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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

Abstract

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.

Keywords

Optimization Convex optimization Radiation therapy Treatment planning 

Notes

Acknowledgements

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.

References

  1. Adler JR Jr., Chang SD, Murphy MJ, Doty J, Geis P, Hancock SL (1998) The cyberKnife: a frameless robotic system for radiosurgery. Stereotact Funct Neurosurg 69(1–4):124–128Google Scholar
  2. Ahmed S, Gozbasi O, Savelsbergh M, Crocker I, Fox T, Schreibmann E (2010) An automated intensity-modulated radiation therapy planning system. INFORMS J Comput 22(4):568–583MathSciNetCrossRefzbMATHGoogle Scholar
  3. Aleman DM, Glaser D, Romeijn HE, Dempsey JF (2010) Interior point algorithms: guaranteed optimality for fluence map optimization IMRT. Phys Med Biol 55(18):5467–5482CrossRefGoogle Scholar
  4. Aleman DM, Mišić VV, Sharpe MB (2013) Computational enhancements to fluence map optimization for total marrow irradiation using IMRT. Comput Oper Res 40(9):2167–2177CrossRefzbMATHGoogle Scholar
  5. Bedford JL (2009) Treatment planning for volumetric modulated arc therapy. Med Phys 36(11):5128–5138CrossRefGoogle Scholar
  6. Bortfeld T, Bürkelbach J, Boesecke R, Schlegel W (1990) Methods of image reconstruction from projections applied to conformation radiotherapy. Phys Med Biol 35(10):1423–1434CrossRefGoogle Scholar
  7. Chan TCY, Mahmoudzadeh H, Purdie TG (2014) A robust-CVaR optimization approach with applications to breast cancer therapy. Eur J Oper Res 238(3):876–885MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chen W, Unkelbach J, Trofimov A, Madden T, Kooy H, Bortfeld T, Craft D (2012) Including robustness in multi-criteria optimization for intensity-modulated proton therapy. Phys Med Biol 57(3):591–608CrossRefGoogle Scholar
  9. Cho PS, Lee S, Marks RJ II, Oh S, Sutlief SG, Phillips MH (1998) Optimization of intensity modulated beams with volume constraints using two methods: cost function minimization and projections onto convex sets. Med Phys 25(4):435–443CrossRefGoogle Scholar
  10. Davino C, Furno M, Vistocco D (2013) Quantile regression: theory and applications. Wiley, New YorkzbMATHGoogle Scholar
  11. Deasy JO (1997) Multiple local minima in radiotherapy optimization problems with dose-volume constraints. Med Phys 24(7):1157–1161CrossRefGoogle Scholar
  12. Diamond S, Boyd S (2016) CVXPY: a Python-embedded modeling language for convex optimization. J Mach Learn Res 17(83):1–5MathSciNetzbMATHGoogle Scholar
  13. Domahidi A, Chu E, Boyd S (2013) ECOS: an SOCP solver for embedded systems. In: European control conference, pp 3071–3076Google Scholar
  14. Dong P, Lee P, Ruan D, Long T, Romeijn HE, Yang Y, Low D, Kupelian P, Sheng K (2013) 4\(\pi\) non-coplanar liver sbrt: a novel delivery technique. Int J Radiat Oncol Biol Phys 85(5):1360–1366CrossRefGoogle Scholar
  15. Ehrgott M, Güler Ç, Hamacher HW, Shao L (2008) Mathematical optimization in intensity modulated radiation therapy. 4OR 6(3):199–262MathSciNetCrossRefzbMATHGoogle Scholar
  16. Glide-Hurst C, Bellon M, Foster R, Altunbas C, Speiser M, Altman M, Westerly D, Wen N, Zhao B, Miften M (2013) Commissioning of the Varian TrueBeam linear accelerator: a multi-institutional study. Med Phys 40(3):031719CrossRefGoogle Scholar
  17. Halabi T, Craft D, Bortfeld T (2006a) Dose-volume objectives in multi-criteria optimization. Phys Med Biol 51(15):3809–3818CrossRefGoogle Scholar
  18. Halabi T, Craft D, Bortfeld T (2006b) Dose-volume objectives in multi-criteria optimization. Phys Med Biol 51:3809–3818CrossRefGoogle Scholar
  19. Hamacher HW, Küfer KH (2002) Inverse radiation therapy planning—a multiple objective optimization approach. Discrete Appl Math 118(1):145–161MathSciNetCrossRefzbMATHGoogle Scholar
  20. Hölder A (2003) Designing radiotherapy plans with elastic constraints and interior point methods. Health Care Manag Sci 6(1):5–16CrossRefGoogle Scholar
  21. Langer M, Brown R, Urie M, Leong J, Stracher M, Shapiro J (1990) Large scale optimization of beam weights under dose-volume restrictions. Int J Radiat Oncol Biol Phys 18(4):887–893CrossRefGoogle Scholar
  22. Lee E, Fox T, Crocker I (2000) Optimization of radiosurgery treatment planning via mixed integer programming. Med Phys 27(5):995–1004CrossRefGoogle Scholar
  23. Lee E, Fox T, Crocker I (2003) Integer programming applied to intensity-modulated radiation therapy treatment planning. Ann Oper Res 119(1–4):165–181CrossRefzbMATHGoogle Scholar
  24. Li R, Xing L (2013) An adaptive planning strategy for station parameter optimized radiation therapy (SPORT): segmentally boosted VMAT. Med Phys 40(5):050701CrossRefGoogle Scholar
