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

Abstract

In the field of bone adaptation, it is believed that the morphology of bone is affected by its mechanical loads, and bone has self-optimizing capability; this phenomenon is well known as Wolff's law of the transformation of bone. In this paper, we simulated trabecular bone adaptation in the human proximal femur using topology optimization and quantitatively investigated the validity of Wolff's law. Topology optimization iteratively distributes material in a design domain producing optimal layout or configuration, and it has been widely and successfully used in many engineering fields. We used a two-dimensional micro-FE model with 50 μm pixel resolution to represent the full trabecular architecture in the proximal femur, and performed topology optimization to study the trabecular morphological changes under three loading cases in daily activities. The simulation results were compared to the actual trabecular architecture in previous experimental studies. We discovered that there are strong similarities in trabecular patterns between the computational results and observed data in the literature. The results showed that the strain energy distribution of the trabecular architecture became more uniform during the optimization; from the viewpoint of structural topology optimization, this bone morphology may be considered as an optimal structure. We also showed that the non-orthogonal intersections were constructed to support daily activity loadings in the sense of optimization, as opposed to Wolff's drawing.

Introduction

As an initial study on the internal bone architecture, von Meyer (1867) presented a drawing of the trabecular bone structure that he had observed in the human proximal femur, and, interestingly, his drawing had strong similarities with what Culmann (1866) drew for the principal stress trajectories in a cranelike curved bar. Based on these studies, Wolff (1892) stated that the functional adaptation of bones reorients the trabeculae, so that they align with the new principal stress trajectories when the environmental loads on the bone are changed by trauma or a life pattern change; this theory is known as Wolff's law or trajectorial hypothesis. He also suggested that the bone obtains “maximum” mechanical efficiency with “minimum” mass; many scientists and engineers have often cited a bone as an example of “optimal” structures.

Over the last 30 years experimental and computational techniques have significantly improved, and the current methods and tools allow for the quantitative study of the Wolff’ law. Trabecular structural change under controlled mechanical environments has been the focus of recent experimental study (Chambers et al., 1993; Goldstein et al., 1991; Moalli et al., 2000). In the whole scale of trabecular architecture, Biewener et al. (1996) showed the close correspondence between the alignment of trabeculae and the orientation of peak compressive strains in the calcaneum of potoroos. More recently, Pontzer (2006) revealed that the trabecular bone adapts dynamically to the orientation of peak compressive forces from the experiments with guinea fowl under two different loading regimes. As an alternative approach, the finite element method (FEM) has been used for computational stress analyses of bone structures. Quantitative information about the tissue stress and strain is a key factor for understanding of bone integrity, as reported in recent work that used so-called micro-FE models (Van Rietbergen et al., 1999, Van Rietbergen et al., 2003; Verhulp et al., 2006). With the help of FEM, many researchers (Beaupré et al., 1990a, Beaupré et al., 1990b; Huiskes et al., 1987; Ruimerman et al., 2005) determined the bone remodeling process based on the hypothesis that bone remodeling develops such that the strain energy (in some cases, stress or strain) over the entire domain converges to a certain reference value. Adachi et al. (1997) proposed a trabecular surface remodeling equation using the local stress uniformity as a criterion, and they applied the algorithm to the simulation of trabecular structural changes in human proximal femur under both single and multiple loads (Tsubota et al., 2002).

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.

In this paper, we developed a topology optimization method for simulating trabecular adaptation in the human proximal femur: a perimeter constraint was introduced in order to represent and maintain porosity in cancellous bone. This perimeter constraint was used for the first time in biomechanical bone adaptation, and the numerical results showed a closer correlation with the observed data (a photograph and radiograph) in the literature. Compared to previous numerical studies, the proposed method produced most realistic results. Starting with a nonuniform strain energy distribution in the initial trabecular architecture under three load cases, the bone became to have more uniform strain energy state in the final optimized trabecular architecture. The fully stressed state is the most efficient design of a structure from the viewpoint of the structural optimization theory. Therefore, the results suggest that the bone has “self-optimizing” capability, which is indeed a statement of Wolff's law. However, as opposed to Wolff's drawing (1892), we numerically showed that the non-orthogonal intersections should be constructed to support daily activity loadings in the sense of optimization.