Stability, permanence and positive periodicity in a model of bone remodeling under impulsive PTH control

Abstract

In this paper, a mathematical model of bone remodeling process, which incorporates the effect of impulsive hormone supplementary treatments, is investigated both numerically and theoretically. A three dimensional model proposed in our earlier work in 2003 is first extended to incorporate impulsive treatment of estrogen supplement. It is illustrated that it is possible for the treatment to be interrupted with no apparent drop in its desirable effect on maintaining a normal bone mass. When the parathyroid hormone is assumed to have a very fast dynamics, the model in its reduced two dimensional form is then analyzed in terms of the boundedness, asymptotic stability, permanence, and oscillatory behavior. We show that there is a stable periodic solution, at the vanishing level of osteoclastic cells, when the impulsive period is less than some critical value. The conditions for permanence of the system are then given. Finally, it is shown that as the impulsive period increases beyond a certain critical value, the emergence of stable positive periodic solution may be observed under appropriate conditions on the system parameters. Thus, dynamic behavior of the system is sensitive to the period and amplitude of the hormone supplements so that the variation of these parameters are crucial for the proper management and control of this complex system.