Necessary and sufficient conditions for global stability and uniqueness in finite element simulations of adaptive bone remodeling

Abstract

Conditions which can guarantee the global stability and uniqueness of the solution to a bone remodeling simulation are derived using a specific rate equation based on strain energy density. We modeled bone tissue as isotropic with a constant Poisson ratio and the elastic modulus proportional to volumetric density of calcified tissue raised to the power n. Our remodeling rate equation took the rate of change of volumetric hard tissue density as proportional to the difference between a stimulus (strain energy density divided by volumetric density taken to the power m) and a set point. In previous studies we defined state variables which are conjugate to the remodeling stimulus, and the function which acts as a variational indicator for the remodeling stimulus. In this study, we use the properties of this variational indicator to establish the stability and the uniqueness of the solution to the remodeling rate equations for all possible density distributions. We show that the solution is the global minimum of a weighted sum of the total strain energy and the integral of density to the power m over the remodeling elements. These results are proven for n < m, and we show that taking n > m will eliminate the possibility that a unique solution exists.