Prediction of bone adaptation using damage accumulation.

The adaptation of bones to a change in function has been recognised for many centuries, but only recently have mathematical laws been proposed to describe it. One proposed mathematical law is based on the hypothesis that, after a change in load, the strain in the bone microstructure is regulated to a homeostatic equilibrium. Strain-adaptive remodelling has been used successfully to simulate bone adaptation around orthopaedic implants but the predictive capabilities are constrained because many empirical constants are required in the remodelling law (a reference stimulus, a zone of equilibrium stresses or 'lazy zone' and a parameter transducing macroscopic stresses to a tissue level stimulus). An alternative approach has been proposed. It is that bone adapts to attain an optimal strength by regulating the damage generated in its microstructural elements. The question is raised whether or not a mathematical law to predict the time course of bone adaptation can be derived for damage-adaptive remodelling in a similar way to the mathematical laws based on a strain stimulus. In the present study, the hypotheses required to develop damage-adaptive remodelling laws are proposed and a remodelling law to predict the time course of bone adaptation is derived. It is shown that this is an integral remodelling law which accounts naturally for the stress history to which the tissue has been exposed since formation. A simulation of the adaptive response of a bone diaphysis under a change in torsional load shows that the law gives physically reasonable predictions. The initial remodelling prediction is similar to strain-adaptive remodelling. However, in the later stages of remodelling, the predictions differ from strain-adaptive remodelling in that direct convergence to a homeostatic strain is not predicted. Instead, undershoot (in the case of a reduction in load) and overshoot (in the case of an increase in load) are predicted.