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A multiscale view of nonlinear diffusion in biology: From cells to tissues

D. Burini

Department of Mathematics and Computer Science, Università degli Studi di Perugia, Perugia, Italy

E-mail Address: dilettaburini@alice.it

Corresponding author.

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and

N. Chouhad

Cadi Ayyad University, Ecole Nationale des Sciences Appliquées, Marrakech, Morocco

E-mail Address: chouhadn@gmail.com

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https://doi.org/10.1142/S0218202519400062Cited by:46 (Source: Crossref)

This article is part of the issue:

Special Issue on Active Particle Methods with Focus and Social Dynamics

Abstract

This paper presents a review on the mathematical tools for the derivation of macroscopic models in biology from the underlying description at the scale of cells as it is delivered by a kinetic theory model. The survey is followed by an overview of research perspectives. The derivation is inspired by the Hilbert’s method, known in classic kinetic theory, which is here applied to a broad class of kinetic equations modeling multicellular dynamics. The main difference between this class of equations with respect to the classical kinetic theory consists in the modeling of cell interactions which is developed by theoretical tools of stochastic game theory rather than by laws of classical mechanics. The survey is focused on the study of nonlinear diffusion and source terms.

Communicated by Nicola Bellomo and Franco Brezzi

Keywords:

AMSC: 82−XX, 34C28, 37Fxx

References

1. D. L. Abel and J. T. Trevors, Self-organization versus self-ordering events in life-origin models, Phys. Life Rev. 3 (2006) 221–228. Crossref, Web of Science, Google Scholar
2. F. Andreu, V. Caselles, J. M. Mazón and S. Moll, Finite propagation speed for limited flux diffusion equations, Arch. Ration. Mech. Anal. 182 (2006) 269–297. Crossref, Web of Science, Google Scholar
3. M. Arias, J. Campos and J. Soler, Cross-diffusion and traveling waves in porous-media flux-saturated Keller–Segel models, Math. Models Methods Appl. Sci. 28 (2018) 2103–2129. Link, Web of Science, Google Scholar
4. V. V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows (Springer-Verlag, 2001). Crossref, Google Scholar
5. J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, From Microscopic to Macroscopic Descriptions, Modeling and Simulation in Science, Engineering and Technology (Birkhäuser, 2014), pp. 223–270. Crossref, Google Scholar
6. P. Barbante, A. Frezzotti and L. Gibelli, A kinetic theory description of liquid menisci at the microscale, Kinet. Relat. Models 8 (2015) 235–254. Crossref, Web of Science, Google Scholar
7. N. Bellomo and A. Bellouquid, On multiscale models of pedestrian crowds from mesoscopic to macroscopic, Comm. Math. Sci. 13 (2015) 1649–1664. Crossref, Web of Science, Google Scholar
8. N. Bellomo, A. Bellouquid and N. Chouhad, From a multiscale derivation of nonlinear cross-diffusion models to Keller–Segel models in a Navier–Stokes fluid, Math. Models Methods Appl. Sci. 26 (2016) 2041–2069. Link, Web of Science, Google Scholar
9. N. Bellomo, A. Bellouquid, L. Gibelli and N. Outada, A Quest Towards a Mathematical Theory of Living Systems (Birkhäuser, 2017). Crossref, Google Scholar
10. N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis from binary mixtures of multicellular growing systems, Math. Models Methods Appl. Sci. 20 (2010) 1179–1207. Link, Web of Science, Google Scholar
11. N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Discrete Contin. Dyn. Syst. Ser. B 19 (2014) 1869–1888. Crossref, Web of Science, Google Scholar
12. N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci. 25 (2015) 1663–1763. Link, Web of Science, Google Scholar
13. N. Bellomo and L. Gibelli, Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds, Math. Models Methods Appl. Sci. 25 (2015) 2417–2437. Link, Web of Science, Google Scholar
14. N. Bellomo and S. Y. Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci. 27 (2017) 745–770. Link, Web of Science, Google Scholar
