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Mathematical optimization in intensity modulated radiation therapy

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Abstract

The design of an intensity modulated radiotherapy treatment includes the selection of beam angles (geometry problem), the computation of an intensity map for each selected beam angle (intensity problem), and finding a sequence of configurations of a multileaf collimator to deliver the treatment (realization problem). Until the end of the last century research on radiotherapy treatment design has been published almost exclusively in the medical physics literature. However, since then, the attention of researchers in mathematical optimization has been drawn to the area and important progress has been made. In this paper we survey the use of optimization models, methods, and theories in intensity modulated radiotherapy treatment design.

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References

  • Agmon S (1954) The relaxation method for linear inequalities. Can J Math 6: 382–392

    Google Scholar 

  • Ahuja R, Hamacher H (2004) A network flow algorithm to minimize beam-on-time for unconstrained multileaf collimator problems in cancer radiation therapy. Networks 45(1): 36–41

    Article  Google Scholar 

  • Ahuja R, Magnanti T, Orlin J (1993) Network flows: theory, algorithms and applications. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Alber M, Nüsslin F (2001) A representation of an NTCP function for local complication mechanisms. Phys Med Biol 46: 439–447

    Article  Google Scholar 

  • Alber M, Reemtsen R (2007) Intensity modulated radiation therapy planning by use of a barrier-penalty multiplier method. Optim Meth Softw 22: 391–411

    Article  Google Scholar 

  • Altman M, Chmura S, Deasy J, Roeske J (2006) Optimization of the temporal pattern of radiation: an IMRT based study. Int J Radiat Oncol Biol Phys 66: 898–905

    Google Scholar 

  • Baatar D (2005) Matrix decomposition with time and cardinality objectives: theory, algorithms, and application to multileaf collimator sequencing. PhD Thesis, Department of Mathematics, Technical University of Kaiserslautern

  • Baatar D, Hamacher H (2003) New LP model for multileaf collimators in radiation therapy planning. In: Proceedings of the operations research peripatetic postgraduate programme conference ORP3, Lambrecht, Germany, pp 11–29

  • Baatar D, Hamacher H, Ehrgott M, Woeginger G (2005) Decomposition of integer matrices and multileaf collimator sequencing. Discr Appl Math 152: 6–34

    Article  Google Scholar 

  • Baatar D, Boland N, Brand S, Stuckey P (2007) Minimum cardinality matrix decomposition into consecutive-ones matrices: CP and IP approaches. In: Van Hentenrynck P, Wolsey L(eds) Integration of AI and OR techniques in constraint programming for combinatorial optimization problems—Proceedings of CPAIOR 2007, Brussels, Belgium, vol 4510 of Lecture Notes in Computer Science. Springer, Berlin, pp 1–15

    Google Scholar 

  • Baatar D, Boland N, Hamacher H, Johnston R (2008) A new sequential extraction heuristic for optimizing the delivery of cancer radiation treatment using multileaf collimators. INFORMS J Comput (to appear)

  • Bahr G, Kereiakes J, Horwitz H, Finney R, Galvin J, Goode K (1968) The method of linear programming applied to radiation treatment planning. Radiology 91: 686–693

    Google Scholar 

  • Bansal N, Coppersmith D, Schieber B (2006) Minimizing setup and beam-on times in radiation therapy. In: Díaz J, Jansen K, Rolim J, Zwick U(eds) APPROX—RANDOM. Approximation, randomization, and combinatorial optimization. Algorithms and techniques, 9th International workshop on approximation algorithms for combinatorial optimization problems, APPROX 2006 and 10th International workshop on randomization and computation, RANDOM 2006, Barcelona, Spain, 28–30 August 2006, Proceedings, vol 4110 of Lecture Notes in Computer Science. Springer, Berlin, pp 27–38

    Google Scholar 

  • Bednarz G, Michalski D, Houser C, Huq M, Xiao Y, Anne P, Galvin J (2002) The use of mixed-integer programming for inverse treatment planning with pre-defined field segments. Phys Med Biol 47: 2235–2245

    Article  Google Scholar 

  • Bednarz G, Michalski D, Anne P, Valicenti R (2004) Inverse treatment planning using volume-based objective functions. Phys Med Biol 49: 2503–2514

