Derivation of Macroscopic Equations from Homogeneous Thermostatted Kinetic Equations in the Cancer-Immune System Competition

A model of interactions between cancer cells and immune system cells is considered in the framework of thermostatted kinetic theory. Each cell is supposed to carry an activity accounting for its level of learning. Simulations of the kinetic equations using an adaptation of the direct simulation Monte Carlo method reproduce a clinically observed phenomenon called the 3Es of immunotherapy. This phenomenon consists of an apparent initial elimination of cancer followed by a long pseudo-equilibrium phase and the final escape of cancer from the control of the immune system. In an inhomogeneous system, the simulations also account for pseudo-oscillations of the numbers and mean activities of cancer cells and immune system cells. In this work, we derive the macroscopic equations for the concentrations and mean activities of each cell type from the kinetic equations associated with a homogeneous system. The homogeneous macroscopic equations account for the 3Es but do not include the description of pseudo-oscillations.