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10 results on '"Poulain, Alexandre"'

1. Relaxation of the Cahn–Hilliard equation with singular single-well potential and degenerate mobility

Author

PERTHAME, BENOÎT, POULAIN, ALEXANDRE, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), The authors have received funding from the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme (grant agreement No 740623), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)

Subjects
Asymptotic analysis, Relaxation method, 01 natural sciences, Mathematics - Analysis of PDEs, Singularity, Relaxation system, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH]Mathematics [math], 0101 mathematics, Cahn–Hilliard equation, Physics, 35B40, 35G20, 35Q92, 92C10, Computer simulation, Applied Mathematics, 010102 general mathematics, Degenerate energy levels, Second order equation, Limiting, Degenerate Cahn-Hilliard equation, 2010 Mathematics Subject Classification: 35B40, 35G20, 35Q92, 92C10, 010101 applied mathematics, Classical mechanics, Mathematical biology, Living tissues, Analysis of PDEs (math.AP)
Abstract

The degenerate Cahn–Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn–Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second-order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system: global existence and convergence of the relaxed system to the degenerate Cahn–Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.

Published
2020
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2. Degenerate Cahn-Hilliard and incompressible limit of a Keller-Segel model

Author

Elbar, Charles, Perthame, Benoît, Poulain, Alexandre, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Simula Research Laboratory, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Simula Research Laboratory [Lysaker] (SRL), Sorbonne Université (SU), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics::Fluid Dynamics, Mathematics - Analysis of PDEs, Applied Mathematics, General Mathematics, FOS: Mathematics, Mathematics::Analysis of PDEs, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35B40, 35B45, 35G20, 35Q92, [MATH]Mathematics [math], Nonlinear Sciences::Cellular Automata and Lattice Gases, Nonlinear Sciences::Pattern Formation and Solitons, Analysis of PDEs (math.AP), Quantitative Biology::Cell Behavior
Abstract

The Keller-Segel model is a well-known system representing chemotaxis in living organisms. We study the convergence of a generalized nonlinear variant of the Keller-Segel to the degenerate Cahn-Hilliard system. This analysis is made possible from the observation that the Keller-Segel system is equivalent to a relaxed version of the Cahn-Hilliard system. Furthermore, this latter equivalent system has an interesting application in the modelling of living tissues. Indeed, compressible and incompressible porous medium type equations are widely used to describe the mechanical properties of living tissues. The relaxed degenerate Cahn-Hilliard system, can be viewed as a compressible living tissue model for which the movement is driven by Darcy's law and takes into account the effects of the viscosity as well as surface tension at the surface of the tissue. We study the convergence of the Keller-Segel system to the Cahn-Hilliard equation and some of the analytical properties of the model such as the incompressible limit of our model. Our analysis relies on a priori estimates, compactness properties, and on the equivalence between the Keller-Segel system and the relaxed degenerate Cahn-Hilliard system.

Published
2021
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3. Models of living tissues, numerical simulations and immunotherapy of cancers

Author

Poulain, Alexandre, Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université, Benoît Perthame, Tommaso Lorenzi, and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)

Subjects
Living tissues models, Modèle de Keller-Segel, Keller-Segel model, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Modèles de tissus vivants, Équation de Cahn-Hilliard dégénérée, Analyse numérique, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Numerical analysis, Degenerate Cahn-Hilliard equation
Abstract

