Limiting models for calcification in fibrous tissues adjacent to orthopedic implants: Variational indicator functions and influences of implant stiffness

Calcification and eventual integration of orthopedic implants into bone is important to many load-bearing devices, and the influence of load and implant stiffness on this process are assessed in this mathematical modelling study. Three research questions are posed in this study. First, can limiting material models provide useful information on the overall behavior of the tissue adjacent to a loaded orthopedic implant? Second, can the limiting models lead to optimization criteria? Third, can an optimization approach be used to differentiate between the four prospective remodeling rate equations which are proposed? The answers are yes, yes, and no, respectively. A two degree of freedom lumped parameter model for axial loading of an intramedullary implant is considered. Two limiting composite material models are used, and the strain energy density in the calcified and non-calcified phases are assessed as stimuli for calcification. The rate equations posed here assume that the calcified material volume fraction decreases at high strain-energy densities, and increases at small strain-energy densities. In all four cases (both models, both phases) the steady states for these rate equations find equilibrium points of indicator functions which are a weighted sum of total strain energy and the mass of calcified tissue in the layer considered. The weights on strain-energy density and mass differ in each case. This shows that for appropriate choices of parameters, all four models can yield the same results, and it also shows that an optimization approach does not uniquely determine the appropriate rate equation in these cases. The rate equations showed complicated dynamic behavior and a phase-plane analysis was used which led to upper bounds on load, which depended on implant stiffness and distal support. The predictions of the four cases studied are compared.