Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-12T15:14:04.232Z Has data issue: false hasContentIssue false
  • English
  • Français

A hyperbolic model of chemotaxison a network: a numerical study

Published online by Cambridge University Press:  10 January 2014

G. Bretti,
R. Natalini and
M. Ribot
G. Bretti
Affiliation:
Istituto per le Applicazioni del Calcolo “M. Picone” – Consiglio Nazionale delle Ricerche, Via dei Taurini 19, Rome, Italy. g.bretti@iac.cnr.it; roberto.natalini@cnr.it
R. Natalini
Affiliation:
Istituto per le Applicazioni del Calcolo “M. Picone” – Consiglio Nazionale delle Ricerche, Via dei Taurini 19, Rome, Italy. g.bretti@iac.cnr.it; roberto.natalini@cnr.it
M. Ribot
Affiliation:
Laboratoire J.A.Dieudonné, UMR 7351 CNRS, Université Nice Sophia Antipolis, Nice, France; ribot@unice.frProject Team COFFEE, INRIA Sophia-Antipolis, France

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.

Keywords

Hyperbolic system on networkinitial-boundary value problemtransmission conditionsasymptotic behaviorfinite difference schemeschemotaxis

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 1 , January 2014, pp. 231 - 258
Copyright
© EDP Sciences, SMAI, 2014

