On the Dynamics of Tumor-Immune System Interactions and Combined Chemo- and Immunotherapy

  • Alberto d’Onofrio
  • Urszula Ledzewicz
  • Heinz Schättler
Part of the SIMAI Springer Series book series (SEMA SIMAI)

Abstract

Tumor-immune system interplay is extremely complex, and, as such, it represents a big challenge for mathematical oncology. Here we investigate a simple general family of models for this important interplay by considering both the delivery of a cytotoxic chemotherapy and of immunotherapy. Then methods of geometrical optimal control are applied to a special case (the Stepanova model) in order to infer (under suitable constraints) the best combination of drugs scheduling to transfer — through therapy —the system from an initial condition in the malignant region of the state space into a benign region. Our findings suggest that chemotherapy is always needed first to reduce a large tumor volume before the immune system can become effective.

References

  1. 1.
    Agarwala, S.A. (Guest Editor): New Applications of Cancer Immunotherapy. Sem. Oncol. 29(3), Special Issue, Suppl. 7 (2003)Google Scholar
  2. 2.
    Bellomo, N., Delitala, M.: From the mathematical kinetic, and stochastic game theory for active particles to modelling mutations, onset, progression and immune competition of cancer cells. Phys. Life Rev. 5, 183–206 (2008)CrossRefGoogle Scholar
  3. 3.
    Bonnard, B., Chyba, M.: Singular Trajectories and their Role in Control Theory. Mathématiques & Applications, vol. 40, Springer, Paris (2003)Google Scholar
  4. 4.
    Caravagna, G., d’Onofrio, A., Milazzo, P., Barbuti, R.: Antitumour Immune Surveillance Through Stochastic Oscillations. J. Theor. Biol. 265, 336–345 (2010)CrossRefGoogle Scholar
  5. 5.
    de Pillis, L.G., Radunskaya, A.E., Wiseman, C.L.: A validated mathematical model of cellmediated immune response to tumor growth. Cancer Res.65, 7950–7958 (2005)CrossRefGoogle Scholar
  6. 6.
    d’Onofrio, A.: A general framework for modelling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedial inferences. Physica D 208, 202–235 (2005)Google Scholar
  7. 7.
    d’Onofrio, A.: The role of the proliferation rate of effectors in the tumor-immune system competition. Math. Mod. Meth. Appl. Sci. 16, 1375–1401 (2006)Google Scholar
  8. 8.
    d’Onofrio, A.: Tumor evasion from immune control: strategies of a MISS to become a MASS. Chaos, Solitons and Fractals 31, 261–268 (2007)Google Scholar
  9. 9.
    d’Onofrio, A.: Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy. Math. Comput. Modelling 47, 614–637 (2008)Google Scholar
  10. 10.
    d’Onofrio, A.: Bounded-noise-induced transitions in a tumor-immune system interplay. Phys. Rev. E 81, (2010), 021923 (2010)Google Scholar
  11. 11.
    Dunn, G.P., Old L.J., Schreiber, R.D.: The three ES of cancer immunoediting. Ann. Rev. Immunol. 22, 322–360 (2004)CrossRefGoogle Scholar
  12. 12.
    Forys, U., Waniewski J., Zhivkov, P.: Anti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapy. J. Biol. Syst. 14, 13–30 (2006)CrossRefMATHGoogle Scholar
  13. 13.
    Guiot, C., Degiorgis, P.G., Delsanto, P.P., Gabriele, P., Deisboecke T.S.: Does tumor growth follow a “universal law”? J. Theor. Biol. 225, 147–151 (2003)CrossRefGoogle Scholar
  14. 14.
    Hart, D., Shochat, E., Agur, Z.: The growth law of primary breast cancer as inferred from mammography screening trials data. Br. J. Cancer 78, 382–387 (1999)CrossRefGoogle Scholar
  15. 15.
    Kaminski, J.M., Summers, J.B., Ward, M.B., Huber, M.R., Minev, B.: Immunotherapy and prostate cancer. Canc. Treat. Rev. 29, (2004), 199–209 (2004)Google Scholar
  16. 16.
    Kennedy, B.J.: Cyclic leukocyte oscillations in chronic myelogenous leukemia during hydroxyurea therapy. Blood 35, (1970), 751–760 (1970)Google Scholar
  17. 17.
    Kindt, T.J., Osborne, B.A., Goldsby, R.A.: Kuby Immunology. W.H. Freeman, New York (2006)Google Scholar
  18. 18.
    Kirschner, D., Panetta, J.C.: Modeling immunotherapy of the tumor-immune interaction. J. Math. Biol. 37, 235–252 (1998)CrossRefMATHGoogle Scholar
  19. 19.
    Koebel, C.M., Vermi, W., Swann, J.B., Zerafa, N., Rodig, S.J., Old, L.J., Smyth, M.J., Schreiber, R.D.: Adaptive immunity maintains occult cancer in an equilibrium state. Nature 450, 903–907(2007)CrossRefGoogle Scholar
  20. 20.
    Kogan, Y., Forys, U., Shukron, O., Kronik, N., Agur, Z.: Cellular immunotherapy for high grade gliomas: mathematical analysis deriving efficacious infusion rates based on patient requirements. SIAM J. Appl. Math. 70, (2010), 1953–1976 (2010)Google Scholar
