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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.

    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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  • 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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    2. Andrea K. Cooper, Peter S. Kim, 2014, Chapter 2, 978-1-4939-1792-1, 21, 10.1007/978-1-4939-1793-8_2
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    6. Evans K. Afenya, Rachid Ouifki, Baba I. Camara, Suneel D. Mundle, Mathematical modeling of bone marrow – peripheral blood dynamics in the disease state based on current emerging paradigms, part I, 2016, 274, 00255564, 83, 10.1016/j.mbs.2016.01.010
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    8. Wafa F. Alfwzan, Dumitru Baleanu, Ali Raza, Muhammad Rafiq, Nauman Ahmed, Dynamical analysis of a class of SEIR models through delayed strategies, 2023, 13, 2158-3226, 10.1063/5.0159942
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