A mathematical model of vascular tumor treatment by chemotherapy

From the viewpoint of biological stoichiometry, a mathematical model of vascular tumor treatment with chemotherapy techniques is proposed utilizing a system of delayed differential equations representing the change in mass of healthy cells, competing parenchyma cells, chemotherapy, and the number of blood vessels within the tumor. In the absence of treatment, mathematical analysis of the model equations with regard to invariance of nonnegativity, boundedness of solutions, nature of equilibria, permanence, and global stability are analyzed. It is shown that the system can be permanent, but whenever the boundary equilibrium is stable, the interior equilibrium of the system cannot be globally stable for at least small values of time delay. Further, in this case, persistence cannot occur at least for small values of the time delay. Necessary and sufficient conditions for Hopf bifurcation to occur are also obtained by using the time delay as a bifurcation parameter. Finally, based on all these qualitative behaviors of the model, a continuous treatment for tumor growth is considered. The analysis is carried out both analytically and numerically.