Dynamics of tumor–immune system interaction: a kinetic approach

A kinetic model describing the competition between tumor cells and immune system is investigated. The starting point is represented by the kinetic equations for the evolution of dominance in populations of interacting organisms, taking nonconservative effects of proliferation and destruction into account. Four interacting populations are considered, representing, respectively, tumors cells, cells of the host environment, cells of the immune system, and interleukines, which are capable to modify the tumor-immune system inter- action, and to contribute to destroy tumor cells. The internal state variable (activity) measures the capability of a cell of prevailing in a binary interaction. A closed set of autonomous ODEs is then derived by a moment procedure, representing a sort of macroscopic continuity equations in the sense of kinetic theory. Under very reasonable assumptions on the microscopic interaction parameters, two three-dimensional reduced systems of ODEs are obtained in a partial quasi-steady state approximation. In the first approximation the host environment plays the role of background; in the second one this role is assumed by the interleukines. The qualitative analysis of these evolution problems is then performed in the framework of the theory of dynamical systems. Essential steps are determination of fixed points, as stationary living conditions of the considered organism, and of their stability, as well as possible occurrence of bifurcations for varying parameters. The analysis shows that the reduced system describing interactions between tumor, immune system and interleukines can be investigated in the framework of the theory of the asymptotically autonomous differential systems, and then its asymptotic behaviours can be deduced from an easier two dimensional limit system. An important feature of the second reduced system for the interactions between tumor, immune system and host environment is the presence, also in this context, of a backward bifurcation, which is usually related to epidemic models.