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

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


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.



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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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneyCamperdownAustralia

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