ON THE ASYMPTOTIC THEORY FROM MICROSCOPIC TO MACROSCOPIC GROWING TISSUE MODELS: AN OVERVIEW WITH PERSPECTIVES

This paper proposes a review and critical analysis of the asymptotic limit methods focused on the derivation of macroscopic equations for a class of equations modeling complex multicellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of biological functions and proliferative/destructive events. The asymptotic analysis deals with suitable parabolic, hyperbolic, and mixed limits. The review includes the derivation of the classical Keller–Segel model and flux limited models that prevent non-physical blow up of solutions.

[1]  R. Weinberg,et al.  The Biology of Cancer , 2006 .

[2]  L. Segel,et al.  Traveling bands of chemotactic bacteria: a theoretical analysis. , 1971, Journal of theoretical biology.

[3]  Evelyn Fox Keller.  ASSESSING THE KELLER-SEGEL MODEL: HOW HAS IT FARED? , 1980 .

[4]  H. Othmer,et al.  Models of dispersal in biological systems , 1988, Journal of mathematical biology.

[5]  Spatial and spatio-temporal patterns in a cell-haptotaxis model , 1989, Journal of mathematical biology.

[6]  Rosenau.  Free-energy functionals at the high-gradient limit. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[7]  Rosenau.  Tempered diffusion: A transport process with propagating fronts and inertial delay. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[8]  R. Clark.  Biology of dermal wound repair. , 1993, Dermatologic clinics.

[9]  Philip K. Maini,et al.  A mathematical model for fibro-proliferative wound healing disorders , 1996 .

[10]  Juan Soler,et al.  On the Vlasov–Poisson–Fokker–Planck Equations with Measures in Morrey Spaces as Initial Data☆ , 1997 .

[11]  F. Ritort,et al.  Exactly Solvable Phase Oscillator Models with Synchronization Dynamics , 1998, cond-mat/9803055.

[12]  Philip K. Maini,et al.  Simple modelling of extracellular matrix alignment in dermal wound healing I. cell flux induced alignment , 1998 .

[13]  J. Sherratt,et al.  Extracellular matrix-mediated chemotaxis can impede cell migration , 1998, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[14]  J A Sherratt,et al.  Mathematical modelling of anisotropy in fibrous connective tissue. , 1999, Mathematical biosciences.

[15]  J. Hopfield,et al.  From molecular to modular cell biology , 1999, Nature.

[16]  Juan Soler,et al.  PARABOLIC LIMIT AND STABILITY OF THE VLASOV–FOKKER–PLANCK SYSTEM , 2000 .

[17]  L. Bonilla,et al.  High-field limit of the Vlasov-Poisson-Fokker-Planck system: A comparison of different perturbation methods , 2000, cond-mat/0007164.

[18]  D. Hanahan,et al.  The Hallmarks of Cancer , 2000, Cell.

[19]  Hans G. Othmer,et al.  The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes , 2000, SIAM J. Appl. Math..

[20]  J. Folkman,et al.  Clinical translation of angiogenesis inhibitors , 2002, Nature Reviews Cancer.

[21]  K. Painter,et al.  Volume-filling and quorum-sensing in models for chemosensitive movement , 2002 .

[22]  J. Folkman.  Role of angiogenesis in tumor growth and metastasis. , 2002, Seminars in oncology.

[23]  Hans G. Othmer,et al.  The Diffusion Limit of Transport Equations II: Chemotaxis Equations , 2002, SIAM J. Appl. Math..

[24]  N. Bellomo,et al.  From a class of kinetic models to the macroscopic equations for multicellular systems in biology , 2003 .

[25]  Yann Brenier,et al.  Extended Monge-Kantorovich Theory , 2003 .

[26]  C. Villani,et al.  Optimal Transportation and Applications , 2003 .

[27]  B. Perthame,et al.  Kinetic Models for Chemotaxis and their Drift-Diffusion Limits , 2004 .

[28]  B. Perthame.  Mathematical tools for kinetic equations , 2004 .

[29]  Juan Soler,et al.  Low-Field Limit for a Nonlinear Discrete Drift-Diffusion Model Arising in Semiconductor Superlattices Theory , 2004, SIAM J. Appl. Math..

[30]  A. Bellouquid,et al.  Kinetic (cellular) models of cell progression and competition with the immune system , 2004 .

[31]  V. Caselles,et al.  The Cauchy problem for a strongly degenerate quasilinear equation , 2005 .

