Over the last decades, great strides have been made by the mathematical and medical communities towards the understanding of solid tumor growth. The recently achieved novelties arise from two leading factors: on the one hand, the flourishing of mathematical models for biological systems, and on the other hand, the more and more accurate computational methods and numerical solvers rose in the last decades. Despite the deep and challenging aim of understanding the hidden mechanisms behind the disease, the scientists' factual goal is to provide robust methods that may help the practitioners suiting the best therapy for every single patient. In this direction, the mathematical approach to tumour growth models might bring new lymph and hope to this arduous journey. In this talk, we aim at providing some mathematical insights for a particular model whose key assumption is that the tumour cells are submerged in a nutrient-rich environment which is the primary source of nourishment for the tumorous cells: this is a reasonable assumption at least for young tumours (avascular tumours). This paradigm leads us to analyse a four-species PDE system (tumour cells, healthy cells, nutrient-rich concentration, nutrient-poor concentration) which couples a Cahn-Hilliard type equation with source term for the tumour with a reaction-diffusion equation for the surrounding nutrient. To begin with, we fix some ideas concerning modelling. Then, we address some selected results related to the well-posedness for that system, and lastly, based on the analytic results presented, we discuss a specific optimal control problem.
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