Kinetic and Moment Models for Cell Motion in Fiber Structures

  • Raul Borsche
  • Axel KlarEmail author
  • Florian Schneider
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


This review focuses on kinetic and macroscopic models for the migration of cells in fiber structures. Typical applications of cell migration models in such geometries are tumor cell invasion into tissue, or tissue-engineering and the movement of fibroblasts on artificial scaffolds during wound healing.



The second author is supported by DFG grant 1105/27, by BMBF grant 05M16UKB, GlioMaTh and by the DAAD PhD program MIC.


  1. 1.
    Anile, A.M., Pennisi, S., Sammartino, M.: A thermodynamical approach to Eddington factors. Journal of Mathematical Physics 32(2), 544 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    B. A. C. Harley H. Kim, M.H.Z.I.V.Y.D.A.L., Gibson, L.J.: Microarchitecture of three-dimensional scaffolds influences cell migration behavior via junction interactions. Biophysical Journal 29, 4013–4024 (2008)CrossRefGoogle Scholar
  3. 3.
    Bellomo, N., Bellouquid, A., Nieto, J., Soler, J.: Complexity and mathematical tools toward the modeling of multicellular growing systems. Mathematical and Computer Modeling 51, 441–451 (2010)zbMATHCrossRefGoogle Scholar
  4. 4.
    Bellomo, N., Bellouquid, A., Nieto, J., Soler, J.: Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems. Mathematical Models and Methods in Applied Sciences 20(7), 1179–1207 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bellomo, N., Bellouquid, A., Nieto, J., Soler, J.: On the asymptotic theory from microscopic to macroscopic growing tissue models: an overview with perspectives. Mathematical Models and Methods in Applied Sciences 22(1), 1130001 (27 pages), (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25(9), 1663–1763 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Borsche, R., Göttlich, S., Klar, A., Schillen, P.: The scalar Keller-Segel model on networks. Math. Models Methods Appl. Sci. 24(2), 221–247 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Borsche, R., Kall, J., Klar, A., Pham, T.: Kinetic and related macroscopic models for chemotaxis on networks. Mathematical Models and Methods in Applied Sciences 26(06), 1219–1242 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Borsche, R., Klar, A.: Kinetic layers and coupling conditions for macroscopic equations on networks. SIAM Sci. Computing 40 (2018)Google Scholar
  10. 10.
    Borsche, R., Klar, A.: Kinetic layers and coupling conditions for nonlinear scalar equations on networks. Nonlinearity 31, 3512–3541 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Borsche, R., Klar, A., Pham, T.H.: Nonlinear flux-limited models for chemotaxis on networks. Networks & Heterogeneous Media 12(3), 381–401 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bournaveas, N., Calvez, V.: The one-dimensional Keller-Segel model with fractional diffusion of cells. Nonlinearity 23(4), 923–935 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bretti, G., Natalini, R., Ribot, M.: A hyperbolic model of chemotaxis on a network: a numerical study. ESAIM: M2AN 48(1), 231–258 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Brunner, T.A.: Forms of approximate radiation transport. SAND2002-1778, Sandia National Laboratory (July) (2002)Google Scholar
  15. 15.
    Brunner, T.A., Holloway, J.: One-dimensional Riemann solvers and the maximum entropy closure. Journal of Quantitative Spectroscopy and Radiative Transfer 69(5), 543–566 (2001)CrossRefGoogle Scholar
  16. 16.
    Burger, M., Di Francesco, M., Dolak-Struss, Y.: The Keller-Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion. SIAM J. Math. Anal. 38(4), 1288–1315 (2006). MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Camilli, F., Corrias, L.: Parabolic models for chemotaxis on weighted networks. J. Math. Pures Appl. 108, 459–480 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Chalub, F., Markowich, P., Perthame, B., Schmeiser, C.: Kinetic models for chemotaxis and their drift-diffusion limits. Monatsh. Math. 142, 123–141 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Chavanis, P.: Jeans type instability for a chemotactic model of cellular aggregation. Eur. Phys. J. B 52, 433–443 (2006)CrossRefGoogle Scholar
