Mathematical modeling and numerical solutions for functionally dependent bone remodeling.

The phenomenon of bone remodeling is a complex biological process which is dependent on genetic, hormonal, metabolic, and age-related factors as well as functional requirements. The possibility of successfully developing a mathematical model to describe and predict the adaptive response of bone to load will be significantly increased after identification of the nature of the transducer(s) which senses functional requirements and provides signals for the cellular processes responsible for bone synthesis and bone removal. In spite of the present limitations in knowledge about the functional dependence of bone remodeling, a phenomenological model has been developed that assumes that the output signal from the (as yet unspecified) transducer is a remodeling potential that can be modulated by genetic, hormonal, and metabolic factors. An attempt has been made to cast the mathematical model in such a form that the constants and variables appearing in the equations are not mere abstractions, but can be related to biological parameters. In order to use the adaptive hypothesis with specific structural model examples, a numerical procedure has been developed to determine the strain distribution, predict the remodeling (assuming that the remodeling rate is related to the strain history), and update the model by changing the geometry and material properties in response to the remodeling. This numerical procedure is repeatedly iterated to determine the structural architecture at subsequent times. The numerical approach allows use of the remodeling concepts with models of irregular geometry, inhomogeneous material distribution, and anisotropic material properties.