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A multiscale view of nonlinear diffusion in biology: from cells to tissues. (English) Zbl 1427.35291

Summary: 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.

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C17 Cell movement (chemotaxis, etc.)
82C40 Kinetic theory of gases in time-dependent statistical mechanics

Software:

Chemotaxis
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Full Text: DOI

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[1] Abel, D. L. and Trevors, J. T., Self-organization versus self-ordering events in life-origin models, Phys. Life Rev.3 (2006) 221-228.
[2] Andreu, F., Caselles, V., Mazón, J. M. and Moll, S., Finite propagation speed for limited flux diffusion equations, Arch. Ration. Mech. Anal.182 (2006) 269-297. · Zbl 1142.35455
[3] Arias, M., Campos, J. and Soler, J., Cross-diffusion and traveling waves in porous-media flux-saturated Keller-Segel models, Math. Models Methods Appl. Sci.28 (2018) 2103-2129. · Zbl 1411.35157
[4] Aristov, V. V., Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows (Springer-Verlag, 2001). · Zbl 0983.82011
[5] Banasiak, J. and Lachowicz, M., Methods of Small Parameter in Mathematical Biology, (Birkhäuser, 2014), pp. 223-270. · Zbl 1309.92012
[6] Barbante, P., Frezzotti, A. and Gibelli, L., A kinetic theory description of liquid menisci at the microscale, Kinet. Relat. Models8 (2015) 235-254. · Zbl 1362.82044
[7] Bellomo, N. and Bellouquid, A., On multiscale models of pedestrian crowds from mesoscopic to macroscopic, Comm. Math. Sci.13 (2015) 1649-1664. · Zbl 1329.90029
[8] Bellomo, N., Bellouquid, A. and Chouhad, N., 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. · Zbl 1353.35038
[9] Bellomo, N., Bellouquid, A., Gibelli, L. and Outada, N., A Quest Towards a Mathematical Theory of Living Systems (Birkhäuser, 2017). · Zbl 1381.92001
[10] Bellomo, N., Bellouquid, A., Nieto, J. and Soler, J., Multiscale biological tissue models and flux-limited chemotaxis from binary mixtures of multicellular growing systems, Math. Models Methods Appl. Sci.20 (2010) 1179-1207. · Zbl 1402.92065
[11] Bellomo, N., Bellouquid, A., Nieto, J. and Soler, J., On the multiscale modeling of vehicular traffic: From kinetic to hydrodynamics, Discrete Contin. Dyn. Syst. Ser. B19 (2014) 1869-1888. · Zbl 1302.35372
[12] Bellomo, N., Bellouquid, A., Tao, Y. and Winkler, M., Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci.25 (2015) 1663-1763. · Zbl 1326.35397
[13] Bellomo, N. and Gibelli, L., Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds, Math. Models Methods Appl. Sci.25 (2015) 2417-2437. · Zbl 1325.91042
[14] Bellomo, N. and Ha, S. Y., A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci.27 (2017) 745-770. · Zbl 1366.82077
[15] Bellomo, N. and Winkler, M., A degenerate chemotaxis system with flux limitation: Maximally extended solutions and absence of gradient blow-up, Comm. Partial Differential Equations42 (2017) 436-473. · Zbl 1430.35166
[16] Bellouquid, A. and Chouhad, N., Kinetic models of chemotaxis towards the diffusive limit: Asymptotic analysis, Math. Models Methods Appl. Sci.39 (2016) 3136-3151. · Zbl 1342.35403
[17] Bellouquid, A. and De Angelis, E., From kinetic models of multicellular growing systems to macroscopic biological tissue models, Nonlinear Anal. Real World. Appl.12 (2011) 1111-1122. · Zbl 1203.92020
[18] Bellouquid, A., De Angelis, E. and Fermo, L., Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach, Math. Models Methods Appl. Sci.22 (2012) 1140003. · Zbl 1243.35157
[19] Bellouquid, A., De Angelis, E. and Knopoff, D., From the modeling of the immune hallmarks of cancer to a black swan in biology, Math. Models Methods Appl. Sci.23 (2013) 949-978. · Zbl 1303.92040
[20] Bellouquid, A. and Delitala, M., Mathematical Modeling of Complex Biological Systems (Birkhäuser, 2006). · Zbl 1178.92002
[21] Bellouquid, A., Nieto, J. and Urrutia, L., About the kinetic description of fractional diffusion equations modeling chemotaxis, Math. Models Methods Appl. Sci.26 (2016) 249-268. · Zbl 1333.35298
[22] Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Oxford Univ. Press, 1994).
