We have developed a set of mathematical models for the invasion process of tumor $\sqrt{a^2+b^2}$ cell populations into a half-space of collagen gel. In particular, we aimed for a quantitative understanding of the characteristic shape of the invasion profiles, their temporal evolution, and their dependence on the initial cell surface density.

(1)On the microscopic scale, collagen is a highly inhomogeneous fiber network. The detailed procedure by which individual tumor cells migrate through this porous network (involving steps such as finding adhesion ligands, forming and disintegrating focal adhesion contacts, up- and down-regulating acto-myosin traction forces, ..) is not well understood at present.

Moreover, it depends on experimentally inaccessible local conditions. It is therefore reasonable to describe cell migration as a stochastic process, i.e. essentially as a random walk. The effective diffusion constant of this random walk summarizes the complex interactions of the cell with its surrounding material in a coarse-grained way. For simplicity, we have Huhu further assumed that the collagen gel is statistically homogeneous and isotropic, i.e. the diffusion of (isolated, non-cooperating) cells of a certain type is equally fast at any position within the gel and does not depend on the spatial directions.

- A typical distribution, for fixed time t_0, consists of up to three distinctive layers, or zones. These zones are most easily visable in a semi-logarithmic plot of P(z,t_0).

Within a narrow layer close to the surface, P(z,t_0) is rapidly decaying, i.e. according to a faster-than-exponential law.

- Within a broad intermediate zone, the cummulative probability is decaying exponentially, P(z,t_0) ~ e^(-z/z_0) with a characteristic length scale z_0.

4.) For very large depths, at the "front zone", P(z,t_0) decays apparently according to a faster-than-exponential law. Note that, for all finite populations, there is always a single cell at the foremost front of the distribution. Statistics is becoming extremely poor close to that region.

6.) As time passes, the characteristic length scale z_0 of the exponential zone increases.

7.) The mean squared invasion depth of the cell population is increasing with time t. The functional dependence of on t is non-linear, indicating an anomaleous diffusion process.

8.) The invasion profiles P(z,t) depend strongly on the 2D density of tumor cells initially plated onto the gel surface:

9.) For very small population densities, the surface layer, marked by the rapid drop of cumulative invasion propability, disappears.

10.) For increasingly higher densities, the drop of cumulative probability at small depths becomes more pronounced, indicating a dense aggregate of tumor cells immediately below the surface. Beyond this aggregate, only a relatively small fraction of cells is invading into the deep bulk of the gel. However, the characteristic length scale z_0 of the invaded fraction tends to increase with initial cell density.