Towards Microscopic and Nonlocal Models of Tumour Invasion of Tissue

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


Microscopic Model Macroscopic Model Tissue Invasion Nonlocal Model Markov Jump Process 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arlotti, L., Bellomo, N., Lachowicz, M.: Kinetic equations modelling population dynamics. Transport Theory Statist. Phys.,29, 125–139 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arlotti, L., Bellomo, N., De Angelis, E., Lachowicz, M.,: Generalized Kinetic Models in Applied Sciences, World Sci., River Edge, NJ (2003)MATHGoogle Scholar
  3. 3.
    Bellomo, N., Bellouquid, A., Delitala, M.: Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition. Math. Models Methods Appl. Sci.,14, 1683–1733 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bellomo, N., Forni, G.: Dynamics of tumor interaction with the host immune system. Math. Comput. Modelling,20, 107–122 (1994)MATHCrossRefGoogle Scholar
  5. 5.
    Bellomo, N., Forni, G.: Looking for new paradigms towards a biological-mathematical theory of complex multicellular systems. Math. Models Methods Appl. Sci.,16, 1001–1029 (2006)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases, Springer, New York (1994)MATHGoogle Scholar
  7. 7.
    Chaplain, M.A.J., Anderson, A.R.A.: Mathematical modelling of tissue invasion. In: Preziosi, L. (ed): Cancer Modelling and Simulation, 269–297, Chapman – Hall/CRT, London (2003)Google Scholar
  8. 8.
    Chaplain, M.A.J., Lolas, G.: Spatio-temporal heterogeneity arising in a mathematical model of cancer invasion of tissue. Math. Models Methods Appl. Sci.,15, 1685–1734 (2005)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chaplain, M.A.J.: Modelling tumour growth. In Multiscale Problems in Life Sciences. From Microscopic to Macroscopic. Eds. Capasso, V., Lachowicz, M., CIME Courses, Springer Lecture Notes in Mathematics (2008), to appear.Google Scholar
  10. 10.
    Corrias, L., Perthame, B., Zaag, H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math.,72, 1–28 (2004)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986)MATHGoogle Scholar
  12. 12.
    Filbert, F., Laurençcot, P., Perthame, B.: Derivation of hyperbolic models for chemosensitive movement. J. Math. Biol.,50, 189–207 (2005)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Jäager, E., Segel, L.: On the distribution of dominance in a population of interacting anonymous organisms. SIAM J. Appl. Math.,52, 1442–1468 (1992)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Lachowicz, M.: From microscopic to macroscopic description for generalized kinetic models. Math. Models Methods Appl. Sci.,12, 985–1005 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lachowicz, M.: Describing competitive systems at the level of interacting individuals. In: Proceedings of the Eight Nat.Confer. Appl. Math. Biol. Medicine, _Lajs, Poland (25–28 Sept.), 95–100 (2002)Google Scholar
  16. 16.
    Lachowicz, M.: From microscopic to macroscopic descriptions of complex systems. Comp. Rend. Mecanique (Paris),331, 733–738 (2003)MATHCrossRefGoogle Scholar
  17. 17.
    Lachowicz, M.: On bilinear kinetic equations. Between micro and macro descriptions of biological populations. Banach Center Publ.,63, 217–230 (2004)MathSciNetGoogle Scholar
  18. 18.
    Lachowicz, M.: General population systems. Macroscopic limit of a class of stochastic semigroups. J. Math. Anal. Appl.,307, 585–605 (2005)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lachowicz, M.: Micro and meso scales of description corresponding to a model of tissue invasion by solid tumours. Math. Models Methods Appl. Sci.,15, 1667–1683 (2005)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lachowicz, M.: Links Between Microscopic and Macroscopic Descriptions. In Multiscale Problems in Life Sciences. From Microscopic to Macroscopic. Eds. Capasso, V., Lachowicz, M., CIME Courses, Springer Lecture Notes in Mathematics (2008), to appear.Google Scholar
  21. 21.
    Lachowicz, M., Pulvirenti, M.: A stochastic particle system modeling the Euler equation. Arch. Rational Mech. Anal.,109, 81–93 (1990)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lachowicz, M., Wrzosek, D.: Nonlocal bilinear equations. Equilibrium solutions and diffusive limit. Math. Models Methods Appl. Sci.,11, 1375–1390 (2001)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Lolas, G.: Mathematical modelling of the urokinase plasminogen activation system and its role in cancer invasion of tissue. Ph.D. Thesis, Department of Mathematics, University of Dundee (2003)Google Scholar
  24. 24.
    Morale, D., Capasso, V., Oelschläger, K.: An interacting particle system modelling aggregation behaviour: from individuals to populations. J. Math. Biol.,50, 49–66 (2005)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Morales-Rodrigo, C.: Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours. Math. Comput. Model., to appearGoogle Scholar
  26. 26.
    Perthame, B.: PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic. Appl. Math.,49, 539–564 (2004)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    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
  28. 28.
    Szyma’nska, Z., Morales Rodrigo, C., Lachowicz, M., Chaplain, M.A.J.: Mathematical modelling of cancer invasion of tissue: Nonlocal interactions, to appear.Google Scholar
  29. 29.
    Wagner, W.: A functional law of large numbers for Boltzmann type stochastic particle systems. Stochastic Anal. Appl.,14, 591–636 (1996)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and Mechanics Faculty of Mathematics, Informatics and MechanicsUniversity of Warsawul. Banacha, 2Poland

Personalised recommendations