  20. 20.
    Chertock, A., Kurganov, A., Wang, X., Wu, Y.: On a chemotaxis model with saturated chemotactic flux. Kinet. Relat. Models 5(1), 51–95 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Childress, S., Percus, J.: Nonlinear aspects of chemotaxis. Math. Biosci. 56, 217–237 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Colombo, R.M., Garavello, M.: On the Cauchy problem for the p-system at a junction. SIAM J. Math. Anal. 39(5), 1456–1471 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Colombo, R.M., Guerra, G.: On general balance laws with boundary. J. Differential Equations 248(5), 1017–1043 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Coons, S.: Anatomy and growth patterns of diffuse gliomas. In: M. Berger, C. Wilson (eds.) The gliomas, pp. 210–225. W.B. Saunders Company, Philadelphia (1999)Google Scholar
  25. 25.
    Corbin, G., Hunt, A., Schneider, F., Klar, A., Surulescu, C.: Higher-order models for glioma invasion: from a two-scale description to effective equations for mass density and momentum. M3AS 28, 1771–1800 (2018)Google Scholar
  26. 26.
    Coulombel, J., Golse, F., Goudon, T.: Diffusion approximation and entropy-based moment closure for kinetic equations. Asymptotic Analysis 45(1), 1–34 (2005)MathSciNetzbMATHGoogle Scholar
  27. 27.
    D’Abaco, G., Kaye, A.: Integrins: Molecular determinants of glioma invasion. Journal of Clinical Neuroscience 14, 1041–1048 (2007)CrossRefGoogle Scholar
  28. 28.
    Daumas-Duport, C., Varlet, P., Tucker, M., Beuvon, F., Cervera, P., Chodkiewicz, J.: Oligodendrogliomas. part i: Patterns of growth, histological diagnosis, clinical and imaging correlations: A study of 153 cases. Journal of Neuro-Oncology 34, 37–59 (1997)CrossRefGoogle Scholar
  29. 29.
    Dubroca, B., Klar, A.: Half-moment closure for radiative transfer equations. Journal of Computational Physics 180, 584–596 (2002)zbMATHCrossRefGoogle Scholar
  30. 30.
    Engwer, C., Hillen, T., Knappitsch, M., Surulescu, C.: Glioma follow white matter tracts: a multiscale DTI-based model. Journal of Mathematical Biology 71, 551–582 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Engwer, C., Hunt, A., Surulescu, C.: Effective equations for anisotropic glioma spread with proliferation: a multiscale approach. IMA Journal of Mathematical Medicine and Biology 33, 435–459 (2016)zbMATHCrossRefGoogle Scholar
  32. 32.
    Engwer, C., Knappitsch, M., Surulescu, C.: A multiscale model for glioma spread including cell-tissue interactions and proliferation. Journal of Engineering Mathematics 13, 443–460 (2016)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Fermo, L., Tosin, A.: A fully-discrete-state kinetic theory approach to traffic flow on road networks. Math. Models Methods Appl. Sci. 25(3), 423–461 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Filbet, F., Laurençot, P., Perthame, B.: Derivation of hyperbolic models for chemosensitive movement. J Math Biol. 50(2), 189–207 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Frank, M., Dubroca, B., Klar, A.: Partial moment entropy approximation to radiative heat transfer. Journal of Computational Physics 218(1), 1–18 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Frank, M., Hensel, H., Klar, A.: A fast and accurate moment method for the Fokker-Planck equation and applications to electron radiotherapy. SIAM Journal on Applied Mathematics 67(2), 582–603 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Garrett, C.K., Hauck, C.: A comparison of moment closures for linear kinetic transport equations: the line source benchmark. Transport Theory and Statistical Physics 42, 203–235 (2013)zbMATHCrossRefGoogle Scholar
  38. 38.
    Gerstner, E., Chen, P.J., Wen, P., Jain, R., Batchelor, T., Sorensen, G.: Infiltrative patterns of glioblastoma spread detected via diffusion MRI after treatment with cediranib. Neuro-Oncology 12(5), 466–472 (2010)Google Scholar
  39. 39.
    Giese, A., Kluwe, L., H., M., E., M., Westphal, M.: Migration of human glioma cells on myelin. Neurosurgery 38, 755–764 (1996)CrossRefGoogle Scholar
  40. 40.
    Giese, A., Westphal, M.: Glioma invasion in the central nervous system. Neurosurgery 39, 235–252 (1996)CrossRefGoogle Scholar
  41. 41.
