Robust treatment planning with conditional value at risk chance constraints in intensity-modulated proton therapy:

Yu An, Jianming Liang, Steven E. Schild, Martin Bues, Wei Liu

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


Background and purpose: Intensity-modulated proton therapy (IMPT) is highly sensitive to range uncertainties and uncertainties caused by setup variation. The conventional inverse treatment planning of IMPT based on the planning target volume (PTV) is not often sufficient to ensure robustness of treatment plans. We applied a probabilistic framework (chance-constrained optimization) in IMPT planning to hedge against the influence of uncertainties. Material and methods: We retrospectively selected one patient with lung cancer, one patient with head and neck (H&N) cancer, and one with prostate cancer for this analysis. Using their original images and prescriptions, we created new IMPT plans using two methods: (1) a robust chance-constrained treatment planning method with the clinical target volume (CTV) as the target; (2) the margin-based method with PTV as the target, which was solved by commercial software, CPLEX, using linear programming. For the first method, we reformulated the model into a tractable mixed-integer programming problem and sped up the calculation using Benders decomposition. The dose-volume histograms (DVHs) from the nominal and perturbed dose distributions were used to assess and compare plan quality. DVHs for all uncertain scenarios along with the nominal DVH were plotted. The width of the "bands" of DVHs was used to quantify the plan sensitivity to uncertainty. The newly developed Benders decomposition method was compared with a commercial solution to demonstrate its computational efficiency. The trade-off between nominal plan quality and plan robustness was investigated. Results: Our chance-constrained model outperformed the PTV method in terms of tumor coverage, tumor dose homogeneity, and plan robustness. Our model was shown to produce IMPT plans to meet the dose-volume constraints of organs at risk (OARs) and had better sparing of OARs than the PTV method in the three clinical cases included in this study. The chance-constrained model provided a flexible tool for users to balance between plan robustness and plan quality. In addition, our in-house developed method was found to be much faster than the commercial solution. Conclusion: With explicit control of plan robustness, the chance-constrained robust optimization model generated superior IMPT plans compared to the PTV-based method.

Original languageEnglish (US)
Pages (from-to)28-36
Number of pages9
JournalMedical physics
Issue number1
StatePublished - Jan 2017


  • Benders decomposition
  • chance constraints
  • conditional value at risk (CVaR)
  • intensity-modulated proton therapy (IMPT)
  • robustness

ASJC Scopus subject areas

  • Biophysics
  • Radiology Nuclear Medicine and imaging


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