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 ofan $M$-phase specific drug on the development of cancer, includingthe resting phase $G_0$ and the immune response. The cell cycle ofcancer cells is split into the mitotic phase (M-phase), thequiescent phase ($G_0$-phase) and the interphase ($G_1,\ S,\G_2$ phases). We include a time delay for the passage through theinterphase, and we assume that the immune cells interact with allcancer cells. We study analytically and numerically the stabilityof the cancer-free equilibrium and its dependence on the modelparameters. We find that quiescent cells can escape the $M$-phasedrug. The dynamics of the $G_0$ phase dictates the dynamics ofcancer as a whole. Moreover, we find oscillations through a Hopfbifurcation. Finally, we use the model to discuss the efficiencyof cell synchronization before treatment (synchronization method).

Keywords

• hopf bifurcation.
• cancer growth
• cycle-phase-specific drugs
• time delay