References

Aregba-Driollet, D., Briani, M. and Natalini, R., Asymptotic high-order schemes for 2 × 2 dissipative hyperbolic systems. SIAM J. Numer. Anal. 46 (2008) 869–894. Google Scholar
Barocas, V. and Tranquillo, R., An anisotropic biphasic theory of tissue-equivalent mechanics: The interplay among cell traction, fibrillar network deformation, fibril alignment, and cell contact guidance. J. Biomech. Eng. 119 (1997) 137–145. Google Scholar
G. Bastin, J-M. Coron, B. d’Andréa-Novel, P. Suvarov, A. Vande Wouwer and A. Kienle, Stability of switched hyperbolic systems: the example of SMB chromatography” submitted IEEE-CDC (2013).
Bretti, G., Natalini, R. and Ribot, M., A numerical scheme for a hyperbolic relaxation model on networks. Numerical Analysis and Applied Mathematics ICNAAM 2011, AIP Conf. Proc. 1389 (2011) 1412–1415. Google Scholar
A. Chauviere, L. Preziosi, Mathematical framework to model migration of cell population in extracellular matrix, in Cell mechanics. From single scale-based models to multiscale modeling. A. Chauviere, L. Preziosi and C. Verdier Eds., Taylor & Francis Group, CRC Press publisher, (2010) 285–318.
R. Dáger and E. Zuazua, Wave propagation, observation and control in 1 - d flexible multi-structures, vol. 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin (2006).
Dolak, Y. and Hillen, T., Cattaneo models for chemosensitive movement. Numerical solution and pattern formation. J. Math. Biol. 46 (2003) 153–170; corrected version after misprinted p. 160 in J. Math. Biol. 46 (2003) 461–478. Google ScholarPubMed
Filbet, F., Laurençot, P. and Perthame, B., Derivation of hyperbolic models for chemosensitive movement. J. Math. Biol. 50 (2005) 189–207. Google ScholarPubMed
Gamba, A., Ambrosi, D., Coniglio, A., de Candia, A., Di Talia, S., Giraudo, E., Serini, G., Preziosi, L. and Bussolino, F., Percolation, morphogenesis, and Burgers dynamics in blood vessels formation. Phys. Rev. Lett. 90 (2003) 118101.1–118101.4. Google ScholarPubMed
M. Garavello and B. Piccoli, Traffic flow on networks. Conservation laws models, vol. 1 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006).
Gosse, L., Asymptotic-preserving and well-balanced schemes for the 1D Cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes. J. Math. Anal. Appl. 388 (2012) 964–983. Google Scholar
Gosse, L., Well-balanced numerical approximations display asymptotic decay toward Maxwellian distributions for a model of chemotaxis in a bounded interval. SIAM J. Sci. Comput. 34 (2012) A520–A545. Google Scholar
Greenberg, J.M. and Alt, W., Stability results for a diffusion equation with functional drift approximating a chemotaxis model. Trans. Amer. Math. Soc. 300 (1987) 235–258. Google Scholar
I. Guaraldo, Some analytical results for hyperbolic chemotaxis model on networks Ph.D. thesis, Università di Roma “La Sapienza” (2012).
Guarguaglini, F. and Natalini, R., Nonlinear transmission problems for quasilinear diffusion problems. Networks and Heterogeneous media 2 (2007) 359–381. Google Scholar
Guarguaglini, F., Natalini, R., Mascia, C. and Ribot, M., Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis. Discrete Contin. Dyn. Syst. Ser. B 12 (2009) 39–76. Google Scholar
Häcker, A., A mathematical model for mesenchymal and chemosensitive cell dynamics. J. Math. Biol. 64 (2012) 361–401. Google Scholar
Haut, B. and Bastin, G., A second order model of road junctions in fluid models of traffic networks. Netw. Heterog. Media 2 (2007) 227–253. Google Scholar
Hillen, T., Hyperbolic models for chemosensitive movement. Special issue on kinetic theory. Math. Mod. Methods Appl. Sci. 12 (2002) 1007–1034. Google Scholar
Hillen, T., Rohde, C. and Lutscher, F., Existence of weak solutions for a hyperbolic model of chemosensitive movement. J. Math. Anal. Appl. 26 (2001) 173–199. Google Scholar
Hillen, T. and Stevens, A., Hyperbolic models for chemotaxis in 1-D. Nonlinear Anal. Real World Appl. 1 (2000) 409–433. Google Scholar
Kedem, O. and Katchalsky, A., Thermodynamic analysis of the permeability of biological membrane to non-electrolytes. Biochimica et Biophysica Acta 27 (1958) 229–246. Google ScholarPubMed
Keller, E.F. and Segel, L.A., Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970) 399–415. Google ScholarPubMed
Harley, B.A.C., Kim, H., Zaman, M.H., Yannas, I.V., Lauffenburger, D.A. and Gibson, L.J., Microarchitecture of Three-Dimensional Scaffolds Inuences Cell Migration Behavior via Junction Interactions. Biophys. J. 29 (2008) 4013–4024. Google Scholar
Horstmann, D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein. 105 (2003) 103–165. Google Scholar
Mandal, B.B. and Kundu, S.C., Cell proliferation and migration in silk broin 3D scaffolds. Biomaterials 30 (2009) 2956–2965. Google Scholar
J.D. Murray, Mathematical biology. I. An introduction, 3rd edn., vol. 17 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York (2002); II. Spatial models and biomedical applications, 3rd edn., vol. 18 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York (2003).
Natalini, R., Convergence to equilibrium for the relaxation approximation of conservation laws. Commun. Pure Appl. Math. 49 (1996) 795–823. Google Scholar
Natalini, R. and Ribot, M., An asymptotic high order mass-preserving scheme for a hyperbolic model of chemotaxis. SIAM J. Num. Anal. 50 (2012) 883–905. Google Scholar
B. Perthame, Transport equations in biology, Frontiers in Mathematics. Birkhäuser (2007).
Preziosi, L. and Tosin, A., Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J. Math. Biol. 58 (2009) 625–656. Google Scholar
Segel, L.A., A theoretical study of receptor mechanisms in bacterial chemotaxis. SIAM J. Appl. Math. 32 (1977) 653–665. Google Scholar
C. Spadaccio, A. Rainer, S. De Porcellinis, M. Centola, F. De Marco, M. Chello, M. Trombetta, J.A. Genovese, A G-CSF functionalized PLLA scaffold for wound repair: an in vitro preliminary study. Conf. Proc. IEEE Eng. Med. Biol. Soc. (2010).
Valein, J. and Zuazua, E., Stabilization of the wave equation on 1-D networks. SIAM J. Control Optim. 48 (2009) 2771–2797. Google Scholar