Complexity and mathematical tools toward the modelling of multicellular growing systems

This paper deals with a multiscale modelling approach to complex biological systems constituted by several interacting entities. The methodology is based on mathematical kinetic theory for active particles and is focused on the modelling of complex multicellular systems under therapeutic actions at the cellular level and mutations with onset of new populations. Asymptotic hyperbolic methods are developed to derive models at the macroscopic scale of tissues from the underlying description at the level of cells for a open system with variable number of populations.

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

[2]  E. Angelis,et al.  Mathematical models of therapeutical actions related to tumour and immune system competition , 2005 .

[3]  T. Jessell,et al.  A hedgehog-insensitive form of patched provides evidence for direct long-range morphogen activity of sonic hedgehog in the neural tube. , 2001, Molecular cell.

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

[5]  Jack T. Trevors,et al.  Self-organization vs. self-ordering events in life-origin models , 2006 .

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

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

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

[9]  Qing Nie,et al.  Do morphogen gradients arise by diffusion? , 2002, Developmental cell.

[10]  Maria Letizia Bertotti,et al.  On the existence of limit cycles of opinion formation processes under time periodic influence of persuaders , 2008 .

[11]  C. Woese.  A New Biology for a New Century , 2004, Microbiology and Molecular Biology Reviews.

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

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

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

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

[16]  Maria Letizia Bertotti,et al.  Conservation laws and asymptotic behavior of a model of social dynamics , 2008 .

[17]  Juan Soler,et al.  From the mathematical kinetic theory for active particles on the derivation of hyperbolic macroscopic tissue models , 2009, Math. Comput. Model..

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

[19]  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 .

[20]  Marcello Edoardo Delitala,et al.  MODELLING EPIDEMICS AND VIRUS MUTATIONS BY METHODS OF THE MATHEMATICAL KINETIC THEORY FOR ACTIVE PARTICLES , 2009 .

[21]  Krishanu Saha,et al.  Signal dynamics in Sonic hedgehog tissue patterning , 2006, Development.

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

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

[24]  Barbara Stecca,et al.  Melanomas require HEDGEHOG-GLI signaling regulated by interactions between GLI1 and the RAS-MEK/AKT pathways , 2007, Proceedings of the National Academy of Sciences.

[25]  N. Bellomo,et al.  Looking for new paradigms towards a biological-mathematical theory of complex multicellular systems , 2006 .

[26]  Barbara Stecca,et al.  The Gli code: an information nexus regulating cell fate, stemness and cancer. , 2007, Trends in cell biology.

[27]  Carlo Bianca,et al.  On the modelling of space dynamics in the kinetic theory for active particles , 2010, Math. Comput. Model..

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

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

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

[31]  A. Ruiz i Altaba,et al.  A GLI1-p53 inhibitory loop controls neural stem cell and tumour cell numbers , 2009, The EMBO journal.

[32]  G. Ajmone Marsan.  On the modelling and simulation of the competition for a secession under media influence by active particles methods and functional subsystems decomposition , 2009, Comput. Math. Appl..

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

[34]  E. Angelis,et al.  Qualitative analysis of a mean field model of tumor-immune system competition , 2003 .

[35]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

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

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

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