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We study here, spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system and phase-space analysis of a certain mathematical model of tumor growth with an immune responses. Mathematical modeling of this process is viewed as a potentially powerful tool in the development of improved treatment regimens. We study some features of local and global behavior the multipoint problem for the three-dimensional tumour growth model system obtained by de Pillis and Radunskaya in 2003, with dynamics described in terms of densities of three cells populations: tumor cells, healthy host cells and effector immune cells. We found sufficient conditions, under which trajectories from the positive domain of feasible multipoint initial conditions tend to one of equilibrium points. The addition of a drug term to the system can move the solution trajectory into a desirable basin of attraction. We show that the solutions of the model with a time-varying drug term approach can be evaluated in a more fruitful way and down to earth style from the point of practical importance than the solutions of the system without drug treatment, in the condition of stimulated immune processes, only.

The local and global dynamics model of a cancer tumor growth