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A General Framework for Multiscale Modeling of Tumor–Immune System Interactions

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Mathematical Oncology 2013

Abstract

In this paper we review methods that allow the construction of a consistent set of models that may describe the interactions between a tumor and the immune system on microscopic, mesoscopic, and macroscopic scales. The presented structures may be a basis for a description on the sub–cellular, cellular, and macroscopic levels. Important open problems are indicated.

To Nicola Bellomo in occasion of his Anniversary.

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References

  1. M. Al-Tameemi, M. Chaplain, A. d’Onofrio, Evasion of tumours from the control of the immune system: consequences of brief encounters. Biol Direct. 7(31), 1–22 (2012)

    Google Scholar 

  2. L. Arlotti, N. Bellomo, E. De Angelis, M. Lachowicz, Generalized Kinetic Models in Applied Sciences ( World Scienctific, New Jersay (2003)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz, N. Bellomo, On a class of integro–differential equations modeling complex systems with nonlinear interactions. Appl. Math. Lett. 25, 490–495 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, Multiscale biological tissue models and flux–limited chemotaxis for multicellular growing systems. Math. Models Methods Appl. Sci. 20, 1179–1207 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives. Math. Models Methods Appl. Sci. 22, 1130001 (2012)

    Article  MathSciNet  Google Scholar 

  7. N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, Modelling chemotaxis from L 2–closure moments in kinetic theory of active particles. Discrete Contin. Dyn. Systems B 18, 847–863 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bellomo, N., Carbonaro, B.: Toward a mathematical theory of living system focusing on developmental biology and evolution: A review and prospectives. Physics of Life, Reviews 8, 1–18 (2011)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  10. N. Bellomo, G. Forni, Looking for new paradigms towards a biological-mathematical theory of complex multicellular systems. Math. Models Methods Appl. Sci. 16, 1001–1029 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. N. Bellomo, D. Knopoff, J. Soler, On the difficult interplay between life, “complexity” and mathematical sciences. Math. Models Methods Appl. Sci. 23, 1861–1913 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Bellomo, M. Lachowicz, J. Polewczak, G. Toscani, Mathematical Topics in Nonlinear Kinetic Theory II: The Enskog Equation (World Scientific, Singapore 1991)

    Book  MATH  Google Scholar 

  13. N. Bellomo, N.K. Li, P.K. Maini, On the foundations of cancer modelling: Selected topics, speculations and perspectives. Math. Models Methods Appl. Sci. 18, 593–646 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. N. Bellomo, L. Preziosi, Modelling and mathematical problems related to tumour evolution and its interaction with immune system. Math. Comput. Model. 32, 413–452 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. N., Bellomo, A. Bellouquid, E. de Angelis, The modelling of immune competition by generalised kinetic (Boltzmann) models: review and research perspectives. Math. Comput. Modelling 37, 65–86 (2003)

    Google Scholar 

  16. M., Bodnar, U., Foryś, Z. Szymańska, Model of AIDS–related tumour with time delay, Appl. Math. (Warsaw) 36, 263–278 (2009)

    Google Scholar 

  17. V. Capasso, D. Morale, in Rescaling Stochastic Processes: Asymptotics. eds. by J. Banasiak, V. Capasso, M.A.J. Chaplain, M. Lachowicz, J. Miȩkisz, Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic, Lecture Notes Math. vol. 1940, (Springer, Berlin 2008) pp. 91–146.

    Google Scholar 

  18. Capasso, V., Morale, D.: Asymptotic behavior of a system of stochastic particles subject to nonlocal interactions. Stoch. Anal. Appl. 27, 574–603 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  20. N. Champagnat, R. Ferrière, S. Méléard, Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Th. Popul. Biol. 69, 297–321 (2006)

    Article  MATH  Google Scholar 

  21. N. Champagnat, R. Ferrière, S. Méléard, From individual stochastic processes to macroscopic models in adaptive evolution. Stochastic Models 24 2–44 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. M.A.J. Chaplain, in Modelling aspects of cancer growth: insight from mathematical and numerical analysis and computational simulation, eds. by J. Banasiak, V. Capasso, M.A.J. Chaplain, M. Lachowicz, J. Miȩkisz, Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic, Lecture Notes Math. vol. 1940, (Springer, Berlin 2008) pp. 147–200.

