Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Chertock, Alina; Degond, Pierre; Hecht, Sophie; Vincent, Jean‐Paul

doi:10.3934/mbe.2019290

Cited by 7 publications

(12 citation statements)

References 43 publications

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“…We highlight that similar difficulties arise when the pressure is not given in form of a power law but blows up at a finite threshold, cf. [12,13,17]. All these results have one thing in commontheir minute study of the equation satisfied by the population pressure, cf.…”

Section: Introductionmentioning

confidence: 84%

Incompressible limit for a two-species model with coupling through Brinkman's law in any dimension

Dębiec

Perthame

Schmidtchen

et al. 2021

Journal de Mathématiques Pures et Appliquées

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We study the incompressible limit for a two-species model with applications to tissue growth in the case of coupling through the so-called Brinkman's law in any space dimensions. The coupling through this elliptic equation accounts for viscosity effects among the individual species. In a recent paper DĘBIEC & SCHMIDTCHEN established said result in one spacial dimension, with their proof hinging on being able to establish uniform BV-bounds. This approach is fundamentally different from the one-species case in arbitrary dimension, established by PERTHAME & VAUCHELET. Their result relies on a kinetic reformulation to obtain strong compactness of the pressure. In this paper we fill this gap in the literature and present the incompressible limit for the system in arbitrary space dimension. The difficulty stems from jump discontinuities in the pressure not only at the boundary of the support of the two species but also at internal layers giving rise to the question as to how compactness can be obtained. The answer is a combination of techniques consisting of the application of the compactness method of BRESCH & JABIN, an adaptation of the aforementioned kinetic reformulation, and several parallels to the one dimensional strategy. The main result of this paper establishes a rigorous bridge between the population dynamics of growing tissue at a density level and a geometric model thereof.

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Section: Introductionmentioning

confidence: 84%

Incompressible limit for a two-species model with coupling through Brinkman's law in any dimension

Dębiec

Perthame

Schmidtchen

et al. 2021

Journal de Mathématiques Pures et Appliquées

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“…The numerical simulations are performed using a finite volume method similar as the one proposed in [12,14]. The scheme used for the conservative part is a classical explicit upwind scheme.…”

Section: Numerical Simulations 41 Numerical Schemementioning

confidence: 99%

Incompressible limit of a continuum model of tissue growth for two cell populations

Degond¹,

Hecht²,

Vauchelet³

2020

Networks &Amp; Heterogeneous Media

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This paper investigates the incompressible limit of a system modelling the growth of two cells population. The model describes the dynamics of cell densities, driven by pressure exclusion and cell proliferation. It has been shown that solutions to this system of partial differential equations have the segregation property, meaning that two population initially segregated remain segregated. This work is devoted to the incompressible limit of such system towards a free boundary Hele Shaw type model for two cell populations.

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“…Regarding the second approach, it is closer to the biological vision of the tissue and allows to study its motion and dynamics. There is a well-developed technique to establish a link between the two approaches, the so-called incompressible limit, which implies that the pressure becomes stiff [57,58,44,52,14,23,50,49].…”

Section: The Aronson-bénilan Estimate and Regularity Theory Of The Po...mentioning

confidence: 99%

“…For the DHV pressure law, i.e., assuming that the pressure blows up at a finite threshold, cf. [44,23,31], similar mathematical difficulties arise: in order to pass to the limit, strong restrictions have to be imposed. While the incompressible limit for multiple species remains an interesting open problem for the Darcy law, including viscosity of cells in the model, i.e., altering the velocity, u, in Eq.…”

Section: The Aronson-bénilan Estimate and Regularity Theory Of The Po...mentioning

confidence: 99%

The Aronson-Bénilan Estimate in Lebesgue Spaces

Bevilacqua¹,

Perthame²,

Schmidtchen³

2020

Preprint

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In a celebrated three-pages long paper in 1979, Aronson and Bénilan obtained a remarkable estimate on second order derivatives for the solution of the porous media equation. Since its publication, the theory of porous medium flow has expanded relentlessly with applications including thermodynamics, gas flow, ground water flow as well as ecological population dynamics. The purpose of this paper is to clarify the use of recent extensions of the Aronson and Bénilan estimate in L p spaces, of some modifications and improvements, as well as to show certain limitations of their strategy.

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Incompressible limit of a continuum model of tissue growth with segregation for two cell populations

Cited by 7 publications

References 43 publications

Incompressible limit for a two-species model with coupling through Brinkman's law in any dimension

Incompressible limit for a two-species model with coupling through Brinkman's law in any dimension

Incompressible limit of a continuum model of tissue growth for two cell populations

The Aronson-Bénilan Estimate in Lebesgue Spaces

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