A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse
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Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, T6G 2G1
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Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, Edmonton, T6G 2G1
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Received:
01 July 2006
Accepted:
29 June 2018
Published:
01 February 2007
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MSC :
92B05.
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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).
Citation: Wenxiang Liu, Thomas Hillen, H. I. Freedman. A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse[J]. Mathematical Biosciences and Engineering, 2007, 4(2): 239-259. doi: 10.3934/mbe.2007.4.239
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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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