We study two classes of mathematical models currently used for the modeling in time and space of tumors: the Cahn-Hilliard equation for living tissues and the Keller-Segel model. The numerical methods we propose aim to represent these equations efficiently and accurately while preserving their properties. For the Cahn-Hilliard equation, our study is based on a relaxation method for which we prove the convergence to the original model. Even though the physical effects modeled by these equations are close to the ones studied in fluid dynamics, the equations used to model living tissues are different in order to represent the active behavior of cells. The resulting equations contain numerous singularities and degeneracies, which result in technical difficulties to analyze and simulate them efficiently.Our relaxation method has been introduced to facilitate the implementation of our numerical schemes. Hence, we propose numerical schemes that are easy to implement in already existing finite element software. In order to preserve the properties of the equations during numerical simulations, we design numerical schemes based on the Scalar Auxiliary Variable method. Based on these numerical works, we present two studies of biological phenomena.; Nous étudions deux types de modèles couramment utilisés pour la représentation en temps et en espace des tumeurs: l’équation de Cahn-Hilliard pour les tissus vivants et le modèle de Keller-Segel. Les méthodes numériques que nous développons cherchent à représenter de manière précise et efficace ces équations tout en préservant leurs propriétés. Pour l'équation de Cahn-Hilliard, notre étude s’appuie sur une méthode de relaxation dont nous prouvons la convergence vers le modèle initial. Même si elles représentent mathématiquement des phénomènes physiques proches de ceux étudiés en dynamique des fluides, les équations utilisées pour les tissus vivants sont souvent différentes pour rendre compte du caractère actif des cellules. Les équations résultantes contiennent de nombreuses singularités et dégénérescences qui sont difficiles à analyser théoriquement et simuler numériquement de manière efficace. La méthode de relaxation a été introduite pour faciliter l’implémentation de nos schémas numériques; nous proposons ainsi des schémas numériques éléments finis simples à adapter dans les codes pré-existants. Afin de préserver les propriétés des équations continues lors des simulations numériques, nous proposons des schémas numériques basés sur la Méthode de Variable Auxiliaire. Sur la base de ces travaux numériques, nous présentons l’étude de deux phénomènes biologiques.

Published
2021

4. Effective interface conditions for a model of tumour invasion through a membrane

Author

Ciavolella, Giorgia, David, Noemi, Poulain, Alexandre, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Dipartimento di Matematica [Rome], Università degli Studi di Roma Tor Vergata [Roma], Dipartimento di Matematica [Bologna], Alma Mater Studiorum Università di Bologna [Bologna] (UNIBO), Giorgia Ciavolella and Alexandre Poulain have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement N° 740623). The work of Giorgio Ciavolella was also partially supported by GNAMPA-INdAM.Noemi David has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie (grant agreement N° 754362)., European Project: 740623,H2020 Pilier ERC,ADORA(2017), European Project: 754362,MathInParis(2017), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)

Subjects
Membrane boundary conditions, 2010Mathematics Subject Classification.35B45, 35K57, 35K65, 35Q92, 76N10, 76S05, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Effective interface, [MATH]Mathematics [math], Porous medium equation, Nonlinear reaction-diffusion equations, Tumour growth models, Quantitative Biology::Cell Behavior
Abstract

Motivated by biological applications on tumour invasion through thin membranes, we study a porous-medium type equation where the density of the cell population evolves under Darcy's law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problem and the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al. (2019), which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator which allows to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.

Published
2021

5. Effective interface conditions for a porous medium type problem

Author

Ciavolella, Giorgia, David, Noemi, Poulain, Alexandre, Ciavolella, Giorgia, Asymptotic approach to spatial and dynamical organizations - ADORA - - H2020 Pilier ERC2017-09-01 - 2022-08-31 - 740623 - VALID, International Doctoral Training in Mathematical Sciences in Paris - MathInParis - 2017-09-01 - 2022-08-31 - 754362 - VALID, Modélisation Mathématique pour l'Oncologie (MONC), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], UNICANCER-UNICANCER-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université de Lyon, Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Université de Lille, Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Giorgia Ciavolella and Alexandre Poulain have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement N° 740623). The work of Giorgio Ciavolella was also partially supported by GNAMPA-INdAM.Noemi David has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie (grant agreement N° 754362)., European Project: 740623,H2020 Pilier ERC,ADORA(2017), European Project: 754362,MathInParis(2017), Modélisation mathématique, calcul scientifique (MMCS), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL)

Subjects
Membrane boundary conditions, Effective interface, [MATH] Mathematics [math], Porous medium equation, Nonlinear reaction-diffusion equations, Tumour growth models, Quantitative Biology::Cell Behavior, Mathematics - Analysis of PDEs, 2010Mathematics Subject Classification.35B45, 35K57, 35K65, 35Q92, 76N10, 76S05, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH]Mathematics [math], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 35B45, 35K57, 35K65, 35Q92, 76N10, 76S05, Analysis of PDEs (math.AP)
Abstract

Motivated by biological applications on tumour invasion through thin membranes, we study a porous-medium type equation where the density of the cell population evolves under Darcy's law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problemand the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al., (2019), which are compatible with nonlinear generalized Kedem-Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator which allows to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.

