Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint

Hecht, Sophie; Vauchelet, Nicolas

doi:10.4310/cms.2017.v15.n7.a6

Cited by 19 publications

(37 citation statements)

References 23 publications

(43 reference statements)

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“…This allows us to define the quotients c (1) γ,ε and c (2) γ,ε . On the other hand, the positivity ensures that n γ,ε is a strong solution of (14). In fact, the regularisation allows us to get rid of the degenerate parabolicity of the equation, and then solutions are classical [32][Theorem 3.1].…”

Section: Preliminaries and Regularisationmentioning

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%

“…A difficulty is that we cannot hope to derive any general comparison result at the level of the system. For that reason, the control from below relies on the observation that, thanks to the definition of p γ , (14) can be rewritten in a porous medium equation form:…”

Section: Preliminaries and Regularisationmentioning

confidence: 99%

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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“…Proof of Theorem 2.2. Having established strong convergence of the sequence (p k ) k , and weak convergence of (n k ) k due to the a priori estimates, we can pass to the limit in the relation (14) n…”

Section: Incompressible Limit and Complementarity Relationmentioning

confidence: 99%

“…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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“…In these models the tissue is described by a domain and its growth and movement are driven by the motion of the boundary. The link between these two types of model has been been made via an incompressible limit in [21,22,24,26,27,28]. This link is interesting as both models have their advantages.…”

Section: Introductionmentioning

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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Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint

Cited by 19 publications

References 23 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 Tumour Model with Coupling Through Brinkman’s Law in One Dimension

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

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