15. N. Bellomo and M. Winkler, A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up, Comm. Partial Differential Equations 42 (2017) 436–473. Crossref, Web of Science, Google Scholar
16. A. Bellouquid and N. Chouhad, Kinetic models of chemotaxis towards the diffusive limit: Asymptotic analysis, Math. Models Methods Appl. Sci. 39 (2016) 3136–3151. Crossref, Web of Science, Google Scholar
17. A. Bellouquid and E. De Angelis, From kinetic models of multicellular growing systems to macroscopic biological tissue models, Nonlinear Anal. Real World. Appl. 12 (2011) 1111–1122. Crossref, Web of Science, Google Scholar
18. A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci. 22 (2012) 1140003. Link, Web of Science, Google Scholar
19. A. Bellouquid, E. De Angelis and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology, Math. Models Methods Appl. Sci. 23 (2013) 949–978. Link, Web of Science, Google Scholar
20. A. Bellouquid and M. Delitala, Mathematical Modeling of Complex Biological Systems (Birkhäuser, 2006). Google Scholar
21. A. Bellouquid, J. Nieto and L. Urrutia, About the kinetic description of fractional diffusion equations modeling chemotaxis, Math. Models Methods Appl. Sci. 26 (2016) 249–268. Link, Web of Science, Google Scholar
22. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Oxford Univ. Press, 1994). Crossref, Google Scholar
23. Y. Brenier, Extended Monge–Kantorovich Theory, Optimal Transportation and Applications (Springer-Verlag, 2003), pp. 91–121. Google Scholar
24. D. Burini and N. Chouhad, Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles, Math. Models Methods Appl. Sci. 27 (2017) 1327–1353. Link, Web of Science, Google Scholar
25. D. Burini, S. De Lillo and G. Fioriti, On the well posedness of the initial value problem in a kinetic traffic flow model, J. Comput. Theor. Transp. 45 (2016) 528–539. Crossref, Web of Science, Google Scholar
26. D. Burini, S. De Lillo and G. Fioriti, Influence of drivers ability in a discrete vehicular traffic model, Int. J. Mod. Phys. C 28 (2017) 1750030. Link, Web of Science, Google Scholar
27. J. Campos and J. Soler, Qualitative behavior and traveling waves for flux-saturated porous media equations arising in optimal mass transportation, Nonlinear Analysis 137 (2016) 266–290. Crossref, Web of Science, Google Scholar
28. C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Diluted Gas (Springer-Heidelberg, 1993). Google Scholar
29. F. A. Chalub, Y. Dolak-Struss, P. Markowich, D. Oeltz, C. Schmeiser and A. Soref, Model hierarchies for cell aggregation by chemotaxis, Math. Models Methods Appl. Sci. 16 (2006) 1173–1198. Link, Web of Science, Google Scholar
30. F. A. Chalub, P. Markovich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift–diffusion limits, Monatsh. Math. 142 (2004) 123–141. Crossref, Web of Science, Google Scholar
31. L. Chen and A. Juengel, Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal. 36 (2004) 301–322. Crossref, Web of Science, Google Scholar
32. L. Chen and A. Juengel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations 224 (2006) 39–59. Crossref, Web of Science, Google Scholar
33. D. Chowdhury, A. Schadschneider and K. Nishinari, Physics of transport and traffic phenomena in biology: From molecular motors and cells to organisms, Phys. Life Rev. 2 (2005) 318–352. Crossref, Web of Science, Google Scholar
34. W. Cintra, C. Morales-Rodrigo and A. Suarez, Coexistence states in a cross-diffusion system of a predator–prey model with predator satiation term, Math. Models Methods Appl. Sci. 28 (2018) 2131–2159. Link, Web of Science, Google Scholar
35. E. L. Cooper, Evolution of immune system from self/not self to danger to artificial immune system, Phys. Life Rev. 7 (2010) 55–78. Crossref, Web of Science, Google Scholar
36. E. De Angelis, On the mathematical theory of post-Darwinian mutations, selection, and evolution, Math. Models Methods Appl. Sci. 24 (2014) 2723–2742. Link, Web of Science, Google Scholar
37. S. De Lillo, M. Delitala and M. Salvatori, Modeling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles, Math. Models Methods Appl. Sci. 19 (2009) 1405–1425. Link, Web of Science, Google Scholar
38. G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer. 23 (2016) 369–520. Crossref, Web of Science, Google Scholar
