Micro-finite element simulation of trabecular-bone post-yield behaviour--effects of material model, element size and type.

Micro-finite element (micro-FE) analysis became a standard tool for the evaluation of trabecular bone mechanical properties. The accuracy of micro-FE models for linear analyses is well established. However, the accuracy of recently developed nonlinear micro-FE models for simulations of trabecular bone failure is not known. In this study, a trabecular bone specimen was compressed beyond the apparent yield point. The experiment was simulated using different micro-FE meshes with different element sizes and types, and material models based on cortical bone. The results from the simulations were compared with experimental results to study the effects of the different element and material models. It was found that a decrease in element size from 80 to 40 mum had little effect on predicted post-yield behaviour. Element type and material model had significant effects. Nevertheless, none of the established material models for cortical bone were able to predict the typical descent in the load-displacement curve seen during compression of trabecular bone.

Indirect determination of trabecular bone effective tissue failure properties using micro-finite element simulations.

Trabecular bone strength is marked not only by the onset of local yielding, but also by post-yield behavior. To study and predict trabecular bone elastic and yield properties, micro-finite element (micro-FE) models were successfully applied. However, trabecular bone strength predictions require micro-FE models incorporating post-yield behavior of trabecular bone tissue. Due to experimental difficulties, such data is currently not available. Here we used micro-FE modeling to determine failure behavior of trabecular bone tissue indirectly, by iteratively fitting FE simulation to experimental results. Failure parameters were fitted to an isotropic plasticity model based on Hill's yield function, using materially and geometrically nonlinear micro-FE models of seven bovine trabecular bone specimens. The predictive value of the averaged effective tissue properties was subsequently tested. The results showed that compression softening had to be included on the tissue level in order to accurately describe the apparent-level behavior of the bone specimens. A sensitivity study revealed that the simulated response was less sensitive to variations in the post-yield properties of the bone tissue than variations in the elastic and yield properties. Due to fitting of the tissue properties, apparent-level behavior could be accurately reproduced for each specimen separately. Predictions based on the averaged and fixed tissue properties were less accurate, due to inter-specimen variations in the tissue properties.

Load distribution in the healthy and osteoporotic human proximal femur during a fall to the side.

Due to remodeling of bone architecture, an optimal structure is created that minimizes bone mass and maximizes strength. In the case of osteoporotic vertebral bodies, however, this process can create over-adaptation, making them vulnerable for non-habitual loads. In a recent study, micro-finite element models of a healthy and an osteoporotic human proximal femur were analyzed for the stance phase of gait. In the present study, tissue stresses and strains were calculated with the same proximal femur micro-finite element models for a simulated fall to the side onto the greater trochanter. Our specific objectives were to determine the contribution of trabecular bone to the strength of the proximal femurs for this non-habitual load. Further, we tested the hypothesis that the trabecular structure of osteoporotic bone is over-adapted to habitual loads. For that purpose, we calculated the load distributions and estimated the apparent yield and ultimate loads from linear analyses. Two different methods were used for this purpose, which resulted in very similar values, all in a realistic range. Distributions of maximal principal strain and effective strain in the entire model suggest that the contributions to bone strength of the trabecular and cortical structures are similar. However, a thick cortical shell is preferred over a dense trabecular core in the femoral neck. When the load applied to the osteoporotic femur was reduced to approximately 61% of the original value, strain distributions were created similar in value to those obtained for the healthy femur. Since a comparable reduction factor was found for habitual load cases, it was concluded that the osteoporotic femur was not 'over-adapted'.

Comparison of micro-level and continuum-level voxel models of the proximal femur.

Continuum-level finite element (FE) models became standard computational tools for the evaluation of bone mechanical behavior from in vivo computed tomography scans. Such scans do not account for the anisotropy of the bone. Instead, local mechanical properties in the continuum-level FE models are assumed isotropic and are derived from bone density, using statistical relationships. Micro-FE models, on the other hand, incorporate the anisotropic structure in detail. This study aimed to quantify the effects of assumed isotropy, by comparing continuum-level voxel models of a healthy and a severely osteoporotic proximal femur with recently analyzed micro-FE models of the same bones. The micro-model element size was coarsened to generate continuum FE models with two different element sizes (0.64 and 3.04 mm) and two different density-modulus relationships found in the literature for wet and ash density. All FE models were subjected to the same boundary conditions that simulated a fall to the side, and the stress and strain distributions, model stiffness and yield load were compared. The results indicated that the stress and strain distributions could be reproduced well with the continuum models. The smallest differences between the continuum-level model and micro-level model predictions of the stiffness and yield load were obtained with the coarsest element size. Better results were obtained for both continuum-element sizes when isotropic moduli were based on ash density rather than wet density.

A three-dimensional digital image correlation technique for strain measurements in microstructures.

A three-dimensional digital image correlation technique is presented for strain measurements in open-cell structures such as trabecular bone. The technique uses high-resolution computed tomography images for displacement measurements in the solid structure. In order to determine the local strain-state within single trabeculae, a tetrahedronization method is used to fill the solid structure with tetrahedrae. Displacements are calculated at the nodes of the tetrahedrae. The displacement data is subsequently converted to a deformation tensor in each of the tetrahedral element centers with a least-squares estimation method. Because the trabeculae are represented by a mesh, it is possible to deform this mesh according to the deformation tensor and, at the same time, visualize the calculated local strain in the deformed mesh with a finite element post-processing tool. In this way, the deformation of a single trabecula from an aluminum foam sample was determined and validated with rendered images of the three-dimensional sample. A precision analysis showed that a rigid translation or rotation does not affect the accuracy. Typical values for the standard deviation in the displacement and strain components are 2.0 microm and 0.01, respectively. Presently, the precision limits the technique to strain measurements beyond the yield strain.