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, Huiskes et al. (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.

Computational study of Wolff's law with trabecular architecture in the human proximal femur using topology optimization

“…Bone remodeling simulation was also conducted by means of structural topology optimization: Hollister et al (1994) used the homogenization method to obtain the architecture of microtrabecular bone, and Bagge (2000) used topology optimization with compliance minimization to determine the initial material distribution for a femur. Jang et al (2008) studied the analogy between the two seemingly different approaches: strain energy density (SED)-based bone remodeling algorithm and the topology optimization method. They showed that topology optimization and the bone remodeling equations produced equivalent solutions.…”

Dynamical systems, SIMP, bone remodeling and time dependent loads

“…Rather, we like to clarify and strengthen the connection between models of evolutionary problems such as bone remodeling and damage evolution, on the one hand, and structural optimization formulations, on the other. Such a connection is hinted at in several publications, see, e.g., Achtziger et al [1] and Jang et al [8], but since the dynamical systems concept has been missing, this has not been achieve to a full extend in our view. These issues were further discussed in our previous paper [11].…”

“…In fact, when the previous remark holds, (8) is equivalent to the following definition of forces as (generalized) derivative of a potential −R ∈ ∂f e (ρ),…”

Lazy zone bone remodeling theory and its relation to topology optimization

“…For the numerical integration of (13), including (14) as a special case, we discretize time into steps of length ∆t. Given a solution ρ n at time t n , we find the…”

“…Research aimed at applying topology optimization theory to bone remodeling can be found in, e.g., recent work by Jang and Kim [11,12,13], while studies with the converse aim, i.e., applying bone remodeling theories in structural and topology optimization, can be found in Penninger et al [19] and Andreaus et al [1]. The theoretical basis for this connection of theories is independently discussed in Jang et al [14] and Klarbring and Torstenfelt [15,16]. To be more specific, in the latter works we showed that there is a one-to-one correspondence between a dynamical systems approach to SIMP topology optimization and the apparent density bone remodeling theory of Harrigan and Hamilton [8,9].…”