39. Y. Dolak and C. Schmeiser, Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol. 51 (2005) 595–615. Crossref, Web of Science, Google Scholar
40. C. Escudero, The fractional Keller–Segel model, Nonlinearity 19 (2006) 2909–2918. Crossref, Web of Science, Google Scholar
41. F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol. 50 (2005) 189–207. Crossref, Web of Science, Google Scholar
42. A. Gerisch and K. J. Painter, Mathematical modeling of cell adhesion and its applications to developmental biology and cancer invasion, in Cell Mechanics: From Single Scale-Based Models to Multiscale Modeling, Chap. 12 (2010), pp. 319–350. Crossref, Google Scholar
43. A. N. Gorban and I. Karlin, Hilbert’s 6th problem: Exact and approximate hydrodynamic manifolds for kinetic equations, Bull. Amer. Math. Soc. 51 (2014) 186–246. Web of Science, Google Scholar
44. D. Hanahan and R. A. Weinberg, The hallmarks of cancer, Cell 100 (2000) 57–70. Crossref, Web of Science, Google Scholar
45. J. Haskovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller–Segel system, J. Stat. Phys. 135 (2009) 133–151. Crossref, Web of Science, Google Scholar
46. J. Haskovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller–Segel system, Comm. Partial Differential Equations 36 (2011) 940–960. Crossref, Web of Science, Google Scholar
47. D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902) 437–479. Crossref, Google Scholar
48. T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump processes, SIAM J. Appl. Math. 61 (2000) 751–775. Crossref, Web of Science, Google Scholar
49. T. Hillen and K. J. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol. 58 (2009) 183–217. Crossref, Web of Science, Google Scholar
50. D. Horstmann, From 1970 until present: The Keller–Segel model in chemotaxis and its consequences: Jahresber I, Jahresber. Deutsch. Math.-Verein. 105 (2003) 103–165. Google Scholar
51. B. Hu and Y. Tao, To the exclusion of blow-up in a three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci. 26 (2016) 2111–2128. Link, Web of Science, Google Scholar
52. A. Juengel, The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity 28 (2015) 1963–2001. Crossref, Web of Science, Google Scholar
53. A. Juengel, Entropy Methods for Diffusive Partial Differential Equations, Cross-Diffusion Systems, Springer Briefs in Mathematics (Springer, 2016), pp. 69–108. Crossref, Google Scholar
54. E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970) 399–415. Crossref, Web of Science, Google Scholar
55. E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol. 30 (1971) 225–234. Crossref, Web of Science, Google Scholar
56. J. Klafter, B. S. White and M. Levandowsky, Microzooplankton feeding behavior and the Lévy walk, in Biological Motion (Springer, 1990), pp. 281–296. Crossref, Google Scholar
57. A. Klar and R. Wegener, A hierarchy of models for multilane vehicular traffic I: Modeling, SIAM J. Appl. Math. 59 (1999) 983–1001. Crossref, Web of Science, Google Scholar
58. A. Klar and R. Wegener, Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math. 60 (2000) 1749–1766. Crossref, Web of Science, Google Scholar
59. M. Lachowicz, On the initial layer and the existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl. Sci. 9 (1987) 27–70. Crossref, Web of Science, Google Scholar
60. M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumors, Math. Methods Appl. Sci. 15 (2005) 1667–1683. Link, Web of Science, Google Scholar
61. J. Lankeit, Long-term behavior in a chemotaxis fluid system with logistic source, Math. Models Methods Appl. Sci. 26 (2016) 2071–2109. Link, Web of Science, Google Scholar
62. J. Liao, Global solution for a kinetic chemotaxis model with internal dynamics and its fast adaptation limit, J. Differential Equations 259 (2015) 6432–6458. Crossref, Web of Science, Google Scholar
63. K. Lin, C. Mu and D. Zhou, Stabilization in a higher-dimensional attraction–repulsion chemotaxis system if repulsion dominates over attraction, Math. Models Methods Appl. Sci. 28 (2018) 1105–1134. Link, Web of Science, Google Scholar
64. A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci. 20 (2010) 449–476. Link, Web of Science, Google Scholar
