Bifurcation analysis on a hybrid systems model of intermittent hormonal therapy for prostate cancer

Hybrid systems are widely used to model dynamical phenomena that are characterized by interplay between continuous dynamics and discrete events. An example of biomedical application is modeling of disease progression of prostate cancer under intermittent hormonal therapy, where continuous tumor dynamics is switched by interruption and reinstitution of medication. In the present paper, we study a hybrid systems model representing intermittent androgen suppression (IAS) therapy for advanced prostate cancer. Intermittent medication with switching between on-treatment and off-treatment periods is intended to possibly prevent a prostatic tumor from developing into a hormone-refractory state and is anticipated as a possible strategy for delaying or hopefully averting a cancer relapse which most patients undergo as a result of long-term hormonal suppression. Clinical efficacy of IAS therapy for prostate cancer is still under investigation but at least worth considering in terms of reduction of side effects and economic costs during off-treatment periods. In the model of IAS therapy, it depends on some clinically controllable parameters whether a relapse of prostate cancer occurs or not. Therefore, we examine nonlinear dynamics and bifurcation structure of the model by exploiting a numerical method to clarify bifurcation sets in the hybrid system. Our results suggest that adjustment of the normal androgen level in combination with appropriate medication scheduling could enhance the possibility of relapse prevention. Moreover, a two-dimensional piecewise-linear system reduced from the original model highlights the origin of nonlinear phenomena specific to the hybrid system.