Mathematical Modelling for the Role of CD4T Cells in Tumor-Immune Interactions.

Mathematical modelling has been used to study tumor-immune cell interaction. Some models were proposed to examine the effect of circulating lymphocytes, natural killer cells, and CD8T cells, but they neglected the role of CD4T cells. Other models were constructed to study the role of CD4T cells but did not consider the role of other immune cells. In this study, we propose a mathematical model, in the form of a system of nonlinear ordinary differential equations, that predicts the interaction between tumor cells and natural killer cells, CD4T cells, CD8T cells, and circulating lymphocytes with or without immunotherapy and/or chemotherapy. This system is stiff, and the Runge-Kutta method failed to solve it. Consequently, the "Adams predictor-corrector" method is used. The results reveal that the patient's immune system can overcome small tumors; however, if the tumor is large, adoptive therapy with CD4T cells can be an alternative to both CD8T cell therapy and cytokines in some cases. Moreover, CD4T cell therapy could replace chemotherapy depending upon tumor size. Even if a combination of chemotherapy and immunotherapy is necessary, using CD4T cell therapy can better reduce the dose of the associated chemotherapy compared to using combined CD8T cells and cytokine therapy. Stability analysis is performed for the studied patients. It has been found that all equilibrium points are unstable, and a condition for preventing tumor recurrence after treatment has been deduced. Finally, a bifurcation analysis is performed to study the effect of varying system parameters on the stability, and bifurcation points are specified. New equilibrium points are created or demolished at some bifurcation points, and stability is changed at some others. Hence, for systems turning to be stable, tumors can be eradicated without the possibility of recurrence. The proposed mathematical model provides a valuable tool for designing patients' treatment intervention strategies.