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A new standard quadratic optimization approach to beam angle optimization for fixed-field intensity modulated radiation therapy

  • Department of Engineering and Applied Physics, University of Science and Technology of China, Hefei 230026, China

Cite this:
https://doi.org/10.52396/JUST-2021-0091
  • Received Date: 31 March 2021
  • Rev Recd Date: 29 June 2021
  • Publish Date: 31 August 2021
    Abstract

  • Abstract

    Beam angle optimization (BAO) largely determines the performance of the fixed-field intensity modulated radiation therapy (IMRT), and it is usually considered as non-convex optimization and a non-deterministic polynomial(NP) hard problem. In this work, BAO is reformulated into a highly efficient framework of standard quadratic optimization.The maximum of beamlet intensities for each incident field as the surrogate variable indicates whether one radiation field has been selected. By converting the function of maximum value in the objective into a set of linear constraints, the problem is solved as standard quadratic optimization via reweighting l1-norm scheme.The performance of the proposed BAO has been verified on a digital phantom and two patients.And the conclusion is drawn: the proposed convex optimization framework is able to find an optimal set of beam angles, leading to improved dose sparing on organs-at-risk (OARs) in the fixed-field IMRT.

    Abstract

    Beam angle optimization (BAO) largely determines the performance of the fixed-field intensity modulated radiation therapy (IMRT), and it is usually considered as non-convex optimization and a non-deterministic polynomial(NP) hard problem. In this work, BAO is reformulated into a highly efficient framework of standard quadratic optimization.The maximum of beamlet intensities for each incident field as the surrogate variable indicates whether one radiation field has been selected. By converting the function of maximum value in the objective into a set of linear constraints, the problem is solved as standard quadratic optimization via reweighting l1-norm scheme.The performance of the proposed BAO has been verified on a digital phantom and two patients.And the conclusion is drawn: the proposed convex optimization framework is able to find an optimal set of beam angles, leading to improved dose sparing on organs-at-risk (OARs) in the fixed-field IMRT.
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    • References(42)
  • [1]
    Bortfeld T, Bürkelbach J, Boesecke R, et al. Methods of image reconstruction from projections applied to conformation radiotherapy. Physics in Medicine & Biology, 1990, 35(10): 1423-1434.
    [2]
    Pugachev A, Li J G, Boyer A L, et al. Role of beam orientation optimization in intensity-modulated radiation therapy. International Journal of Radiation Oncology·Biology·Physics, 2001, 50(2): 551-560.
    [3]
    Bortfeld T. The number of beams in IMRT—theoretical investigations and implications for single-arc IMRT. Physics in Medicine & Biology, 2010, 55(1): 83-97.
    [4]
    Sultan A, Saher A. Optimization of Beam Orientation in Intensity Modulated Radiation Therapy Planning. Kaiserslautern, Germany: Technical University of Kaiserslautern, 2006.
    [5]
    Liu H, Dong P, Xing L. A new sparse optimization scheme for simultaneous beam angle and fluence map optimization in radiotherapy planning. Physics in Medicine & Biology, 2017, 62(16): 6428-6445.
    [6]
    Bangert M, Unkelbach J. Accelerated iterative beam angle selection in IMRT. Medical Physics, 2016, 43(3): 1073-1082.
    [7]
    Stein J, Mohan R, Wang X H, et al. Number and orientations of beams in intensity‐modulated radiation treatments. Medical Physics, 1997, 24(2): 149-160.
    [8]
    Bortfeld T. IMRT: a review and preview. Physics in Medicine & Biology, 2006, 51(13): R363-R379.
    [9]
    Yang R J, Dai J R, Yang Y, et al. Beam orientation optimization for intensity-modulated radiation therapy using mixed integer programming. Physics in Medicine & Biology, 2006, 51(15): 3653-3666.
    [10]
    Bortfeld T, Schlegel W. Optimization of beam orientations in radiation therapy: some theoretical considerations. Physics in Medicine & Biology, 1993, 38(2): 291-304.
    [11]
    Xing L, Pugachev A, Li J, et al. 190 A medical knowledge based system for the selection of beam orientations in intensity-modulated radiation therapy (IMRT). International Journal of Radiation Oncology·Biology·Physics, 1999, 45(3): 246-247.
    [12]
    Pugachev A, Xing L. Incorporating prior knowledge into beam orientaton optimization in IMRT. International Journal of Radiation Oncology·Biology·Physics, 2002, 54(5): 1565-1574.
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    Rowbottom C G, Nutting C M, Webb S. Beam-orientation optimization of intensity-modulated radiotherapy: clinical application to parotid gland tumours. Radiotherapy and Oncology, 2001, 59(2): 169-177.