[23] Brenier, Y., Extended Monge-Kantorovich Theory, Optimal Transportation and Applications (Springer-Verlag, 2003), pp. 91-121. · Zbl 1064.49036
[24] Burini, D. and Chouhad, N., Hilbert method toward a multiscale analysis from kinetic to macroscopic models for active particles, Math. Models Methods Appl. Sci.27 (2017) 1327-1353. · Zbl 1372.35302
[25] Burini, D., De Lillo, S. and Fioriti, G., On the well posedness of the initial value problem in a kinetic traffic flow model, J. Comput. Theor. Transp.45 (2016) 528-539. · Zbl 07503234
[26] Burini, D., De Lillo, S. and Fioriti, G., Influence of drivers ability in a discrete vehicular traffic model, Int. J. Mod. Phys. C28 (2017) 1750030. · Zbl 07503234
[27] Campos, J. and Soler, J., Qualitative behavior and traveling waves for flux-saturated porous media equations arising in optimal mass transportation, Nonlinear Analysis137 (2016) 266-290. · Zbl 1386.35198
[28] Cercignani, C., Illner, R. and Pulvirenti, M., The Mathematical Theory of Diluted Gas (Springer-Heidelberg, 1993). · Zbl 0813.76001
[29] Chalub, F. A., Dolak-Struss, Y., Markowich, P., Oeltz, D., Schmeiser, C. and Soref, A., Model hierarchies for cell aggregation by chemotaxis, Math. Models Methods Appl. Sci.16 (2006) 1173-1198. · Zbl 1094.92009
[30] Chalub, F. A., Markovich, P., Perthame, B. and Schmeiser, C., Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math.142 (2004) 123-141. · Zbl 1052.92005
[31] Chen, L. and Juengel, A., Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal.36 (2004) 301-322. · Zbl 1082.35075
[32] Chen, L. and Juengel, A., Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations224 (2006) 39-59. · Zbl 1096.35060
[33] Chowdhury, D., Schadschneider, A. and Nishinari, K., Physics of transport and traffic phenomena in biology: From molecular motors and cells to organisms, Phys. Life Rev.2 (2005) 318-352.
[34] Cintra, W., Morales-Rodrigo, C. and Suarez, A., 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. · Zbl 1416.35104
[35] Cooper, E. L., Evolution of immune system from self/not self to danger to artificial immune system, Phys. Life Rev.7 (2010) 55-78.
[36] De Angelis, E., On the mathematical theory of post-Darwinian mutations, selection, and evolution, Math. Models Methods Appl. Sci.24 (2014) 2723-2742. · Zbl 1328.92050
[37] De Lillo, S., Delitala, M. and Salvatori, M., Modeling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles, Math. Models Methods Appl. Sci.19 (2009) 1405-1425. · Zbl 1175.92035
[38] Dimarco, G. and Pareschi, L., Numerical methods for kinetic equations, Acta Numer.23 (2016) 369-520. · Zbl 1398.65260
[39] Dolak, Y. and Schmeiser, C., Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms, J. Math. Biol.51 (2005) 595-615. · Zbl 1077.92003
[40] Escudero, C., The fractional Keller-Segel model, Nonlinearity19 (2006) 2909-2918. · Zbl 1121.35068
[41] Filbet, F., Laurençot, P. and Perthame, B., Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol.50 (2005) 189-207. · Zbl 1080.92014
[42] Gerisch, A. and Painter, K. J., 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. · Zbl 1404.92077
[43] Gorban, A. N. and Karlin, I., Hilbert’s 6th problem: Exact and approximate hydrodynamic manifolds for kinetic equations, Bull. Amer. Math. Soc.51 (2014) 186-246. · Zbl 1294.35052
[44] Hanahan, D. and Weinberg, R. A., The hallmarks of cancer, Cell100 (2000) 57-70.
