  The competition between cancer cells and immune system cells in inhomogeneous
conditions is described at cell scale within the framework of the thermostatted
kinetic theory. Cell learning is reproduced by increased cell activity during
favorable interactions. The cell activity fluctuations are controlled by a
thermostat. The direction of cell velocity is changed according to stochastic
rules mimicking a dense fluid. We develop a kinetic Monte Carlo algorithm
inspired from the direct simulation Monte Carlo (DSMC) method initially used
for dilute gases. The evolution of an initially localized tumor is analyzed.
Qualitatively different behaviors are observed as the field regulating activity
fluctuations decreases. For high field values, i.e. efficient thermalization,
cancer is controlled. For small field values, cancer rapidly and monotonously
escapes from immunosurveillance. For the critical field value separating these
two domains, the 3E's of immunotherapy are reproduced, with an apparent initial
elimination of cancer, a long quasi-equilibrium period followed by large
fluctuations, and the final escape of cancer, even for a favored production of
immune system cells. For field values slightly smaller than the critical value,
more regular oscillations of the number of immune system cells are
spontaneously observed in agreement with clinical observations. The
antagonistic effects that the stimulation of the immune system may have on
oncogenesis are reproduced in the model by activity-weighted rate constants for
the autocatalytic productions of immune system cells and cancer cells. Local
favorable conditions for the launching of the oscillations are met in the
fluctuating inhomogeneous system, able to generate a small cluster of immune
system cells with larger activities than those of the surrounding cancer cells.
