• Aldous criterion
• cancer model
• macroscopic limit
• ODEs
• Skorohod space

This work focuses on the description of the growth of the primary tumor of metastatic colorectal cancer (mCRC), and related angiogenesis before and during a treatment with fluorouracil and/or bevacizumab. It relies on a model, based on ordinary differential equations (ODEs), that describes the variation of a few number of quantities which summarize the phenomena. The spatial structure is neglected, preferring simplicity to precision. Nevertheless the model succeed in reproducing reasonable results, both from a qualitative and a quantitative points of view.
In Chapter 1 there is a detailed description of the mathematical model, which summarizes the number of normoxic and hypoxic cells, the level of vascularization due to angiogenesis and the density of VEGF. We assumed that tumor grows following a spherical shape. At the beginning, the growth is exponential, then it depleted and only the boundary region of this sphere proliferates. The situation is partially restored by the angiogenesis, which starts when tumor size is around 1mm^3. The randomization of some parameters, through Monte Carlo method, allow us to compute progression free survival (PFS) medians and Kaplain-Meier curves, and calibrate our model through comparison with clinical data.
In Chapter 2 we recall the theoretical framework to study the stochastic processes with càdlàg paths. In particular, it is showed the construction and the main results related to the Skorohod space D[0,1]. It is recalled an important criterion, due to Aldous, which produces sufficient conditions for the tightness of a family of probability measures which are the laws of processes with paths in D[0,1].
In Chapter 3 we show a connection between microscopic and macroscopic scales, linking a simple discrete model with a continuous one. Specifically, we construct a simple counting process whose macroscopic limit represents the exponential growth of a cancer in the first stages of its growth. To this end, we show that the family of the laws associated to the discrete model is tight, thanks to an application of the Aldous' criterion.