Analogy of Strain Energy Density Based Bone-Remodeling Algorithm and Structural Topology Optimization

Abstract

In bone-remodeling studies, it is believed that the morphology of bone is affected by its internal mechanical loads. From the 1970s, high computing power enabled quantitative studies in the simulation of bone remodeling or bone adaptation. Among them, (1987, “Adaptive Bone Remodeling Theory Applied to Prosthetic Design Analysis,” J. Biomech. Eng., 20, pp. 1135–1150) proposed a strain energy density based approach to bone remodeling and used the apparent density for the characterization of internal bone morphology. The fundamental idea was that bone density would increase when strain (or strain energy density) is higher than a certain value and bone resorption would occur when the strain (or strain energy density) quantities are lower than the threshold. Several advanced algorithms were developed based on these studies in an attempt to more accurately simulate physiological bone-remodeling processes. As another approach, topology optimization originally devised in structural optimization has been also used in the computational simulation of the bone-remodeling process. The topology optimization method systematically and iteratively distributes material in a design domain, determining an optimal structure that minimizes an objective function. In this paper, we compared two seemingly different approaches in different fields—the strain energy density based bone-remodeling algorithm (biomechanical approach) and the compliance based structural topology optimization method (mechanical approach)—in terms of mathematical formulations, numerical difficulties, and behavior of their numerical solutions. Two numerical case studies were conducted to demonstrate their similarity and difference, and then the solution convergences were discussed quantitatively.