Observations of convergence and uniqueness of node-based bone remodeling simulations

Some investigators have indicated that mathematical theories and computational models of bone adaptation may not converge and that the density solutions from such simulations are dependent on the initial density distribution. In this study, two-dimensional finite element models were used to investigate the effect of initial density distribution on the final density distribution produced using a node-based bone remodeling simulation. The first model was a generic long bone, and the second was a proximal femur, For each model, we conducted time-dependent, node-based, linear rate-law bone remodeling simulations. Five initial density conditions were used with the generic long bone and three with the proximal femur. Remodeling simulations were performed, and the largest average nodal density differences at the end of the simulations were 0.000010 g/cm3 and 0.000006 g/cm3 for the generic long bone and proximal femur models, respectively. Results illustrate that, for a given set of loads and a given finite element model, the node-based bone adaptation algorithm can yield a unique density distribution. In conjunction with previous studies, this finding suggests that uniqueness of the density solution is dependent on both the mathematical theory and the computational implementation.