Optimized Evaluation of Bone Tissue Material Properties by Inverse Finite Element Analysis and Femur Fracture Testing

Abstract

The main objective of this research is to characterize bone inhomogeneous elastic, yield, and post-yield behaviors, using a computational-experimental approach. The current study uses the force-displacement results of one hundred four cadaveric femora that were previously tested to fracture in a fall on the hip loading configuration. Recorded force-displacement data are used to determine stiffness, yield force, and femoral strength values. Finite element (FE) models of the femora are developed from the quantitative computed tomography scans captured before the fracture tests. A power law, or a sigmoid function, is used to determine the elastic modulus from the ash densities for each case modeled. The models are used for FE simulations that mimic the experiments. Inverse finite element analysis is employed to identify the unknown coefficients in the bone density-elasticity relationships. Optimization algorithms are used to minimize the error function between the experimental and FE estimated results in a large subset of female femora. The results of the obtained relationships show a good agreement with the experimental data. This contributes to a coefficient of determination of 70%, which is higher than those of previously proposed density-elasticity relationships on the same set of femora. The parts of the bones with the densities up to 0.5 g/cm3, play an important role in the deformation of the neck and the head of the femur. While power law and sigmoid function show similar correlation in the prediction of stiffness, distribution of stresses and strains are notably different, showing different response in the yield and post-yield behavior. To simulate the material damage, a power density-yield strain relationship is used as the failure criterion in FE models, assuming a ductile and a brittle material behavior for the bone. The unknown coefficients in the density-yield strain relationship are identified for the ductile and brittle material models. The ductile material model shows a more realistic post-yield behavior iv than the brittle model, but it is computationally expensive and may face convergence issues due to nonlinearities. The brittle material model, on the other hand, estimates the bone strength fairly and, due to its simplicity, it seems more applicable for clinical use.