Three-dimensional micro-level computational study of Wolff's law via trabecular bone remodeling in the human proximal femur using design space topology optimization
Introduction
Internal bone architecture, with its unique complex material matrix, has long been studied to determine the underlying principles of its adaptation and “remodeling”. Wolff, 1892, Wolff, 1986 proposed that trabecular bone in the proximal femur functionally adapts to external mechanical loading stimuli, orientating to align with the principal stress trajectories. Wolff had observed the “self-optimizing” property of bone and theorized that bone achieves maximum mechanical efficiency with minimal mass: a naturally optimum structure. These hypotheses known as Wolff's Law has been analyzed, critiqued, and refined using clinical, experimental, and analytical means. In the past few decades, computational techniques have been developed that utilize the finite element (FE) method to simulate this “bone remodeling” process.
Carter and Hayes (1977), Carter (1984) proposed some of the earliest theoretical frameworks for adaptive bone remodeling, including developing the apparent bone density elasticity formulation utilizing an exponential penalization factor, similar to the penalized material model used in topology optimization (Bendsøe, 1989). They completed several of the first single load continuum-level analyses using peak effective stress for remodeling (Carter, 1987, Carter et al., 1987, Carter et al., 1989). Huiskes et al. (1987) proposed a time-dependent strain energy density (SED) approach for remodeling, which has become one of the dominant methods for computational bone remodeling. Additionally, several stress-based methods have also been developed (Beaupré et al., 1990, Adachi et al., 1997). With increasing computing power, recent studies have been able to use micro-FE models that accurately represent trabecular bone architecture; however, these are still predominantly two-dimensional models that inadequately describe actual femur loading conditions and feature simplified geometries (Tsubota et al., 2002, Jang and Kim, 2008, Jang and Kim, 2009). Previous three-dimensional studies have typically used a macro-level or hierarchical multi-scale approach to reduce computational cost (Bagge, 2000, Coelho et al., 2009). These methods provide adequate insight into overall mechanical properties and density distribution but cannot accurately represent the individual trabecular architecture. Three-dimensional micro-FE models of the proximal femur have been created using computed tomography (CT) scans to complete a single stress analysis; such simulations can only be achieved by utilizing computationally efficient algorithms and parallel computing (van Rietbergen et al., 2003, Verhulp et al., 2006). Computational bone remodeling requires numerous FE analyses in an iterative fashion and additional optimization calculations; as a result, the computational cost is much greater. Only one study has achieved three-dimensional results using a micro-level resolution using local-stress criteria (Tsubota et al., 2009). The common feature of these various numerical approaches is that they are empirically derived specifically to emulate the bone remodeling process. These phenomenological approaches can predict the progress of bone remodeling; however, they do not prove that bone is an optimum structure, which is an essential assumption in Wolff's Law.
This study utilized design space optimization (DSO), a rigorous mathematical structural optimization technique developed by Kim and Kwak (2002), in order to determine the global optimum structure of the cancellous bone in the proximal femur. DSO is a specialized topology optimization algorithm that attempts to distribute a finite amount of material into the areas of highest loading to achieve an optimal strength to weight ratio by minimizing the global strain energy (SE). This method has the unique ability to describe the intermediate structural adaptation progress in the time domain and incorporate multi-disciplinary and multi-objective models (Jang and Kim, 2009).
The objective of this study is to conduct the first micro-level three-dimensional FE bone remodeling simulation of the proximal femur using DSO topology optimization to address Wolff's hypothesis of self optimization using proven mathematical theory. All loading and geometric simplifications required for previous studies were addressed by utilizing accurate three-dimensional loads and an accurate femur model based on clinical and CT scan data. We analyzed the results by comparing them to natural bone via cross-sectional imagery and by using new novel methods, such as simulated radiographs and quantitative comparison to morphologic parameters, to fully address the hypothesis of natural optimality using structural optimization, while identifying key limitations and future areas of improvement.
Section snippets
Modeling and simulation conditions
The femur model was created using CT scans from a 90.8 kg, 183 cm male (Heiner and Brown, 2001). A micro-FE model was constructed using 23.3 million finite elements, representing over 84 million degrees of freedom, with an individual cubic voxel size of 175 μm. The initial cancellous region was populated with randomly positioned tori, consistent with previous studies, to create a primarily isotropic cancellous structure, as shown in Fig. 1 (Jang and Kim, 2008, Tsubota et al., 2002, Tsubota et al.,
Results
Ward's classification (Ward, 1838, Whitehouse and Dyson, 1974) of the trabecular architecture contains four main loading groups: the principal compressive group (PC), the principal tensile group (PT), the secondary compressive group (SC), and the secondary tensile group (ST) (Fig. 3). An additional area of interest known as Ward's Triangle, in the central neck region, is also noted; this area is known to be a location of minimum bone density in the proximal femur (Tobin, 1955).
The initial
Discussion
A three-dimensional FE bone remodeling simulation in the proximal femur was completed using the DSO topology optimization method. The computational simulation utilized a large-scale micro-finite element model with two daily loading regimes, walking and stair climbing. The resulting adapted femur architecture reveals similar trabecular patterns in the four anisotropic loading groups as discussed above. Measurement of the simulation arch angles in the neck region reveals an average range from 84°
Conflict of interest statement
There are no conflicts of interest.
Acknowledgements
This work is funded through the Nature Sciences and Research Council of Canada (NSERC). The authors thank Dr. K. Svanberg at KTH (Stockholm, Sweden) for providing the MMA code for academic research. The authors also thank Dr. J. Skedros, Dr. C. Lovejoy, and Dr. F. Baruffaldi for providing the various images of human proximal femur.
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2022, Journal of BiomechanicsCitation Excerpt :Obtaining the similar solutions with surface remodeling algorithms, topology optimization (TO) algorithm is more efficient to predict bone remodeling, due to better iterative convergence (Bagge, 2000; Coelho et al., 2009). With the prerequisite condition of perimeter constraint, the micro-level trabeculae structure of femoral was reconstructed by means of TO algorithm (Boyle and Kim, 2011b; Jang and Kim, 2008; Park et al., 2018). Wu et al. introduced a local density constraint into TO algorithm to reproduce bone-like cancellous structure and claimed the superior robustness and manufacturability of the artificial trabeculae structure (Wu et al., 2018).