Chaos in the discrete-time algorithm for bone-density remodeling rate equations.
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.
Cowin
S C
SC
Department of Mechanical Engineering, School of Engineering, City College, New York, NY.
Arramon
Y P
YP
Luo
G M
GM
Sadegh
A M
AM
D000222:Adaptation, Physiological; D000465:Algorithms; D015519:Bone Density / Q000502:physiology*; D001857:Bone Matrix / Q000502:physiology; D016723:Bone Remodeling / Q000502:physiology*; D004548:Elasticity; D006801:Humans; D016015:Logistic Models; D008954:Models, Biological; D017711:Nonlinear Dynamics; D013314:Stress, Mechanical; D013997:Time Factors