الفهرس 
INTRODUCTIONOnly 14 pages are availabe for public view
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Abstract
Mathematical models have been proposed to investigate tumor-immune cell interaction. The proposed mathematical models are in the form of a system of nonlinear ordinary differential equations which predicts the interaction between tumor cells and natural killer cells, CD4+T cells, CD8+T cells, and circulating lymphocytes. The model is extended to include the effect of immunotherapy and/or chemotherapy. Remarkably, both roles of CD4+T cells as a helper (augmented role) and as a killer to the tumor are considered in the proposed models. Due to the sharp changes for the variables of the model, the Runge-Kutta method failed to solve it. Consequently, the “Adams predictor-corrector” method is used. Results reveal that the patient’s immune system can overcome small tumors, however, if the tumor is large then therapy is needed. In some cases, chemotherapy can eliminate the tumor, however, it can be replaced with immunotherapy so could spare patients the agony of chemotherapy-induced side effects and toxicities. In other cases, chemotherapy has to be used but immunotherapy could reduce the chemotherapy doses. Results also show that CD4+T therapy could replace combined CD8 +T and cytokines therapy in some cases. In some other cases when combined CD8 +T and cytokines could not eradicate the tumor adding CD4+T to the therapy succeeded in eradicating the tumor. An exact solution cannot be found to the proposed system like many other nonlinear systems. Yet, an approximate analytical solution is explored. The power series method is used to obtain a series solution. However, this solution has a small radius of convergence, therefore, the Padé approximant method is used to extend the domain of convergence. Hence, the obtained approximate analytical solution is valid over a large interval and has a remarkable accuracy when compared with the numerical solution. Stability analysis is performed and it has been found that all equilibrium points are unstable for the considered patients’ data. A condition for preventing tumor recurrence after treatment, for the zero tumor equilibrium points, has been deduced. Then, a bifurcation analysis is performed to study the effect of varying system parameters on the stability of equilibrium points, 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. Sensitivity analysis is performed, for two sets of human data, to identify the most effective body parameters in eliminating tumor cell population. Both numerical sensitivity coefficient method and sensitivity function method are implemented, and the results are compared. While the first method identifies the most effective parameters at a specific instant of time, like many published works, the second method recognizes these parameters over a wide time interval. The required order of the most effective parameters is identified, so this sensitivity analysis answers the question: Which parameter can be changed to get the largest effect on the tumor cells population? The obtained results provide a valuable tool to identify the parameters that would be increased or decreased before starting a treatment. Parameter estimation is performed to fit the model with experimental data. After that, the fitting result is verified. Using the estimated values of the parameters, numerical simulations are performed for the model with and without the treatment. Sensitivity analysis is performed on the system with the estimated values of the parameters to detect the order of the most effective parameters on the tumor cells population. The proposed models offer powerful tools for predicting the response of the body against cancer without any treatment and in presence of immunotherapy and/or chemotherapy. Thus, the models provide a valuable tool for planning patients’ treatment intervention strategies.