    Google Scholar 

  23. M.A.J. Chaplain, M. Lachowicz, Z. Szymańska, D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell–cell adhesion and cell-matrix adhesion. Math. Models Methods Appl. Sci. 24, 719–743 (2011)

    Article  Google Scholar 

  24. M. Cobbold, H. De La Pea, A. Norris, J.M. Polefrone, J. Qian, A.M. English, K.L. Cummings, S. Penny, J.E. Turner, J. Cottine, J.G. Abelin, S.A. Malaker, A.L. Zarling, H.W. Huang, O. Goodyear, S.D. Freeman, J. Shabanowitz, G. Pratt, C. Craddock, M.E Williams, D.F. Hunt, V.H. Engelhard, MHC Class I-Associated Phosphopeptides Are the Targets of Memory-like Immunity in Leukemia. Sci. Transl. Med. 203, 203–125 (2013)

    Google Scholar 

  25. Criaco, D., Dolfin, M.,. Restuccia, L: Approximate smooth solution pf a mathematical model for the activation and clonal expansion of T cells. Math. Biosci. Engineer. 10, 59–73 (2013)

    MathSciNet  MATH  Google Scholar 

  26. D. Criaco, M. Dolfin, A phenomenological approach to the dynamics of activation and clonal expansion of T cells. Math. Computer Model. 53, 314–329 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. A. d’Onofrio, A general framework for modeling tumour–immune system competition and immunotherapy: mathematical analysis and biomedical inferences. Physica D 208, 220–235 (2005)

    Google Scholar 

  28. d’Onofrio, A.: Tumor–immune system interaction: modeling the tumor–stimulated proliferation of effectors and immunotherapy. Math. Models Methods Appl. Sci. 16, 1375–1401 (2006)

    Google Scholar 

  29. A. d’Onofrio, Tumor evasion from immune control: Strategies of a MISS to become a MASS. Chaos, Solitons and Fractals 31, 261–268 (2007)

    Google Scholar 

  30. A. d’Onofrio, Metamodeling tumor–immune system interaction, tumor evasion and immunotherapy. Math. Comput. Model. 47,614–637 (2008)

    Google Scholar 

  31. A. d’Onofrio, F. Gatti, P. Cerrai, L. Freschi, Delay–induced oscillatory dynamics of tumour–immune system interaction. Math. Comput. Model. 51, 572–591 (2010)

    Google Scholar 

  32. A. d’Onofrio, A. Ciancio, Simple biophysical model of tumor evasion from immune system control. Phys Rev E Stat Nonlin Soft Matter Phys. 84 (2011)

    Google Scholar 

  33. P. Ehrlich, Über den jetzigen Stand der Karzinomforschung. Ned. Tijdschr. Geneeskd. 5, 273–290 (1909)

    Google Scholar 

  34. A. Eibeck, W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations. Annals Appl. Prob. 13, 845–889 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  36. D., Finkelshtein, Y., Kondratiev, O. Kutoviy, Semigroup approach to birth–and–death stochastic dynamics in continuum. J. Funct. Anal. 262, 1274–1308 (2012)

    Google Scholar 

  37. A. Gerisch, M.A.J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: local and non-local models and the effect of adhesion. J. Theoret. Biol. 250, 684–704 (2008)

    Article  MathSciNet  Google Scholar 

  38. J. Gołab, M. Jakóbisiak, W. Lasek (eds.), Immunologia, Wydawnictwo Naukowe PWN, Warszawa (2002), in Polish

    Google Scholar 

  39. D., Hanahan, R.A. Weinberg, Hallmarks of cancer: the next generation. Cell 144, 646–674 (2011)

    Google Scholar 

  40. H.L. Hartwell, J.J. Hopfield, S. Leibner, A.W. Murray, From molecular to modular cell biology. Nature 402, c47–c52 (1999)

    Article  Google Scholar 

  41. R. B. Herberman (ed.), NK Cells and Other Natural Effector Cells (Academic Press, New York 1982)

    Google Scholar 

  42. C.A. Janeway, P. Travers, M. Walport, Immunobiology: The Immune System in Health and Disease ( Garland Science, New York 2001)

    Google Scholar 

  43. B., Joshi, X., Wang, S., Banerjee, H., Tian, A., Matzavinos, M.A.J. Chaplain, On immunotherapies and cancer vaccination protocols: A mathematical modelling approach. J. Theor. Biol. 259, 820–827 (2009)

    Google Scholar 

  44. D., Kirschner, J.C. Panetta, Modeling immunotherapy of the tumor - immune interaction. J. Math. Biol. 37, 235–252 (1998)

    Google Scholar 

  45. V.A., Kuznetsov, I.A. Makalkin, M.A. Taylor, A.S. Perelson, Nonlinear dynamics of immunogenic tumours: Parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56, 295–321 (1994)

    Google Scholar 

  46. M. Lachowicz, in Links between microscopic and macroscopic descriptions eds. by J. Banasiak, V. Capasso, M.A.J. Chaplain, M. Lachowicz, J. Miȩkisz Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic, Lecture Notes Math vol. 1940, (Springer, Berlin 2008) pp. 201–268

    Google Scholar 

  47. Lachowicz, M.: in Towards microscopic and nonlocal models of tumour invasion of tissue eds. by In: N. Bellomo, M. Chaplain, E. De Angelis, ( Birkhauser, Boston 2008) pp. 49–63.