  21. 21.
    Kuznetsov, V.A., Makalkin, I.A., Taylor, M.A., Perelson, A.S.: Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56, 295–321 (1994)CrossRefMATHGoogle Scholar
  22. 22.
    Ledzewicz, U., d’Onofrio, A., Schattler, H.: Tumor development under combination treatments with anti-angiogenic therapies. in: Ledzewicz, U., Schattler, H., Friedman, A., Kashdan, E. (eds.) Mathematical Methods and Models in Biomedicine, Lecture Notes on Mathematical Modeling in the Life Sciences, Vol. 1, pp. 301–327. Springer, Heidelberg (2012)Google Scholar
  23. 23.
    Ledzewicz, U., Naghnaeian, M., Schattler, H.: Dynamics of tumor-immune interactions under treatment as an optimal control problem. Proc. of the 8th AIMS Conf., Dresden, Germany, pp. 971–980(2010)Google Scholar
  24. 24.
    Ledzewicz, U., Naghnaeian, M., Schättler, H.: Optimal response to chemotherapy for a mathematical model of tumor-immune dynamics. J. Math. Biol. 64, 557–577 (2012)CrossRefMATHGoogle Scholar
  25. 25.
    Matzavinos, A., Chaplain, M., Kuznetsov, V.A.: Mathematical modelling of the spatiotemporal response of cytotoxic T-lymphocytes to a solid tumour. Math. Med. Biol. 21, (2004), 1-34(2004)CrossRefMATHGoogle Scholar
  26. 26.
    Norton, L.: A Gompertzian model of human breast cancer growth. Cancer Res. 48, (1988), 7067–7071 (1988)Google Scholar
  27. 27.
    Pardoll, D.: Does the immune system see tumors as foreign or self? Ann. Rev. Immunol. 21, (2003), 807–839 (2003)Google Scholar
  28. 28.
    Peckham, M., Pinedo, H.M., Veronesi, U.: The Oxford Textbook of Oncology. Oxford University Press, Oxford (1995)Google Scholar
  29. 29.
    Rao, A.V., Benson, D.A., Huntington, G.T., Francolin, C., Darby, C.L., Patterson M.A.: User’s Manual for GPOPS: A MATLAB Package for Dynamic Optimization Using the Gauss Pseudospectral Method. University of Florida Report, http://www.gpops.org (2008)Google Scholar
  30. 30.
    Schättler, H., Ledzewicz, U.: Geometric Optimal Control: Theory, Methods and Examples. Springer, Heidelberg (2012)CrossRefMATHGoogle Scholar
  31. 31.
    Schättler, H., Ledzewicz, U., Faraji, M.: Optimal controls for a mathematical model of tumorimmune interactions under chemotherapy with immune boost. Disc. Cont. Dyn. Syst. Ser. B, (2013), to appearGoogle Scholar
  32. 32.
    Schmielau, J., Finn, O.J.: Activated granulocytes and granulocyte-derived hydrogen peroxide are the underlying mechanism of suppression of T-cell function in advanced cancer patients. Cancer Res. 61, 4756–4760 (2001)Google Scholar
  33. 33.
    Skipper, H.E.: On mathematical modeling of critical variables in cancer treatment (goals: better understanding of the past and better planning in the future). Bull. Math. Biol. 48, 253–278 (1986)CrossRefGoogle Scholar
  34. 34.
    Stepanova, N.V.: Course of the immune reaction during the development of a malignant tumour. Biophysics 24, 917–923 (1980)Google Scholar
  35. 35.
    Stewart, T.J., Abrams, S.I.: How tumours escape mass destruction. Oncogene 27, 5894–5903 (2008)CrossRefGoogle Scholar
  36. 36.
    Swann, J.B., Smyth, M.J.: Immune surveillance of tumors. J. Clin. Inv. 117, 1137–1146 (2007)CrossRefGoogle Scholar
  37. 37.
    de Vladar, H.P., González, J.A.: Dynamic response of cancer under the influence of immunological activity and therapy. J. Theor. Biol. 227, 335–348 (2004)CrossRefGoogle Scholar
  38. 38.
    Vodopick, H., Rupp, E.M., Edwards, C.L., Goswitz, F.A., Beauchamp, J.J.: Spontaneous cyclic leukocytosis and thrombocytosis in chronic granulocytic leukemia. New Engl. J. Med. 286, (1972), 284–290 (1972)Google Scholar
  39. 39.
    Wheldon, T.E.: Mathematical Models in Cancer Research. Hilger Publishing, BostonPhiladelphia (1988)MATHGoogle Scholar
  40. 40.
    Whiteside, T.L.: Tumor-induced death of immune cells: its mechanisms and consequences. Sem. Canc. Biol. 12, 43–50 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2012

Authors and Affiliations

  • Alberto d’Onofrio
    • 1
  • Urszula Ledzewicz
    • 2
  • Heinz Schättler
    • 3
  1. 1.Department of Experimental OncologyEuropean Institute of OncologyMilanItaly
  2. 2.Department of Mathematics and StatisticsSouthern Illinois University EdwardsvilleEdwardsvilleUSA
  3. 3.Department of Electrical and Systems EngineeringWashington UniversitySt. LouisUSA

Personalised recommendations