Modeling the autophagic effect in tumor growth: a cross diffusion model and its free boundary limit

In this paper, we propose two macroscopic tumor growth models to incorporate and investigate the spatial effects of autophagy. The cells are classified into two phases: normal cells and autophagic cells, whose dynamics are also coupled with the nutrients. First, we construct a fluid mechanical model describing the evolution of cell densities, where the drift is determined by the negative gradient of the joint pressure, and the reaction terms manifest the unique mechanism of autophagy. Next, in the incompressible limit, such a cell density model naturally connects to a free boundary model, describing the geometric motion of the tumor region. Analyzing the free boundary model in a special case, we show that the ratio of the two phases of cells exponentially converges to a well-mixed limit. Within this well-mixed limit, we obtain an analytical solution of the free boundary model which indicates the exponential growth of the tumor size in the presence of autophagy in contrast to the linear growth without it. We also provide extensive numerical simulations to illustrate the properties of the tumor models and explore with specially designed experiments the effects of autophagy, such as the enhancement of survivability.