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Mathematical Biosciences and Engineering (MBE)
 

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

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

 
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Maria Vittoria Barbarossa - Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany (email)
Christina Kuttler - Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany (email)
Jonathan Zinsl - Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany (email)

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.

Keywords:  Delay differential equations, tumor, cell cycle, phase-specific drugs, $G_0$, $G_1$, immune system, chemotherapy, immunotherapy.
Mathematics Subject Classification:  Primary: 34K17, 92B05; Secondary: 92C50.

Received: May 2011;      Accepted: October 2011;      Available Online: March 2012.

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