    [14]
    Breedveld S, Storchi P R M, Voet P W J, et al. iCycle: Integrated, multicriterial beam angle, and profile optimization for generation of coplanar and noncoplanar IMRT plans. Medical Physics, 2012, 39(2): 951-963.
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    Woudstra E, Storchi P R M. Constrained treatment planning using sequential beam selection. Physics in Medicine & Biology, 2000, 45(8): 2133-2149.
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    D'Souza W D, Meyer R R, Shi L. Selection of beam orientations in intensity-modulated radiation therapy using single-beam indices and integer programming. Physics in Medicine & Biology, 2004, 49(15): 3465-3481.
    [17]
    Meedt G, Alber M, Nüsslin F. Non-coplanar beam direction optimization for intensity-modulated radiotherapy. Physics in Medicine & Biology, 2003, 48(18): 2999-3019.
    [18]
    Pugachev A, Xing L. Pseudo beam’s-eye-view as applied to beam orientation selection in intensity-modulated radiation therapy. International Journal of Radiation Oncology·Biology·Physics, 2001, 51(5): 1361-1370.
    [19]
    Djajaputra D, Wu Q, Wu Y, et al. Algorithm and performance of a clinical IMRT beam-angle optimization system. Physics in Medicine & Biology, 2003, 48(19): 3191-3212.
    [20]
    Hou Q, Wang J, Chen Y, et al. Beam orientation optimization for IMRT by a hybrid method of the genetic algorithm and the simulated dynamics. Medical Physics, 2003, 30(9): 2360-2367.
    [21]
    Li Y J, Yao J, Yao D Z. Automatic beam angle selection in IMRT planning using genetic algorithm. Physics in Medicine & Biology, 2004, 49(10): 1915-1932.
    [22]
    Li Y J, Yao D Z, Yao J, et al. A particle swarm optimization algorithm for beam angle selection in intensity-modulated radiotherapy planning. Physics in Medicine & Biology, 2005, 50(15):3491-3514.
    [23]
    Schreibmann E, Lahanas M, Xing L, et al. Multiobjective evolutionary optimization of the number of beams, their orientations and weights for intensity-modulated radiation therapy. Physics in Medicine & Biology, 2004, 49(5): 747-770.
    [24]
    Schreibmann E, Xing L. Feasibility study of beam orientation class‐solutions for prostate IMRT. Medical Physics, 2004, 31(10): 2863-2870.
    [25]
    Ezzell G A. Genetic and geometric optimization of three‐dimensional radiation therapy treatment planning. Medical Physics, 1996, 23(3): 293-305.
    [26]
    Zhu L, Lee L, Ma Y, et al. Using total-variation regularization for intensity modulated radiation therapy inverse planning with field-specific numbers of segments. Physics in Medicine & Biology, 2008, 53(23): 6653-6672.
    [27]
    Zhu L, Xing L. Search for IMRT inverse plans with piecewise constant fluence maps using compressed sensing techniques. Medical Physics, 2009, 36(5): 1895-1905.
    [28]
    Jia X, Men C, Lou Y, et al. Beam orientation optimization for intensity modulated radiation therapy using adaptive l2, 1-minimization. Physics in Medicine & Biology, 2011, 56(19): 6205-6222.
    [29]
    Yuan M, Lin Y. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2006, 68: 49-67.
    [30]
    Bach F, Jenatton R, Mairal J, et al. Optimization with sparsity-inducing penalties. Foundations and Trends© in Machine Learning, 2012, 4(1): 1-106.
    [31]
    Simon N, Friedman J, Hastie T, et al. A sparse-group lasso. Journal of Computational and Graphical Statistics, 2013, 22(2): 231-245.
    [32]
    Meier L, Van De Geer S, Bühlmann P. The group lasso for logistic regression. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2008, 70(1): 53-71.
    [33]
    Bach F R. Consistency of the group lasso and multiple kernel learning. The Journal of Machine Learning Research, 2008, 9: 1179-1225.
    [34]
    Huang J, Zhang T. The benefit of group sparsity. The Annals of Statistics, 2010, 38(4): 1978-2004.
    [35]
    O’Connor D, Yu V, Nguyen D, et al. Fraction-variant beam orientation optimization for non-coplanar IMRT. Physics in Medicine & Biology, 2018, 63(4): 045015.
    [36]
    Kawrakow I. Improved modeling of multiple scattering in the Voxel Monte Carlo model. Medical Physics, 1997, 24(4): 505-517.
    [37]
    Xing L, Li J G, Donaldson S, et al. Optimization of importance factors in inverse planning. Physics in Medicine & Biology, 1999, 44(10): 2525-2536.
