A tumor growth model with autophagy: the reaction-(cross-)diffusion system and its free boundary limit

TL;DR

This paper develops a reaction-(cross-)diffusion tumor growth model incorporating autophagy, linking it to a free boundary system, and demonstrates that autophagy leads to exponential tumor growth, supported by analytical and numerical results.

Contribution

It introduces a novel reaction-(cross-)diffusion model with autophagy effects and connects it to a free boundary limit, providing analytical solutions and growth insights.

Findings

Autophagy causes exponential tumor growth.

The free boundary model converges to a well-mixed cell ratio.

Numerical simulations confirm analytical predictions.

Abstract

In this paper, we propose a tumor growth model 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 reaction-(cross-)diffusion system 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 system, 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 system which indicates the exponential growth of the tumor size in the presence of autophagy in contrast to the linear growth without it. Numerical simulations are also provided to illustrate the analytical properties and to explore more scenarios.