  The basic process of the innate immune system when phagocyte (white blood
cell) engulf or swallow a target particle (bacterium or dead cell), is called
phagocytosis. We apply the phase field approach in the spirit of [1], that
couples the order parameter $u$ with 3D polarization (orientation) vector field
$\textbf{P}$ of the actin network of the phagocyte cytoskeleton. We derive a
single closed scalar integro-differential equation governing the 3D phagocyte
membrane dynamics during bead engulfment, 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 of the
motionless rigid spherical bead. This equation is easily solved numerically.
The simulation manifests the pedestal and the cup phases but not the final
complete bead internalization.
