In this study, we develop a biologically informed mathematical model of breast cancer that integrates key tumor, cytokine, and immune-cell interactions. To balance mechanistic detail with computational efficiency, a stepwise modeling strategy is adopted. In the first phase, the dynamics of two major cytokines, Interleukin-2, (IL-2) and Interferon- γ (IFN- γ ), are formulated together with their regulatory effects on tumor cells. Simulations show that cytokine activity alone is insufficient to control tumor progression. Motivated by this result, the model is expanded to incorporate essential immune cell populations, including natural killer (NK) cells, CD4 + helper T cells, and CD8 + cytotoxic T cells, with their interactions described through nonlinear feedback mechanisms such as Michaelis–Menten kinetics. Unlike classical compartmental models that treat cytokines or chemotherapy as external factors, the proposed framework introduces a novel two-phase cytokine–immune structure that mechanistically bridges molecular and cellular scales within a unified system. Despite this enhanced coupling, the immune response alone fails to eliminate tumor cells for realistic initial tumor burdens. To overcome this limitation, a constant-dose chemotherapy component is incorporated and solved using the non-standard finite difference (NSFD) method. The combined immune–chemotherapy model demonstrates complete tumor clearance even for an initial tumor size of 1 0 9 cells. Finally, sensitivity analysis identifies Q t (chemotherapy-induced tumor cell death rate) and r t (tumor growth rate) as the most influential parameters, offering valuable guidance for optimizing treatment strategies. It should be noted that model validation is performed using rescaled, population-averaged tumor growth data, and therefore the reported fit supports reproduction of population-level trends rather than individualized patient-specific prediction. Explanation of the immune system mechanisms, from tumor cell recognition to elimination, illustrating immune cell interactions and cytokine dynamics in the proposed breast cancer model, and demonstrating tumor elimination with chemotherapy integration. • A simplified yet biologically grounded mathematical model is developed to capture the key anti-tumor immune dynamics involving CD4 + helper T cells, CD8 + cytotoxic T cells, NK cells, and tumor cells, along with cytokines IL-2 and IFN- γ . The model omits immunosuppressive and indirect regulatory components to reduce dimensionality and computational stiffness, enabling efficient simulation and parameter analysis while preserving core tumor–immune interactions. • We consider our model in two phases. The first phase is based on cytokines, and the second phase is based on cells. In the first phase, we graphically show that the concentrations of interleukin-2 and interferon- γ cytokines stabilize after a certain period of time, and the coefficients obtained from this phase are then used in the cell-based model in the second phase. • The system based on immune and tumor cells can be solved using any numerical method. In this paper, the fourth-order Runge–Kutta method is used. • The proposed cell-based model was validated using clinical data. It accurately predicted tumor growth achieving a strong fit with experimental observations ( R 2 = 0 . 9065 ). • Using MATLAB, we demonstrate that the cell-based model alone is unable to eliminate tumor cells. Without the addition of chemotherapy, the tumor cell population increases significantly. • After incorporating chemotherapy, we solved the model using the non-standard finite difference method and showed graphically that chemotherapy can eliminate a tumor population of 1 × 1 0 9 cells in less than 40 days. • We applied both local (OFAT) and global (PRCC) sensitivity analyses to identify the key parameters influencing tumor dynamics. The tumor growth rate ( r t ) and the chemotherapy-related parameter ( Q t ) were the most influential in both approaches. While OFAT captured local effects, PRCC revealed nonlinear interactions and highlighted additional influential parameters, including immune response and drug toxicity factors. Combining both methods provides a more comprehensive understanding of the model’s behavior.
Gholami et al. (Sun,) studied this question.