Skip to main content
Springer Nature Link
Log in

Optimal Control for a Mathematical Model of Glioma Treatment with Oncolytic Therapy and TNF-\(\alpha \) Inhibitors

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

A mathematical model for combination therapy of glioma with oncolytic therapy and TNF-\(\alpha \) inhibitors is analyzed as an optimal control problem. In the objective, a weighted average between the tumor volume and the total amount of viruses given is minimized. It is shown that optimal controls representing the virus administration are generically of the bang-bang type, i.e., the virus should be applied at maximal allowed dose with possible rest periods. On the other hand, optimal controls representing the dosage of TNF-\(\alpha \) inhibitors follow a continuous regimen of concatenations between pieces that lie on the boundary and in the interior of the control set.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Csatary, L.K., Gosztonyi, G., Szeberenyi, J., Fabian, Z., Liszka, V., Bodey, B., Csatary, C.M.: MTH-68/H oncolytic viral treatment in human highgrade gliomas. J. Neurooncol. 67, 83–93 (2004)

    Article  Google Scholar 

  2. Lorence, R.M., Pecora, A.L., Major, P.P., Hotte, S.J., Laurie, S.A., Roberts, M.S., Groene, W.S., Bamat, M.K.: Overview of phase I studies of intravenous administration of PV701, an oncolytic virus. Curr. Opin. Mol. Ther. 5, 618–624 (2003)

    Google Scholar 

  3. Markert, J.M.: Conditionally replicating herpes simplex virus mutant, G207 for the treatment of malignant glioma: results of a phase I trial. Gene Ther. 7, 867–874 (2000)

    Article  Google Scholar 

  4. Chiocca, E.A.: Oncolytic viruses. Nat. Rev. Cancer 2, 938–50 (2002)

    Article  Google Scholar 

  5. Fulci, G., Breymann, L., Gianni, D., Kurozomi, K., Rhee, S.S., Yu, J., Kaur, B., Louis, D.N., Weissleder, R., Caligiuri, M.A., Chiocca, E.A.: Cyclophosphamide enhances glioma virotherapy by inhibiting innate immune responses. Proc. Natl. Acad. Sci. PNAS 103(34), 12873–12878 (2006)

    Article  Google Scholar 

  6. Friedman, A., Tian, J., Fulci, G., Chioca, E., Wang, J.: Glioma virotherapy: effects of innate immune suppression and increased viral replication capacity. Cancer Res. 66(4), 2314–2319 (2006)

    Article  Google Scholar 

  7. Auffinger, B., Ahmed, A.U., Lesniak, M.S.: Oncolytic virotherapy for malignant glioma: translating laboratory insights into clinical practice. Front. Oncol. (2013). https://doi.org/10.3389/fonc.2013.00032

  8. Ratajczyk, E., Ledzewicz, U., Leszczyński, M., Friedman, A.: The role of TNF-alpha inhibitor in Glioma virotherapy: a mathematical model. Math. Biosci. Eng. 14(1), 305–319 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ratajczyk, E., Ledzewicz, U., Leszczyński, M., Schättler, H.: Treatment of Glioma with virotherapy and TNF-\(\alpha \) inhibitors: analysis as a dynamical system. Discrete Contin. Dyn. Syst. Ser. B, 23(1), to appear (2018)

  10. Bajzer, Z., Carr, T., Josić, K., Russell, S.J., Dingli, D.: Modeling of cancer virotherapy with recombinant viruses. J. Theor. Biol. 252(1), 109–122 (2008)

    Article  MathSciNet  Google Scholar 

  11. Crivelli, J.J., Földes, J., Kim, P.S., Wares, J.: A mathematical model for cell-cycle specific cancer virotherapy. J. Biol. Dyn. 6, 104–120 (2012)

    Article  MathSciNet  Google Scholar 

  12. El-alami Laaroussi, A., El Hia, M.E., Rachik, M., Benlahmar, E., Rachik, Z.: Analysis of a mathematical model for treatment of cancer with oncolytic virotherapy. Appl. Math. Sci 8, 929–940 (2014)

    Article  Google Scholar 

  13. 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. App. Math. 70, 1953–1976 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wares, J.R., Crivelli, J.J., Yun, C., Choi, I., Gevertz, J.L., Kim, P.S.: Treatment strategies for combining immunostimulatory oncolytic virus therapeutics with dendritic cell injections. Math. Biosci. Eng. 12(6), 1237–1256 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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)

    Article  MATH  Google Scholar 

  16. dOnofrio, A.: A general framework for modelling tumor-immune system competition and immunotherapy. Phys. D 208, 202–235 (2005)

