Several classical transport-diffusion systems arise as simple models in chemotaxis (motion of bacterias interacting through a chemical signal) and angiogenesis (development of capillary blood vessels from an exhogeneous chemoattractive signal by solid tumors). The name of Keller-Segel is usally attached to these systems which are coupled through a nonlinear transport term depending on the gradient of the chemoattractant and that can exhibit various qualitative behavior (collapse, rings dynamics).

After a presentation of such models, we will focuss on a microscopic picture based on a kinetic modelling of the interaction and introduced by Alt, Dunbar, Ohtmer and studied by Hadeler, Stevens...etc. It has the form of a nonlinear scattering equation. We will show that, in opposition to the parabolic model, the kinetic system admits global solutions that converge in finite time to the Keller-Segel model, as a scaling parameter vanishes.

The kinetic model has the advantage to present a unified picture for various macroscopic models of cell movement and sustend biological assumptions on the individual interactions.