Skip to main content
SpringerLink
Log in

Optimal Control of Cancer Treatments: Mathematical Models for the Tumor Microenvironment

  • Chapter
Analysis and Geometry in Control Theory and its Applications

Part of the book series: Springer INdAM Series ((SINDAMS,volume 11))

Abstract

Mathematical models for cancer treatment are analyzed as optimal control problems when selected aspects of the tumor microenvironment are taken into account. The significance of treatment protocols that administer agents at less than maximum tolerated dose rates is analyzed in this context. When angiogenic signaling or tumor immune system interactions are included in the model, singular controls that administer therapeutic agents at less than maximum dose become optimal. Their relations to metronomic dosing are discussed.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Anderson, A., Chaplain, M.: Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60, 857–899 (1998)

    Article  MATH  Google Scholar 

  2. André, N., Carré, M., Pasquier, E.: Metronomics: towards personalized chemotherapy? Nat. Rev. Clin. Oncol. 11(7), 413–31 (2014)

    Article  Google Scholar 

  3. André, N., Padovani, L., Pasquier, E.: Metronomic scheduling of anticancer treatment: the next generation of multitarget therapy? Fut. Oncol. 7(3), 385–394 (2011)

    Article  Google Scholar 

  4. Arakelyan, L., Vainstain, V., Agur, Z.: A computer algorithm describing the process of vessel formation and maturation, and its use for predicting the effects of anti-angiogenic and anti-maturation therapy on vascular tumour growth. Angiogenesis 5, 203–214 (2003)

    Article  Google Scholar 

  5. Bellomo, N., Preziosi, L.: Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math. Comput. Modell. 32, 413–452 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bocci, G., Nicolaou, K., Kerbel, R.S.: Protracted low-dose effects on human endothelial cell proliferation and survival in vitro reveal a selective antiangiogenic window for various chemotherapeutic drugs. Cancer Res. 62, 6938–6943 (2002)

    Google Scholar 

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

    Google Scholar 

  8. Bressan, A., Piccoli, A.: Introduction to the Mathematical Theory of Control. American Institute of Mathematical Sciences, Springfield (2007)

    MATH  Google Scholar 

  9. Browder, T., Butterfield, C.E., Kräling, B.M., Shi, B., Marshall, B., O’Reilly, M., Folkman, J.: Antiangiogenic scheduling of chemotherapy improves efficacy against experimental drug-resistant cancer. Cancer Res. 60, 1878–1886 (2000)

    Google Scholar 

  10. Cesari, L.: Optimization – Theory and Applications. Springer, New York (1983)

    Book  MATH  Google Scholar 

  11. Davis, S., Yancopoulos, G.D.: The angiopoietins: Yin and Yang in angiogenesis. Cur. Topics Microbiol. Immunol. 237, 173–185 (1999)

    Google Scholar 

  12. Ergun, A., Camphausen, K., Wein, L.M.: Optimal scheduling of radiotherapy and angiogenic inhibitors. Bull. Math. Biol. 65, 407–424 (2003)

    Article  Google Scholar 

  13. Forys, U., Keifetz, Y., Kogan, Y.: Critical-point analysis for three-variable cancer angiogenesis models. Math. Biosci. Eng. (MBE) 2(3), 511–525 (2005)

    Google Scholar 

  14. 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 

  15. Friedman, A.: Cancer as multifaceted disease. Math. Model. Nat. Phenom. 7, 1–26 (2012)

    Article  Google Scholar 

  16. Grigorieva, E.V., Khailov, E.N., Bondarenko, N., Korobeinikov, A.: Modeling and optimal control for antiretroviral therapy. J. Biol. Syst. 22(2) (2014). doi:10.1142/s0218339014400026

    Google Scholar 

  17. Hahnfeldt, P., Panigrahy, D., Folkman, J., Hlatky, L.: Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res. 59, 4770–4775 (1999)

    Google Scholar 

  18. Hahnfeldt, P., Folkman, J., Hlatky, L.: Minimizing long-term burden: the logic for metronomic chemotherapy dosing and its angiogenic basis. J. Theor. Biol. 220, 545–554 (2003)

