Chaos in the discrete-time algorithm for bone-density remodeling rate equations.

Abstract

We compare the predictions of the differential equation form of a class of bone-density stress adaptation models with their associated discrete-time computational algorithms. Although our considerations apply to the class of adaptation models based on bulk or apparent bone-density remodeling, we focus attention on a particular model in this class, a model employed by Weinans et al. [Trans. Orthop. Res. Soc. 14, 310 (1989); Trans. First World Congress of Biomechanics, Vol. II, p. 75 (1990)]. We show that the discrete-time computational algorithm of that stress adaptation model has a well-known chaos mechanism for stress values of practical interest. Further, we obtain a condition on the discrete-time step that prevents the transition to chaos, and conditions that insure monotonic convergence. This chaos mechanism is only present in the discrete-time computational algorithm; we show that the corresponding differential equation form of the bone-density stress adaptation model is smooth, monotonic and nonchaotic.