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


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.


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

Mathematics Subject Classification

49K15 92C50 


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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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