Optimization and Engineering

, Volume 20, Issue 1, pp 277–300 | Cite as

A convex optimization approach to radiation treatment planning with dose constraints

  • Anqi FuEmail author
  • Barıṣ Ungun
  • Lei Xing
  • Stephen Boyd
Research Article


We present a method for handling dose constraints as part of a convex programming framework for inverse treatment planning. Our method uniformly handles mean dose, maximum dose, minimum dose, and dose-volume (i.e., percentile) constraints as part of a convex formulation. Since dose-volume constraints are non-convex, we replace them with a convex restriction. This restriction is, by definition, conservative; to mitigate its impact on the clinical objectives, we develop a two-pass planning algorithm that allows each dose-volume constraint to be met exactly on a second pass by the solver if its corresponding restriction is feasible on the first pass. In another variant, we add slack variables to each dose constraint to prevent the problem from becoming infeasible when the user specifies an incompatible set of constraints or when the constraints are made infeasible by our restriction. Finally, we introduce ConRad, a Python-embedded open-source software package for convex radiation treatment planning. ConRad implements the methods described above and allows users to construct and plan cases through a simple interface.


Optimization Convex optimization Radiation therapy Treatment planning 



We thank Michael Folkerts for providing the anonymized dataset for the head and neck VMAT reweighting case, and Peng Dong for the anonymized dataset for the prostate IMRT case. This research was supported by the Stanford Graduate Fellowship, Stanford Bio-X Bowes Fellowship, and NIH Grant 5R01CA176553.


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Copyright information

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

Authors and Affiliations

  • Anqi Fu
    • 1
    Email author
  • Barıṣ Ungun
    • 2
  • Lei Xing
    • 3
  • Stephen Boyd
    • 1
  1. 1.Department of Electrical EngineeringStanford UniversityStanfordUSA
  2. 2.Department of BioengineeringStanford UniversityStanfordUSA
  3. 3.Department of Radiation OncologyStanford School of MedicineStanfordUSA

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