From Kinetic Theory for Active Particles to Modelling Immune Competition

Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


Immune Cell Neoplastic Cell Abnormal Cell Active Particle Probable Output 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bellomo, N., De Angelis, E., Preziosi, L.: Multiscale modelling and mathematical problems related to tumor evolution and medical therapy. J. Theor. Medicine,5, 111–136 (2003).MATHCrossRefGoogle Scholar
  2. 2.
    Bellomo, N., Bellouquid, A., Delitala, M.: Mathematical topics in the modelling complex multicellular systems and tumor immune cells competition. Math. Mod. Meth. Appl. Sci.,14, 1683–1733 (2004).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bellouquid, A., Delitala, M.: Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach. Birkhäuser, Boston (2006).MATHGoogle Scholar
  4. 4.
    Arlotti, L., Bellomo, N., De Angelis, E.: Generalized kinetic (Boltzmann) models: mathematical structures and applications. Math. Mod. Meth. Appl. Sci.,12, 579–604 (2002).CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bellomo, N.: Modeling Complex Living Systems. A Kinetic Theory and Stochastic Game Approach. Birkhäuser, Boston (2008).MATHGoogle Scholar
  6. 6.
    Bellomo, N., Forni, G.: Dynamics of tumor interaction with the host immune system. Math. Comp. Mod.,20, 107–122 (1994).MATHCrossRefGoogle Scholar
  7. 7.
    Arlotti, L., Lachowicz, M., Gamba, A.: A kinetic model of tumor/immune system cellular interactions. J. Theor. Medicine,4, 39–50 (2002).MATHCrossRefGoogle Scholar
  8. 8.
    De Angelis, E., Jabin, P.E.: Qualitative analysis of a mean field model of tumorimmune system competition. Math. Meth. Appl. Sci.,28, 2061–2083 (2005).MATHCrossRefGoogle Scholar
  9. 9.
    Kolev, M.: Mathematical modeling of the competition between acquired immunity and cancer. Appl. Math. Comp. Science,13, 289–297 (2003).MATHMathSciNetGoogle Scholar
  10. 10.
    Bellouquid, A., Delitala, M.: Kinetic (cellular) models of cell progression and competition with the immune system. Z. Angew. Math. Phys.,55, 295–317 (2004).MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Derbel, L.: Analysis of a new model for tumor-immune system competition including long time scale effects. Math. Mod. Meth. Appl. Sci.,14, 1657–1682 (2004).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kolev, M.: A mathematical model of cellular immune response to leukemia. Math. Comp. Mod.,41, 1071–1082 (2005).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kolev, M., Kozlowska, E., Lachowicz, M.: Mathematical model of tumor invasion along linear or tubular structures. Math. Comp. Mod.,41, 1083–1096 (2005).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bellouquid, A., Delitala, M.: Mathematical methods and tools of kinetic theory towards modelling of complex biological systems. Math. Mod. Meth. Appl. Sci.,15, 1619–1638 (2005).CrossRefMathSciNetGoogle Scholar
  15. 15.
    Brazzoli, I., Chauviere, A.: On the discrete kinetic theory for active particles. Modelling the immune competition. Comput. and Math. Meth. in Medicine,7, 143–157 (2006).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Adam, J., Bellomo, N. (Eds.): A Survey of Models on Tumor Immune Systems Dynamics. Birkhäuser, Boston (1997).Google Scholar
  17. 17.
    Preziosi, L.: Modeling Cancer Growth. CRC Press-Chapman Hall, Boca Raton, FL (2003).Google Scholar
  18. 18.
    Bellomo, N., Maini, P.K.: Preface of the Special issue on cancer modeling (II). Math. Mod. Meth. Appl. Sci.,16, iii–vii (2006).CrossRefMathSciNetGoogle Scholar
  19. 19.
    Bellomo, N., Sleeman, B.D.: Preface of the Special issue on multiscale cancer modelling. Comput. and Math. Meth. in Medicine,7, 67–70 (2006).CrossRefMathSciNetGoogle Scholar
  20. 20.
