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 Winkler (Commun Phys 2:82, 2019). Our model couples the order parameter u with 3D polarization (orientation) vector field 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 Dobereiner (Phys Rev Lett 93:108105, 2004). Also we find agreement with the universal power law (Cuvelier in Curr Biol 17:694-699, 2007) which suggest that cell adhesion or contact area versus time behave as similar to t(1/2) in the early stage of cell spreading dynamics, and slow down at the next stages.
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