  A connection is established between discrete stochastic model describing
microscopic motion of fluctuating cells, and macroscopic equations describing
dynamics of cellular density. Cells move towards chemical gradient (process
called chemotaxis) with their shapes randomly fluctuating. Nonlinear diffusion
equation is derived from microscopic dynamics in dimensions one and two using
excluded volume approach. Nonlinear diffusion coefficient depends on cellular
volume fraction and it is demonstrated to prevent collapse of cellular density.
A very good agreement is shown between Monte Carlo simulations of the
microscopic Cellular Potts Model and numerical solutions of the macroscopic
equations for relatively large cellular volume fractions. Combination of
microscopic and macroscopic models were used to simulate growth of structures
similar to early vascular networks.
