A 3D-1D multiphase model for growing vascular tumors with angiogenesis

In this work we aim to describe both the angiogenesis and the growth of a vascular tumor. In the first part of the thesis, we focus only on the formation of new blood vessels, providing an hybrid model to outline the evolution of pressure, chemicals and endothelial cells, that lead the formation of the nascent vasculature. It is important to underline the fact that, in this framework, we neglect the existence of any cancer cell population and we focus only on the growing network dynamics. On the mathematical point of view, this kind of problem deals with coupling mechanism between the interstitial tissue and the network, because blood and chemicals are allowed to flow across the vessels walls. Therefore, we provide a suitable numerical method that resolves the problem introduced above, exploiting the minimization of a constrained cost functional. In the second part, instead, we introduce the tumor growth dynamics considering a multiphase model for a cell population and a liquid phase, in which chemicals are dissolved. In this framework, the model takes into account the fact that cells need oxygen to survive, so they induce the formation of new blood vessels that bring nutrients. Since the main novelty of the second part is the introduction of the cell dynamics, we provide a numerical method for the resolution of its governing law, while the technique proposed in the first part for the coupling mechanism is still valid. To conclude, both in the first and in the second part, we present some numerical experiments to investigate the behavior of the corresponding dynamics, in order to understand the biological reliability of the model.