65. J. Nieto and L. Urrutia, A multiscale model of cell mobility: From a kinetic to a hydrodynamic description, J. Math. Anal. Appl. 433 (2016) 1055–1071. Crossref, Web of Science, Google Scholar
66. H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol. 26 (1988) 263–298. Crossref, Web of Science, Google Scholar
67. H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math. 62 (2002) 1222–1250. Crossref, Web of Science, Google Scholar
68. N. Outada, N. Vauchelet, T. Akrid and M. Khaladi, From kinetic theory of multicellular systems to hyperbolic tissue equations: Asymptotic limits and computing, Math. Models Methods Appl. Sci. 26 (2016) 2709–2734. Link, Web of Science, Google Scholar
69. P. Y. H. Pang and Y. Wang, Global boundedness of solutions to a chemotaxis–haptotaxis model with tissue remodeling, Math. Models Methods Appl. Sci. 28 (2018) 2211–2235. Link, Web of Science, Google Scholar
70. L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods (Oxford Univ. Press, 2014). Google Scholar
71. C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953) 311–338. Crossref, Google Scholar
72. J. Paulsonn, Models of stochastic gene expression, Phys. Life Rev. 2 (2005) 157–175. Crossref, Web of Science, Google Scholar
73. S. Paveri Fontana, On Boltzmann like treatments for traffic flow, Transp. Res. 9 (1975) 225–235. Google Scholar
74. R. Pinnau, C. Totzeck, O. Tse and S. Martin, A consensus-based model for global optimization and its mean-field limit, Math. Models Methods Appl. Sci. 27 (2017) 183–204. Link, Web of Science, Google Scholar
75. I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic (Elsevier, 1971). Google Scholar
76. M. Pulvirenti, Il sesto problema di Hilbert e le moderne teorie cinetiche, Boll. Unione Mat. Ital. 7 (2004) 545–562. Google Scholar
77. L. Saint-Raymond, Hydrodynamic Limits of the Boltzmann Equation, Lecture Notes in Mathematics, No. 1971 (Springer, 2009). Crossref, Google Scholar
78. J. G. Skellam, Random dispersal in theoretical populations, Biometrika 38 (1951) 196–218. Crossref, Web of Science, Google Scholar
79. M. Slemrod, From Boltzmann to Euler: Hilbert’s 6th problem revisited, Comp. Math. Appl. 63 (2013) 1477–1501. Google Scholar
80. C. Stinner, C. Surulescu and A. Uatay, Global existence of a go-or-grow multiscale model for tumor invasion with therapy, Math. Models Methods Appl. Sci. 26 (2016) 2163–2201. Link, Web of Science, Google Scholar
81. E. Tannenbaum and E. I. Shakhnovich, Semiconservative replication, genetic repair, and many-gened genomes: Extending the quasi-species paradigm to living systems, Phys. Life Rev. 2 (2005) 290–317. Crossref, Web of Science, Google Scholar
82. Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal. Real World Appl. 12 (2011) 418–435. Crossref, Web of Science, Google Scholar
83. Y. Tao and M. Winkler, Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system, Math. Models Methods Appl. Sci. 27 (2017) 1645–1683. Link, Web of Science, Google Scholar
84. J. I. Tello and D. Wrzosek, Predator–prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci. 26 (2016) 2129–2162. Link, Web of Science, Google Scholar
85. M. Verbeni et al., Morphogenetic action through flux-limited spreading, Phys. Life Rev. 10 (2013) 457–475. Crossref, Web of Science, Google Scholar
86. Y. Wang, Global weak solutions in a three-dimensional Keller–Segel–Navier–Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci. 27 (2017), 2745–2780. Link, Web of Science, Google Scholar
87. R. A. Weinberg, The Biology of Cancer (Garland Sciences — Taylor and Francis, 2007). Google Scholar
88. M. Winkler, Chemotactic cross-diffusion in complex frameworks, Math. Models Methods Appl. Sci. 26 (2016) 2035–2040. Link, Web of Science, Google Scholar
89. M. Winkler, How far do chemotaxis-driven forces influence regularity in the Navier–Stokes system? Trans. Amer. Math. Soc. 369 (2017) 3067–3125. Crossref, Web of Science, Google Scholar
90. S. Wu, J. Wang and J. Shi, Dynamics and pattern formation of a diffusive predator–prey model with predator-taxis, Math. Models Methods Appl. Sci. 28 (2018) 2275–2312. Link, Web of Science, Google Scholar