From kinetic models of multicellular growing systems to macroscopic biological tissue models

Nonlinear Analysis: Real World Applications

Volumn 12, Issue 2, 2011, Pages 1111-1122

  Bellouquid, A ^a   De Angelis, E ^b  

^a National School of Applied Sciences   (Morocco)

^b POLITECNICO DI TORINO   (Italy)

Author keywords

Asymptotic limits; Hyperbolic limits; Kinetic theory; Living systems; Multicellular systems

Indexed keywords

ASYMPTOTIC LIMITS; BIOLOGICAL FUNCTIONS; BIOLOGICAL TISSUE MODEL; BIOLOGICAL TISSUES; CELLULAR INTERACTION; EQUILIBRIUM SOLUTIONS; GROWING TISSUES; HYPERBOLIC LIMITS; HYPERBOLIC MODELS; KINETIC MODELS; LIVING SYSTEMS; MACROSCOPIC EQUATION; MULTICELLULAR SYSTEMS; NON-LINEARITY;

ASYMPTOTIC ANALYSIS; BIOLOGICAL SYSTEMS; CONTINUUM MECHANICS; CONTROL NONLINEARITIES; ELASTICITY; HISTOLOGY; KINETIC THEORY; NONLINEAR EQUATIONS;

TISSUE;

EID: 78249272318     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2010.09.005     Document Type: Article

References (64)