    Article  Google Scholar 

  • Benson HP (1998) An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J Global Optim 13: 1–24

    Article  Google Scholar 

  • Billups S, Kennedy J (2003) Minimum-support solutions for radiotherapy planning. Ann Oper Res 119: 229–245

    Article  Google Scholar 

  • Boland N, Hamacher H, Lenzen F (2004) Minimizing beam-on time in cancer radiation treatment using multileaf collimators. Networks 43(4): 226–240

    Article  Google Scholar 

  • Bortfeld T (1999) Optimized planning using physical objectives and constraints. Semin Radiat Oncol 9: 20–34

    Article  Google Scholar 

  • Bortfeld T, Burkelbach J, Boesecke R, Schlegel W (1990) Method of image reconstructions from projections applied to conformation therapy. Phys Med Biol 35(10): 1423–1434

    Article  Google Scholar 

  • Bortfeld T, Boyer AL, Schlegel W, Kahler DL, Waldron TJ (1994) Realisation and verification of three-dimensional conformal radiotherapy with modulated fields. Int J Radiat Oncol Biol Phys 30: 899–908

    Google Scholar 

  • Bortfeld T, Boyer A, Kahler D, Waldron T (1994) X-ray field compensation with multileaf collimators. Int J Radiat Oncol Biol Phys 28(3): 723–730

    Google Scholar 

  • Bortfeld T, Stein J, Preiser K (1997) Clinically relevant intensity modulated optimization using physical criteria. In: Leavitt D(eds) XIIth International conference on the use of computers in radiation therapy. USA Madison Medical Physics Publishing, Salt Lake City, Utah, pp 1–4

    Google Scholar 

  • Boyer AL, Yu CX (1999) Intensity-modulated radiation therapy with dynamic multileaf collimators. Semin Radiat Oncol 9(1): 48–59

    Article  Google Scholar 

  • Brahme A (1988) Optimization of stationary and moving beam radiation therapy techniques. Radiother Oncol 12: 129–140

    Article  Google Scholar 

  • Brahme A (2001) Individualizing cancer treatment: biological optimization models in treatment and planning. Int J Radiat Oncol Biol Phys 49: 327–337

    Article  Google Scholar 

  • Brahme A, Agren AK (1987) Optimal dose distribution for eradication of heterogeneous tumours. Acta Oncol 26: 77–85

    Google Scholar 

  • Burkard R (2002) Open problem session. In: Oberwolfach conference on combinatorial optimization, 24–29 November 2002

  • Carlsson F, Forsgren A (2006) Iterative regularization in intensity-modulated radiation therapy optimization. Med Phys 33: 225–234

    Article  Google Scholar 

  • Carlsson F, Forsgren A, Rehbinder H, Eriksson K (2006) Using eigenstructure of the Hessian to reduce the dimension of the intensity modulated radiation therapy optimization problem. Ann Oper Res 148: 81–94

    Article  Google Scholar 

  • Censor Y, Altschuler M, Powlis W (1988a) A computational solution of the inverse problem in radiation therapy treatment planning. Appl Math Comput 25: 57–87

    Article  Google Scholar 

  • Censor Y, Altschuler M, Powlis W (1988b) On the use of Cimmino’s simultaneous projections method for computing a solution of the inverse problem in radiation therapy treatment planning. Inv Probl 4: 607–623

    Article  Google Scholar 

  • Censor Y, Ben-Israel A, Xiao Y, Galvin J (2008) On linear infeasibility arising in intensity-modulated radiation therapy inverse planning. Lin Algebra Appl 428: 1406–1420

    Article  Google Scholar 

  • Chen D, Hu X, Luan S, Wang C, Naqvi S, Yu C (2004a) Generalized geometric approaches for leaf sequencing problems in radiation therapy. In: Procedings of the 15th Annual international symposium on algorithms and computation (ISAAC), Hong Kong, December 2004, vol 3341 of Lecture Notes in Computer Science. Springer, Berlin, pp 271–281

  • Chen D, Hu X, Luan S, Wang C, Wu X (2004b) Geometric algorithms for static leaf sequencing problems in radiation therapy. Int J Comput Geometry Appl 14(5): 311–339

