Mathematical modelling of tumor-immune interactions in breast cancer

The dynamic interplay between tumors and immune system is pivotal to the progression of breast cancer. To systematically investigate how interactions between tumor cells and immune cells shape breast cancer evolution, we developed a mathematical model that incorporates tumor cells, dendritic cells (DCs), natural killer (NK) cells, regulatory T cells (Tregs) and CD8+ T cells. We first established analytical conditions for the local stability of the tumor-free equilibrium, identifying key constraints on tumor growth imposed by immune activity. The existence of a positive equilibrium solution further suggests the potential coexistence of tumor and immune cells. Numerical simulations demonstrate that effective tumor control is achieved under a high baseline level of CD8+ T cell precursors coupled with a low level of regulatory T cell precursors. These results highlight the important role of balancing immunostimulatory and immunosuppressive forces within the tumor microenvironment. Through bifurcation analysis, we identified regimes of bistability in which both high-tumor and low-tumor equilibria coexist with dynamic features that may underlie divergent clinical outcomes and present a critical challenge for clinical therapeutic intervention. Moreover, simulations of tumor-immune dynamics in virtual cohorts reveal that tumor control hinges on CD8+ T cell infiltration, whereas regulatory T cell abundance is a potent predictor of immune escape. Finally, we formulated an optimal control framework to design adaptive CD8+ T cell injection protocols. Numerical solutions demonstrate that such optimized strategies achieve superior tumor reduction compared with constant dosing, despite using the same total injection dose of CD8+ T cells and identical treatment intervals. Collectively, our findings provide a mechanistic understanding of breast cancer progression and establish a theoretical foundation for developing personalized therapeutic strategies to optimize clinical outcomes.