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

We propose a mathematical kinetic framework to investigate interactions between tumor cells and the immune system, focusing on the spatial dynamics of tumor progression and immune responses. We develop two kinetic models: one describes a conservative scenario where immune cells switch between active and passive states without proliferation, while the other incorporates immune cell proliferation and apoptosis. By considering specific assumptions about the microscopic processes, we derive macroscopic systems featuring linear diffusion, nonlinear cross-diffusion, and nonlinear self-diffusion. Our analysis provides insights into equilibrium configurations and stability, revealing clear correspondences among the macroscopic models derived from the same kinetic framework. Using dynamical systems theory, we examine the stability of equilibrium states and conduct numerical simulations to validate our findings. These results highlight the significance of spatial interactions in tumor-immune dynamics, paving the way for a structured exploration of therapeutic strategies and further investigations into immune responses in various pathological contexts.