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

Wenxiang Liu; Thomas Hillen; H. I. Freedman

doi:10.3934/mbe.2007.4.239

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2007, Volume 4, Issue 2: 239-259. Doi: 10.3934/mbe.2007.4.239

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

1.
Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, T6G 2G1
2.
Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, Edmonton, T6G 2G1

Received: July 2006

Revised: September 2006

Published: February 2007

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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 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).

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Mathematics Subject Classification: 92B05.

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

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