Anti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapy

A two-dimensional system of ordinary differential equations is used to characterize the basic types of phase portraits of the immune system - tumor interactions model, and to study the impact of anti-immune activity by tumor on the outcome of immunotherapy. The focus is on specific (acquired) immunity and different forms of immunotherapy as active therapy with in vivo stimulation of the immunity and passive one with infusion of ex vivo produced specific immunity. The analysis is performed for two families of stimulation function, which describes the dynamics of the stimulation of the immune system by tumor antigens: (1) antigen dependent and (2) antigen per one immunity unit dependent functions, with Michaelis-Menten and sigmoid functions in each family. We show that there are no limit cycles in the system and that anti-immune activity by tumor changes all equilibrium points from global to local ones. In the latter case, the immune system has no control over the growth of large tumors. Furthermore, if the immunity is weak, the immune system cannot eradicate even small tumors. The weak immunity and stimulation strength result in unrestricted tumor growth. The patterns of asymptotic behavior of the system do not depend on the type of the stimulation function, but do depend on its parameters. Our results reflect the basic clinical and experimental knowledge about immunotherapy and its effectiveness and yield new suggestions for an efficient immunotherapy.