    Article  Google Scholar 

  • Chen D, Hu X, Luan S, Wu X, Yu C (2005) Optimal terrain construction problems and applications in intensity-modulated radiation therapy. Algorithmica 42: 265–288

    Article  Google Scholar 

  • Chen D, Hu X, Luan S, Wang C, Naqvi S, Yu C (2006) Generalized geometric approaches for leaf sequencing problems in radiation therapy. Int J Comput Geom Appl 16(2-3): 175–204

    Article  Google Scholar 

  • Chen W, Herman G, Censor Y (2008) Algorithms for satisfying dose volume constraints in intensity-modulated radiation therapy. In: Censor Y, Jiang M, Louis A (eds) Mathematical methods in biomedical imaging and intensity-modulated radiation therapy (IMRT) Edizioni della Normale, Pisa (to appear)

  • Chen Y, Michalski D, Houser C, Galvin J (2002) A deterministic iterative least-squares algorithm for beam weight optimization in conformal radiotherapy. Phys Med Biol 47: 1647–1658

    Article  Google Scholar 

  • Cho P, Lee S, Marks R, Oh S, Sutlief S, Phillips M (1998) Optimization of intensity modulated beams with volume constraints using two methods: cost function minimization and projections onto convex sets. Med Phys 25: 435–443

    Article  Google Scholar 

  • Choi B, Deasy J (2002) The generalized equivalent uniform dose function as a basis for intensity-modulated treatment planning. Phys Med Biol 47: 3579–3589

    Article  Google Scholar 

  • Chue M, Zinchenko Y, Henderson S, Sharpe M (2005) Robust optimization for intensity modulated radiation therapy treatment planning under uncertainty. Phys Med Biol 50: 5463–5477

    Article  Google Scholar 

  • Collins M, Kempe D, Saia J, Young M (2007) Nonnegative integral subset representations of integer sets. Inf Process Lett 101(3): 129–133

    Article  Google Scholar 

  • Convery D, Rosenbloom M (1992) The generation of intensity-modulated fields for conformal radiotherapy by dynamic collimation. Phys Med Biol 37(6): 1359–1374

    Article  Google Scholar 

  • Convery D, Webb S (1998) Generation of discrete beam-intensity modulation by dynamic multileaf collimation under minimum leaf separation constraints. Phys Med Biol 43: 2521–2538

    Article  Google Scholar 

  • Cotrutz C, Xing L (2002) Using voxel-dependent importance factors for interactive DVH-based dose optimisation. Phys Med Biol 47: 1659–1669

    Article  Google Scholar 

  • Cotrutz C, Xing L (2003) Segment-based dose optimisation using a genetic algorithm. Phys Med Biol 48: 2987–2998

    Article  Google Scholar 

  • Cotrutz C, Lahanas M, Kappas C, Baltas D (2001) A multiobjective gradient-based dose optimization algorithm for external beam conformal radiotherapy. Phys Med Biol 46: 2161–2175

    Article  Google Scholar 

  • Craft D (2007) Local beam angle optimization with linear programming and gradient search. Phys Med Biol 52: 127–135

    Article  Google Scholar 

  • Craft D, Halabi T, Bortfeld T (2005) Exploration of tradeoffs in intensity-modulated radiotherapy. Phys Med Biol 50: 5857–5868

    Article  Google Scholar 

  • Craft D, Halabi T, Shih H, Bortfeld T (2006) Approximating convex Pareto surfaces in multiobjective radiotherapy planning. Med Phys 33: 3399–3407

    Article  Google Scholar 

  • Crooks S, McAven L, Robinson D, Xing L (2002) Minimizing delivery time and monitor units in static IMRT by leaf-sequencing. Phys Med Biol 47: 3105–3116

    Article  Google Scholar 

  • Crooks SM, Xing L (2002) Application of constrained least-squares techniques to IMRT treatment planning. Int J Radiat Oncol Biol Phys 54(4): 1217–1224

    Article  Google Scholar 

  • Dai J, Zhu Y (2003) Conversion of dose–volume constraints to dose limits. Phys Med Biol 48: 3927–3941

    Article  Google Scholar 

  • Das I, Dennis J (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct Multidisciplin Optim 14: 63–69

