Appears in collection : 2022 - T1 - WS1 - Tissue growth and movement

We introduce a mathematical model intended to describe by means of a system of partial differential equations the early stages of the interactions between effector immune cells and tumor cells. The model is structured in size and space: on the one hand the tumor development is governed by growth and cell division mechanisms, and, on the other hand the immune cells, activated by the presence of the tumor, are directed towards the tumor micro-environment by chemotactic mechanisms. Remarkably, the model exhibits a possible control of the tumor growth by the immune response; nevertheless, the control is not complete in the sense that the asymptotic states keep residual tu- mors and activated immune cells, a situation corresponding to the equilibrium phase predicted by biologists.The equilibrium state can be interpreted in terms of an eigenvalue problem coupled to a constrained drift-diffusion equation, an interpretation which is fruitful to set up numerical approaches in order to investigate the features of the equilibrium.

The model can be extended to take into account protumor effects of the immune response which can be very important to switch to the escape phase, with an uncontrolled growth of the tumor. Such a modeling can be useful to guide the design of combined immunotherapy strategies.