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January  2012, 11(1): 243-260. doi: 10.3934/cpaa.2012.11.243

A congestion model for cell migration

Julien Dambrine 1, , Nicolas Meunier 1, , Bertrand Maury 2, and  Aude Roudneff-Chupin 2,

1. 

MAP5, UFR de Mathématiques et Informatique, Université Paris Descartes, 45 rue des Saints-Pères 75270 Paris cedex 06, France, France
2. 

Laboratoire de Mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay Cedex

Received  February 2010 Revised  September 2010 Published  September 2011

This paper deals with a class of macroscopic models for cell migration in a saturated medium for two-species mixtures. Those species tend to achieve some motion according to a desired velocity, and congestion forces them to adapt their velocity. This adaptation is modelled by a correction velocity which is chosen minimal in a least-square sense. We are especially interested in two situations: a single active species moves in a passive matrix (cell migration) with a given desired velocity, and a closed-loop Keller-Segel type model, where the desired velocity is the gradient of a self-emitted chemoattractant.
We propose a theoretical framework for the open-loop model (desired velocities are defined as gradients of given functions) based on a formulation in the form of a gradient flow in the Wasserstein space. We propose a numerical strategy to discretize the model, and illustrate its behaviour in the case of a prescribed velocity, and for the saturated Keller-Segel model.
Keywords: chemotaxis, optimal transport., Congestion, aggregation.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Julien Dambrine, Nicolas Meunier, Bertrand Maury, Aude Roudneff-Chupin. A congestion model for cell migration. Communications on Pure & Applied Analysis, 2012, 11 (1) : 243-260. doi: 10.3934/cpaa.2012.11.243
References:
[1]

L. Ambrosio, N. Gigli and G. Savare, Gradient flows in metric spaces in the space of probability measures,, Lectures in Mathematics, (2005).   Google Scholar

[2]

L. Ambrosio and G. Savare, "Gradient Flows of Probability Measures,", Handbook of Differential Equations, 3 (2007).   Google Scholar

[3]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,, Numer. Math., 84 (2001), 375.  doi: 10.1007/s002110050002.  Google Scholar

[4]

A. L. Dalibard and B. Perthame, Existence of solutions of the hyperbolic Keller-Segel model,, Trans. Amer. Math. Soc., 361 (2009), 2319.   Google Scholar

[5]

E. De Giorgi, New problems on minimizing movements,, Boundary Value Problems for PDE and Applications (eds. C. Baiocchi and J. L. Lions), (1993), 81.   Google Scholar

[6]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity,, SIAM J. Appl. Math., 66 (2005), 286.  doi: 10.1137/040612841.  Google Scholar

[7]

N. Gigli and F. Otto, Entropic Burgers' equation via a minimizing movement scheme based on the Wasserstein metric,, submitted., ().   Google Scholar

[8]

R. Jordan and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.  doi: 10.1137/S0036141096303359.  Google Scholar

[9]

E. F Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[10]

R. Kimmel and J. Sethian, Fast marching methods for computing distance maps and shortest paths,, Technical Report, 669 (1996).   Google Scholar

[11]

B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type,, Math. Mod. Meth. Appl. Sci., ().   Google Scholar

[12]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Equations, 26 (1977), 347.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[13]

F. Otto and Weinan E., Thermodynamically driven incompressible fluid mixtures,, J. Chem. Phys., 107 (1997).  doi: 10.1063/1.474153.  Google Scholar

[14]

B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539.  doi: 10.1007/s10492-004-6431-9.  Google Scholar

[15]

G. Peyre, Toolbox Fast Marching - A toolbox for Fast Marching and level sets computations,, software, (2008).   Google Scholar

[16]

M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations,", Texts in App. Math., 13 (2004).   Google Scholar

[17]

C. Villani, "Topics in Optimal Transportation,", Grad. Stud. Math., 58 (2003).   Google Scholar

[18]

C. Villani, Optimal transport, old and new,, Grundlehren der mathematischen Wissenschaften, 338 (2009).   Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savare, Gradient flows in metric spaces in the space of probability measures,, Lectures in Mathematics, (2005).   Google Scholar

[2]

L. Ambrosio and G. Savare, "Gradient Flows of Probability Measures,", Handbook of Differential Equations, 3 (2007).   Google Scholar

[3]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,, Numer. Math., 84 (2001), 375.  doi: 10.1007/s002110050002.  Google Scholar

[4]

A. L. Dalibard and B. Perthame, Existence of solutions of the hyperbolic Keller-Segel model,, Trans. Amer. Math. Soc., 361 (2009), 2319.   Google Scholar

[5]

E. De Giorgi, New problems on minimizing movements,, Boundary Value Problems for PDE and Applications (eds. C. Baiocchi and J. L. Lions), (1993), 81.   Google Scholar

[6]

Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity,, SIAM J. Appl. Math., 66 (2005), 286.  doi: 10.1137/040612841.  Google Scholar

[7]

N. Gigli and F. Otto, Entropic Burgers' equation via a minimizing movement scheme based on the Wasserstein metric,, submitted., ().   Google Scholar

[8]

R. Jordan and F. Otto, The variational formulation of the Fokker-Planck equation,, SIAM J. Math. Anal., 29 (1998), 1.  doi: 10.1137/S0036141096303359.  Google Scholar

[9]

E. F Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[10]

R. Kimmel and J. Sethian, Fast marching methods for computing distance maps and shortest paths,, Technical Report, 669 (1996).   Google Scholar

[11]

B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type,, Math. Mod. Meth. Appl. Sci., ().   Google Scholar

[12]

J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space,, J. Differential Equations, 26 (1977), 347.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[13]

F. Otto and Weinan E., Thermodynamically driven incompressible fluid mixtures,, J. Chem. Phys., 107 (1997).  doi: 10.1063/1.474153.  Google Scholar

[14]

B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539.  doi: 10.1007/s10492-004-6431-9.  Google Scholar

[15]

G. Peyre, Toolbox Fast Marching - A toolbox for Fast Marching and level sets computations,, software, (2008).   Google Scholar

[16]

M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations,", Texts in App. Math., 13 (2004).   Google Scholar

[17]

C. Villani, "Topics in Optimal Transportation,", Grad. Stud. Math., 58 (2003).   Google Scholar

[18]

C. Villani, Optimal transport, old and new,, Grundlehren der mathematischen Wissenschaften, 338 (2009).   Google Scholar

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