×

zbMATH — the first resource for mathematics

Derivation of hyperbolic models for chemosensitive movement. (English) Zbl 1080.92014
Summary: A Chapman-Enskog expansion is used to derive hyperbolic models for chemosensitive movements as a hydrodynamic limit of a velocity-jump process. On the one hand, it connects parabolic and hyperbolic chemotaxis models since the former arise as diffusion limits of a similar velocity-jump process. On the other hand, this approach provides a unified framework which includes previous models obtained by ad hoc methods or methods of moments. Numerical simulations are also performed and are motivated by recent experiments with human endothelial cells on matrigel. Their movements lead to the formation of networks that are interpreted as the beginning of a vasculature. These structures cannot be explained by parabolic models but are recovered by numerical experiments on hyperbolic models. Our kinetic model suggests that some kind of local interactions might be enough to explain them.

MSC:
92C17 Cell movement (chemotaxis, etc.)
35L10 Second-order hyperbolic equations
82D99 Applications of statistical mechanics to specific types of physical systems
65C20 Probabilistic models, generic numerical methods in probability and statistics
Software:
Chemotaxis
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alt, W.: Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9, 147-177 (1980) · Zbl 0434.92001
[2] Chalub, F.A.C.C., Markovich, P., Perthame, B., Schmeiser, C.: Kinetic models for chemotaxis and their drift-diffusion limits. To appear in Monatsh. Math. · Zbl 1052.92005
[3] Chaplain, M.A.J.: A vascular growth, angiogenesis and vascular growth in solid tumors: the mathematical modelling of the stages of tumor development. Math. Comput. Modelling 23, 47-87 (1996) · Zbl 0859.92012
[4] Dolak, Y., Hillen, T.: Cattaneo models for chemotaxis, numerical solution and pattern formation. J. Math. Biol. 46, 461-478 (2003) · Zbl 1062.92501
[5] Dolak, Y., Schmeiser, C.: Kinetic models for chemotaxis: Hydrodynamic limits and the back-of-the-wave problem. ANUM preprint · Zbl 1077.92003
[6] Filbet, F., Shu, C.-W.: Approximation of Hyperbolic Models for Chemosensitive Movement. MAPMO preprint · Zbl 1141.35396
[7] Gajewski, H., Zacharias, K.: Global behavior of a reaction diffusion system modelling chemotaxis. Math. Nachr. 195, 77-114 (1998) · Zbl 0918.35064
[8] Gamba, A., Ambrosi, D., Coniglio, A., de Candia, A., Di Talia, S., Giraudo, E., Serini, G., Preziosi, L., Bussolino, F.: Percolation, morphogenesis, and Burgers dynamics in blood vessels formation. Phys. Rev. Lett. 90, 118101 (2003)
[9] Godlewski, E., Raviart, P.-A.: Numerical approximation of hyperbolic systems of conservation laws. Appl. Math. Sci. 118, Springer, New York, 1996 · Zbl 0860.65075
[10] Herrero, M.A., Medina, E., Velázquez, J.J.L.: Finite-time aggregation into a single point in a reaction-diffusion system. Nonlinearity 10, 1739-1754 (1997) · Zbl 0909.35071
[11] Hillen, T., Othmer, H.G.: The diffusion limit of transport equations derived from velocity jump processes. SIAM J. Appl. Math. 61, 751-775 (2000) · Zbl 1002.35120
[12] Hillen, T.: Transport equations and chemosensitive movement. Habilitation Thesis, University of Tübingen, 2001
[13] Hillen, T.: Hyperbolic models for chemosensitive movement. Math. Models Methods Appl. Sci. 12, 1007-1034 (2002) · Zbl 1163.35491
[14] Horstmann, D.: From 1970 until now: The Keller-Segel model in chemotaxis and its consequences I. Jahresberichte der DMV, 105, 103-165 (2003) · Zbl 1071.35001
[15] Horstmann, D.: From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II. Jahresberichte der DMV, 106, 51-69 (2004) · Zbl 1072.35007
[16] Hwang, H.J., Kang, K., Stevens, A.: Global solutions of nonlinear transport equations for chemosensitive movement. To appear in SIAM J. Math. Anal. · Zbl 1099.82018
[17] Keller, E.F., Segel, L.A.: Traveling band of chemotactic bacteria: A theoretical analysis. J. Theor. Biol. 30, 235-248 (1971) · Zbl 1170.92308
[18] LeVeque, R.: Numerical methods for conservation laws. Birkhäuser, Basel, 1992 · Zbl 0847.65053
[19] Levine, H.A., Nilsen-Hamilton, M., Sleeman, B.D.: Mathematical modelling of the onset of capillary formation initiating angiogenesis. J. Math. Biol. 42, 195-238 (2001) · Zbl 0977.92013
[20] Liu, T.P.: Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108, 153-175 (1987) · Zbl 0633.35049
[21] Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Appl. Sci. 5, 581-601 (1995) · Zbl 0843.92007
[22] Makino, T., Perthame, B.: Sur les solutions à symétrie sphérique de l?équation d?Euler-Poisson pour l?évolution d?étoiles gazeuses. Japan J. Appl. Math. 7, 165-170 (1990) · Zbl 0743.35048
[23] Marrocco, A.: 2D simulation of chemotaxis bacteria aggregation. ESAIM:M2AN, 37 (4), 617-630 (2003) · Zbl 1065.92006
[24] Mirshahi, M.: Personnal communication
[25] Nieto, J., Poupaud, F., Soler, J.: High-field limit for the Vlasov-Poisson-Fokker-Planck system. Arch. Rational Mech. Anal. 158, 29-59 (2001) · Zbl 1038.82068
[26] Othmer, H.G., Hillen, T.: The diffusion limit of transport equations II: chemotaxis equations. SIAM J. Appl. Math. 62, 1222-1250 (2002) · Zbl 1103.35098
[27] Othmer, H.G., Dunbar, S.R., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263-298 (1988) · Zbl 0713.92018
[28] Patlak, C.S.: Random walk with persistence and external bias. Bull. Math. Biol. Biophys. 15, 311-338 (1953) · Zbl 1296.82044
[29] Perthame, B.: Kinetic formulation of conservation laws. Oxford Univ. Press, 2002 · Zbl 1030.35002
[30] Senba, T., Suzuki, T.: Chemotactic collapse in parabolic-elliptic systems of mathematical biology. Adv. Diff. Eqns. 6, 21-50 (2001) · Zbl 0999.92005
[31] Serini, G., Ambrosi, D., Giraudo, E., Gamba, A., Preziosi, L., Bussolino, F.: Modeling the early stages of vascular network assembly. The EMBO J. 22, 1771-1779 (2003)
[32] Stevens, A.: Derivation of chemotaxis-equations as limit dynamics of moderately interacting stochastic many particle systems. SIAM J. Appl. Math. 61, 183-212 (2000) · Zbl 0963.60093
[33] Velázquez, J.J.L.: Stability of some mechanisms of chemotactic aggregation. SIAM J. Appl. Math. 62, 1581-1633 (2002) · Zbl 1013.35004
[34] Zeldovich, Ya.B.: Gravitational instability : an approximate theory for large density perturbations. Astron. Astrophys. 5, 84 (1970)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.