[32]  Juan Soler,et al.  Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system. , 2005 .

[33]  A. Bellouquid,et al.  Mathematical methods and tools of kinetic theory towards modelling complex biological systems , 2005 .

[34]  B. Perthame,et al.  Derivation of hyperbolic models for chemosensitive movement , 2005, Journal of mathematical biology.

[35]  V. Caselles,et al.  A Strongly Degenerate Quasilinear Equation: the Parabolic Case , 2005 .

[36]  M. Lachowicz.  MICRO AND MESO SCALES OF DESCRIPTION CORRESPONDING TO A MODEL OF TISSUE INVASION BY SOLID TUMOURS , 2005 .

[37]  Yan Guo,et al.  Pattern formation (I): The Keller–Segel model , 2005 .

[38]  C. Schmeiser,et al.  Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms , 2005, Journal of mathematical biology.

[39]  F. Andreu,et al.  A strongly degenerate quasilinear elliptic equation , 2005 .

[40]  J. Vázquez.  The Porous Medium Equation: Mathematical Theory , 2006 .

[41]  V. Caselles,et al.  Finite Propagation Speed for Limited Flux Diffusion Equations , 2006 .

[42]  A. Bellouquid,et al.  Mathematical Modeling of Complex Biological Systems: A Kinetic Theory Approach , 2006 .

[43]  J. Vázquez.  The Porous Medium Equation , 2006 .

[44]  Juncheng Wei,et al.  Stationary solutions to a Keller-Segel chemotaxis system , 2006, Asymptot. Anal..

[45]  Martin Burger,et al.  The Keller-Segel Model for Chemotaxis with Prevention of Overcrowding: Linear vs. Nonlinear Diffusion , 2006, SIAM J. Math. Anal..

[46]  G. Pettet,et al.  A Mathematical Model of Integrin-mediated Haptotactic Cell Migration , 2006, Bulletin of mathematical biology.

[47]  C. Schmeiser,et al.  MODEL HIERARCHIES FOR CELL AGGREGATION BY CHEMOTAXIS , 2006 .

[48]  F. A. Chalub,et al.  Global convergence of a kinetic model of chemotaxis to a perturbed Keller-Segel model , 2006 .

[49]  B. Perthame,et al.  Existence of solutions of the hyperbolic Keller-Segel model , 2006, math/0612485.

[50]  N. Bellomo,et al.  On the onset of non-linearity for diffusion models of binary mixtures of biological materials by asymptotic analysis , 2006 .

[51]  James Briscoe,et al.  Interpretation of the sonic hedgehog morphogen gradient by a temporal adaptation mechanism , 2007, Nature.

[52]  N. Bellomo,et al.  MULTICELLULAR BIOLOGICAL GROWING SYSTEMS: HYPERBOLIC LIMITS TOWARDS MACROSCOPIC DESCRIPTION , 2007 .

[53]  Eduard Feireisl,et al.  On convergence to equilibria for the Keller–Segel chemotaxis model , 2007 .

[54]  Nicola Bellomo,et al.  From microscopic to macroscopic description of multicellular systems and biological growing tissues , 2007, Comput. Math. Appl..

[55]  Mark Alber,et al.  Continuous macroscopic limit of a discrete stochastic model for interaction of living cells. , 2007, Physical review letters.

[56]  Mirosław Lachowicz,et al.  Lins Between Microscopic and Macroscopic Descriptions , 2008 .

[57]  N. Bellomo,et al.  Complex multicellular systems and immune competition: new paradigms looking for a mathematical theory. , 2008, Current topics in developmental biology.

[58]  Nicola Bellomo,et al.  From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells ✩ , 2008 .

[59]  Benoit Perthame,et al.  Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces , 2008, Math. Comput. Model..

[60]  P. Biler,et al.  Blowup of solutions to generalized Keller–Segel model , 2008, 0812.4982.

[61]  J. M. Mazón,et al.  Some regularity results on the ‘relativistic’ heat equation , 2008 .

[62]  T. Hillen,et al.  Shock formation in a chemotaxis model , 2008 .

[63]  H. Fischer,et al.  Mathematical Modeling of Complex Biological Systems , 2008, Alcohol research & health : the journal of the National Institute on Alcohol Abuse and Alcoholism.

[64]  G. Parisi,et al.  Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study , 2007, Proceedings of the National Academy of Sciences.

