Modeling Tumor–Immune Dynamics

Abstract

Mathematical models of tumor–immune interactions provide an analytical framework in which to address specific questions regarding tumor–immune dynamics and tumor treatment options. We present a mathematical model, in the form of a system of ordinary differential equations (ODEs), that governs cancer growth on a cell population level. In addition to a cancer cell population, the model includes a population of Natural Killer (NK) and CD8+ T immune cells. Our goal is to understand the dynamics of immune-mediated tumor rejection, in addition to exploring results of applying combination immune, vaccine and chemotherapy treatments. We characterize the ODE system dynamics by locating equilibrium points, determining stability properties, performing a bifurcation analysis, and identifying basins of attraction. These system characteristics are useful, not only for gaining a broad understanding of the specific system dynamics, but also for helping to guide the development of combination therapies. Additionally, a parameter sensitivity analysis suggests that the model can predict which patients may respond positively to treatment. Numerical simulations of mixed chemo-immuno and vaccine therapy using both mouse and human parameters are presented. Simulations of tumor growth using different levels of immune stimulating ligands, effector cells, and tumor challenge, are able to reproduce data from published studies. We illustrate situations for which neither chemotherapy nor immunotherapy alone are sufficient to control tumor growth, but in combination the therapies are able to eliminate the entire tumor.