We have developed a set of mathematical models for the invasion process of tumor 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.
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 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.
B. Figures of merit for testing invasion models
The most detailed description of a (real or simulated) invasion experiment consists of a complete 3D trajectory
\vec{R}_n(t)=(x_n(t),y_n(t),z_n(t))
for each individual cell n. So far, we have mainly focused on the z-coordinates.
In analogy to the time-averaged "mean squared displacement", a standard tool for analyzing stationary random walks, we can define a population-averaged "mean squared invasion depth":
\left\langle z^2 \right\rangle (t) = (1/N) \; \sum_{n=1}^N (z_n(t))^2,
where n runs over all N cells of the population.
The time-dependent z-density distribution (unit: 1/m)) is defined by
n(z,t) = \sum_{n=1}^N \delta(z-z_n(t)).
Since n(z,t) is usually a very noisy quantity, we work in the following with the (equivalent) cumulative probability distribution
P(z,t) = \int_z^\infty n(z,t) dz\; /\; \int_0^\infty n(z,t) dz.
This dimensionless quantity can be interpreted as the probability to find a cell at depth z, or deeper. For any fixed time t_0, the function P(z,t_0) starts with the value 1 at the top surface (z=0) of the collagen slice and monotonically decreases towards zero for larger depths z.
C. Experimental key features
Measured cumulative probability distributions reveal a large variability, depending on the cell type and on the material parameters of the used collagen gel. However, there are a few general properties that we observed in almost all cases investigated so far:
1.) 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).
2.) Within a narrow layer close to the surface, P(z,t_0) is rapidly decaying, i.e. according to a faster-than-exponential law.
3.) 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
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.
D. Various investigated models
We have considered a variety of different invasion models and explored their ability to account for the above experimental key features.
The most basic possibility would be a homogeneous population of non-cooperative tumor cells, where all individuals migrate with the same effective diffusion constant, independently from each other. However, the simple Gaussian invasion profiles, resulting from standard diffusion into a half-space, are clearly inconsistent with the complex layered structure of the measured P(z,t) distributions, in particular with the zone of exponential decay (experimental feature 3).
Heterogeneous cell populations with a spectrum of vastly different diffusion constants could produce close-to-exponential cumulative probabilities. Yet, this requires fine-tuning of the assumed parameter distribution. In addition, heterogeneity cannot account for the observed cell density dependence.
It is well-known that standard diffusion combined with a process of reverse drift, back to the surface, at constant velocity, produces exponential profiles. One then has to explain the biological cause of this slight trend in the random walk of the tumor cells. Among the possibilities are mesoscopic spatial gradients in the distribution of local properties (topological, rheological, or chemical) within the gel. Again, such external causes cannot account for the observed cell density dependence.
We have also considered a model in which the cells themselves emit a chemo-attractant, which independently diffuses within the gel and degrades at a certain rate. For certain parameter ranges, the resulting invasion profiles are indeed consistent with the observations. However, the values of the diffusion constants required for a good fit to the data turned out to be not realistic.
E. The "Density-Dependend Diffusion" model
At present, our best model for the invasion process is based on the assumption of a situation-dependent diffusion constant of the tumor cells. This idea is motivated by the observation of the cell aggregate (experimental feature 10) that formes in a narrow layer below the gel surface when the initial cell surface density was large. Within this layer, neighboring cells are very close to each other. The aggregate remains rather stable over long times, meaning that the diffusion constant of cells in an aggregated state is small. On the other hand, once individual cells escape from the aggregate into the bulk of the gel, their diffusion constant becomes much higher, indicated by the large characteristic length scale of the exponential zone.
We therefore tested a simple model in which the individual cells switch their diffusion constant between a small and a large value, depending on the presence or absence of at least one neighboring cell within a certain detection range. The full model has been implemented as a multi-dimenional Monte-Carlo simulation. Additionally, in a mean field approximation, the corresponding 1D Fokker-Planck equation has been solved numerically.
The three profiles shown in the figure below correspond to three separate invasion experiments, differing only in the initial cell surface density (The red, green and blue curve correspond to a total number of 33, 498 and 4348 cells in the total field of view). In all three cases, the tumor cell positions within the gel have been measured after the same time delay (relative to planting the cells onto the gel surface). The characteristic density dependent changes of the invasion behavior, especially the features 9 and 10 listed above, are clearly visible.
The next figure shows the results of corresponding computer experiments, obtained by the Monte Carlo simulation of the DDD model. No attempt has been made to fit the parameters to the specific experiments. Nevertheless, all major features and trends are qualitatively reproduced:
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