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

Abstract

Recently, the mesoscale of cortical bone has been given particular attention in association with novel experimental techniques such as nanoindentation, micro-computed X-ray tomography and quantitative scanning acoustic microscopy (SAM). A need has emerged for reliable mathematical models to interpret the related microscopic and mesoscopic data in terms of effective elastic properties. In this work, a new model of cortical bone elasticity is developed and used to assess the influence of mesoscale porosity on the induced anisotropy of the material. Only the largest pores (Haversian canals and resorption cavities), characteristic of the mesoscale, are considered. The input parameters of the model are derived from typical mesoscale experimental data (e.g. SAM data). We use the method of asymptotic homogenization to determine the local effective elastic properties by modelling the propagation of low-frequency elastic waves through an idealized material that models the local mesostructure. We use a novel solution of the cell problem developed by Parnell & Abrahams. This solution is stable for the physiological range of variation of mesoscopic porosity and elasticity found in bone. Results are computed efficiently (in seconds) and the solutions can be implemented easily by other workers. Parametric studies are performed in order to assess the influence of mesoscopic porosity, the assumptions regarding the material inside the mesoscale pores (drained or undrained bone) and the shape of pores. Results are shown to be in good qualitative agreement with existing schemes and we describe the potential of the scheme for future use in modelling more complex microstructures for cortical bone. In particular, the scheme is shown to be a useful tool with which to predict the qualitative changes in anisotropy due to variations in the structure at the mesoscale.