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A congestion model for cell migration

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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.
    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • [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. OttoEntropic 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. SantambrogioA macroscopic crowd motion model of gradient flow type, Math. Mod. Meth. Appl. Sci., to appear.

    [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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