American Institute of Mathematical Sciences

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2007, 4(2): 239-259. doi: 10.3934/mbe.2007.4.239

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

Wenxiang Liu 1, , Thomas Hillen 2, and  H. I. Freedman 1,

1. 

Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada, Canada
2. 

Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, Edmonton, T6G 2G1, Canada

Received  July 2006 Revised  September 2006 Published  February 2007

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 interphase ($G_1,\ S,\ G_2$ 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).
Keywords: Hopf bifurcation., cancer growth, cycle-phase-specific drugs, time delay.
Mathematics Subject Classification: 92B0.
Citation: Wenxiang Liu, Thomas Hillen, H. I. Freedman. A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse. Mathematical Biosciences & Engineering, 2007, 4 (2) : 239-259. doi: 10.3934/mbe.2007.4.239
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