    Gimbutas, Z., Greengard, L.: A fast and stable method for rotating spherical harmonic expansions. Journal of Computational Physics 228(16), 5621–5627 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Guarguaglini, F.R., Natalini, R.: Global smooth solutions for a hyperbolic chemotaxis model on a network. SIAM J. Math. Anal. 47(6), 4652–4671 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Hauck, C.D.: High-order entropy-based closures for linear transport in slab geometry. Communications in Mathematical Sciences 9(1), 187–205 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Herty, M., Moutari, S.: A macro-kinetic hybrid model for traffic flow on road networks. Comput. Methods Appl. Math. 9(3), 238–252 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Hillen, T.: Hyperbolic models for chemosensitive movement. Mathematical Models and Methods in Applied Sciences 12(07), 1007–1034 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Hillen, T., Othmer, H.G.: The diffusion limit of transport equations derived from velocity jump processes. Siam Journal on Applied Mathematics 61, 751–775 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Hillen, T., Painter, K.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58(1-2), 183–217 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26(3), 399–415 (1970). MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Keller, E.F., Segel, L.A.: Model for chemotaxis. Journal of Theoretical Biology 30, 225–234 (1971)zbMATHCrossRefGoogle Scholar
  50. 50.
    Kershaw, D.S.: Flux Limiting Nature’s Own Way: A New Method for Numerical Solution of the Transport Equation. Tech. rep., LLNL Report UCRL-78378 (1976)Google Scholar
  51. 51.
    Klar, A., Schneider, F., Tse, O.: Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker–Planck equations. Kinetic and Related Models 7(3), 509–529 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Le Bihan, D., Mangin, J.F., Poupon, C., Clark, C., Pappata, S., Molko, N., Chabriat, H.: Diffusion tensor imaging: concepts and applications. Journal of magnetic resonance imaging 13(4), 534–546 (2001)CrossRefGoogle Scholar
  53. 53.
    Levermore, C.D.: Relating Eddington factors to flux limiters. Journal of Quantitative Spectroscopy and Radiative Transfer 31(2), 149–160 (1984)CrossRefGoogle Scholar
  54. 54.
    Levermore, C.D.: Moment closure hierarchies for kinetic theories. Journal of Statistical Physics 83, 1021–1065 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Mandal, B.B., Kundu., S.: Cell proliferation and migration in silk fibroin 3D scaffolds. Biomaterials 30, 2956–2965 (2009)CrossRefGoogle Scholar
  56. 56.
    Mark, J.C.: The spherical harmonics method, Part {I}. Tech. Rep. MT 92, National Research Council of Canada (1944)Google Scholar
  57. 57.
    Olbrant, E., Hauck, C.D., Frank, M.: A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer. Journal of Computational Physics 231(17), 5612–5639 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Painter, K., Hillen, T.: Mathematical modelling of glioma growth: the use of diffusion tensor imaging (DTI) data to predict the anisotropic pathways of cancer invasion. Journal of Theoretical Biology 323, 25–39 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Pomraning, G.C.: The equations of radiation hydrodynamics. Pergamon Press (1973)Google Scholar
  60. 60.
    Ritter, J., Klar, A., Schneider, F.: Partial-moment minimum-entropy models for kinetic chemotaxis equations in one and two dimensions. J. Comp. Applied Math. 306, 300–315 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Schneider, F., Alldredge, G., Frank, M., Klar, A.: Higher Order Mixed-Moment Approximations for the Fokker–Planck Equation in One Space Dimension. SIAM Journal on Applied Mathematics 74(4), 1087–1114 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Schneider, F., Kall, J., Alldredge, G.: A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry. Kinetic and Related Models 9(1), 193–215 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Schneider, F., Kall, J., Roth, A.: First-order quarter- and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic and Related Models 10 (4), 1127–1161 (2017)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.TU KaiserslauternKaiserslauternGermany

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