Journal of Global Optimization

, Volume 61, Issue 3, pp 407–428 | Cite as

A new mathematical approach for handling DVH criteria in IMRT planning

  • Alexander ScherrerEmail author
  • Filka Yaneva
  • Tabea Grebe
  • Karl-Heinz Küfer


The appropriate handling of planning criteria on the cumulative dose-volume histogram (DVH) is a highly problematic issue in intensity-modulated radiation therapy (IMRT) plan optimization. The nonconvexity of DVH criteria and globality of the resulting optimization problems complicate the design of suitable optimization methods, which feature numerical efficiency, reliable convergence and optimality of the results. This work examines the mathematical structure of DVH criteria and proves the valuable properties of isotonicity/antitonicity, connectedness, invexity and sufficiency of the Karush–Kuhn–Tucker condition. These properties facilitate the use of efficient and goal-oriented optimization methods. An exemplary algorithmic realization with feasible direction methods gives rise to a functional framework for interactive IMRT planning on DVH criteria. Numerical examples on real world planning cases prove its practical capability.


Intensity-modulated radiation therapy (IMRT) Cumulative dose-volume histogram (DVH) Antitone/isotone, connected and invex functions Sufficient KKT condition Feasible direction methods Reduced gradient method of Wolfe 

Mathematics Subject Classification

26B35 49M37 65K05 90C26 



The fundamental research was partly supported by the U.S. National Institutes of Health, Grant No. 2R01CA103904-05 and German Federal Ministry of Education and Research (BMBF), Grant No. 01 IS 08002D.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Alexander Scherrer
    • 1
    Email author
  • Filka Yaneva
    • 2
  • Tabea Grebe
    • 1
  • Karl-Heinz Küfer
    • 1
  1. 1.Department of OptimizationFraunhofer Institute for Industrial Mathematics (ITWM)KaiserslauternGermany
  2. 2.Department of Medical Biometry and Computer ScienceUniversity Hospital HeidelbergHeidelbergGermany

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