Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion

Colson, Chloé; Sánchez‐Garduño, Faustino; Byrne, Helen; Maini, Philip K.; Lorenzi, Tommaso

doi:10.1098/rspa.2021.0593

“…12,13 A model of cell invasion through ECM is presented in El-Hachem et al 14 This consists of a system of two coupled PDEs, with a nonlinear cross-species density-dependent diffusion term and logistic growth, whereby proliferation of the cell population is limited by the presence of cells and ECM. A similar model is considered in Colson et al, 15 where proliferation depends only on the presence of cells. An obvious question to ask is how the predictions of such models may be affected by a consistent description of the role of volume-filling effects (i.e., cells and ECM take up some given volume, preventing cell invasion) across both proliferative and diffusive mechanisms of cell dynamics.…”

Section: Introductionmentioning

confidence: 98%

Traveling waves in a coarse‐grained model of volume‐filling cell invasion: Simulations and comparisons

Crossley,

Maini,

Lorenzi

et al. 2023

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Many reaction–diffusion models produce traveling wave solutions that can be interpreted as waves of invasion in biological scenarios such as wound healing or tumor growth. These partial differential equation models have since been adapted to describe the interactions between cells and extracellular matrix (ECM), using a variety of different underlying assumptions. In this work, we derive a system of reaction–diffusion equations, with cross‐species density‐dependent diffusion, by coarse‐graining an agent‐based, volume‐filling model of cell invasion into ECM. We study the resulting traveling wave solutions both numerically and analytically across various parameter regimes. Subsequently, we perform a systematic comparison between the behaviors observed in this model and those predicted by simpler models in the literature that do not take into account volume‐filling effects in the same way. Our study justifies the use of some of these simpler, more analytically tractable models in reproducing the qualitative properties of the solutions in some parameter regimes, but it also reveals some interesting properties arising from the introduction of cell and ECM volume‐filling effects, where standard model simplifications might not be appropriate.

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“…One of the most interesting features about this model is the appearance of compactly supported solutions, which give rise to the sharp invasion fronts observed in tissue formation experiments [10][11][12][13]. Of course, there are additional effects which can play an important role in collective cell motility and have been modelled using extensions of the mentioned equations, such as cell-cell adhesion [14][15][16][17], viscoelastic forces [1,18,19], interactions with the extracellular matrix [20][21][22], heterogeneity in cell size [23,24] and cell-cycle dynamics [25].…”

Section: Introductionmentioning

confidence: 99%

Quantifying tissue growth, shape and collision via continuum models and Bayesian inference

Falcó

¹

,

Cohen

²

,

Carrillo

³

et al. 2023

J. R. Soc. Interface.

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Although tissues are usually studied in isolation, this situation rarely occurs in biology, as cells, tissues and organs coexist and interact across scales to determine both shape and function. Here, we take a quantitative approach combining data from recent experiments, mathematical modelling and Bayesian parameter inference, to describe the self-assembly of multiple epithelial sheets by growth and collision. We use two simple and well-studied continuum models, where cells move either randomly or following population pressure gradients. After suitable calibration, both models prove to be practically identifiable, and can reproduce the main features of single tissue expansions. However, our findings reveal that whenever tissue–tissue interactions become relevant, the random motion assumption can lead to unrealistic behaviour. Under this setting, a model accounting for population pressure from different cell populations is more appropriate and shows a better agreement with experimental measurements. Finally, we discuss how tissue shape and pressure affect multi-tissue collisions. Our work thus provides a systematic approach to quantify and predict complex tissue configurations with applications in the design of tissue composites and more generally in tissue engineering.

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“… 2019 ), as well as analysing travelling wave solutions of these types of multi-species mathematical models (Colson et al. 2021 ; El-Hachem et al. 2021b ; Gallay and Mascia 2021 ).…”

Section: Introductionmentioning

confidence: 99%

“…This second limitation has been addressed by introducing more complicated mathematical models, such as the celebrated Gatenby-Gawlinski model of tumour invasion (Gatenby and Gawlinski 1996), which explicitly describes how a population of tumour cells degrades and invades into a population of surrounding healthy tissue by explicitly modelling both populations and their interactions. Since the Gatenby-Gawlinski framework was proposed in 1996, subsequent studies have since analysed the relationship between individual-level mechanisms and the resulting population-level continuum descriptions (Painter et al 2003), calibrating these mathematical models to match experimental measurements of melanoma invasion (Browning et al 2019), as well as analysing travelling wave solutions of these types of multi-species mathematical models (Colson et al 2021;El-Hachem et al 2021b;Gallay and Mascia 2021).…”

Section: Introductionmentioning

confidence: 99%

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

El‐Hachem

¹

,

McCue

²

,

Simpson

³

2022

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We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, $${\hat{u}}(\hat{{\mathbf {x}}},{\hat{t}})$$ u ^ ( x ^ , t ^ ) that is coupled to the concentration of an immobile extracellular substrate, $${\hat{s}}(\hat{{\mathbf {x}}},{\hat{t}})$$ s ^ ( x ^ , t ^ ) . Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to $${\hat{s}}$$ s ^ , while a logistic growth term models cell proliferation. The extracellular substrate $${\hat{s}}$$ s ^ is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, $$c = c_{\mathrm{min}}$$ c = c min , as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, $$c > c_{\mathrm{min}}$$ c > c min . We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub.

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Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion

Cited by 9 publications

References 49 publications

Traveling waves in a coarse‐grained model of volume‐filling cell invasion: Simulations and comparisons

Traveling waves in a coarse‐grained model of volume‐filling cell invasion: Simulations and comparisons

Quantifying tissue growth, shape and collision via continuum models and Bayesian inference

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

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