A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse

Abstract

In this paper we use a mathematical model to study the effect of an M-phase specific drug on the development of cancer, including the resting phase G(0) and the immune response. The cell cycle of cancer cells is split into the mitotic phase (M-phase), the quiescent phase (G(0)-phase) and the inter phase (G(1); S; G(2) phases). We include a time delay for the passage through the interphase, and we assume that the immune cells interact with all cancer cells. We study analytically and numerically the stability of the cancer-free equilibrium and its dependence on the model parameters. We find that quiescent cells can escape the M-phase drug. The dynamics of the G(0) phase dictates the dynamics of cancer as a whole. Moreover, we find oscillations through a Hopf bifurcation. Finally, we use the model to discuss the efficiency of cell synchronization before treatment (synchronization method).