    Bellomo, N., De Lillo, S., Salvatori, C.: Mathematical tools of the kinetic theory of active particles with some reasoning on the modelling progression and heterogeneity. Math. Comp. Mod.,45, 564–578 (2007).CrossRefMATHGoogle Scholar
  21. 21.
    De Angelis, E., Delitala, M.: Modelling complex systems in applied sciences methods and tools of the mathematical kinetic theory for active particles. Math. Comp. Mod.,43, 1310–1328 (2006).MATHCrossRefGoogle Scholar
  22. 22.
    Bellomo, N., Forni, G.: Looking for new paradigms towards a biologicalmathematical theory of complex multicellular systems. Math. Mod. Meth. Appl. Sci.,16, 1001–1029 (2006).MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Bellomo, N., Forni, G.: Complex multicellular systems and immune competition: new paradigms looking for a mathematical theory. Current Topics in Developmental Biology,81, 485–502 (2007).CrossRefGoogle Scholar
  24. 24.
    Hartwell, H.L., Hopfield, J.J., Leibner, S., Murray, A.W.: From molecular to modular cell biology. Nature,402, c47–c52 (1999).CrossRefGoogle Scholar
  25. 25.
    Reed, R.: Why is mathematical biology so hard? Notices of the American Mathematical Society,51, 338–342 (2004).MATHMathSciNetGoogle Scholar
  26. 26.
    Woese, C.R.: A new biology for a new century. Microbiology and Molecular Biology Reviews,68, 173–186 (2004).CrossRefGoogle Scholar
  27. 27.
    Lollini, P.L., Motta, S., Pappalardo, F.: Modelling the immune competition. Math. Mod. Meth. Appl. Sci.,16, 1091–1124 (2006).MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Hillen, T., Othmer, H.: The diffusion limit of transport equations derived from velocity jump processes. SIAM J. Appl. Math.,61, 751–775 (2000).MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Stevens, A.: The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math.,61, 183–212 (2000).MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Bellomo, N., Bellouquid, A.: From a class of kinetic models to macroscopic equations for multicellular systems in biology. Discrete Contin. Dyn. Syst. B,4, 59–80 (2004).MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Lachowicz, M.: Micro and meso scales of description corresponding to a model of tissue invasion by solid tumours. Math. Mod. Meth. Appl. Sci.,15, 1667–1684 (2005).MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Bellomo, N., Bellouquid, A.: On the mathematical kinetic theory of active particles with discrete states—The derivation of macroscopic equations. Math. Comp. Mod.,44, 397–404 (2006).MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Chalub, F., Dolak-Struss, Y., Markowich, P., Oeltz, D., Schmeiser, C., Soref, A.: Model hierarchies for cell aggregation by chemotaxis. Math. Mod. Meth. Appl. Sci.,16, 1173–1198 (2006).MATHCrossRefGoogle Scholar
  34. 34.
    Bellomo, N., Bellouquid, A., Nieto, J., Soler, J.: Multicellular biological growing systems: hyperbolic limits towards macroscopic description. Math. Mod. Meth. Appl. Sci.,17, 1675–1692 (2007).MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Delitala, M., Forni, G.: From the mathematical kinetic theory of active particles to modelling genetic mutations and immune competition. Internal Report, Dept. Mathematics, Politecnico, Torino (2008).Google Scholar
  36. 36.
    Nowak, M.A., Sigmund, K.: Evolutionary dynamics of biological games. Science,303, 793–799 (2004).CrossRefGoogle Scholar
  37. 37.
    Komarova, N.: Stochastic modelling of loss- and gain-of-function mutations in cancer. Math. Mod. Meth. Appl. Sci.,17, 1647–1673 (2007).MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Weinberg, R.A.: The Biology of Cancer. Garland Sciences-Taylor and Francis, New York (2007).Google Scholar

Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.Ecole Nationale des Sciences AppliquéesUniversity Cadi AyyadSafiMaroc
  2. 2.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly

Personalised recommendations