    [38]
    Bortfeld T. Optimized planning using physical objectives and constraints. Seminars in Radiation Oncology, 1999, 9(1): 20-34.
    [39]
    Bortfeld T R, Kahler D L, Waldron T J, et al. X-ray field compensation with multileaf collimators. International Journal of Radiation Oncology·Biology·Physics, 1994, 28(3): 723-730.
    [40]
    Zhu L, Niu T, Petrongolo M. Iterative CT reconstruction via minimizing adaptively reweighted total variation. Journal of X-ray Science and Technology, 2014, 22(2): 227-240.
    [41]
    Candès E J, Wakin M B, Boyd S P. Enhancing sparsity by reweighted l1 minimization. Journal of Fourier analysis and applications, 2008, 14: 877-905.
    [42]
    Grant M, Boyd S, Ye Y. CVX: Matlab software for disciplined convex programming. http://cvxr.com/cvx/.     (Continued on p.627)
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    [1]
    Bortfeld T, Bürkelbach J, Boesecke R, et al. Methods of image reconstruction from projections applied to conformation radiotherapy. Physics in Medicine & Biology, 1990, 35(10): 1423-1434.
    [2]
    Pugachev A, Li J G, Boyer A L, et al. Role of beam orientation optimization in intensity-modulated radiation therapy. International Journal of Radiation Oncology·Biology·Physics, 2001, 50(2): 551-560.
    [3]
    Bortfeld T. The number of beams in IMRT—theoretical investigations and implications for single-arc IMRT. Physics in Medicine & Biology, 2010, 55(1): 83-97.
    [4]
    Sultan A, Saher A. Optimization of Beam Orientation in Intensity Modulated Radiation Therapy Planning. Kaiserslautern, Germany: Technical University of Kaiserslautern, 2006.
    [5]
    Liu H, Dong P, Xing L. A new sparse optimization scheme for simultaneous beam angle and fluence map optimization in radiotherapy planning. Physics in Medicine & Biology, 2017, 62(16): 6428-6445.
    [6]
    Bangert M, Unkelbach J. Accelerated iterative beam angle selection in IMRT. Medical Physics, 2016, 43(3): 1073-1082.
    [7]
    Stein J, Mohan R, Wang X H, et al. Number and orientations of beams in intensity‐modulated radiation treatments. Medical Physics, 1997, 24(2): 149-160.
    [8]
    Bortfeld T. IMRT: a review and preview. Physics in Medicine & Biology, 2006, 51(13): R363-R379.
    [9]
    Yang R J, Dai J R, Yang Y, et al. Beam orientation optimization for intensity-modulated radiation therapy using mixed integer programming. Physics in Medicine & Biology, 2006, 51(15): 3653-3666.
    [10]
    Bortfeld T, Schlegel W. Optimization of beam orientations in radiation therapy: some theoretical considerations. Physics in Medicine & Biology, 1993, 38(2): 291-304.
    [11]
    Xing L, Pugachev A, Li J, et al. 190 A medical knowledge based system for the selection of beam orientations in intensity-modulated radiation therapy (IMRT). International Journal of Radiation Oncology·Biology·Physics, 1999, 45(3): 246-247.
    [12]
    Pugachev A, Xing L. Incorporating prior knowledge into beam orientaton optimization in IMRT. International Journal of Radiation Oncology·Biology·Physics, 2002, 54(5): 1565-1574.
    [13]
    Rowbottom C G, Nutting C M, Webb S. Beam-orientation optimization of intensity-modulated radiotherapy: clinical application to parotid gland tumours. Radiotherapy and Oncology, 2001, 59(2): 169-177.
    [14]
    Breedveld S, Storchi P R M, Voet P W J, et al. iCycle: Integrated, multicriterial beam angle, and profile optimization for generation of coplanar and noncoplanar IMRT plans. Medical Physics, 2012, 39(2): 951-963.
    [15]
    Woudstra E, Storchi P R M. Constrained treatment planning using sequential beam selection. Physics in Medicine & Biology, 2000, 45(8): 2133-2149.
    [16]
    D'Souza W D, Meyer R R, Shi L. Selection of beam orientations in intensity-modulated radiation therapy using single-beam indices and integer programming. Physics in Medicine & Biology, 2004, 49(15): 3465-3481.
    [17]
    Meedt G, Alber M, Nüsslin F. Non-coplanar beam direction optimization for intensity-modulated radiotherapy. Physics in Medicine & Biology, 2003, 48(18): 2999-3019.
