- As the simplest possible model, we assume that the cells perform a (biased) random walk with statistically independent spatial components (dx,dy,dz) of the displacements.
- We are only interested in the temporal evolution of the probability density function P(z,t) of the cell's z-position. Since the 3 coordinates are independent random processes, we consider the whole problem as a 1D-diffusion problem with special boundary conditions and with possible trends (drift terms).
- The trends could, for example, describe the effect of a chemo-attractant in the medium, creating a bias of diffusion into the positive z-direction (later called the "right" direction). A chemo-repellant would cause a drift to the "left".

The corresponding Fokker-Planck equation for the temporal evolution of P(z,t) could be solved analytically. Instead we discretize the problem spatially (z-segments in the figure) and temporally and integrate the resulting master equation numerically. The surface is described by the special segment Z=0.

For given resolutions (dz, dt), we define 4 dimensionless parameters:

- muR = fraction of particles in segment Z that move to the segment Z+1 to the right within one time step.

- muL=fraction moving to the left.

- muV=fraction of particles on the surface (Z=0) that move into the volume.

- muS=fraction of particles in volume segment (Z=1) that move to the surface.

P_z^{(t)}

denote the normalized probability density (PDF) of particles in segment z at time step t.The initial condition is

P_z^{(0)}=\delta_{z0}.

The master equation reads for the surface segment

P_0^{(t+1)}=(1-\mu_V) P_0^{(t)} + \mu_S P_1^{(t)},

for the volume segment Z=1

P_1^{(t+1)}=(1-\mu_S-\mu_R) P_1^{(t)} + \mu_V P_0^{(t)} + \mu_L P_2^{(t)},

and for all other segments Z>1

P_z^{(t+1)}=(1-\mu_L-\mu_R) P_z^{(t)} + \mu_R P_{z-1}^{(t)} + \mu_L P_{z+1}^{(t)}.

First, we shall treat the passage of the cell through the s pecial boundary between the last segment Z=1 and the surface segment Z=0 in the same way as for all other segments:

\mu_V=\mu_R \; , \; \mu_S=\mu_L.

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Simulation results for symmetric conditions:

\mu_R=\mu_L=0.1

Here, the density profiles are approximately Half-Gaussians, peaked at the surface, with temporally increasing variance. The surface density decays at long times according to a t^(-1/2) powerlaw.

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Simulation results with "chemo-attractant":

\mu_L=0.1 \;,\; \mu_R=0.2

Here, the peak of the Gaussian profiles is drifting to the right. The surface density decays exponentially.

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Simulation results with "chemo-repellant":

\mu_L=0.2 \;,\; \mu_R=0.1

Here, the profiles quickly approach a stationary, exponential distribution. The surface density falls to a stationary, finite value.

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We will next compare those simulations with measured cell invasion data.

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