DYNAMIC BEHAVIORS OF A PERIODIC SYSTEM WITH THRESHOLD POLICY-GUIDED PERIODIC AND INTERMITTENT THERAPY OF TUMOR

Abstract

Recent clinical studies provide strong evidence that adaptive tumor therapy is a promising threshold policy-guided periodic and intermittent treatment for prolonging treatment efficacy particularly for prostate cancer, despite the observed tumor size fluctuations in the patients. Understanding plausible patterns of tumor size fluctuations under different treatment scenarios is important to evaluate the therapy outcomes. Here we use state-of-the-art modeling and analytic frameworks of periodic switching systems and state-dependent switching systems to develop a dynamic model that mimics the adaptive therapy, and we describe dynamical behaviors of the model system with a particular focus on the patterns of periodic fluctuation. Under the threshold policy, patients may not be given therapies during the a priori treatment period. We show that this threshold-guided treatment will give rise to a new type of periodic solutions with complex structures, characterized as (m,k)T-periodic solutions. We examine, both theoretically and numerically, the existence, as well as the local and global stability of such periodic solutions, including the boundary periodic solutions with one vanished compartment and positive (m,k)T-periodic solutions. It is hoped this study also shows the great potential in developing a class of models to describe threshold-triggered periodic and intermittent control in many fields, which in turn may generate many new dynamic behaviors of nonsmooth dynamic systems.