A homogenization sampling procedure for calculating trabecular bone effective stiffness and tissue level stress

Abstract

A homogenization sampling procedure is introduced which allows computation of effective trabecular bone stiffness and individual trabecula level stress based on precise models of trabecular bone architecture. Three-dimensional digitized images of 53 trabecular bone specimens with a resolution of 50 μm per voxel were directly converted into three-dimensional finite element meshes by making each voxel an 8-node isoparametric brick element. Owing to the large mesh of 8000 elements, an element-by-element preconditioned conjugate gradient (EBEPCG) program was written to solve the local homogenization finite element equations. Predicted effective stiffness measures correlated well with experimental results ( R 2 > 0.73). The predicted effective stiffnesses tended to under estimate the experimental values. Average absolute errors in effective stiffness estimates ranged between 31 and 38% for the sampling procedure compared to a range 49–150% for a regression fit to volume fraction squared. Trabecula level stress ranged between −200 and +300 times that predicted by analyzing trabecular bone as a continuum. Both tensile and compressive tissue stresses were engendered by a continuum compressive stress. Trabecula level strain energy density (SED) ranged between 0 and 100 times the continuum SED value for two trabecular specimens. In conclusion, the homogenization sampling procedure consistently predicted the influence of trabecular bone architecture on effective stiffness. It can also provide trabecular tissue stress and strain estimates for arbitrary global loading of whole bones. Tissue stresses and strains showed large variations compared to corresponding continuum level quantities.