NECESSARY AND SUFFICIENT CONDITIONS FOR GLOBAL STABILITY AND UNIQUENESS IN FINITE-ELEMENT SIMULATIONS OF ADAPTIVE BONE REMODELING

Conditions which can guarantee the global stability and uniqueness of the solution to a bone remodeling simulation are derived using a speci fic rate equation based on strain energy density. We modeled bone tiss ue as isotropic with a constant Poisson ratio and the elastic modulus proportional to volumetric density of calcified tissue raised to the p ower n. Our remodeling rate equation took the rate of change of volume tric hard tissue density as proportional to the difference between a s timulus (strain energy density divided by volumetric density taken to the power m) and a set point. In previous studies we defined state var iables which are conjugate to the remodeling stimulus, and the functio n which acts as a variational indicator for the remodeling stimulus. I n this study, we use the properties of this variational indicator to e stablish the stability and the uniqueness of the solution to the remod eling rate equations for all possible density distributions. We show t hat 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 remo deling elements. These results are proven for n < m, and we show that taking n > m will eliminate the possibility that a unique solution exi sts.