Modeling tumor cell migration: From microscopic to macroscopic models

Abstract

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.

Modelling biological invasions: Individual to population scales at interfaces

“…With cðx; tÞ again denoting a rescaled cell density with unit carrying capacity, these models are typically of the form where the summation convention is used and the diffusion coefficient is a positive, bounded and C ∞ function of the cell density (Deroulers et al, 2009;Maini et al, 2004;Sherratt and Murray, 1990;Sherratt and Marchant, 1996). Suppose in (C.1) there is an interfacial transition in the diffusion coefficient centred at a manifold C of length scale ϵ in its normal direction, which is smaller than any other length scale in the continuum model, including the radius of curvature of C. Then, on the length scale of the transition region, the manifold C can be …”

“…This has been generalised in numerous investigations, for example in off-lattice models (Lipkova et al, 2011), whereby the framework at the individual level does not rely upon a discretisation of space and/or time, as well as the incorporation of numerous physical and biological features. Examples in this popular field (Chowdhury et al, 2005;Othmer and Stevens, 1997;Hillen and Othmer, 2000;Deroulers et al, 2009) include the consideration of exclusion processes (Landman and Fernando, 2011), motility biases such as chemotaxis (Hillen and Painter, 2009), competing cell populations (Penington et al, 2011), contact interactions (Lushnikov et al, 2008;Painter and Sherratt, 2003) and cell-cell adhesions (Stein et al, 2007).…”

Experimental and Modelling Investigation of Monolayer Development with Clustering

“…The mean-field assumption can be invoked explicitly, such as in the case of considering a lattice-based discrete random walk model where an approximate partial differential equation (pde) or ordinary differential equation (ode) description is derived by assuming that the occupancy status of lattice sites are independent (e.g. Deroulers et al 2009;Penington et al 2011;Penington et al 2012;Plank and Simpson 2012;. Alternatively, the mean-field assumption can be invoked implicitly, such as in the case of applying standard ode or pde descriptions of collective cell behaviour without necessarily considering the underlying discrete process (e.g.…”

Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences

“…Despite this simplification, analogous models have already been proven to be capable of shedding some light on the mechanisms which underlie patterns of evolution and adaptation in asexual populations [14][15][16][17][18][19]. From the mathematical point of view, our work follows earlier papers on the derivation of deterministic mesoscopic models from stochastic agent-based models [20][21][22][23][24][25] and the analysis of integrodifferential equations that arise in models of evolutionary dynamics within phenotype-structured populations [26][27][28][29][30][31][32][33].…”

“…Experience in a variety of contexts has demonstrated the value of relating discrete-time, discretespace, agent-based stochastic models to deterministic continuum models [22][23][24][25]. Apart from a few special fortuitous examples, such as the unbiased simple exclusion process on a regular lattice, the correspondence is approximate rather than exact, and is exhibited using some form of "mean-field" argument, in which correlations between the locations of distinct agents are either ignored or are treated in some truncated or other approximate way [37].…”