    Google Scholar 

  48. M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems. Prob. Engin. Mech. 26, 54–60 (2011)

    Article  Google Scholar 

  49. Lachowicz, M.: Individually–based Markov processes modeling nonlinear systems in mathematical biology. Nonlinear Analysis Real World Appl. 12, 2396–2407 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Lachowicz, A. Quartarone, A general framework for modeling tumor–immune system competition at the mesoscopic level. Appl. Math. Letters 25, 2118–2122 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  51. Lachowicz, M., Quartarone, A., Ryabukha, T.V.: Stability of solutions of kinetic equations corresponding to the replicator dynamics, Kinetic Relat. Models (to appear)

    Google Scholar 

  52. M. Lachowicz, T.V. Ryabukha, Equilibrium solutions for microscopic stochastic systems in population dynamics. Math. Biosci. Engin. 10, 777–786 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  54. O. Lejeune, M.A.J. Chaplain, I. El Akili, Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumours. Math. Comput. Modelling 47 649–662 (2008)

    Google Scholar 

  55. G.W. Litman, J.P., Cannon, L.J. Dishaw, Reconstructing immune phylogeny: new perspectives. Nat. Rev. Immunol. 5, 866–879 (2005)

    Google Scholar 

  56. A. Matzavinos, M.A.J. Chaplain, Travelling wave analysis of a model of the immune response to cancer. Biologies 327, 995–1008 (2004)

    Article  Google Scholar 

  57. A. Matzavinos, M.A.J. Chaplain, V. Kuznetsov, Mathematical modelling of the spatio–temporal response of cytotoxic T–lymphocytes to a solid tumour. Math. Med. Biol. 21, 1–34 (2004)

    Article  MATH  Google Scholar 

  58. G. Mayer,Microbiology and Immunology On–Line Textbook (USC School of Medicine, 2006)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  60. M.R. Owen, J.A. Sherratt, Pattern formation and spatiotemporal irregularity in a model for macrophage–tumour interaction. J. Theor. Biol. 189, 63–80 (1997)

    Article  Google Scholar 

  61. L.G. de Pillis, A.E. Radunskaya, C.L. Wiseman, A validated mathematical model of cell–mediated immune response to tumor growth. Cancer Res. 65, 7950–7958 (2005)

    Google Scholar 

  62. A. Quartarone, Stability of solutions of some mesoscopic equations in Biomathematics. Ph.D. Thesis. University of Messina. Department of Mathematics and Informatics. Messina (2013)

    Google Scholar 

  63. L. Saint–Raymond, Hydrodynamic Limits of the Boltzmann Equation, 1 Lecture Notes Mathematics, vol. 1971, (Springer, Berlin 2009)

    Google Scholar 

  64. Z. Szymańska, Analysis of immunotherapy models in the context of cancer dynamics. Int. J. Appl. Math. Comput. Sci. 13, 407–418 (2003)

    MathSciNet  MATH  Google Scholar 

  65. Z. Szymańska, C. Morales Rodrigo, M., Lachowicz, M.A.J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions. Math. Models Methods Appl. Sci. 19, 257–281 (2009)

    Google Scholar 

  66. M.W. Teng, J.B. Swann, C.M. Koebel, R.D. Schreiber, M.J. Smyth, Immune–mediated dormancy: an equilibrium with cancer. J. Leukoc. Biol. 84, 988–983 (2008)

    Article  Google Scholar 

  67. C.M. Vajdic, van Leeuwen, M.T.: Cancer incidence and risk factors after solid organ transplantation. Int. J. Cancer. 125, 1747–1754 (2009)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  69. R.A. Weinberg, The Biology of Cancer. Garland Sciences (Taylor & Francis Group, New York 2007)

    Google Scholar 

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Acknowledgment

M.D. acknowledges a support from the National Group for Mathematical Physics through the grant Visiting Professors 2013. M.L. acknowledges a support from the National Science Centre of Poland through grant N N201 605640. Z.S. acknowledges a support from the National Science Centre of Poland through grant DEC-2011/01/D/ST1/04133.

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Authors and Affiliations

  1. Department of Civil, Computer, Construction and Environmental Engineering and of Applied Mathematics (DICIEAMA), University of Messina, Contrada Di Dio, Vill. S. Agata, 98166, Messina, Italy

    Marina Dolfin

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

    Mirosław Lachowicz

  3. ICM, University of Warsaw, ul. Pawińskiego 5a, 02–106, Warsaw, Poland

    Zuzanna Szymańska

Authors
  1. Marina Dolfin
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  2. Mirosław Lachowicz
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  3. Zuzanna Szymańska
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Correspondence to Mirosław Lachowicz .

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Editors and Affiliations

  1. International Prevention Research Institute, Lyon, France

    Alberto d'Onofrio

  2. Istituto di Analisi dei Sistemi ed Informatica "Antonio Ruberti"—CNR, Rome, Italy

    Alberto Gandolfi

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Dolfin, M., Lachowicz, M., Szymańska, Z. (2014). A General Framework for Multiscale Modeling of Tumor–Immune System Interactions. In: d'Onofrio, A., Gandolfi, A. (eds) Mathematical Oncology 2013. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-0458-7_5

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