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Towards Microscopic and Nonlocal Models of Tumour Invasion of Tissue

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Selected Topics in Cancer Modeling

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References

  1. Arlotti, L., Bellomo, N., Lachowicz, M.: Kinetic equations modelling population dynamics. Transport Theory Statist. Phys.,29, 125–139 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arlotti, L., Bellomo, N., De Angelis, E., Lachowicz, M.,: Generalized Kinetic Models in Applied Sciences, World Sci., River Edge, NJ (2003)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bellomo, N., Forni, G.: Dynamics of tumor interaction with the host immune system. Math. Comput. Modelling,20, 107–122 (1994)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases, Springer, New York (1994)

    MATH  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986)

    MATH  Google Scholar 

  12. Filbert, F., Laurençcot, P., Perthame, B.: Derivation of hyperbolic models for chemosensitive movement. J. Math. Biol.,50, 189–207 (2005)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  14. Lachowicz, M.: From microscopic to macroscopic description for generalized kinetic models. Math. Models Methods Appl. Sci.,12, 985–1005 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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. Lachowicz, M.: From microscopic to macroscopic descriptions of complex systems. Comp. Rend. Mecanique (Paris),331, 733–738 (2003)

    Article  MATH  Google Scholar 

  17. Lachowicz, M.: On bilinear kinetic equations. Between micro and macro descriptions of biological populations. Banach Center Publ.,63, 217–230 (2004)

    MathSciNet  Google Scholar 

  18. Lachowicz, M.: General population systems. Macroscopic limit of a class of stochastic semigroups. J. Math. Anal. Appl.,307, 585–605 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Lachowicz, M., Pulvirenti, M.: A stochastic particle system modeling the Euler equation. Arch. Rational Mech. Anal.,109, 81–93 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. Lachowicz, M., Wrzosek, D.: Nonlocal bilinear equations. Equilibrium solutions and diffusive limit. Math. Models Methods Appl. Sci.,11, 1375–1390 (2001)

    Article  MathSciNet  Google Scholar 

  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. Morale, D., Capasso, V., Oelschläger, K.: An interacting particle system modelling aggregation behaviour: from individuals to populations. J. Math. Biol.,50, 49–66 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Morales-Rodrigo, C.: Local existence and uniqueness of regular solutions in a model of tissue invasion by solid tumours. Math. Comput. Model., to appear

    Google Scholar 

  26. Perthame, B.: PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic. Appl. Math.,49, 539–564 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Wagner, W.: A functional law of large numbers for Boltzmann type stochastic particle systems. Stochastic Anal. Appl.,14, 591–636 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha, 2, 02-097 Warsaw, Poland

    Miroslaw Lachowicz

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  1. Miroslaw Lachowicz
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Correspondence to Miroslaw Lachowicz .

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Lachowicz, M. (2008). Towards Microscopic and Nonlocal Models of Tumour Invasion of Tissue. In: Selected Topics in Cancer Modeling. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4713-1_3

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