Published
2021

6. Scalar auxiliary variable finite element scheme for the parabolic-parabolic Keller-Segel model

Author

Poulain, Alexandre, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU), Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), The author has received funding from the European Research Council (ERC) under the European Union’s Horizon2020 research and innovation programme (grant agreement No 740623), European Project: 740623,H2020 Pilier ERC,ADORA(2017), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)

Subjects
Keller-Segel, 35K20, 35Q92, 65M12, 35K55, Mathematics::Analysis of PDEs, Numerical Analysis (math.NA), 2010 Mathematics Subject Classification: 35K20, 35Q92, 65M12, 35K55, Quantitative Biology::Cell Behavior, Gradient flow, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Energy stability, Mathematics - Numerical Analysis, [MATH]Mathematics [math], Living tissues, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We describe and analyze a finite element numerical scheme for the parabolic-parabolic Keller-Segel model. The scalar auxiliary variable method is used to retrieve the monotonic decay of the energy associated with the system at the discrete level. This method relies on the interpretation of the Keller-Segel model as a gradient flow. The resulting numerical scheme is efficient and easy to implement. We show the existence of a unique non-negative solution and that a modified discrete energy is obtained due to the use of the SAV method. We also prove the convergence of the discrete solutions to the ones of the weak form of the continuous Keller-Segel model.

Published
2020

7. A Positivity-Preserving Finite Element Scheme for the Relaxed Cahn-Hilliard Equation with Single-Well Potential and Degenerate Mobility

Author

Bubba, Federica, Poulain, Alexandre, Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Abstract

We propose and analyse a finite element approximation of the Cahn-Hilliard equation regularised in space with single-well potential of Lennard-Jones type and degenerate mobility. The Cahn-Hilliard model has recently been applied to model evolution and growth for living tissues: although the choices of degenerate mobility and singular potential are biologically relevant, they induce difficulties regarding the design of a numerical scheme. We propose a finite element scheme in one and two dimensions and we show that it preserves the physical bounds of the solutions thanks to an upwind approach adapted to the finite elements method. Moreover, we show well-posedness, energy stability properties and convergence of solutions to the numerical scheme. Finally, numerical simulations in one and two dimensions are presented.

Published
2019

8. Convergence, error analysis and longtime behavior of the scalar auxiliary variable method for the nonlinear Schrödinger equation

Author

Poulain, Alexandre, Schratz, Katharina, Sorbonne Université (SU), Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)

Subjects
Applied Mathematics, General Mathematics, 2010 Mathematics Subject Classification. 35Q55, 65M70, 65M12, 65M15, Pseudospectral method, 35Q55, 65M70, 65M12, 65M15, Numerical Analysis (math.NA), 7. Clean energy, Pseudospectralmethod, Computational Mathematics, Error bounds, Scalar Auxiliary Variable, Nonlinear Schrödinger equation, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, [MATH]Mathematics [math], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We carry out the convergence analysis of the scalar auxiliary variable (SAV) method applied to the nonlinear Schrödinger equation, which preserves a modified Hamiltonian on the discrete level. We derive a weak and strong convergence result, establish second-order global error bounds and present longtime error estimates on the modified Hamiltonian. In addition, we illustrate the favorable energy conservation of the SAV method compared to classical splitting schemes in certain applications.

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9. Multi-compartmental model of glymphatic clearance of solutes in brain tissue

Author

Alexandre Poulain, Jørgen Riseth, Vegard Vinje, Poulain, Alexandre, Simula Research Laboratory (SRL), Laboratoire Paul Painlevé (LPP), and Université de Lille-Centre National de la Recherche Scientifique (CNRS)

Subjects
[SDV] Life Sciences [q-bio], Multidisciplinary, [SDV]Life Sciences [q-bio], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH] Mathematics [math], [MATH]Mathematics [math], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]
Abstract

The glymphatic system is the subject of numerous pieces of research in biology. Mathematical modelling plays a considerable role in this field since it can indicate the possible physical effects of this system and validate the biologists’ hypotheses. The available mathematical models that describe the system at the scale of the brain (i.e. the macroscopic scale) are often solely based on the diffusion equation and do not consider the fine structures formed by the perivascular spaces. We therefore propose a mathematical model representing the time and space evolution of a mixture flowing through multiple compartments of the brain. We adopt a macroscopic point of view in which the compartments are all present at any point in space. The equations system is composed of two coupled equations for each compartment: One equation for the pressure of a fluid and one for the mass concentration of a solute. The fluid and solute can move from one compartment to another according to certain membrane conditions modelled by transfer functions. We propose to apply this new modelling framework to the clearance of 14C-inulin from the rat brain.