    Google Scholar 

  17. Eisen, M.: Mathematical Models in Cell Biology and Cancer Chemotherapy Lecture Notes in Biomathematics, vol. 30. Springer, Berlin (1979)

    Book  MATH  Google Scholar 

  18. Swan, G.W.: General applications of optimal control theory in cancer chemotherapy, IMA. J. Math. Appl. Med. Biol. 5, 303–316 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  19. Swan, G.W.: Role of optimal control in cancer chemotherapy. Math. Biosci. 101, 237–284 (1990)

    Article  MATH  Google Scholar 

  20. Swierniak, A.: Optimal treatment protocols in leukemia - modelling the proliferation cycle. In: Proceedings of 12th IMACS World Congress, Paris, 4, pp. 170–172 (1988)

  21. Swierniak, A., Ledzewicz, U., Schättler, H.: Optimal control for a class of compartmental models in cancer chemotherapy. Int. J. Appl. Math. Comp. Sci. 13, 357–368 (2003)

    MathSciNet  MATH  Google Scholar 

  22. Ledzewicz, U., Schättler, H.: Analysis of a cell-cycle specific model for cancer chemotherapy. J. Biol. Syst. 10, 183–206 (2002)

    Article  MATH  Google Scholar 

  23. Ledzewicz, U., Schättler, H.: Optimal bang-bang controls for a 2-compartment model in cancer chemotherapy. J. Optim. Theory Appl. 114, 609–637 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ledzewicz, U., Schättler, H.: Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM J. Control Optim. 46(3), 1052–1079 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ledzewicz, U., Schättler, H.: A synthesis of optimal controls for a model of tumor growth under angiogenic inhibitors. In: Proceedings of 44th IEEE Conference on Decision and Control, Sevilla, Spain, pp. 934–939 (2005)

  26. dOnofrio, A., Ledzewicz, U., Maurer, H., Schättler, H.: On optimal delivery of combination therapy for tumors. Math. Biosci. 222, 13–26 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ledzewicz, U., Schättler, H.: Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments. J. Optim. Theory Appl. 153, 195224 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. dOnofrio, A., Ledzewicz, U., Schättler, H.: On the dynamics of tumor immune system interactions and combined chemo- and immunotherapy. In: dOnofrio, A., Cerrai, P., Gandolfi, A. (eds.) New Challenges for Cancer Systems Biomedicine, SIMAI Springer series, Vol. 1, pp. 249–266 (2012)

  29. de Pillis, L.G., Radunskaya, A.: A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. J. Theor. Med. 3, 79–100 (2001)

    Article  MATH  Google Scholar 

  30. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Macmillan, New York (1964)

    Google Scholar 

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

    MATH  Google Scholar 

  32. Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control. American Institute of Mathematical Sciences, San Jose (2007)

    MATH  Google Scholar 

  33. Schättler, H., Ledzewicz, U.: Geometric Optimal Control. Springer, New York (2012)

    Book  MATH  Google Scholar 

  34. Bryson, A.E., Ho, Y.C.: Applied Optimal Control. Hemisphere Publ. Co., Washington (1975)

    Google Scholar 

  35. Ratajczyk, E.: Dynamics and Optimal Control of a Mathematical Model for Combination Therapy of Glioma. Ph.D. thesis at Lodz University of Technology, Lodz, Poland (2017)

  36. Golubitsky, M., Guillemin, V.: Stable Mappings and their Singularities. Springer, New York (1973)

    Book  MATH  Google Scholar 

  37. Schättler, H., Ledzewicz, U.: Optimal Control for Mathematical Models of Cancer Therapies. Springer, New York (2015)

    Book  MATH  Google Scholar 

  38. Leszczyński, M., Ratajczyk, E., Ledzewicz, U., Schättler, H.: Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics. Opuscula Math. 37(3), 403–419 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

U. Ledzewicz’s research was partially supported by the National Science Foundation under collaborative research Grants No. DMS 1311733. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Lodz University of Technology, 90-924, Lodz, Poland

    Elzbieta Ratajczyk & Urszula Ledzewicz

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

    Urszula Ledzewicz

  3. Department of Electrical and Systems Engineering, Washington University, St. Louis, MO, 63130-4899, USA

    Heinz Schättler

Authors
  1. Elzbieta Ratajczyk
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Urszula Ledzewicz
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Heinz Schättler
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Urszula Ledzewicz.

Additional information

Alberto d’Onofrio.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ratajczyk, E., Ledzewicz, U. & Schättler, H. Optimal Control for a Mathematical Model of Glioma Treatment with Oncolytic Therapy and TNF-\(\alpha \) Inhibitors. J Optim Theory Appl 176, 456–477 (2018). https://doi.org/10.1007/s10957-018-1218-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-018-1218-4

Keywords

Mathematics Subject Classification