[45] Haskovec, J. and Schmeiser, C., Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys.135 (2009) 133-151. · Zbl 1173.82021
[46] Haskovec, J. and Schmeiser, C., Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system, Comm. Partial Differential Equations36 (2011) 940-960. · Zbl 1229.35111
[47] Hilbert, D., Mathematical problems, Bull. Amer. Math. Soc.8 (1902) 437-479. · JFM 33.0976.07
[48] Hillen, T. and Othmer, H. G., The diffusion limit of transport equations derived from velocity jump processes, SIAM J. Appl. Math.61 (2000) 751-775. · Zbl 1002.35120
[49] Hillen, T. and Painter, K. J., A users guide to PDE models for chemotaxis, J. Math. Biol.58 (2009) 183-217. · Zbl 1161.92003
[50] Horstmann, D., From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: Jahresber I, Jahresber. Deutsch. Math.-Verein.105 (2003) 103-165. · Zbl 1071.35001
[51] Hu, B. and Tao, Y., 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. · Zbl 1351.35076
[52] Juengel, A., The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity28 (2015) 1963-2001. · Zbl 1326.35175
[53] Juengel, A., Entropy Methods for Diffusive Partial Differential Equations, (Springer, 2016), pp. 69-108.
[54] Keller, E. F. and Segel, L. A., Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol.26 (1970) 399-415. · Zbl 1170.92306
[55] Keller, E. F. and Segel, L. A., Model for chemotaxis, J. Theor. Biol.30 (1971) 225-234. · Zbl 1170.92307
[56] Klafter, J., White, B. S. and Levandowsky, M., Microzooplankton feeding behavior and the Lévy walk, in Biological Motion (Springer, 1990), pp. 281-296.
[57] Klar, A. and Wegener, R., A hierarchy of models for multilane vehicular traffic I: Modeling, SIAM J. Appl. Math.59 (1999) 983-1001. · Zbl 1009.90019
[58] Klar, A. and Wegener, R., Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math.60 (2000) 1749-1766. · Zbl 0953.90007
[59] Lachowicz, M., On the initial layer and the existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl. Sci.9 (1987) 27-70. · Zbl 0632.45005
[60] Lachowicz, M., Micro and meso scales of description corresponding to a model of tissue invasion by solid tumors, Math. Methods Appl. Sci.15 (2005) 1667-1683. · Zbl 1078.92036
[61] Lankeit, J., Long-term behavior in a chemotaxis fluid system with logistic source, Math. Models Methods Appl. Sci.26 (2016) 2071-2109. · Zbl 1354.35059
[62] Liao, J., Global solution for a kinetic chemotaxis model with internal dynamics and its fast adaptation limit, J. Differential Equations259 (2015) 6432-6458. · Zbl 1329.35037
[63] Lin, K., Mu, C. and Zhou, D., Stabilization in a higher-dimensional attraction-repulsion chemotaxis system if repulsion dominates over attraction, Math. Models Methods Appl. Sci.28 (2018) 1105-1134. · Zbl 1391.35200
[64] Marciniak-Czochra, A. and Ptashnyk, M., Boundedness of solutions of a haptotaxis model, Math. Models Methods Appl. Sci.20 (2010) 449-476. · Zbl 1194.35043
[65] Nieto, J. and Urrutia, L., A multiscale model of cell mobility: From a kinetic to a hydrodynamic description, J. Math. Anal. Appl.433 (2016) 1055-1071. · Zbl 1354.92010
[66] Othmer, H. G., Dunbar, S. R. and Alt, W., Models of dispersal in biological systems, J. Math. Biol.26 (1988) 263-298. · Zbl 0713.92018
[67] Othmer, H. G. and Hillen, T., The diffusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math.62 (2002) 1222-1250. · Zbl 1103.35098
[68] Outada, N., Vauchelet, N., Akrid, T. and Khaladi, M., From kinetic theory of multicellular systems to hyperbolic tissue equations: Asymptotic limits and computing, Math. Models Methods Appl. Sci.26 (2016) 2709-2734. · Zbl 1356.35130
[69] Pang, P. Y. H. and Wang, Y., Global boundedness of solutions to a chemotaxis-haptotaxis model with tissue remodeling, Math. Models Methods Appl. Sci.28 (2018) 2211-2235. · Zbl 1416.35052
[70] Pareschi, L. and Toscani, G., Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods (Oxford Univ. Press, 2014). · Zbl 1330.93004
[71] Patlak, C. S., Random walk with persistence and external bias, Bull. Math. Biophys.15 (1953) 311-338. · Zbl 1296.82044
[72] Paulsonn, J., Models of stochastic gene expression, Phys. Life Rev.2 (2005) 157-175.
