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

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.