    Article  Google Scholar 

  19. Hanahan, D., Bergers, G., Bergsland, E.: Less is more, regularly: metronomic dosing of cytotoxic drugs can target tumor angiogenesis in mice. J. Clin. Invest. 105, 1045–1047 (2000)

    Article  Google Scholar 

  20. Hao, Y.B., Yi, S.Y., Ruan, J., Zhao, L., Nan, K.J.: New insights into metronomic chemotherapy-induced immunoregulation. Cancer Lett. 354(2), 220–226 (2014)

    Article  Google Scholar 

  21. Jain, R.K.: Normalizing tumor vasculature with anti-angiogenic therapy: a new paradigm for combination therapy. Nat. Med. 7, 987–989 (2001)

    Article  Google Scholar 

  22. Jain, R.K., Munn, L.L.: Vascular normalization as a rationale for combining chemotherapy with antiangiogenic agents. Princ. Pract. Oncol. 21, 1–7 (2007)

    Google Scholar 

  23. Kamen, B., Rubin, E., Aisner, J., Glatstein, E.: High-time chemotherapy or high time for low dose? J. Clin. Oncol. 18, Editorial, 2935–2937 (2000)

    Google Scholar 

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

    Google Scholar 

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

    Article  MATH  Google Scholar 

  26. Klagsburn, M., Soker, S.: VEGF/VPF: the angiogenesis factor found? Curr. Biol. 3, 699–702 (1993)

    Article  Google Scholar 

  27. Klement, G., Baruchel, S., Rak, J., Man, S., Clark, K., Hicklin, D.J., Bohlen, P., Kerbel, R.S.: Continuous low-dose therapy with vinblastine and VEGF receptor-2 antibody induces sustained tumor regression without overt toxicity. J. Clin. Invest. 105, R15–R24 (2000)

    Article  Google Scholar 

  28. 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)

    Article  MATH  Google Scholar 

  29. Ledzewicz, U., Bratton, K., Schättler, H.: A 3-compartment model for chemotherapy of heterogeneous tumor populations. Acta Appl. Math. (2014). doi:10.1007/s10440-014-9952-6

    MATH  Google Scholar 

  30. Ledzewicz, U., Faraji Mosalman, M.S., Schättler, H.: Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discret. Cont. Dyn. Syst. Ser. B 18, 1031–1051 (2013). doi:10.3934/dcdsb.2013.18.1031.

    Article  MATH  Google Scholar 

  31. Ledzewicz, U., Munden, J., Schättler, H.: Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models. Discret. Cont. Dyn. Syst. Ser. B 12, 415–438 (2009)

    Article  MATH  Google Scholar 

  32. 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). doi:10.1007/s00285-011-0424-6

    Article  MathSciNet  MATH  Google Scholar 

  33. Ledzewicz, U., d’Onofrio, A., Schättler, H.: Tumor development under combination treatments with anti-angiogenic therapies. In: Ledzewicz, U., Schättler, H., Friedman, A., Kashdan, E. (eds.) Mathematical Methods and Models in Biomedicine. Lecture Notes on Mathematical Modeling in the Life Sciences, pp. 301–327. Springer, New York (2012)

    Google Scholar 

  34. 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 

  35. Ledzewicz, U., Schättler, H.: Drug resistance in cancer chemotherapy as an optimal control problem. Discret. Cont. Dyn. Syst. Ser. B 6, 129–150 (2006)

    MATH  Google Scholar 

  36. Ledzewicz, U., Schättler, H.: Anti-angiogenic therapy in cancer treatment as an optimal control problem. SIAM J. Control Optim. 46, 1052–1079 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ledzewicz, U., Schättler, H.: Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis. J. Theory Biol. 252, 295–312 (2008)

    Article  Google Scholar 

  38. Ledzewicz, U., Schättler, H.: On the optimality of singular controls for a class of mathematical models for tumor antiangiogenesis. Discret. Cont. Dyn. Syst. Ser. B 11, 691–715 (2009)

