Rigorous results of limiting behaviors of total tumor size under cyclic intermittent therapy for the system of reversible phenotype-switchable tumor cells

Abstract

We are keenly interested in finding the limiting behaviors of total tumor size when tumor cells are subject to the periodic repetition of therapy and rest periods, called intermittent cyclic therapy. We hypothesize that each tumor cell can take either therapy-sensitive or therapy-tolerant phenotype, its phenotype transition is mainly driven by the presence or absence of environmental stress, and such a transition is reversible. Even though those aforementioned hypotheses make the model system simple, most of prior papers attempted to numerically find the optimal therapeutic scheduling that minimizes total tumor size, and there is no rigorous proof of the limiting behaviors of total tumor size to my knowledge. Here we present such long-waited mathematically rigorous results. In the first part of the paper, we present the derivation of total tumor size reduction criterion and prove two theorems of two different limiting behaviors of total tumor size under two different therapy strategies, one leading to an asymptotic finite tumor size according to an iterated map method and anther leading to asymptotically diminishing of total tumor size. In the second part of the paper, we discuss the effects of the intratumoral competition between sensitive and tolerant phenotypes on the total tumor size reduction criterion