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Mathematical Models of Tumor-Immune System Dynamics pp 21–46Cite as

A Cellular Automata and a Partial Differential Equation Model of Tumor–Immune Dynamics and Chemotaxis

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 107))

Abstract

Immunotherapy is a newly emerging approach to cancer treatment that seeks to stimulate a body’s immune defenses, especially T cells, to combat and potentially eliminate tumors. Relevant tumor–immune interactions depend on stochasticity, since the dynamics involve a small and decreasing number of cells, and spatiotemporal heterogeneity, since the dynamics occur in a localized tumor environment. To account for these two aspects of the system, we develop mathematical models of an anti-tumor immune response using a cellular automaton and a system of partial differential equations. We explicitly model immune cell recruitment to the tumor via cytokine secretion and chemotaxis of immune cells. Our models exhibit three types of behavior: tumor elimination, oscillation, and uncontrolled tumor growth that depend substantially on the strength of immune cell chemotaxis, or recruitment, to the tumor site.

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Acknowledgements

AKC was supported by the KE Bullen Scholarship III awarded to honours students in the School of Mathematics and Statistics at the University of Sydney, and PSK was supported by the Australian Research Council Discovery Early Career Research Award (DE120101113).

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sydney, Camperdown, NSW, 2006, Australia

    Andrea K. Cooper & Peter S. Kim

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  1. Andrea K. Cooper
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  2. Peter S. Kim
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Correspondence to Peter S. Kim .

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Editors and Affiliations

  1. Mathematics Department, The College of Saint Rose, Albany, New York, USA

    Amina Eladdadi

  2. School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales, Australia

    Peter Kim

  3. Mathematical Sciences School, Queensland University of Technology, Brisbane, Queensland, Australia

    Dann Mallet

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Cooper, A.K., Kim, P.S. (2014). A Cellular Automata and a Partial Differential Equation Model of Tumor–Immune Dynamics and Chemotaxis. In: Eladdadi, A., Kim, P., Mallet, D. (eds) Mathematical Models of Tumor-Immune System Dynamics. Springer Proceedings in Mathematics & Statistics, vol 107. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1793-8_2

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