A congestion model for cell migration

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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

Abstract
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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 and 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, ETH Zürich, (2005).
[2]	L. Ambrosio and G. Savare, "Gradient Flows of Probability Measures," Handbook of Differential Equations, Evolutionary Equations (ed. by C.M. Dafermos and E. Feireisl, Elsevier), 3, 2007.
[3]	J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2001), 375-393. doi: 10.1007/s002110050002.
[4]	A. L. Dalibard and B. Perthame, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.
[5]	E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications (eds. C. Baiocchi and J. L. Lions), Masson, (1993), 81-98.
[6]	Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math., 66 (2005), 286-308. doi: 10.1137/040612841.
[7]	N. Gigli and F. Otto, Entropic Burgers' equation via a minimizing movement scheme based on the Wasserstein metric,, submitted., ().
[8]	R. Jordan and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.
[9]	E. F Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.
[10]	R. Kimmel and J. Sethian, Fast marching methods for computing distance maps and shortest paths, Technical Report, CPAM, Univ. of California, Berkeley, 669 (1996).
[11]	B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type,, Math. Mod. Meth. Appl. Sci., ().
[12]	J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7.
[13]	F. Otto and Weinan E., Thermodynamically driven incompressible fluid mixtures, J. Chem. Phys., 107 (1997), 10177. doi: 10.1063/1.474153.
[14]	B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564. doi: 10.1007/s10492-004-6431-9.
[15]	G. Peyre, Toolbox Fast Marching - A toolbox for Fast Marching and level sets computations, software, (2008).
[16]	M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations," Texts in App. Math., 13, Springer-Verlag, New York, 2004.
[17]	C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58 AMS, Providence, 2003.
[18]	C. Villani, Optimal transport, old and new, Grundlehren der mathematischen Wissenschaften, 338 (2009).

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, ETH Zürich, (2005).
[2]	L. Ambrosio and G. Savare, "Gradient Flows of Probability Measures," Handbook of Differential Equations, Evolutionary Equations (ed. by C.M. Dafermos and E. Feireisl, Elsevier), 3, 2007.
[3]	J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2001), 375-393. doi: 10.1007/s002110050002.
[4]	A. L. Dalibard and B. Perthame, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.
[5]	E. De Giorgi, New problems on minimizing movements, Boundary Value Problems for PDE and Applications (eds. C. Baiocchi and J. L. Lions), Masson, (1993), 81-98.
[6]	Y. Dolak and C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math., 66 (2005), 286-308. doi: 10.1137/040612841.
[7]	N. Gigli and F. Otto, Entropic Burgers' equation via a minimizing movement scheme based on the Wasserstein metric,, submitted., ().
[8]	R. Jordan and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17. doi: 10.1137/S0036141096303359.
[9]	E. F Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.
[10]	R. Kimmel and J. Sethian, Fast marching methods for computing distance maps and shortest paths, Technical Report, CPAM, Univ. of California, Berkeley, 669 (1996).
[11]	B. Maury, A. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type,, Math. Mod. Meth. Appl. Sci., ().
[12]	J. J. Moreau, Evolution problem associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374. doi: 10.1016/0022-0396(77)90085-7.
[13]	F. Otto and Weinan E., Thermodynamically driven incompressible fluid mixtures, J. Chem. Phys., 107 (1997), 10177. doi: 10.1063/1.474153.
[14]	B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564. doi: 10.1007/s10492-004-6431-9.
[15]	G. Peyre, Toolbox Fast Marching - A toolbox for Fast Marching and level sets computations, software, (2008).
[16]	M. Renardy and R. C. Rogers, "An Introduction to Partial Differential Equations," Texts in App. Math., 13, Springer-Verlag, New York, 2004.
[17]	C. Villani, "Topics in Optimal Transportation," Grad. Stud. Math., 58 AMS, Providence, 2003.
[18]	C. Villani, Optimal transport, old and new, Grundlehren der mathematischen Wissenschaften, 338 (2009).

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