AN ADAPTIVE STRUCTURE TOPOLOGY OPTIMIZATION APPROACH APPLIED TO VERTEBRAL BONE ARCHITECTURE

Bone is a highly adaptive biological structure. Following Wolff's law, bone realigns and grows to adapt to its mechanical environment. This leads to structural heterogeneity of trabecular bone and orthotropic symmetry of the elastic properties. Determining the bone alignment and material properties for living patients is difficult and involves implantation of force and displacement sensors on the bone to determine the compliance and stiffness properties. Micro computed tomography along with finite element modeling have been limited to the vertebrae of donor cadavers to evaluate trabecular architecture, material properties, and density. Here, an adaptive structure topology optimization algorithm is presented and used to predict trabecular architecture. The algorithm predicts the optimal structure by minimizing the global compliance. The lumbar 1 (L1) vertebra is used as an example. Loads common to L 1 vertebrae are applied and bone volume fraction measurements that can be taken easily from living patients through bone mineral density scans are used as the only inputs. The mathematical model is an adaptation of "99 Line Topology Optimization Code Written in Matlab" developed by Sigmund (2001). Bone is locally assumed to be isotropic with an elastic modulus of 13 GPa and the Poisson ratio of 0.3 applied to each element. The resulting structural heterogeneity results in global orthotropic relations. The model uses bone volume fraction and the loading orientation as inputs and gives the corresponding ideal bone structure geometry as an output. The trabecular structure can be predicted solely from the results of a bone mineral density scan. Finite element analysis of the optimized structure is then conducted and the global material properties are determined. While this model is for two-dimensional examples representing planes within the vertebral bone, it is extended to three-dimensional modeling to develop the cortical bone geometry and define the total volume. Matlab is then used to run the topology optimization simulation. The ideal structure is defined by optimizing for a prescribed displacement field of the system following the implementation of a gradient descent optimization method. The results are compared to published values from a combined experimental and numerical procedure. The procedure on sectioned vertebrae reported average ratios between elastic moduli of E1/E2 = 5.2, E1/E3 = 8.8, and E2/E3 = 1.4. Results between the models and the previously published data yield similar transversely isotropic symmetry in the elastic moduli of trabecular bone. However, the elastic moduli ratios are not quite in agreement. Improving the accuracy of the boundary conditions and loading of the finite element model may improve the correlation between the optimization models and published data.