechanical properties of bones strongly depend on its porosity [15,16], cortical bone would naturally be considered as a functionally graded material. Thus the model employed in this study involves an anisotropic elastic plate sandwiched between two homogeneous fluids and having varied elastic properties in the through-thickness direction.…”