Mathematical model for repair of fatigue damage and stress fracture in osteonal bone

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Abstract
This paper assembles current concepts about bone fatigue and osteonal remodeling into a mathematical theory of the repair of fatigue damage and the etiology of stress fracture. The model was used to address three questions. (a) How does the half-life of fatigue damage compare with the duration of the remodeling cycle? (b) Does the porosity associated with the remodeling response contribute to stress fracture? (c) To what extent is a periosteal callus response necessary to augment repair by remodeling? To develop the theory, existing experimental data were used to formulate mathematical relationships between loading, damage, periosteal bone formation, osteonal remodeling, porosity, and elastic modulus. The resulting nonlinear relationships were numerically solved in an iterative fashion using a computer, and the behavior of the model was studied for various loading conditions and values of system parameters. The model adapted to increased loading by increasing remodeling to repair the additional damage and by adding new bone periosteally to reduce strain. However, if too much loading was encountered, the porosity associated with increased remodeling caused the system to become unstable: i.e., damage, porosity, and strain increased at a very high rate and without limit. It is proposed that this phenomenon is the equivalent of a stress fracture and that its biological and mechanical elements are significant in the etiology of stress fractures. Additional experiments must be done to test the model and provide better values for its parameters. However, the instability characteristic is relatively insensitive to changes in model parameters.