On the derivation of macroscopic hyperbolic equations for binary multicellular growing mixtures

Computers and Mathematics with Applications

Volumn 57, Issue 5, 2009, Pages 744-756

  Bellomo, N ^a   Bellouquid, A ^b  

^a POLITECNICO DI TORINO   (Italy)

^b National School of Applied Sciences   (Morocco)

Author keywords

Asymptotic limits; Growing tissues; Hyperbolic scaling; Kinetic theory; Mixtures; Multicellular systems

Indexed keywords

ASYMPTOTIC ANALYSIS; HISTOLOGY; KINETIC THEORY; MIXTURES; PROGRAMMING THEORY;

ACTIVE PARTICLES; ASYMPTOTIC LIMITS; BIOLOGICAL ACTIVITIES; BOLTZMANN; CELL MOVEMENT; GROWING TISSUES; HYPERBOLIC SCALING; MULTICELLULAR SYSTEMS; TIME SCALING;

BINARY MIXTURES;

EID: 58749111614     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2008.09.024     Document Type: Article

References (28)

1. Bellomo N. Modelling Complex Living Systems - A Kinetic Theory and Stochastic Game Approach (2008), Birkhäuser, Boston
2. Bellomo N., and Forni G. Complex multicellular systems and immune competition: New paradigms looking for a mathematical theory. Current Topics in Developmental Biology 81 (2008) 485-502
3. Bellouquid A., and Delitala M. Mathematical methods and tools of kinetic theory towards modelling of complex biological systems. Math. Mod. Meth. Appl. Sci. 15 (2005) 1639-1666
4. Bellouquid A., and Delitala M. Modelling Complex Biological Systems - A Kinetic Theory Approach (2006), Birkhäuser, Boston
5. Bellomo N., Li N.K., and Maini P.K. On the foundations of cancer modelling: Selected topics, speculations, and perspectives. Math. Mod. Meth. Appl. Sci. 18 (2008) 593-646
6. In: Mollica F., Preziosi L., and Rajagopal K.R. (Eds). Modelling Biological Tissues (2006), Birkäuser, Boston
7. Tiuryn J., Wójtowicz D., and Rudnicky R. A discrete model of small paralog families. Math. Models Methods Appl. Sci. 17 (2007) 933-956
8. Komarova N. Stochastic modeling of loss- and gain-of-function mutation in cancer. Math. Mod. Meth. Appl. Sci. 17 (2007) 1647-1674
9. Weinberg R.A. The Biology of Cancer (2007), Garland Sciences - Taylor and Francis, New York
10. Bellomo N., and Bellouquid A. From a class of kinetic models to macroscopic equations for multicellular systems in biology. Discrete and Continuous Dynamical Systems B 4 (2004) 59-80
11. Bellomo N., and Bellouquid A. On the onset of nonlinearity for diffusion models of binary mixtures of biological materials by asymptotic analysis. Int. J. Non-linear Mech. 41 (2006) 281-293
12. Bellomo N., Bellouquid A., and Herrero M.A. From microscopic to macroscopic description of multicellular systems and biological growing tissues. Comput. Math. Appl. 53 (2007) 647-663
13. Erban R., and Othmer H. From individual to collective behaviour in bacterial chemotaxis. SIAM J. Appl. Math. 65 (2005) 361-391
14. Hillen T., and Othmer H. The diffusion limit of transport equations derived from velocity jump processes. SIAM J. Appl. Math. 61 (2000) 751-775
15. Othmer H., and Hillen T. The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62 (2002) 1222-1250
16. Bellomo N., Bellouquid A., Nieto J., and Soler J. Multicellular growing systems: Hyperbolic limits towards macroscopic description. Math. Mod. Meth. Appl. Sci. 17 (2007) 1675-1693
17. Dolak Y., and Schmeiser C. Kinetic models for chemotaxis: Hydrodynamic limits and spatio temporal mechanisms. J. Math. Biol. 51 (2005) 595-615
18. Filbet F., Laurençot P., and Perthame B. Derivation of hyperbolic models for chemosensitive movement. J. Math. Biol. 50 (2005) 189-207
19. Albeverio S., and Alt W. Stochastic dynamics of viscolelastic skeins: Condensation waves and continuum limit. Math. Mod. Meth. Appl. Sci. 18 Suppl. (2008) 1149-1192
20. Degond P., and Motsch S. Continuum limit of self-driven particles with orientation interaction, A statistical model of criminal behavior. Math. Mod. Meth. Appl. Sci. 18 Suppl. (2008) 1193-1217
21. Stevens A. The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particles systems. SIAM J. Appl. Math. 61 (2000) 183-212
22. Hartwell H.L., Hopfield J.J., Leibner S., and Murray A.W. From molecular to modular cell biology. Nature 402 (1999) c47-c52
23. Bellouquid A., and Delitala M. Kinetic (cellular) models of cell progression and competition with the immune system. Z. Agnew. Math. Phys. 55 (2004) 295-317
24. Dogbè C. The fluid dynamic limits for gas mixture I. Formal derivation. Math. Mod. Meth. Appl. Sci. 18 (2008) 1633-1672
25. Friedman A. Mathematical analysis and challenges arising from models of tumor growth. Math. Mod. Meth. Appl. Sci. 17 (2007) 1751-1772
26. Tao Y., and Guo O. A free boundary problem modelling cancer radiovirotherapy. Math. Mod. Meth. Appl. Sci. 17 (2007) 1241-1259
27. Tao Y., and Zang H. A parabolic-hyperbolic free boundary problem modeling tumor treatment with virus. Math. Mod. Meth. Appl. Sci. 17 (2007) 63-80
28. Frank M., Herty H., and Schäfer M. Optimal treatment planning in radiotherapy based on Boltzmann transport calculations. Math. Mod. Meth. Appl. Sci. 18 (2008) 573-592