[65]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[66]  Hans G Othmer,et al.  Multi-scale models of cell and tissue dynamics , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[67]  Christian Schmeiser,et al.  The two-dimensional Keller-Segel model after blow-up , 2009 .

[68]  N Bellomo,et al.  Complexity analysis and mathematical tools towards the modelling of living systems. , 2009, Physics of life reviews.

[69]  Miguel A. Herrero,et al.  Modelling vascular morphogenesis: current views on blood vessels development , 2009 .

[70]  C. Schmeiser,et al.  Stochastic Particle Approximation for Measure Valued Solutions of the 2D Keller-Segel System , 2009 .

[71]  Yann Brenier,et al.  Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations , 2008, J. Nonlinear Sci..

[72]  Nicola Bellomo,et al.  On the derivation of macroscopic tissue equations from hybrid models of the kinetic theory of multicellular growing systems — The effect of global equilibrium☆ , 2009 .

[73]  T. Ogawa,et al.  ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO DRIFT-DIFFUSION SYSTEM WITH GENERALIZED DISSIPATION , 2009 .

[74]  Mariya Ptashnyk,et al.  BOUNDEDNESS OF SOLUTIONS OF A HAPTOTAXIS MODEL , 2010 .

[75]  K. Painter,et al.  Mathematical modeling of cell adhesion and its applications to developmental biology and cancer invasion , 2010 .

[76]  G. Parisi,et al.  FROM EMPIRICAL DATA TO INTER-INDIVIDUAL INTERACTIONS: UNVEILING THE RULES OF COLLECTIVE ANIMAL BEHAVIOR , 2010 .

[77]  Pascal Silberzan,et al.  Mathematical Description of Bacterial Traveling Pulses , 2009, PLoS Comput. Biol..

[78]  N. Bellomo,et al.  Complexity and mathematical tools toward the modelling of multicellular growing systems , 2010, Math. Comput. Model..

[79]  Andrea L. Bertozzi,et al.  LOCAL EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A PDE MODEL FOR CRIMINAL BEHAVIOR , 2010 .

[80]  Shigeru Kondo,et al.  Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation , 2010, Science.

[81]  A. Stevens,et al.  Qualitative Behavior of a Keller–Segel Model with Non-Diffusive Memory , 2010 .

[82]  Juan Soler,et al.  MULTISCALE BIOLOGICAL TISSUE MODELS AND FLUX-LIMITED CHEMOTAXIS FOR MULTICELLULAR GROWING SYSTEMS , 2010 .

[83]  B. Perthame,et al.  Travelling plateaus for a hyperbolic Keller–Segel system with attraction and repulsion: existence and branching instabilities , 2010, 1009.6090.

[84]  Marcello Delitala,et al.  On the modelling of genetic mutations and immune system competition , 2011, Comput. Math. Appl..

[85]  Christian Schmeiser,et al.  Convergence of a Stochastic Particle Approximation for Measure Solutions of the 2D Keller-Segel System , 2011 .

[86]  Kevin J. Painter,et al.  Spatio-temporal chaos in a chemotaxis model , 2011 .

[87]  J. Soler,et al.  QUALITATIVE PROPERTIES OF THE SOLUTIONS OF A NONLINEAR FLUX-LIMITED EQUATION ARISING IN THE TRANSPORT OF MORPHOGENS , 2011 .

[88]  Tong Li,et al.  Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis , 2011 .

[89]  Youshan Tao.  Global existence for a haptotaxis model of cancer invasion with tissue remodeling , 2011 .

[90]  C. Bianca.  MATHEMATICAL MODELING FOR KELOID FORMATION TRIGGERED BY VIRUS: MALIGNANT EFFECTS AND IMMUNE SYSTEM COMPETITION , 2011 .

[91]  Piotr Biler,et al.  Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis , 2009, Journal of mathematical biology.

[92]  B. Perthame,et al.  Waves for an hyperbolic Keller-Segel model and branching instabilities , 2010 .

[93]  Jos'e M. Maz'on,et al.  On a nonlinear flux-limited equation arising in the transport of morphogens , 2011, 1107.5770.

[94]  Mathematical models for morphogenesis: linear or nonlinear diffusion: comment on "Morphogenetic action through flux-limited spreading" by Verbeni, Sánchez, Mollica, Siegl-Cachedenier, Carleton, Guerrero, Ruiz i Altaba, Soler. , 2013, Physics of life reviews.