Modeling tumor cell migration: From microscopic to macroscopic models

It has been shown experimentally that contact interactions may influence the migration of cancer cells. Previous works have modelized this thanks to stochastic, discrete models (cellular automata) at the cell level. However, for the study of the growth of real-size tumors with several million cells, it is best to use a macroscopic model having the form of a partial differential equation (PDE) for the density of cells. The difficulty is to predict the effect, at the macroscopic scale, of contact interactions that take place at the microscopic scale. To address this, we use a multiscale approach: starting from a very simple, yet experimentally validated, microscopic model of migration with contact interactions, we derive a macroscopic model. We show that a diffusion equation arises, as is often postulated in the field of glioma modeling, but it is nonlinear because of the interactions. We give the explicit dependence of diffusivity on the cell density and on a parameter governing cell-cell interactions. We discuss in detail the conditions of validity of the approximations used in the derivation, and we compare analytic results from our PDE to numerical simulations and to some in vitro experiments. We notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses, supercooled liquids, and jamming systems.

Figure 1

(Color online) 3D view of the first lattice geometry we used to test our results: a cylinder (lattice with periodic boundary conditions in one direction) connected to a full reservoir of cells (a source) on its left end and to an empty reservoir of cells (a sink) on its right end. Here, the system is made of $n_L+1=16$ rings of sites (including the two rings in the reservoirs), and each ring is made of $n_H=16$ sites. After some transient regime, a steady current of cells establishes along the cylinder. A typical configuration of the cells in the steady state for interaction parameter $p=0.8$ is shown.

Figure 2

(Color online) Dependence on the interaction parameter $p$ of the stationary profiles $\rho (x∕L)$ of the site occupation probability along the cylinder between the full reservoir (left, $x=0$, site occupation probability $\rho =1$) and the empty reservoir (right, $x=L$, $\rho =0$)—see Fig. 1. Since the cell concentration $c$ is proportional to the site occupation probability $\rho$, the profiles of $c$ would be the same. For each value of $p$, we plot the simulation results for the cellular automaton (circles) and our prediction from the analytic approximation Eq. (21) (solid lines). From bottom to top, $p=0.01$, 0.05, 0.2, 0.4, 0.7, and 1. The error bars are smaller than the circles’ size. The cylinder is made of $64\times 64$ sites except for $p=0.01$ and 0.05 where there are $512\times 512$ sites.

Figure 3

(Color online) Finite-size effects for the stationary profile of the cell concentration between the two reservoirs. Main curve: dots: profile $\rho (x∕L)$ of the site occupation probability from simulation results for $p=0.01$ and system sizes $n_L=n_H-1=15$, 31, 63, 127, 255, and 511 (from bottom to top). It shows a strong finite-size dependence (for $p=0.05$, not shown, a weaker finite-size dependence is found). The solid line is the analytic approximation, clearly in disagreement with the simulation results. Bottom inset: for $p=0.01$, the extrapolation of the occupation probability $\rho$ at position $x=L∕4$ to an infinite lattice shows that the discrepancy between the simulation data (dots) and the analytical approximation (dashed line) is not due to finite-size effects. Indeed, there is convergence toward a value of the site occupation probability that is clearly below the analytic approximation. Top inset: profile $\rho (x∕L)$ for $p=1$, with numerical data for the sizes $n_L=15$, 31, and 63 only (dots) and analytic prediction (solid line). The finite-size effects are very weak and there is a good agreement between simulation data for large systems and the analytic prediction. This is true more generally for all values of $p$ away from 0 (say, larger than 0.1).

Figure 4

(Color online) Stationary current of cell density between the full and the empty reservoirs, across the cylinder (geometry defined in Fig. 1), as a function of the interaction parameter $p$. To be able to compare different system sizes (bottom to top: $nH-1=n_L=15$, 63, and 255), we actually plot the current density $j$ multiplied by the system length $L$ (dots). For $n_L=255$, only the points that differ significantly from the other sizes are represented. The error bars are smaller than the symbol’s sizes, except for $n_L=255$, where they are drawn. Solid line: analytic approximation, Eq. (20) times $L$. Like for the profile of the cell concentration, there is a large disagreement between the analytic approximation and the simulation results when $p$ is close to 0 and a small disagreement when $p$ is close to 1.

Figure 5

(Color online) Schematic explanation of the correlations that appear for $p$ close to 0 near the full reservoir (left) and for $p$ close to 1 near the empty reservoir (right). Left: when two cells sit next to one another close to the full reservoir and if $p$ is close to 0, they prevent one another from moving in one of the two directions that would be allowed if they were alone (black arrows represent permitted moves). Thus accompanied cells stay longer close to the reservoir than single ones, and positive correlations occur (see text). Right: when one cell sits close to the empty reservoir and if $p$ is close to 1, it cannot move unless it has a neighboring cell. Thus single cells stay longer close to the empty reservoir than accompanied ones, and negative correlations occur (see text).

Figure 6

Density profiles from the automaton (dotted line) and from the numerical solution of the diffusion equation (full line) for $p=1$, 0.5, and 0.05. Error bars for the automaton density profile are smaller than the markers and thus are not represented.

Figure 7

Experimental pattern of migration after $36\mathrm{h}$. The real size of the image is $1.90\times 1.77\mathrm{mm}$.

Figure 8

Comparison between the numerical solution of the nonlinear diffusion equation (full lines) and experimental data, at $t=12\mathrm{h}$ (crosses) and $t=36\mathrm{h}$ (circles). $D_{*}=1240\pm 100\mu {\mathrm{m}}^2{\mathrm{h}}^{-1}$ with $a=35.2\mu \mathrm{m}$.