Abstract
This paper deals with a review and critical analysis on the mathematical kinetic theory of active particles applied to the modelling of the very early stage of cancer phenomena, specifically mutations, onset, progression of cancer cells, and their competition with the immune system. The mathematical theory describes the dynamics of large systems of interacting entities whose microscopic state includes not only geometrical and mechanical variables, but also specific biological functions. Applications are focused on the modelling of complex biological systems where two scales at the level of genes and cells interact generating the heterogeneous onset of cancer phenomena. The analysis also refers to the derivation of tissue level models from the underlying description at the lower scales. The review is constantly linked to a critical analysis focused on various open problems including the ambitious objective of developing a mathematical theory for complex biological systems.
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