Graduate Student University of Calgary Calgary, Alberta, Canada
Introduction: The rabbit has become an important model in orthopaedic biomechanics that offers several benefits over traditional rodent models. Their natural Haversian remodelling makes the rabbit particularly useful for understanding the role of mechanobiology in cortical bone turnover. Computed-tomography (CT)-based finite element (FE) analysis is a valuable tool for simulating the mechanical behaviour of bone, but an accurate density-elasticity relationship is necessary to convert CT derived density values to Young’s moduli for meaningful predictions. This work aims to develop and validate an accurate density-elasticity relationship for rabbit hindlimb bones using mathematical optimization.
Materials and
Methods: Two tibiae and two femora were harvested from rabbit hindlimbs. Radiopaque markers were then fixed to six anatomical positions-of-interest for each bone, and CT scans (0.625 x 0.531 x 0.531 mm) were obtained with a hydroxyapatite equivalent bone density (ρ; g/cm^3) calibration phantom within the field of view. After CT imaging, strain gauge rosettes were mounted to the six anatomical positions-of-interest and bones were mechanically tested in uniaxial compression between -10 N and -500 N for 20 loading cycles at a frequency of 1 Hz. The maximum and minimum principal strains were computed for each strain gauge rosette.
The CT images were processed into FE models composed of approximately 100,000 ten-node quadratic tetrahedron elements. The loading and boundary conditions were chosen to match the experimental conditions. Elements were assigned heterogeneous, isotropic, linear elastic material properties based on the underlying ρ according to: 𝐸 = 𝐴ρ^𝐵 where E is the Young’s modulus (MPa), and A and B are constant parameters. Here, a Nelder-Mead optimization routine was used to solve for the final values of the constant parameters by minimizing the mean- squared-error between the FE predicted and experimentally measured principal strains. The optimization was run for the tibiae and femora separately as well as for both bones combined. After optimizations, the relationships between predicted and experimental measurements were examined using linear regression and Bland-Altman plots.
Results, Conclusions, and Discussions: Regression statistics and optimization results for constants A and B are provided in Table 1. The lowest error (RMSE = 159 με) and strongest agreement (r^2 = 0.97) between FE predicted and experimentally measured principal strains was observed for the tibia only optimization. This relationship did illustrate an intercept that was significantly different from zero; however, compared to a previously reported density-elasticity relationship for the rabbit tibia, our relationship exhibits a slope that is not significantly different from one and explains 17% more of the variation in experimental measurements. When both bones were used for the optimization, the regression exhibited an X=Y type of relationship with a strong linear correlation. Bland- Altman analysis demonstrated that the FE predictions exhibited strong agreement with experimental measurements (Fig. 1). While the femur only optimization performed well, we suspect the comparably weaker correlation was the result of local geometric variability in the region of strain gauge placement. The sensitivity to measurement location will be addressed with a larger sample size as we continue to expand on this work.
This research represents the first optimized density-elasticity relationship for rabbit hindlimb bones. The strong performance of all groups suggested that it may be appropriate to use a single density-elasticity relationship for the entire hindlimb. Work is currently ongoing to use the underlying density-elasticity relationship in CT-based FE models of microdamage accumulation in response to cyclic loading. Ultimately, these CT-based FE models will enable future work to examine the influence of mechanobiology in the bone remodelling response of cortical bone.