Published
2022
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10. Treatment-induced shrinking of tumour aggregates: a nonlinear volume-filling chemotactic approach

Author

Luis Almeida, Lisa Oliver, Diane Peurichard, François M. Vallette, Alexandre Poulain, Gissell Estrada-Rodriguez, Universitat Politècnica de Catalunya. Departament de Matemàtiques, Modelling and Analysis for Medical and Biological Applications (MAMBA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Apoptosis and Tumor Progression (CRCINA-ÉQUIPE 9), Centre de Recherche en Cancérologie et Immunologie Nantes-Angers (CRCINA), Institut National de la Santé et de la Recherche Médicale (INSERM)-Université de Nantes - UFR de Médecine et des Techniques Médicales (UFR MEDECINE), Université de Nantes (UN)-Université de Nantes (UN)-Centre hospitalier universitaire de Nantes (CHU Nantes)-Centre National de la Recherche Scientifique (CNRS)-Université d'Angers (UA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université de Nantes - UFR de Médecine et des Techniques Médicales (UFR MEDECINE), Université de Nantes (UN)-Université de Nantes (UN)-Centre hospitalier universitaire de Nantes (CHU Nantes)-Centre National de la Recherche Scientifique (CNRS)-Université d'Angers (UA), This research was funded by the Plan Cancer/ITMO HTE (tumor heterogeneity and ecosystem) program (MoGlimaging)., Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU), Poulain, Alexandre, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), and Université d'Angers (UA)-Université de Nantes (UN)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Centre hospitalier universitaire de Nantes (CHU Nantes)-Université d'Angers (UA)-Université de Nantes (UN)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Centre hospitalier universitaire de Nantes (CHU Nantes)

Subjects
FOS: Physical sciences, Pattern Formation and Solitons (nlin.PS), 03 medical and health sciences, 0302 clinical medicine, [NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS], Neoplasms, Quimiotaxi, [NLIN.NLIN-PS] Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS], Cell Behavior (q-bio.CB), Extracellular, Organoid, medicine, Tumor Microenvironment, Humans, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 030304 developmental biology, 0303 health sciences, Partial differential equation, Física [Àrees temàtiques de la UPC], Chemistry, Cell growth, Applied Mathematics, Physics, Chemotaxis, Aggregate (data warehouse), Volume-filling approach, Matemàtiques i estadística [Àrees temàtiques de la UPC], Stability analysis, Física, medicine.disease, Agricultural and Biological Sciences (miscellaneous), Nonlinear Sciences - Pattern Formation and Solitons, Nonlinear system, FOS: Biological sciences, Modeling and Simulation, Biophysics, Quantitative Biology - Cell Behavior, Glioblastoma, Matemàtica, 030217 neurology & neurosurgery, Mathematics
Abstract

International audience; Motivated by experimental observations in 3D/organoid cultures derived from glioblastoma, we develop a mathematical model where tumour aggregate formation is obtained as the result of nutrient-limited cell proliferation coupled with chemotaxis-based cell movement. The introduction of a chemotherapeutic treatment induces mechanical changes at the cell level, with cells undergoing a transition from rigid bodies to semi-elastic entities. We analyse the influence of these individual mechanical changes on the properties of the aggregates obtained at the population level by introducing a nonlinear volume-filling chemotactic system of partial differential equations. The elastic properties of the cells are taken into account through the so-called squeezing probability, which allows us to change the packing capacity of the aggregates, depending on the concentration of the treatment in the extracellular microenvironment. We explore two scenarios for the effect of the treatment: firstly, the treatment acts only on the mechanical properties of the cells and, secondly, we assume it also prevents cell proliferation. A linear stability analysis enables us to study the ability of the system to create patterns. We provide numerical simulations in 1D and 2D that illustrate the shrinking of the aggregates due to the presence of the treatment.

Published
2021
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