Representation of the terms included in Eq (1).

The evolution of the order parameter depends on two processes: the endothelial cell proliferation and the movement of endothelial cells of the capillary on the ECM. The latter results from three main mechanisms that are implemented in our phase-field model. First, the constitution of a capillary wall (cells resist being isolated and leaving the vessel); we model this term as a surface tension. Second, the endothelial cells occupy regions of stiff tissue with high strains; this term is proportional to the difference between the tissue rigidities, but also proportional to the the local strain. Finally the cells can be pulled by the adhesion forces.