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Research Papers

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

[+] Author and Article Information
In Gwun Jang

Department of Mechanical and Materials Engineering, Queen’s University, McLaughlin Hall 221, 130 Stuart Street Kingston, ON, K7L 3N6, Canadajangin@me.queensu.ca

Il Yong Kim1

Department of Mechanical and Materials Engineering, Queen’s University, McLaughlin Hall 221, 130 Stuart Street Kingston, ON, K7L 3N6, Canadaiykim@me.queensu.ca

Byung Man Kwak

Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu Daejeon 305-701, South Koreabmkwak@khp.kaist.ac.kr

1

Corresponding author.

J Biomech Eng 131(1), 011012 (Nov 26, 2008) (7 pages) doi:10.1115/1.3005202 History: Received January 14, 2008; Revised July 18, 2008; Published November 26, 2008

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 (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.

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Figures

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Figure 1

Geometric and loading boundary condition for a plate model of bone tissue: (a) Load step 1; (b) load step 2

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Figure 2

Simulation result of structural configuration for a plate model of bone tissue: (a) Strain energy density based bone remodeling; (b) topology optimization

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Figure 3

Geometric and loading boundary condition for a vertebral model

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Figure 4

Simulation result of structural configuration for a vertebral model: (a) Strain energy density based bone remodeling; (b) topology optimization

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Figure 5

Convergence history during simulation of a plate model of bone tissue: (a) Mass change; (b) total strain energy change

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Figure 6

Convergence history during simulation of a vertebral problem: (a) Mass change; (b) total strain energy change

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