Optimality conditions for finite element simulation of adaptive bone remodeling

Bone remodeling in vertebrates is widely quoted as a process which optimizes the use of structural material, subject to mechanical requirements. In vertebrates, bone remodeling continues throughout life and tends to preserve the structure of a particular bone over decades of life. This implies that the processes which remodel bone are stable, at least over a time period spanning many years. Many recent numerical simulations of bone remodeling have used rate equations which have not been carefully assessed for stability and their ability to produce an optimal structure. In this study, we re-state the conditions necessary for stability of a particular bone remodeling rate equation derived in a related study, and we investigate whether the rate equation used can produce an optimal structure. Within the context of a finite element discretization, we show that this rale equation does not produce a structure optimized with respect to density. By making a simple modification to the stable remodeling rate equation, we show that the remodeling stimulus used can produce an optimal structure if the state variable manipulated is density taken to a power. We conclude that if bone is a stable, self-optimizing structure, there are specific requirements for the point-by-point rate of change of bone density in response to mechanical stress. The implications of these requirements for simulations ot adaptive bone remodeling are discussed.