Topology optimization of bone using cubic material design and evolutionary methods based on internal remodeling

Abstract

In the present paper, the influence of the loading conditions on the trabecular architecture of a femur is investigated by using the cubic material design and an evolutionary approach based on internal remodeling. The response of bone to mechanical loading stimuli leads to alterations of the internal architecture of bone, traduced by a modification of its effective density and mechanical properties. The cubic material design approach deals with the minimum compliance problem of femur bone made of a non-homogeneous elastic material with cubic symmetry. The optimal cubic material characteristics are in fact reflected by the properties of the underlying microstructure. Internal remodeling is here also considered and thus belongs to the class of evolutionary topology optimization methods. The effective continuum mechanical properties of the trabecular bone are derived from an initially discrete planar hexagonal structure representative of femur bone microstructure, relying on the discrete homogenization approach. This leads to scaling laws of the effective elastic properties of bone versus density at an intermediate mesoscopic scale. The performed simulations of the optimal trabecular bone architecture illustrate the respective advantages and limitations of the employed methods. The simulation results indicate that the proposed approaches could reasonably mimic the major geometrical and material features of the natural bone.