Skip to main content
Book cover

New Challenges for Cancer Systems Biomedicine pp 249–266Cite as

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

  • Chapter

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.

Keywords

  • Optimal Control Problem
  • Stable Manifold
  • Singular Control
  • Control Trajectory
  • Immune Boost

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-88-470-2571-4_13
  • Chapter length: 18 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   109.00
Price excludes VAT (USA)
  • ISBN: 978-88-470-2571-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   139.99
Price excludes VAT (USA)
Hardcover Book
USD   139.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 1
Fig. 2
Fig. 3

References

  1. Agarwala, S.A. (Guest Editor): New Applications of Cancer Immunotherapy. Sem. Oncol. 29(3), Special Issue, Suppl. 7 (2003)

    Google Scholar 

  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)

    CrossRef  Google Scholar 

  3. Bonnard, B., Chyba, M.: Singular Trajectories and their Role in Control Theory. Mathématiques & Applications, vol. 40, Springer, Paris (2003)

    Google Scholar 

  4. Caravagna, G., d’Onofrio, A., Milazzo, P., Barbuti, R.: Antitumour Immune Surveillance Through Stochastic Oscillations. J. Theor. Biol. 265, 336–345 (2010)

    CrossRef  MathSciNet  Google Scholar 

  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)

    CrossRef  Google Scholar 

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

    Google Scholar 

  10. d’Onofrio, A.: Bounded-noise-induced transitions in a tumor-immune system interplay. Phys. Rev. E 81, (2010), 021923 (2010)

    Google Scholar 

  11. Dunn, G.P., Old L.J., Schreiber, R.D.: The three ES of cancer immunoediting. Ann. Rev. Immunol. 22, 322–360 (2004)

    CrossRef  Google Scholar 

  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)

    CrossRef  MATH  Google Scholar 

  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)

    CrossRef  MathSciNet  Google Scholar 

  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)

    CrossRef  Google Scholar 

  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. Kennedy, B.J.: Cyclic leukocyte oscillations in chronic myelogenous leukemia during hydroxyurea therapy. Blood 35, (1970), 751–760 (1970)

    Google Scholar 

  17. Kindt, T.J., Osborne, B.A., Goldsby, R.A.: Kuby Immunology. W.H. Freeman, New York (2006)

    Google Scholar 

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

    CrossRef  MATH  Google Scholar 

  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)

    CrossRef  Google Scholar 

  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. 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)

    CrossRef  MATH  Google Scholar 

  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. 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. 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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  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)

    CrossRef  MATH  Google Scholar 

  26. Norton, L.: A Gompertzian model of human breast cancer growth. Cancer Res. 48, (1988), 7067–7071 (1988)

    Google Scholar 

  27. Pardoll, D.: Does the immune system see tumors as foreign or self? Ann. Rev. Immunol. 21, (2003), 807–839 (2003)

    Google Scholar 

  28. Peckham, M., Pinedo, H.M., Veronesi, U.: The Oxford Textbook of Oncology. Oxford University Press, Oxford (1995)

    Google Scholar 

  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. Schättler, H., Ledzewicz, U.: Geometric Optimal Control: Theory, Methods and Examples. Springer, Heidelberg (2012)

    CrossRef  MATH  Google Scholar 

  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 appear

    Google Scholar 

  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. 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)

    CrossRef  MathSciNet  Google Scholar 

  34. Stepanova, N.V.: Course of the immune reaction during the development of a malignant tumour. Biophysics 24, 917–923 (1980)

    Google Scholar 

  35. Stewart, T.J., Abrams, S.I.: How tumours escape mass destruction. Oncogene 27, 5894–5903 (2008)

    CrossRef  Google Scholar 

  36. Swann, J.B., Smyth, M.J.: Immune surveillance of tumors. J. Clin. Inv. 117, 1137–1146 (2007)

    CrossRef  Google Scholar 

  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)

    CrossRef  MathSciNet  Google Scholar 

  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. Wheldon, T.E.: Mathematical Models in Cancer Research. Hilger Publishing, BostonPhiladelphia (1988)

    MATH  Google Scholar 

  40. Whiteside, T.L.: Tumor-induced death of immune cells: its mechanisms and consequences. Sem. Canc. Biol. 12, 43–50 (2002)

    CrossRef  Google Scholar 

Download references

Acknowledgements

This material is based upon research supported by the National Science Foundation under collaborative research grants DMS 1008209 and 1008221 (U.L. and H.S.), and by the EU project “p-Medicine: Personalized Medicine” (FP7-ICT-2009.5.3-270089) (A. d’O.). We also would like to thank our students Mohamad Naghnaeian and Mozhdeh Faraji for carrying out the numerical computations and making the figures used in the paper.

Author information

Authors and Affiliations

  1. Department of Experimental Oncology, European Institute of Oncology, Via Ripamonti 435, 20141, Milan, Italy

    Alberto d’Onofrio

  2. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, 62026-1653, Edwardsville, IL, USA

    Urszula Ledzewicz

  3. Department of Electrical and Systems Engineering, Washington University, 631304899, St. Louis, MO, USA

    Heinz Schättler

Authors
  1. Alberto d’Onofrio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Urszula Ledzewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Heinz Schättler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Urszula Ledzewicz .

Editor information

Editors and Affiliations

  1. Department of Experimental Oncology, European Institute of Oncology, Milan, Italy

    Alberto d’Onofrio

  2. Department of Mathematics, University of Pisa, Italy

    Paola Cerrai

  3. Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti“ — CNR, Rome, Italy

    Alberto Gandolfi

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Italia

About this chapter

Cite this chapter

d’Onofrio, A., Ledzewicz, U., Schättler, H. (2012). On the Dynamics of Tumor-Immune System Interactions and Combined Chemo- and Immunotherapy. In: d’Onofrio, A., Cerrai, P., Gandolfi, A. (eds) New Challenges for Cancer Systems Biomedicine. SIMAI Springer Series. Springer, Milano. https://doi.org/10.1007/978-88-470-2571-4_13

Download citation