[73] Fontana, S. Paveri, On Boltzmann like treatments for traffic flow, Transp. Res.9 (1975) 225-235.
[74] Pinnau, R., Totzeck, C., Tse, O. and Martin, S., A consensus-based model for global optimization and its mean-field limit, Math. Models Methods Appl. Sci.27 (2017) 183-204. · Zbl 1388.90098
[75] Prigogine, I. and Herman, R., Kinetic Theory of Vehicular Traffic (Elsevier, 1971). · Zbl 0226.90011
[76] Pulvirenti, M., Il sesto problema di Hilbert e le moderne teorie cinetiche, Boll. Unione Mat. Ital.7 (2004) 545-562.
[77] Saint-Raymond, L., Hydrodynamic Limits of the Boltzmann Equation, , No. 1971 (Springer, 2009). · Zbl 1171.82002
[78] Skellam, J. G., Random dispersal in theoretical populations, Biometrika38 (1951) 196-218. · Zbl 0043.14401
[79] Slemrod, M., From Boltzmann to Euler: Hilbert’s 6th problem revisited, Comp. Math. Appl.63 (2013) 1477-1501. · Zbl 1382.76215
[80] Stinner, C., Surulescu, C. and Uatay, A., Global existence of a go-or-grow multiscale model for tumor invasion with therapy, Math. Models Methods Appl. Sci.26 (2016) 2163-2201. · Zbl 1348.35282
[81] Tannenbaum, E. and Shakhnovich, E. I., Semiconservative replication, genetic repair, and many-gened genomes: Extending the quasi-species paradigm to living systems, Phys. Life Rev.2 (2005) 290-317.
[82] Tao, Y., Global existence for a haptotaxis model of cancer invasion with tissue remodeling, Nonlinear Anal. Real World Appl.12 (2011) 418-435. · Zbl 1205.35144
[83] Tao, Y. and Winkler, M., Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Methods Appl. Sci.27 (2017) 1645-1683. · Zbl 1516.35092
[84] Tello, J. I. and Wrzosek, D., Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci.26 (2016) 2129-2162. · Zbl 1349.92133
[85] Verbeni, M.et al., Morphogenetic action through flux-limited spreading, Phys. Life Rev.10 (2013) 457-475.
[86] Wang, Y., Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci.27 (2017), 2745-2780. · Zbl 1378.92010
[87] Weinberg, R. A., The Biology of Cancer (Garland Sciences — Taylor and Francis, 2007).
[88] Winkler, M., Chemotactic cross-diffusion in complex frameworks, Math. Models Methods Appl. Sci.26 (2016) 2035-2040. · Zbl 1515.35056
[89] Winkler, M., How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?Trans. Amer. Math. Soc.369 (2017) 3067-3125. · Zbl 1356.35071
[90] Wu, S., Wang, J. and Shi, J., Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci.28 (2018) 2275-2312. · Zbl 1411.35171
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