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Alberto d'Onofrio, "Tumour angiogenesis and anti-angiogenesis therapies: a modeling contribution." Short abstract: A model of cellular immune response to cancer is presented. The model is developed with statistical methods analogous to those of kinetic theory. The model is a system of integro-differential equations of Boltzmann type. Results of numerical simulations are presented and compared to experimental data. In this talk we shall illustrate a class of models that describe the mutual interaction between tumour growth and the development of tumour vasculature and that generalize existing models. In this presentation we shall mainly focus on the use of this class of models to investigate the effect of an ''canti-angiogenic therapy'' that induces either tumour vessel loss or inhibition of the proliferation of vessels. Our aim is finding conditions that asymptotically guarantee the eradication of the disease under constant infusion or periodic administration of the drug. Furthermore, since tumour and/or vessel dynamics may exhibit time delays, we shall illustrate the destabilizing effects of such time lags (for example the onset of oscillations through Hopf bifurcations) as well as how they may affect the conditions for the stability of equilibria. Finally, we shall show the possible relevant effects due to another phenomenon of temporal delay: the non instantaneous death of endothelial cells after the uptake of anti-angiogenic drugs. We shall show, by means of an appropriate mathematical model, that the eradication is impossible if this lag exceeds a biologically meaningful threshold related to the pro-angiogenic activity of tumours. |