  25. Lim G, Cao W (2012) A two-phase method for selecting IMRT treatment beam angles: Branch-and-Prune and local neighborhood search. Eur J Oper Res 217(3):609–618MathSciNetCrossRefzbMATHGoogle Scholar
  26. Mackie TR, Holmes T, Swerdloff S, Reckwerdt P, Deasy JO, Yang J, Paliwal B, Kinsella T (1993) Tomotherapy: a new concept for the delivery of dynamic conformal radiotherapy. Med Phys 20(6):1709–1719CrossRefGoogle Scholar
  27. Mageras GS, Mohan R (1993) Application of fast simulated annealing to optimization of conformal radiation treatments. Med Phys 20(3):639–647CrossRefGoogle Scholar
  28. Marks LB, Yorke ED, Jackson A, Haken RKT, Constine LS, Eisbruch A, Bentzen SM, Nam J, Deasy JO (2010) Use of normal tissue complication probability (NTCP) models in the clinic. Int J Radiat Oncol Biol Phys 76(3 Suppl):S10–S19CrossRefGoogle Scholar
  29. O’Donoghue B, Chu E, Parikh N, Boyd S (2016) Conic optimization via operator splitting and homogeneous self-dual embedding. J Optim Theory Appl 169(3):1042–1068MathSciNetCrossRefzbMATHGoogle Scholar
  30. Oelfke U, Bortfeld T (2001) Inverse planning for photon and proton beams. Med Dosim 26(2):113–124CrossRefGoogle Scholar
  31. Oskoorouchi MR, Ghaffari HR, Terlaky T, Aleman DM (2011) An interior point constraint generation algorithm for semi-infinite optimization with health-care application. Oper Res 59(5):1184–1197MathSciNetCrossRefzbMATHGoogle Scholar
  32. Rockafellar R, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21–42CrossRefGoogle Scholar
  33. Romeijn HE, Ahuja RK, Dempsey JF, Kumar A, Li JG (2003) A novel linear programming approach to fluence map optimization for intensity modulated radiation therapy treatment planning. Phys Med Biol 48(21):3521–3542CrossRefGoogle Scholar
  34. Romeijn HE, Dempsey J, Li J (2004) A unifying framework for multi-criteria fluence map optimization models. Phys Med Biol 49(10):1991–2013CrossRefGoogle Scholar
  35. Romeijn HE, Ahuja RK, Dempsey JF, Kumar A (2006) A new linear programming approach to radiation therapy treatment planning problems. Oper Res 54(2):201–216MathSciNetCrossRefzbMATHGoogle Scholar
  36. Rosen II, Lane RG, Morrill SM, Belli JA (1991) Treatment plan optimization using linear programming. Med Phys 18(2):141–152CrossRefGoogle Scholar
  37. Schweikard A, Schlaefer A, Adler J Jr. (2006) Resampling: an optimization method for inverse planning in robotic radiosurgery. Med Phys 33(11):4005–4011CrossRefGoogle Scholar
  38. Shepard DM, Ferris MC, Olivera GH, Mackie TR (1999) Optimizing the delivery of radiation therapy to cancer patients. SIAM Rev 41(4):721–744CrossRefzbMATHGoogle Scholar
  39. Shepard DM, Ferris MC, Ove R, Ma L (2000a) Inverse treatment planning for Gamma Knife radiosurgery. Med Phys 27(9):2146–2149CrossRefGoogle Scholar
  40. Shepard DM, Olivera GH, Reckwerdt PJ, Mackie TR (2000b) Iterative approaches to dose optimization in tomotherapy. Phys Med Biol 45(1):69–90CrossRefGoogle Scholar
  41. Spirou SV, Chui C (1998) A gradient inverse planning algorithm with dose-volume constraints. Med Phys 25(3):321–333CrossRefGoogle Scholar
  42. Webb S (1989) Optimization of conformal radiotherapy dose distribution by simulated annealing. Phys Med Biol 34(10):1349–1370CrossRefGoogle Scholar
  43. Webb S (1992) Optimization by simulated annealing of three-dimensional, conformal treatment planning for radiation fields defined by a multileaf collimator: II. inclusion of two-dimensional modulation of the X-ray intensity. Phys Med Biol 37(8):1689–1704CrossRefGoogle Scholar
  44. Wu Q, Mohan R (2002) Multiple local minima in IMRT optimization based on dose-volume criteria. Med Phys 29(7):1514–1527CrossRefGoogle Scholar
  45. Xing L, Chen GTY (1996) Iterative methods for inverse treatment planning. Phys Med Biol 41(10):2107–2123CrossRefGoogle Scholar
  46. Xing L, Hamilton RJ, Spelbring D, Pelizzari CA, Chen GTY, Boyer AL (1998) Fast iterative algorithms for three-dimensional inverse treatment planning. Med Phys 25(10):1845–1849CrossRefGoogle Scholar
  47. Zarepisheh M, Shakourifar M, Trigila G, Ghomi PS, Couzens S, Abebe A, Noreña L, Shang W, Jiang SB, Zinchenko Y (2013) A moment-based approach for DVH-guided radiotherapy treatment plan optimization. Phys Med Biol 58(6):1869–1887CrossRefGoogle Scholar
  48. Zarepisheh M, Long T, Li N, Tian Z, Romeijn HE, Jia X, Jiang SB (2014) A DVH-guided IMRT optimization algorithm for automatic treatment planning and adaptive radiotherapy replanning. Med Phys 41(6):061711CrossRefGoogle Scholar

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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