1. K.R. Rajagopal, and L. Tao Mechanics of Mixtures Series on Advances in Mathematics for Applied Sciences vol. 35 1995 World Scientific
2. J.D. Humphrey, and K.R. Rajagopal A constrained mixture model for growth and remodeling of soft tissues Math. Models Methods Appl. Sci. 12 2002 407 430
3. F. Mollica, K.R. Rajagopal, L. Preziosi, Modeling of Biological Materials Series on Modeling and Simulation in Science, Engineering and Technology 2007 Birkhuser Boston
4. N. Bellomo, C. Bianca, and M. Delitala Complexity analysis and mathematical tools towards the modelling of living systems Phys. Life Rev. 6 2009 144 176
5. S. De Lillo, M. Delitala, and M.C. Salvatori Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles Math. Models Methods Appl. Sci. 19 2009 1405 1425
6. G. Ajmone Marsan New paradigms towards the modelling of complex systems in behavioral economics Math. Comput. Modelling 50 2009 584 597
7. E. De Angelis, and C.P. Grunfeld Modeling and analytic problems for a generalized Boltzmann equation for a multicomponent reacting gas Nonlinear Anal. RWA 4 2003 189 202
8. T. Goudon, O. Snchez, J. Soler, and L.L. Bonilla Low-field limit for a nonlinear discrete driftdiffusion model arising in semiconductor superlattices theory SIAM J. Appl. Math. 64 2004 1526 1549
9. F. Poupaud, and J. Soler Parabolic limit and stability of the VlasovPoissonFokkerPlanck system Math. Models Methods Appl. Sci. 10 2000 1027 1045
10. L.L. Bonilla, and J. Soler High field limit for the VlasovPoissonFokkerPlank system: a comparison of different perturbation methods Math. Models Methods Appl. Sci. 11 2001 1457 1681
11. T. Goudon, J. Nieto, J. Soler, and F. Poupaud Multidimensional high-field limit of the electrostatic VlasovPoissonFokkerPlanck system J. Differential Equations 213 2005 418 442
12. C. Villani Trend to equilibrium for dissipative equations, functional inequalities and mass transportation Recent Advances in the Theory and Applications of Mass Transport Contemp. Math. vol. 353 2004 Amer. Math. Soc. Providence, RI 95 109
13. B. Perthame Mathematical tools for kinetic equations Bull. Amer. Math. Soc. 41 2004 205 244
14. N. Bellomo, and A. Bellouquid On the onset of nonlinearity for diffusion models of binary mixtures of biological materials by asymptotic analysis Internat. J. Non-Linear Mech. 41 2006 281 293
15. N. Bellomo, A. Bellouquid, J. Nieto, and J. Soler Multicellular growing systems: hyperbolic limits towards macroscopic description Math. Models Methods Appl. Sci. 17 2007 1675 1692
16. 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
17. F.A. Chalub, P. Markovich, B. Perthame, and C. Schmeiser Kinetic models for chemotaxis and their driftdiffusion limits Monatsh. Math. 142 2004 123 141
18. 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
19. R. Erban, and H.G. Othmer From individual to collective behavior in bacterial chemotaxis SIAM J. Appl. Math. 65 2004/05 361 391
20. R. Erban, and H.G. Othmer From signal transduction to spatial pattern formation in E. coli: a paradigm for multiscale modeling in biology Multiscale Model. Simul. 3 2005 362 394
21. F. Filbet, P. Laurenot, and B. Perthame Derivation of hyperbolic models for chemosensitive movement J. Math. Biol. 50 2005 189 207
22. T. Hillen, and H. Othmer The diffusion limit of transport equations derived from velocity-jump processes SIAM J. Appl. Math. 61 2000 751 775
23. H.G. Othmer, and T. Hillen The diffusion limit of transport equations II: chemotaxis equations SIAM J. Appl. Math. 62 2002 1222 1250
24. M.A. Stolarska, Y. Kim, and H.G. Othmer Multiscale models of cell and tissue dynamics Philos. Trans. R. Soc. Lond. Ser. A 367 2009 3525 3553
25. C. Xue, H.G. Othmer, and R. Erban From individual to collective behavior of unicellular organisms: recent results and open problems R.M. Sinclair, K.M. Sriefel, Multiscale Phenomena in Biology AIP Conf. Proc. vol. 1167 2009 3 14
26. N. Bellomo, N.K. Li, and P.K. Maini On the foundations of cancer modelling: selected topics, speculations, and perspectives Math. Models Methods Appl. Sci. 18 2008 593 646
27. H.I. Freedman, and S.T.R. Pinho Stability criteria for the cure state in a cancer model with radiation treatment Nonlinear Anal. RWA 10 2009 2709 2715
28. A. Ito, M. Gokieli, M. Niezgdka, and Z. Szymaska Local existence and uniqueness of solutions to approximate systems of 1D tumor invasion model Nonlinear Anal. RWA 11 2010 3555 3566
29. X. Wei, and C. Guo Global existence for a mathematical model of the immune response to cancer Nonlinear Anal. RWA 11 2010 3903 3911
30. W. Xuemei, and C. Shangbin Existence and uniqueness of global solutions for a mathematical model of antiangiogenesis in tumor growth Nonlinear Anal. RWA 9 2008 1827 1836
31. A. Bellouquid, and M. Delitala Mathematical models and tools of kinetic theory towards modelling complex biological systems Math. Models Methods Appl. Sci. 15 2005 1639 1666
