| The problem of modeling cancer growth and its interaction with other cell types is a very complex one and researchers have usually focused on particular issues. The idea is to gain sufficient insight on the underlying process to predict or control the disease. It has been shown that in order to eradicate tumor cells, the immune system plays an important role, without which cure is an impossible task. For this reason, this work focuses on tumor growth and its interaction with immune cells. We consider the application of a phase-specific drug (e.g. Hydroxy Ara-C, Paclitaxel, etc.), which selectively kills tumor cells at a specific phase of the cell cycle. The inclusion of this type of protocol calls for a model that would distinguish between phases of the cell cycle. In modeling cell phases in a natural way we are led to the use of Delay Differential Equations (DDE).; We first analyze the drug-free system to gain sufficient information about the different dynamics that can arise. This is done analytically using standard techniques for DDE. We perform some stability analysis for the fixed points that arise in the system and demonstrate the occurrence of a Hopf Bifurcation as we vary a parameter of the model, and therefore prove the existence of periodic solutions. We study the nature of the basins of attraction for coexisting stable fixed points with and without drug. Finally, the parameters involved in the model will be estimated for breast cancer using given data and we will use optimal control to design optimal drug protocols.; This model is not intended to be used by clinicians to treat their patients, but rather as a tool for exploring the efficiency of different drugs and their effects. |
