Leukaemia is a type of malignant blood cancer, and the most common cause of cancer-related death is leukaemia. In this study, we present a comprehensive mathematical analysis of T cell-based cancer treatment. We reconstruct a mathematical model for chimeric antigen receptors (CAR) T cell therapy targeting leukaemia and adding new compartmental term cytokine in this model. We create a mathematical model to simulate chimeric antigen receptor (CAR) T cell therapy for leukaemia. We present five nonlinear differential equations compartmental mathematical model (MM) of leukaemia, which includes susceptible blood cells \({S}_{1}(t)\) , infected blood cells \({I}_{1}(t)\) , cancer cells \({C}_{1}(t)\) , immune blood cells \({W}_{1}(t)\) , cytokine cells \({C}_{2}(t)\) and we describe the dynamics of transmission of the disease. We employ optimal control (OC) theory, as CAR T cell therapy impacts both normal and cancer cells. The optimized CAR T-cell dosage serves a pivotal role and is introduced into the system as the controller denoted as \({u}_{1}\) . Conversely, to manage the cytokine release syndrome that leads to singularity, we introduce another controller \({u}_{2}\) . This controller, tocilizumab, acts as an immunosuppressive agent for cytokine release syndrome within the system. We have discussed the simulation results in two ways: one is the optimal control (OC) approach (using the SDRE method), and another is the conventional approach (without control). Moreover, the solutions that have been found using the methods are compared, and we have discussed stability and bifurcation analysis of the results. Finally, we have seen that the optimal control treatment reduces the quantity of cancer cells and the release of cytokines.
