A kinetic derivation of spatial distributed models for tumor-immune system interactions

We propose a mathematical kinetic framework to investigate the complex nonlinear interactions between tumor cells and the immune system, focusing on the spatial dynamics that drive tumor progression and immune responses. We develop two kinetic models: the first represents a conservative scenario where immune cells switch between active and passive states without proliferation, while the second incorporates immune cell proliferation and apoptosis. By considering specific assumptions on the microscopic processes, we formally derive macroscopic systems featuring linear diffusion, nonlinear cross-diffusion, and nonlinear self-diffusion. Using dynamical systems theory, we examine the equilibrium states and their stability, and conduct numerical simulations to validate our theoretical findings. Our analysis reveals clear correspondences between the derived macroscopic systems and the underlying kinetic model. Moreover, these findings underscore the significance of spatial interactions in shaping tumor-immune dynamics, paving the way for a more structured and targeted exploration of immune responses across different pathological contexts.