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

Degond, Pierre; Hecht, Sophie; Vauchelet, Nicolas

doi:10.3934/nhm.2020003

Cited by 20 publications

(39 citation statements)

References 33 publications

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“…The second condition (10) is technical and will be required when deriving some a priori estimates in Section 4. The last set of conditions (11) appears rather technical at first glance, but it is a natural assumption, cf. also [8], as it allows us to handle the points where both initial densities vanish, i.e., vacuum or the absence of any species.…”

Section: Preliminaries and Regularisationmentioning

confidence: 99%

“…Indeed, Helly's selection theorem applies since the quotients are bounded both in L ∞ and in BV by assumption (11). Our approach is then based on the observation that, in the absence of cross-reactions (F 2 = G 1 = 0), a direct manipulation shows that n γ c…”

Section: Segregation Propertymentioning

confidence: 99%

“…Also, they are naturally related to the important segregation property mentioned earlier. Note that the paper [11], for a different pressure law, provides a deep understanding of the internal layer dynamics in the incompressible limit when the initial data is assumed to be initially segregated with a single contact point. This has the advantage of directly applying regularity results of [2].…”

Section: Introductionmentioning

confidence: 99%

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Hele–Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues

Bubba

Perthame

Pouchol

et al. 2019

Arch Rational Mech Anal

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Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a segregation result.Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson-Bénilan estimates cannot be established in our context. We are lead, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an L 1 version in place of the standard upper bound.

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Section: Preliminaries and Regularisationmentioning

confidence: 99%

Section: Segregation Propertymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hele–Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues

Bubba

Perthame

Pouchol

et al. 2019

Arch Rational Mech Anal

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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: 83%

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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“…Let us stress that the same type of difficulties are also encountered when the pressure is not given as a power law, cf. [11,12,14]. A key tool in obtaining existence results and stable (with respect to the parameter k) estimates is to devise and manipulate the equation satisfied by the (joint) population pressure, cf.…”

Section: Introductionmentioning

confidence: 99%

Incompressible Limit for a Two-Species Tumour Model with Coupling Through Brinkman’s Law in One Dimension

Dębiec

Schmidtchen

2020

Acta Appl Math

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We present a two-species model with applications in tumour modelling. The main novelty is the coupling of both species through the so-called Brinkman law which is typically used in the context of visco-elastic media, where the velocity field is linked to the total population pressure via an elliptic equation. The same model for only one species has been studied by Perthame and Vauchelet in the past. The first part of this paper is dedicated to establishing existence of solutions to the problem, while the second part deals with the incompressible limit as the stiffness of the pressure law tends to infinity. Here we present a novel approach in one spatial dimension that differs from the kinetic reformulation used in the aforementioned study and, instead, relies on uniform BV-estimates.

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

Cited by 20 publications

References 33 publications

Hele–Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues

Hele–Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues

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

Incompressible Limit for a Two-Species Tumour Model with Coupling Through Brinkman’s Law in One Dimension

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