    Article  Google Scholar 

  • Das I, Dennis JE (1998) Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8(3): 631–657

    Article  Google Scholar 

  • Das S, Cullip T, Tracton G, Chang S, Marks L, Anscher M, Rosenman J (2003) Beam orientation selection for intensity modulated radiation therapy based on target equivalent uniform dose maximization. Int J Radiat Oncol Biol Phys 55(1): 215–224

    Article  Google Scholar 

  • De Werra D, Costa M, Picouleau C, Ries B (2008) On the use of graphs in discrete tomography. 4OR 6: 101–123

    Article  Google Scholar 

  • Deasy JO (1997) Multiple local minima in radiotherapy optimization problems with dose–volume constraints. Med Phys 24(7): 1157–1161

    Article  Google Scholar 

  • Dirkx M, Heijmen B, van Santvoort J (1998) Leaf trajectory calculation for dynamic multileaf collimation to realize optimized fluence profiles. Phys Med Biol 43: 1171–1184

    Article  Google Scholar 

  • Djajaputra D, Wu Q, Wu Y, Mohan R (2003) Algorithm and performance of a clinical IMRT beam-angle optimization system. Phys Med Biol 48: 3191–3212

    Article  Google Scholar 

  • D’Souza W, Meyer R, Shi L (2004) Selection of beam orientations in intensity-modulated radiation therapy using single-beam indices and integer programming. Phys Med Biol 49: 3465–3481

    Article  Google Scholar 

  • Ehrgott M (2005) Multicriteria optimization, 2nd edn. Springer, Berlin

    Google Scholar 

  • Ehrgott M, Burjony M (2001) Radiation therapy planning by multicriteria optimisation. In: Proceedings of the 36th Annual conference of the Operational Research Society of New Zealand, pp 244–253

  • Ehrgott M, Johnston R (2003) Optimisation of beam directions in intensity modulated radiation therapy planning. OR Spectr 25(2): 251–264

    Article  Google Scholar 

  • Ehrgott M, Hamacher H, Nußbaum M (2007) Decomposition of matrices and static multileaf collimators: a survey. In: Alves C, Pardalos P, Vincente L(eds) Optimization in medicine. Springer, Berlin, pp 27–48

    Google Scholar 

  • Ehrgott M, Holder A, Reese J (2008) Beam selection in radiotherapy design. Lin Algebra Appl 428: 1272–1312

    Article  Google Scholar 

  • Emami B, Lyman J, Brown A (1991) Tolerance of normal tissue to therapeutic irradiation. Int J Radiat Oncol Biol Phys 21: 109–122

    Google Scholar 

  • Engel K (2005) A new algorithm for optimal MLC field segmentation. Discr Appl Math 152: 35–51

    Article  Google Scholar 

  • Engel K, Tabbert E (2005) Fast simultaneous angle, wedge, and beam intensity optimization in inverse radiotherapy planning. Optim Eng 6: 393–419

    Article  Google Scholar 

  • Engelbeen C (2007) Réalisation de profils d’intensité par des collimateurs multilames statiques en radiothérapie. Master’s thesis, Département de Mathématique, Université Libre de Bruxelles

  • Engelbeen C, Fiorini S (2007) Constrained decompositions of integer matrices and their applications to intensity modulated radiation therapy. Technical report, Département de Mathématique, Université Libre de Bruxelles. Available at http://homepages.ulb.ac.be/~sfiorini/papers/decomp_IMRT.pdf

  • Ernst A, Mak V, Mason L (2007) An exact method for the minimum cardinality problem in the planning of IMRT. Technical report, School of Engineering and Information Technology, Deakin University

  • Ezzell GA (1996) Genetic and geometric optimization of three-dimensional radiation therapy. Med Phys 23: 293–305

    Article  Google Scholar 

  • Ferris M, Voelker M (2004) Fractionation in radiation treatment planning. Math Program Ser B 101: 387–413

    Google Scholar 

  • Fippel M, Alber M, Birkner M, Laub W, Nüsslin F, Kawrakow I (2001) Inverse treatment planning for radiation therapy based on fast Monte-Carlo dose calculation. In: Kling A, Barao F, Nakagawa M, Travora L, Vaz P(eds) Advanced Monte Carlo for radiation physics, particle transport simulation and applications: proceedings of the Monte Carlo 2000 conference, Lisbon, 23–26 October 2000. Springer, Berlin, pp 217–222

