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

A congestion model for cell migration

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.
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).

[2]

L. Ambrosio and G. Savare, "Gradient Flows of Probability Measures,", Handbook of Differential Equations, 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. 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.

[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.

[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.

[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. doi: 10.1137/S0036141096303359.

[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.

[10]

R. Kimmel and J. Sethian, Fast marching methods for computing distance maps and shortest paths,, Technical Report, 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. doi: 10.1016/0022-0396(77)90085-7.

[13]

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

[14]

B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539. 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 (2004).

[17]

C. Villani, "Topics in Optimal Transportation,", Grad. Stud. Math., 58 (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, (2005).

[2]

L. Ambrosio and G. Savare, "Gradient Flows of Probability Measures,", Handbook of Differential Equations, 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. 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.

[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.

[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.

[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. doi: 10.1137/S0036141096303359.

[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.

[10]

R. Kimmel and J. Sethian, Fast marching methods for computing distance maps and shortest paths,, Technical Report, 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. doi: 10.1016/0022-0396(77)90085-7.

[13]

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

[14]

B. Perthame, PDE models for chemotactic movements: parabolic, hyperbolic and kinetic,, Appl. Math., 49 (2004), 539. 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 (2004).

[17]

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

[18]

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

[1]

Qinglan Xia. An application of optimal transport paths to urban transport networks. Conference Publications, 2005, 2005 (Special) : 904-910. doi: 10.3934/proc.2005.2005.904

[2]

Wilfrid Gangbo, Andrzej Świech. Optimal transport and large number of particles. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1397-1441. doi: 10.3934/dcds.2014.34.1397

[3]

Robert J. McCann. A glimpse into the differential topology and geometry of optimal transport. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1605-1621. doi: 10.3934/dcds.2014.34.1605

[4]

Paul Pegon, Filippo Santambrogio, Davide Piazzoli. Full characterization of optimal transport plans for concave costs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6113-6132. doi: 10.3934/dcds.2015.35.6113

[5]

Gleb Beliakov. Construction of aggregation operators for automated decision making via optimal interpolation and global optimization. Journal of Industrial & Management Optimization, 2007, 3 (2) : 193-208. doi: 10.3934/jimo.2007.3.193

[6]

Christian Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1533-1574. doi: 10.3934/dcds.2014.34.1533

[7]

Yangang Chen, Justin W. L. Wan. Numerical method for image registration model based on optimal mass transport. Inverse Problems & Imaging, 2018, 12 (2) : 401-432. doi: 10.3934/ipi.2018018

[8]

Klas Modin. Geometry of matrix decompositions seen through optimal transport and information geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 335-390. doi: 10.3934/jgm.2017014

[9]

Karthik Elamvazhuthi, Piyush Grover. Optimal transport over nonlinear systems via infinitesimal generators on graphs. Journal of Computational Dynamics, 2018, 0 (0) : 1-32. doi: 10.3934/jcd.2018001

[10]

Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio. Congestion-driven dendritic growth. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1575-1604. doi: 10.3934/dcds.2014.34.1575

[11]

Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio, Juliette Venel. Handling congestion in crowd motion modeling. Networks & Heterogeneous Media, 2011, 6 (3) : 485-519. doi: 10.3934/nhm.2011.6.485

[12]

Chichia Chiu, Jui-Ling Yu. An optimal adaptive time-stepping scheme for solving reaction-diffusion-chemotaxis systems. Mathematical Biosciences & Engineering, 2007, 4 (2) : 187-203. doi: 10.3934/mbe.2007.4.187

[13]

Brendan Pass. Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1623-1639. doi: 10.3934/dcds.2014.34.1623

[14]

Cédric Villani. Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 559-571. doi: 10.3934/dcds.2011.30.559

[15]

Tehuan Chen, Chao Xu, Zhigang Ren. Computational optimal control of 1D colloid transport by solute gradients in dead-end micro-channels. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1251-1269. doi: 10.3934/jimo.2018052

[16]

Fethallah Benmansour, Guillaume Carlier, Gabriel Peyré, Filippo Santambrogio. Numerical approximation of continuous traffic congestion equilibria. Networks & Heterogeneous Media, 2009, 4 (3) : 605-623. doi: 10.3934/nhm.2009.4.605

[17]

Habibe Zare Haghighi, Sajad Adeli, Farhad Hosseinzadeh Lotfi, Gholam Reza Jahanshahloo. Revenue congestion: An application of data envelopment analysis. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1311-1322. doi: 10.3934/jimo.2016.12.1311

[18]

Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 301-323. doi: 10.3934/dcds.2010.27.301

[19]

Yanghong Huang, Andrea Bertozzi. Asymptotics of blowup solutions for the aggregation equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1309-1331. doi: 10.3934/dcdsb.2012.17.1309

[20]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (3)

[Back to Top]