Predicting Human Metatarsal Fatigue Life with Finite Elements
Mechanical Testing
We divided the metatarsals into four groups to capture the differences in fatigue behavior between high vs. low magnitude loads, and axial vs. bending loads. Fourteen human cadaver specimens provided this study with 28 feet, and the pairs were divided by axial vs. bending loads. Seven of the donors were randomly assigned to the low magnitude, and the rest to the high magnitude group.
To simulate running, the tests were conducted at 4 Hz because the metatarsal experiences loading from the stance phase for approximately 250 ms at running pace. The bones were wrapped in saline-soaked gauze and drip-irrigated by a standard IV setup to ensure hydration throughout the test.
Geometry, Mesh, and Material Properties
Each metatarsal was imaged with high-resolution, quantitative Computed Tomography (CT) (246 μm voxel edge length). These images were segmented semi-automatically by Mimics (v 26.0) to spatially define the bone. We used an adaptive meshing function provided by 3-Matic (v 18.0) with a max element edge length of 2 mm, and improved the quality by subsequently applying the gradient mesh optimization function. The CT’s calibration curve was used to relate voxel brightness to density, in which we relate to orthotropic material properties measured in human bone. Finally, we oriented the bone by its top three principal components with the long axis on z, and the bending (dorsal/plantar) axis on y.
Fung, A. et al., 2017, J Biomech Cezayirlioglu, H. et al., 1985, J Biomech
An example segmentation and the 3D render of the resulting mesh with density scaling on a heatmap.
Model Loading and Boundary Conditions
To imitate the mechanical tests, we fixed the nodes within 25 mm of the proximal metatarsal, and applied a bending or axial load on the metatarsal head’s plantar surface or distal surface, respectively. The load node was kinematically coupled to surface nodes within 5 mm. To avoid boundary artifacts, output data was restricted to elements 5+ mm away from the nodes with a prescribed condition. We used Abaqus 2020 to solve multiple quasi-static loading scenarios for each metatarsal.
Example of a bending load applied to a metatarsal with constraints highlighted.
Outcomes
The maximum von Mises stress and individual strain components were measured for each element, and Tsai-Wu damage criteria was applied during data processing to compare the volume of failed elements estimated by the two criteria. The subtle difference between them lies in the shape of the failure envelope created by their 3D functions. The von Mises envelope depicts a symmetrical cylinder and thus cannot model anisotropic yield properties of bone; however, the stresses developed are still dependent on the anisotropic material properties applied in both scenarios. The Tsai-Wu damage criterion smooths this envelope into a ellipsoid with asymmetrical threshold boundaries governed by compressive and tensile strength in the longitudinal and transverse directions, as well as the shear strain limit.
Validation
Typically, the bones exhibited lower stiffness in the first few cycles than ones later in the test, which we attribute to the viscoelastic creep. Once it reached a steady state, we usually observed a minimum error between FE-predicted displacement and total recorded displacement. The average minimum error from all tests was 0.95 (0.13 stdev, 1.10 max) mm, which is reasonable given the deformation seen in the epoxy from the video above.
An example stiffness and displacement error trace
