Analysis Of S Macrophage-Based Gene Therapy Model

In this paper, we study a mathematical model of the macrophages-based tumor therapy.The model is used to describe the mechanism of a tumor treated by using macrophage cells.This gene therapy that transfers genes based on macrophages is a novel method. Comparedwith general drug therapy or radiation therapy, this method is more significant. Thismacrophage-based gene therapy can be described as a free boundary problem. The modelconsists of four semi-linear parabolic equations which are used to describe the tumor cells,tumor-associated macrophages and tumor substances within the region (including proteins,oil, water, etc.) and four nonlinear elliptic equations which are used to describe the tumoroxygen, the prodrug, the drugs and the chemokines. The surface of this tumor spheroid is afree boundary. We will use theLP estimate, Schauder estimate of elliptic and parabolicequations and Banach fixed point theorem to prove that the problem is locally well-posed.Then we use the extension method and the monotone method in the theory of reactiondiffusion equations to prove that the solution either disappeared within a limited period oftime, or exists globally.