ANALYSIS AND SIMULATION OF A FREE BOUNDARY PROBLEM MODELING TUMOR EVOLUTIONDr.Xuming Xie

Abstract

This dissertation concerns analysis and simulation of a free boundary problem modeling tumor growth. Incorporating immune cell interactions, the model consists of a coupled system of partial differential equations describing the densities of cancer cells and T-cells within a tumor region whose boundary evolves dynamically over time. A new framework is developed by assuming flux boundary conditions for both cancer cells and T-cells at the moving boundary. Using tools in the theory of Partial Differential Equations, we establish the existence, uniqueness, and global behavior of classical solutions under suitable initial conditions. To complement the theoretical results, a fully discrete numerical algorithm based on the finite difference method is derived after rescaling the problem to a fixed domain, the stability of the numerical algorithm is discussed. Numerical simulations are conducted to explore the long-term behavior of the tumor, demonstrating that the tumor may shrink or grow depending on the immune killing rate. The numerical results validate the theoretical predictions and provide deeper insights into the interplay between tumor evolution and immune response. This study contributes to the mathematical understanding of tumor-immune dynamics and highlights the critical role of immune strength in determining tumor fate, offering potential implications for immunotherapy strategies.