Numerical errors and uncertainties in finite-element modeling of trabecular bone

Although micromechanical finite-element models are being increasingly used to help interpret the results of bio-mechanical tests, there has not yet been a systematic study of the numerical errors and uncertainties that occur with these methods. In this work, finite-element models of human L1 vertebra have been used to analyze the sensitivity of the calculated elastic moduli to resolution, boundary conditions, and variations in the Poisson's ratio of the tissue material. Our results indicate that discretization of the bone architecture, inherent in the tomography process, leads to an underestimate in the calculated elastic moduli of about 20% at 20 microm resolution; these errors vary roughly linearly with the size of the image voxels. However, it turns out that there is a cancellation of errors between the softening introduced by the discretization of the bone architecture and the excess bending resistance of eight-node hexahedral finite elements. Our empirical finding is that eight-node cubic elements of the same size as the image voxels lead to the most accurate calculation for a given number of elements, with errors of less than 5% at 20 microm resolution. Comparisons with mechanical testing are also hindered by uncertainties in the grip conditions: our results show that these uncertainties are of comparable magnitude to the systematic differences in mechanical testing methods. Both discretization errors and uncertainties in grip conditions have a smaller effect on relative moduli, used when comparing between different specimens or different load directions, than on an absolute modulus. The effects of variations in the Poisson's ratio of the bone tissue were found to be negligible.