Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells

Maria Vittoria Barbarossa; Christina Kuttler; Jonathan Zinsl; Maria Vittoria Barbarossa; Christina Kuttler; Jonathan Zinsl

doi:10.3934/mbe.2012.9.241

Mathematical Biosciences and Engineering

2012, Volume 9, Issue 2: 241-257. doi: 10.3934/mbe.2012.9.241

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Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells

1.
Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München

Received: 01 May 2011 Accepted: 29 June 2018 Published: 01 March 2012
MSC : Primary: 34K17, 92B05; Secondary: 92C50.

In this work we present a mathematical model for tumor growth based on the biology of the cell cycle. For an appropriate description of the effects of phase-specific drugs, it is necessary to look at the cell cycle and its phases. Our model reproduces the dynamics of three different tumor cell populations: quiescent cells, cells during the interphase and mitotic cells. Starting from a partial differential equations (PDEs) setting, a delay differential equations (DDE) model is derived for an easier and more realistic approach. Our equations also include interactions of tumor cells with immune system effectors. We investigate the model both from the analytical and the numerical point of view, give conditions for positivity of solutions and focus on the stability of the cancer-free equilibrium. Different immunotherapeutic strategies and their effects on the tumor growth are considered, as well.
- tumor,
- cell cycle,
- $G_1$,
- $G_0$,
- immunotherapy.,
- chemotherapy,
- Delay differential equations,
- phase-specific drugs,
- immune system
Citation: Maria Vittoria Barbarossa, Christina Kuttler, Jonathan Zinsl. Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells[J]. Mathematical Biosciences and Engineering, 2012, 9(2): 241-257. doi: 10.3934/mbe.2012.9.241

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Abstract

In this work we present a mathematical model for tumor growth based on the biology of the cell cycle. For an appropriate description of the effects of phase-specific drugs, it is necessary to look at the cell cycle and its phases. Our model reproduces the dynamics of three different tumor cell populations: quiescent cells, cells during the interphase and mitotic cells. Starting from a partial differential equations (PDEs) setting, a delay differential equations (DDE) model is derived for an easier and more realistic approach. Our equations also include interactions of tumor cells with immune system effectors. We investigate the model both from the analytical and the numerical point of view, give conditions for positivity of solutions and focus on the stability of the cancer-free equilibrium. Different immunotherapeutic strategies and their effects on the tumor growth are considered, as well.
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