From Microscopic to Macroscopic Descriptions of Cell Migration on Growing Domains

Baker, Ruth E.; Yates, Christian A.; Erban, Radek

doi:10.1007/s11538-009-9467-x

Cited by 95 publications

(182 citation statements)

References 42 publications

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“…The deduction of Fickian diffusion can proceed as above and has been presented (without the complication of microscale kinetics) on numerous occasions in the literature (e.g. Baker et al, 2010). Following such derivations with logistic kinetics and appropriate rescalings one finds…”

Section: Unbiased Fickian Diffusion In the Continuum Limitmentioning

confidence: 90%

Modelling biological invasions: Individual to population scales at interfaces

Belmonte-Beitia

Woolley

Scott

et al. 2013

Journal of Theoretical Biology

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We explore the influence of spatial heterogeneity in biological systems. We consider how local transport mechanisms impact behaviour near interfaces. We study cell movement between white and grey matter in the brain, as an example. Profound differences in population behaviour emerge in the presence of interface. The commonly used Fickian diffusion transport model cannot predict this dynamics. a r t i c l e i n f o b s t r a c tExtracting the population level behaviour of biological systems from that of the individual is critical in understanding dynamics across multiple scales and thus has been the subject of numerous investigations. Here, the influence of spatial heterogeneity in such contexts is explored for interfaces with a separation of the length scales characterising the individual and the interface, a situation that can arise in applications involving cellular modelling. As an illustrative example, we consider cell movement between white and grey matter in the brain which may be relevant in considering the invasive dynamics of glioma. We show that while one can safely neglect intrinsic noise, at least when considering glioma cell invasion, profound differences in population behaviours emerge in the presence of interfaces with only subtle alterations in the dynamics at the individual level. Transport driven by local cell sensing generates predictions of cell accumulations along interfaces where cell motility changes. This behaviour is not predicted with the commonly used Fickian diffusion transport model, but can be extracted from preliminary observations of specific cell lines in recent, novel, cryo-imaging. Consequently, these findings suggest a need to consider the impact of individual behaviour, spatial heterogeneity and especially interfaces in experimental and modelling frameworks of cellular dynamics, for instance in the characterisation of glioma cell motility.

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Section: Unbiased Fickian Diffusion In the Continuum Limitmentioning

confidence: 90%

Modelling biological invasions: Individual to population scales at interfaces

Belmonte-Beitia

Woolley

Scott

et al. 2013

Journal of Theoretical Biology

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“…An area of great interest to the mathematical biology community is to understand how the averaged properties of a particular discrete process can be described using continous models such as ordinary differential equations and partial differential equations [30,20,1,[40][41][42]. While some previous work has been completed on understanding how to derive approximate differential equation description of systems with different sized agents [43,44], none of these previous studies considered systems that were composed of several types of different sized agents like we have shown in Fig.…”

Section: Discussionmentioning

confidence: 99%

“…Typical mathematical modelling approaches used to interpret tissue growth experiments involve using a continuous logistic-type models [2,[16][17][18]5], which can be calibrated to provide an estimate of the proliferation rate. An important limitation of using continuous models is that they neglect any explicit description of individual cell behaviour [19][20][21][22][23][24][25]. This limitation can be partially overcome by using discrete, random walk models.…”

Section: Introductionmentioning

confidence: 99%

Cell density and cell size dynamics during in vitro tissue growth experiments: Implications for mathematical models of collective cell behaviour

Binder

Simpson

2016

Applied Mathematical Modelling

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a b s t r a c tWe present a detailed experimental data set describing a tissue growth experiment where a population of cells is initially distributed uniformly, at low density, on a two-dimensional substrate, and grows to eventually form a confluent monolayer. Using image processing tools, we provide precise information about temporal changes in the number of cells, the location of cells and the total area occupied by cells. This information shows that the increase in area occupied by the cell population is affected by both the increase in cell number as well as an increase in the average size of the cells. We show that standard approaches to interpret such experiments, where the cell size is typically treated as a constant, can lead to errors. Furthermore we show that a standard, discrete, random walk model of biological cell motility and cell proliferation should not be used to represent our experimental data set since this standard model treats all cells as having a constant size that does not change with time. Instead, we introduce a generalization of the standard model which allows agents in the random walk model to move, proliferate and grow in size, and we show that the data produced by this more general model is consistent with our experimental data set.

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“…In our exclusion processes [15,61], agents move on a fixed d-dimensional lattice, each site of which may contain 0 or 1 agent. For simplicity, we will apply our results only to the bidimensional hexagonal tiling (triangular lattice), but they can be extended straightforwardly.…”

Section: The Modelmentioning

confidence: 99%

Exclusion processes: Short-range correlations induced by adhesion and contact interactions

Ascolani¹,

Badoual²,

Deroulers³

2013

Phys. Rev. E

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We analyze the out-of-equilibrium behavior of exclusion processes where agents interact with their nearest neighbors, and we study the short-range correlations which develop because of the exclusion and other contact interactions. The form of interactions we focus on, including adhesion and contact-preserving interactions, is especially relevant for migration processes of living cells. We show the local agent density and nearest-neighbor two-point correlations resulting from simulations on two dimensional lattices in the transient regime where agents invade an initially empty space from a source and in the stationary regime between a source and a sink. We compare the results of simulations with the corresponding quantities derived from the master equation of the exclusion processes, and in both cases, we show that, during the invasion of space by agents, a wave of correlations travels with velocity v(t) ∼ t −1/2 . The relative placement of this wave to the agent density front and the time dependence of its height may be used to discriminate between different forms of contact interactions or to quantitatively estimate the intensity of interactions. We discuss, in the stationary density profile between a full and an empty reservoir of agents, the presence of a discontinuity close to the empty reservoir. Then, we develop a method for deriving approximate hydrodynamic limits of the processes. From the resulting systems of partial differential equations, we recover the self-similar behavior of the agent density and correlations during space invasion.

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From Microscopic to Macroscopic Descriptions of Cell Migration on Growing Domains

Cited by 95 publications

References 42 publications

Modelling biological invasions: Individual to population scales at interfaces

Modelling biological invasions: Individual to population scales at interfaces

Cell density and cell size dynamics during in vitro tissue growth experiments: Implications for mathematical models of collective cell behaviour

Exclusion processes: Short-range correlations induced by adhesion and contact interactions

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