32. Z. Szymaska, C. Morale Rodrigo, M. Lachowicz, and M.A.J. Chaplain Mathematical modelling of cancer invasion of tissue: the role and effect of nonlocal interactions Math. Models Methods Appl. Sci. 19 2009 257 281
33. N. Bellomo, and A. Bellouquid On the derivation of macroscopic tissue equations from hybrid models of the kinetic theory of multicellular growing systemsthe effect of global equilibrium Nonlinear Anal. Hybrid Syst. 3 2009 215 224
34. A. Bellouquid, and M. Delitala Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach 2006 Birkuser Boston
35. N. Bellomo, and G. Forni Complex multicellular systems and immune competition: new paradigms looking for a mathematical theory Curr. Top. Dev. Biol. 81 2008 485 502
36. H.L. Hartwell, J.J. Hopfield, S. Leibner, and A.W. Murray From molecular to modular cell biology Nature 402 1999 c47 c52
37. A. Bellouquid, and M. Delitala Kinetic (cellular) models of cell progression and competition with the immune system Z. Angew. Math. Phys. 55 2004 295 317
38. K. Drucis, M. Kolev, W. Majda, and B. Zubik-Kowal Nonlinear modeling with mammografic evidence of carcinoma Nonlinear Anal. RWA 11 2010 4326 4334
39. Z. Jackiewicz, C.L. Jorcyk, M. Kolev, and B. Zubik-Kowal Correlation between animal and mathematical models for prostate cancer progression Comput. Math. Methods Med. 10 2009 241 252
40. L. Arlotti, A. Gamba, and M. Lachowicz A kinetic model of tumor-immune system cellular interactions J. Theoret. Med. 4 2002 39 50
41. N. Bellomo, L. Preziosi, and G. Forni On a kinetic (cellular) theory for competition between tumors and the host immune system J. Biol. Syst. 4 1996 479 502
42. C. Bianca, On the modelling of space dynamics in the kinetic theory for active particles, Math. Models Methods Appl. Sci. 21, (2011) (in press).
43. M. Kolev, B. Zubik-Kowal, Numerical solution for a model of tissue invasion and migration of tumor cells, Comput. Math. Methods Med. (2010) (in press).
44. D. Hanahan, and R.A. Weinberg The hallmarks of cancer Cell 100 2000 57 70
45. S.A. Frank Dynamics of Cancer: Inheritance, and Evolution 2007 Princeton University Press
46. R.A. Weinberg The Biology of Cancer 2007 Garland Sciences, Taylor and Francis New York
47. I. Michor, Y. Iwasa, and M.A. Nowak Dynamics of cancer progression Nat. Rev. Cancer 4 2004 197 205
48. M.A. Nowak, and K. Sigmund Evolutionary dynamics of biological games Science 303 2004 793 799
49. E. Gabetta, and E. Regazzini About the gene families in a recent model of genome evolution Math. Models Methods Appl. Sci. 20 2010 1005 1020
50. A. Hastings, P. Arzberger, B. Bolker, S. Collins, A.R. Ives, N.A. Johnson, and M.A. Palmer Quantitative bioscience for the 21st century BioScience 55 2005 511 517
51. J.D. Watson, T.A. Baker, S.P. Bell, A. Gann, M. Levine, and R. Losick Molecular Biology of the Gene 5th ed. 2004 Peason Benjamin Cummings, Cold Spring Harbor Laboratory Press
52. C.R. Woese A new biology for a new century Microbiol. Mol. Biol. Rev. 68 2004 173 186
53. L. Arlotti, and N. Bellomo Solution of a new class of nonlinear kinetic models of population dynamics Appl. Math. Lett. 9 1996 65 70
54. H.G. Othmer, S.R. Dunbar, and W. Alt Models of dispersal in biological systems J. Math. Biol. 26 1988 263 298
55. J.S. Lowengrub, H.B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S.M. Wise, and V. Cristini Nonlinear modelling of cancer: bridging the gap between cells and tumours Nonlinearity 23 2010 R1 R91
56. E. De Angelis, and P.-E. Jabin Mathematical models of therapeutical actions related to tumour and immune system competition Math. Methods Appl. Sci. 28 2005 2061 2083
57. I. Gipson, and T. Inatomi Extra-cellular matrix and growth factors in corneal wound healing Curr. Opin. Ophthalmol. 6 1995 3 10
58. A. Br, and M.A. Herrero From the physical laws of tumor growth to modeling cancer processes Math. Models Methods Appl. Sci. 16 2006 1199 1218
59. A. Friedman Mathematical analysis and challenges arising from models of tumor growth Math. Models Methods Appl. Sci. 17 2007 1751 1772
60. Y. Tao, Q. Guo, and K. Aihara A model at the macroscopic scale of prostate tumor growth under intermittent androgen suppression Math. Models Methods Appl. Sci. 19 2009 2177 2201
61. F. Zhou, J. Wu, and X. Wei Analyticity of solutions to a free boundary problem modeling the growth of multi-layer tumors Nonlinear Anal. RWA 11 2010 2698 2707
62. V. Cristini, and J. Lowengrub Multiscale Modeling of CancerAn Integrated Experimental and Mathematical Modeling Approach 2010 Cambridge University Press
63. M.R. Owen, T. Alarcn, P.K. Maini, and H.M. Byrne Angiogenesis and vascular remodelling in normal and cancerous tissues J. Math. Biol. 58 2009 689 721
64. B.D. Sleeman Mathematical aspects of modelling tumour angiogenesis Nonlinear Dynamics and Evolution Equations Fields Inst. Commun. vol. 48 2006 Amer. Math. Soc. Providence, RI 257 278