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Journal of Optimization Theory and Applications

, Volume 180, Issue 1, pp 321–340 | Cite as

Optimal Combined Radio- and Anti-Angiogenic Cancer Therapy

  • Urszula LedzewiczEmail author
  • Helmut Maurer
  • Heinz Schättler
Article
  • 43 Downloads

Abstract

A mathematical model for combination of radio- and anti-angiogenic therapy is considered as optimal control problem with the objective of minimizing the tumor volume subject to isoperimetric constraints that limit the total radiation dose and the overall amount of anti-angiogenic agents to be given. The dynamics combines a model for tumor development under angiogenic inhibitors with the linear-quadratic model for the damage done by radiation ionization. The system has been investigated analytically as an optimal control problem and explicit expressions for possible singular controls were derived before. In this paper, for varying total radiation doses, examples of numerically computed optimal controls are given that verify and confirm these analytical structures: optimal schedules for the anti-angiogenic agents typically start with a brief full-dose segment, and then use up all inhibitors along a time-varying singular control while optimal radiotherapy schedules intensify the dosing and, after a brief period when the control is singular, end with a maximum dose segment. Singular controls occur for both the anti-angiogenic and radiotherapy dose rates. A discussion of the difficulties in proving the strong local optimality of corresponding trajectories is included.

Keywords

Optimal control Combination therapy Anti-angiogenic treatments Radiotherapy Numerical methods Arc parameterization 

Mathematics Subject Classification

49K15 92C50 

References

  1. 1.
    Kerbel, R.S.: Tumor angiogenesis: past, present and near future. Carcinogensis 21, 505–515 (2000)CrossRefGoogle Scholar
  2. 2.
    Jain, R.K.: Normalizing tumor vasculature with anti-angiogenic therapy: a new paradigm for combination therapy. Nat. Med. 7, 987–989 (2001)CrossRefGoogle Scholar
  3. 3.
    d’Onofrio, A., Ledzewicz, U., Maurer, H., Schättler, H.: On optimal delivery of combination therapy for tumors. Math. Biosci. 222, 13–26 (2009).  https://doi.org/10.1016/j.mbs.2009.08.004 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ergun, A., Camphausen, K., Wein, L.M.: Optimal scheduling of radiotherapy and angiogenic inhibitors. Bull. Math. Biol. 65, 407–424 (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    Chudej, K., Huebner, D., Pesch, H.J.: Numerische Lösung eines mathematischen Modells für eine Optimale Krebskombinationstherapie aus Anti-Angiogenese und Strahlentherapie. ASIM, Dresden (2016)Google Scholar
  7. 7.
    Ledzewicz, U., Schättler, H.: Multi-input optimal control problems for combined tumor anti-angiogenic and radiotherapy treatments. J. Optim. Theory Appl. 153, 195–224 (2012).  https://doi.org/10.1007/s10957-011-9954-8 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Aronna, M.S., Bonnans, J.F., Dmitruk, A.V., Lotito, P.: Quadratic order conditions for bang-singular extremals. Numer. Algebra Control Optim. 2(3), 511–546 (2012).  https://doi.org/10.3934/naco.2012.2.511 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Felgenhauer, U.: Structural stability investigation of bang-singular-bang optimal controls. J. Optim. Theory Appl. JOTA 152(3), 605–631 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Poggiolini, L., Stefani, G.: Bang-singular-bang extremals: sufficient optimality conditions. J. Dyn. Control Syst. 17(4), 469–514 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Poggiolini, L., Stefani, G.: Strong local optimality for bang-bang-singular extremals in single input control problems. IFAC PapersOnLine 50(1), 6128–6133 (2017).  https://doi.org/10.1016/j.ifacol.2017.08.2022 CrossRefGoogle Scholar
  12. 12.
    Ledzewicz, U., Schättler, H.: Anti-angiogenic therapy in cancer treatment as an optimal control problem. SIAM J. Control Optim. 46, 1052–1079 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Schättler, H., Ledzewicz, U.: Optimal Control for Mathematical Models of Cancer Therapies. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  14. 14.
    Fowler, J.F.: The linear-quadratic formula and progress in fractionated radiotherapy. Br. J. Radiol. 62, 679–694 (1989)CrossRefGoogle Scholar
  15. 15.
    Kellerer, A.M., Rossi, H.H.: The theory of dual radiation action. Curr. Top. Radiat. Res. Q. 8, 85–158 (1972)Google Scholar
  16. 16.
    Thames, H.D., Hendry, J.H.: Fractionation in Radiotherapy. Taylor and Francis, London (1987)Google Scholar
  17. 17.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. MacMillan, New York (1964)Google Scholar
  18. 18.
    Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory, Mathématiques & Applications, vol. 40. Springer, Paris (2003)zbMATHGoogle Scholar
  19. 19.
    Bressan, A., Piccoli, B.: Introduction to the Mathematical Theory of Control. American Institute of Mathematical Sciences, Springfield (2007)zbMATHGoogle Scholar
  20. 20.
    Liberzon, D.: Calculus of Variations and Optimal Control Theory. Princeton University Press, Princeton (2012)zbMATHGoogle Scholar
  21. 21.
    Schättler, H., Ledzewicz, U.: Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  22. 22.
    Haynes, G.W.: On the Optimality of a Totally Singular Vector Control. National Aeronautics and Space Administration, Washington D.C (1965)Google Scholar
  23. 23.
    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. Duxbury Press, Brooks-Cole Publishing Company, Plymouth (1993)zbMATHGoogle Scholar
  24. 24.
    Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–57 (2006). (cf. IPOPT home page (C. Laird, C., Wächter, A.): https://projects.coin-or.org/Ipopt)
  25. 25.
    Büskens, C.: Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen, Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany (1998)Google Scholar
  26. 26.
    Büskens, C., Maurer, H.: SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control. J. Comput. Appl. Math. 120, 85–108 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Maurer, H., Büskens, C., Kim, J.H.R., Kaya, Y.: Optimization methods for the verification of second-order sufficient conditions for bang-bang controls. Optim. Control Methods Appl. 26, 129–156 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Vossen, G.: Switching time optimization for bang-bang and singular controls. J. Optim. Theory Appl. 144, 409–429 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ledzewicz, U., Maurer, H., Schättler, H.: Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy. Math. Biosci. Eng. (MBE) 8, 307–328 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Osmolovskii, N.P., Maurer, H.: Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, vol. 24. SIAM Publications, Philadelphia (2012)CrossRefzbMATHGoogle Scholar
  31. 31.
    Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis. Academic Press, New York (1983)zbMATHGoogle Scholar
  32. 32.
    Büskens, C., Maurer, H.: Sensitivity analysis and real-time optimization of parametric nonlinear programming problems. In: Grötschel, M., Krumke, S.O., Rambau, J. (eds.) Online Optimization of Large Scale Systems, pp. 3–16. Springer, Berlin (2001)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Urszula Ledzewicz
    • 1
    • 2
    Email author
  • Helmut Maurer
    • 3
  • Heinz Schättler
    • 4
  1. 1.Southern Illinois University EdwardsvilleEdwardsvilleUSA
  2. 2.Lodz University of TechnologyLodzPoland
  3. 3.Westfälische Wilhelms Universität MünsterMünsterGermany
  4. 4.Washington UniversitySt. LouisUSA

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