    Article  MATH  Google Scholar 

  39. d’Onofrio, A.: A general framework for modelling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedial inferences. Phys. D 208, 202–235 (2005)

    MathSciNet  Google Scholar 

  40. 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)

    Article  MathSciNet  MATH  Google Scholar 

  41. d’Onofrio, A.: Fractal growth of tumors and other cellular populations: linking the mechanistic to the phenomenological modeling and vice versa. Chaos Solitons Fractals 41, 875–880 (2009)

    Article  Google Scholar 

  42. d’Onofrio, A., Gandolfi, A.: Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. Math. Biosci. 191, 159–184 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. d’Onofrio, A., Gandolfi, A., Rocca, A.: The dynamics of tumour-vasculature interaction suggests low-dose, time-dense antiangiogenic schedulings. Cell Prolif. 42, 317–329 (2009)

    Article  Google Scholar 

  44. d’Onofrio, A., Ledzewicz, U., Maurer, H., Schättler, H.: On optimal delivery of combination therapy for tumors. Math. Biosci. 222, 13–26 (2009). doi:10.1016/j.mbs.2009.08.004

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  46. Pasquier, E., Kavallaris, M., André, N.: Metronomic chemotherapy: new rationale for new directions. Nat. Rev. Clin. Oncol. 7, 455–465 (2010)

    Article  Google Scholar 

  47. 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 

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

    Google Scholar 

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

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  51. Schättler, H., Ledzewicz, L., Amini, B.: Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy. J. Math. Biol. (published online 19 June 2015). doi:10.1007s/00285-015-0907y

    Google Scholar 

  52. Schättler, H., Ledzewicz, U., Cardwell, B.: Robustness of optimal controls for a class of mathematical models for tumor anti-angiogenesis. Math. Biosci. Eng. 8, 355–369 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  55. Swierniak, A.: Cell cycle as an object of control. J. Biol. Syst. 3, 41–54 (1995)

    Article  Google Scholar 

  56. Swierniak, A.: Direct and indirect control of cancer populations. Bull. Pol. Acad. Sci. Technol. Sci. 56, 367–378 (2008)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  58. Swierniak, A., Smieja, J.: Cancer chemotherapy optimization under evolving drug resistance. Nonlinear Anal. 47, 375–386 (2000)

    Article  MathSciNet  Google Scholar 

  59. Vinter, R.B.: Optimal Control Theory. Birkhäuser, Boston (2000)

    Google Scholar 

  60. 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)

    Article  Google Scholar 

  61. Weitman, S.D., Glatstein, E., Kamen, B.A.: Back to the basics: the importance of concentration × time in oncology. J. Clin. Oncol. 11, 820–821 (1993)

    Google Scholar 

Download references

Acknowledgements

This material is based upon work supported by the National Science Foundation under collaborative research Grants Nos. DMS 1311729/1311733. 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. Department of Electrical and Systems Engineering, Washington University, St. Louis, MO, 63130, USA

    Heinz Schättler

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

    Urszula Ledzewicz

Authors
  1. Heinz Schättler
    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

Corresponding author

Correspondence to Heinz Schättler .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Université de Bretagne Occidentale, Brest, France

    Piernicola Bettiol

  2. Department of Mathematics, Università di Roma "Tor Vergata", Roma, Italy

    Piermarco Cannarsa

  3. Department of Mathematics, Università di Padova, Padova, Italy

    Giovanni Colombo

  4. Department of Mathematics, Università di Padova, Padova, Italy

    Monica Motta

  5. Department of Mathematics, Università di Padova, Padova, Italy

    Franco Rampazzo

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Schättler, H., Ledzewicz, U. (2015). Optimal Control of Cancer Treatments: Mathematical Models for the Tumor Microenvironment. In: Bettiol, P., Cannarsa, P., Colombo, G., Motta, M., Rampazzo, F. (eds) Analysis and Geometry in Control Theory and its Applications. Springer INdAM Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-06917-3_8

Download citation

Publish with us

Policies and ethics

Search

Navigation