    [18]
    Pugachev A, Xing L. Pseudo beam’s-eye-view as applied to beam orientation selection in intensity-modulated radiation therapy. International Journal of Radiation Oncology·Biology·Physics, 2001, 51(5): 1361-1370.
    [19]
    Djajaputra D, Wu Q, Wu Y, et al. Algorithm and performance of a clinical IMRT beam-angle optimization system. Physics in Medicine & Biology, 2003, 48(19): 3191-3212.
    [20]
    Hou Q, Wang J, Chen Y, et al. Beam orientation optimization for IMRT by a hybrid method of the genetic algorithm and the simulated dynamics. Medical Physics, 2003, 30(9): 2360-2367.
    [21]
    Li Y J, Yao J, Yao D Z. Automatic beam angle selection in IMRT planning using genetic algorithm. Physics in Medicine & Biology, 2004, 49(10): 1915-1932.
    [22]
    Li Y J, Yao D Z, Yao J, et al. A particle swarm optimization algorithm for beam angle selection in intensity-modulated radiotherapy planning. Physics in Medicine & Biology, 2005, 50(15):3491-3514.
    [23]
    Schreibmann E, Lahanas M, Xing L, et al. Multiobjective evolutionary optimization of the number of beams, their orientations and weights for intensity-modulated radiation therapy. Physics in Medicine & Biology, 2004, 49(5): 747-770.
    [24]
    Schreibmann E, Xing L. Feasibility study of beam orientation class‐solutions for prostate IMRT. Medical Physics, 2004, 31(10): 2863-2870.
    [25]
    Ezzell G A. Genetic and geometric optimization of three‐dimensional radiation therapy treatment planning. Medical Physics, 1996, 23(3): 293-305.
    [26]
    Zhu L, Lee L, Ma Y, et al. Using total-variation regularization for intensity modulated radiation therapy inverse planning with field-specific numbers of segments. Physics in Medicine & Biology, 2008, 53(23): 6653-6672.
    [27]
    Zhu L, Xing L. Search for IMRT inverse plans with piecewise constant fluence maps using compressed sensing techniques. Medical Physics, 2009, 36(5): 1895-1905.
    [28]
    Jia X, Men C, Lou Y, et al. Beam orientation optimization for intensity modulated radiation therapy using adaptive l2, 1-minimization. Physics in Medicine & Biology, 2011, 56(19): 6205-6222.
    [29]
    Yuan M, Lin Y. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2006, 68: 49-67.
    [30]
    Bach F, Jenatton R, Mairal J, et al. Optimization with sparsity-inducing penalties. Foundations and Trends© in Machine Learning, 2012, 4(1): 1-106.
    [31]
    Simon N, Friedman J, Hastie T, et al. A sparse-group lasso. Journal of Computational and Graphical Statistics, 2013, 22(2): 231-245.
    [32]
    Meier L, Van De Geer S, Bühlmann P. The group lasso for logistic regression. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2008, 70(1): 53-71.
    [33]
    Bach F R. Consistency of the group lasso and multiple kernel learning. The Journal of Machine Learning Research, 2008, 9: 1179-1225.
    [34]
    Huang J, Zhang T. The benefit of group sparsity. The Annals of Statistics, 2010, 38(4): 1978-2004.
    [35]
    O’Connor D, Yu V, Nguyen D, et al. Fraction-variant beam orientation optimization for non-coplanar IMRT. Physics in Medicine & Biology, 2018, 63(4): 045015.
    [36]
    Kawrakow I. Improved modeling of multiple scattering in the Voxel Monte Carlo model. Medical Physics, 1997, 24(4): 505-517.
    [37]
    Xing L, Li J G, Donaldson S, et al. Optimization of importance factors in inverse planning. Physics in Medicine & Biology, 1999, 44(10): 2525-2536.
    [38]
    Bortfeld T. Optimized planning using physical objectives and constraints. Seminars in Radiation Oncology, 1999, 9(1): 20-34.
    [39]
    Bortfeld T R, Kahler D L, Waldron T J, et al. X-ray field compensation with multileaf collimators. International Journal of Radiation Oncology·Biology·Physics, 1994, 28(3): 723-730.
    [40]
    Zhu L, Niu T, Petrongolo M. Iterative CT reconstruction via minimizing adaptively reweighted total variation. Journal of X-ray Science and Technology, 2014, 22(2): 227-240.
    [41]
    Candès E J, Wakin M B, Boyd S P. Enhancing sparsity by reweighted l1 minimization. Journal of Fourier analysis and applications, 2008, 14: 877-905.
    [42]
    Grant M, Boyd S, Ye Y. CVX: Matlab software for disciplined convex programming. http://cvxr.com/cvx/.     (Continued on p.627)
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