MULTISCALE BIOLOGICAL TISSUE MODELS AND FLUX-LIMITED CHEMOTAXIS FOR MULTICELLULAR GROWING SYSTEMS

This paper deals with the derivation of macroscopic tissue models from the underlying description delivered by a class of equations that models binary mixtures of multicellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of the biological functions and proliferative and destructive events. The asymptotic analysis deals with suitable parabolic and hyperbolic limits, and is specifically focused on the modeling of the chemotaxis phenomena.

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

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

[3]  V. Quaranta,et al.  Integrative mathematical oncology , 2008, Nature Reviews Cancer.

[4]  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.

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

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

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

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

[9]  Thomas Hillen.  QUALITATIVE ANALYSIS OF SEMILINEAR CATTANEO EQUATIONS , 1998 .

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

[11]  P. Chavanis.  Jeans type instability for a chemotactic model of cellular aggregation , 2006, 0810.5286.

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

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

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

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

[16]  M. Hadeler.  Reaction Transport Systems in Biological , 1997 .

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

[18]  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 .

[19]  Frédéric Poupaud,et al.  Parabolic Limit and Stability of the Vlasov-Poisson-Fokker-Planck system , 1998 .

[20]  Karl Peter Hadeler,et al.  Mathematics Inspired by Biology , 2000 .

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

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

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

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

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

[26]  Vicenç Méndez,et al.  Reaction–Transport Systems , 2010 .

[27]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

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

[29]  L. Segel,et al.  Model for chemotaxis. , 1971, Journal of theoretical biology.

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

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

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

[33]  M. B. Rubin,et al.  Hyperbolic heat conduction and the second law , 1992 .

[34]  C. Patlak.  Random walk with persistence and external bias , 1953 .

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

[36]  Juan Soler,et al.  Vanishing Viscosity Regimes and Nonstandard Shock Relations for Semiconductor Superlattices Models , 2011, SIAM J. Appl. Math..

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

[38]  Nicola Bellomo,et al.  Methods and tools of the mathematical Kinetic theory toward modeling complex biological systems , 2006 .

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

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

[41]  C. Sire,et al.  Jeans type analysis of chemotactic collapse , 2007, 0708.3163.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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