Reaction-Diffusion Model of CAR-T Cell Therapy in Solid Tumours with Antigen Escape

Developing effective CAR-T cell therapy for solid tumours remains challenging because of biological barriers such as antigen escape and an immunosuppressive microenvironment. The aim of this study is to develop a mathematical model of the spatio-temporal dynamics of tumour processes in order to assess key factors that limit treatment efficacy. We propose a reaction–diffusion model described by a system of partial differential equations for the densities of tumour cells and CAR-T cells, the concentration of immune inhibitors, and the degree of antigen escape. The methods of investigation include stability analysis and numerical solution of the model using a finite-difference scheme. The simulations show that antigen escape produces a resistant tumour core and relapse after an initial regression; increasing the escape rate from $\gamma =0.001$ to $0.1$ increases the final tumour volume at $t=100$ days from approximately 35.3 a.u. to 36.2 a.u. Parameter mapping further indicates that for $\gamma \le 0.01$ tumour control can be achieved at moderate killing rates ($k_{CT}\approx 1{\mathrm{day}}^{-1}$), whereas for $\gamma \ge 0.05$ comparable control requires $k_{CT}\approx 2$–$5{\mathrm{day}}^{-1}$. Repeated CAR-T administration improves durability: the residual normalised tumour volume at $t=100$ days decreases from approximately 4.5 after a single infusion to approximately 0.9 (double) and approximately 0.5 (triple), with a saturating benefit for further intensification. We conclude that the proposed model is a valuable tool for analysing and optimising CAR-T therapy protocols, and that our results highlight the need for combined strategies aimed at overcoming antigen escape.