Multi-level mathematical models for cell migration and invasion in constraining environments

The aim of this contribution is to put together in a systematic way several approaches operating at different scales that were recently developed to describe the phenomenon of physical limit of migration, that occurs when the environment surrounding cells results restrictive for migration, and to apply it to tumour growth and invasion. In particular, we will present: (i) a mechanical model of the behaviour of a cell within a microchannel that gives a blockage criterium for its penetration; (ii) a cellular Potts model to describe the dependence of the speed of a malignant cell from the mechanical characteristics both of its compartments (i.e., nucleus and cytosol) and of its environment; (iii) a multiphase model embodying such effects; (iv) the proper interface conditions to implement to deal with tumour invasion across matrix membranes and cell linings.