Phase field model for cell spreading dynamics

We suggest a 3D phase field model to describe 3D cell spreading on a flat substrate. The model is a simplified version of a minimal model that was developed in [1]. Our model couples the order parameter $u$ with 3D polarization (orientation) vector field $\textbf{P}$ of the actin network. We derive a closed integro-differential equation governing the 3D cell spreading dynamics on a flat substrate, which includes the normal velocity of the membrane, curvature, volume relaxation rate, a function determined by the molecular effects of the subcell level, and the adhesion effect. This equation is easily solved numerically. The results are in agreement with the early fast phase observed experimentally in [2]. Also we find agreement with the universal power law [3] which suggest that cell adhesion or contact area versus time behave as $\sim t^{1/2}$ in the early stage of cell spreading dynamics, and slow down at the next stages.