    Google Scholar 

  • Gersho A, Gray R (1991) Vector quantization and signal compression. Kluwer, Boston

    Google Scholar 

  • Gong Y (2006) Integer programming methods for beam selection in radiotherapy treatment planning. Master’s Thesis, Department of Engineering Science, The University of Auckland

  • Gunawardena A, D’Souza W, Goadrick L, Meyer R, Sorensen K, Naqvi S, Shi L (2006) A difference-matrix metaheuristic for intensity map segmentation in step-and-shoot imrt delivery. Phys Med Biol 51: 2517–2536

    Article  Google Scholar 

  • Haas O, Burnham K, Mills J (1998) Optimization of beam orientation in radiotherapy using planar geometry. Phys Med Biol 43: 2179–2193

    Article  Google Scholar 

  • Halabi T, Craft D, T B (2006) Dose–volume objectives in multi-criteria optimization. Phys Med Biol 51: 3809–3818

    Article  Google Scholar 

  • Hamacher H, Küfer K-H (2002) Inverse radiation therapy planning—a multiple objective optimization approach. Discr Appl Math 118(1-2): 145–161

    Article  Google Scholar 

  • Hodes L (1974) Semiautomatic optimization of external beam radiation treatment planning. Radiology 110: 191–196

    Google Scholar 

  • Holder A (2003) Designing radiotherapy plans with elastic constraints and interior point methods. Health Care Manage Sci 6: 5–16

    Article  Google Scholar 

  • Holder A (2004) Radiotherapy treatment design and linear programming. In: Brandeau M, Sainfort F, Pierskalla W(eds) Operations research and health care. Kluwer, Norwell, pp 741–774

    Google Scholar 

  • Holder A (2006) Partitioning multiple objective optimal solutions with applications in radiotherapy design. Optim Eng 7: 501–526

    Article  Google Scholar 

  • Holder A, Llagostera D (2008) Optimal treatments for photodynamic therapy. 4OR 6: 167–182

    Article  Google Scholar 

  • Holder A, Salter B (2004) A tutorial on radiation oncology and optimization. In: Greenberg H(eds) Tutorials on emerging methodologies and applications in operations research, Chap 4. Kluwer, Boston

    Google Scholar 

  • Holmes T, Mackie TR (1994) A filtered backprojection dose calculation method for inverse treatment planning. Med Phys 21: 303–313

    Article  Google Scholar 

  • Hou Q, Wang J, Chen Y, Galvin J (2003) Beam orientation optimization for imrt by a hybrid method of the genetic algorithm and the simulated dynamics. Med Phys 30: 2360–2367

    Article  Google Scholar 

  • Hristov D, Fallone B (1997) An active set algorithm for treatment planning optimization. Med Phys 24: 91–106

    Article  Google Scholar 

  • Hristov D, Fallone B (1998) A continuous penalty function method for inverse treatment planning. Med Phys 25(2): 208–223

    Article  Google Scholar 

  • Jackson A, Kutcher GJ (1993) Probability of radiation-induced complications for normal tissues with parallel architecture subject to non-uniform irradiation. Med Phys 20(3): 613–625

    Article  Google Scholar 

  • Jeleń U, Söhn M, Alber M (2005) A finite pencil beam for IMRT dose optimization. Med Phys 50: 1747–1766

    Article  Google Scholar 

  • Jeraj R, Wu C, Mackie T (2003) Optimizer convergence and local minima errors and their clinical importance. Phys Med Biol 48: 2809–2827

    Article  Google Scholar 

  • Kalinowski T (2004) Algorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentation. Technical report, Department of Mathematics, University of Rostock. Preprint 2004/1

  • Kalinowski T (2005) A duality based algorithm for multileaf collimator field segmentation with interleaf collision constraint. Discr Appl Math 152: 52–88

    Article  Google Scholar 

  • Kalinowski T (2008) Reducing the tongue-and-groove underdosage in mlc shape matrix decomposition. Algor Oper Res 3: 165–174

    Google Scholar 

  • Källman P, Lind B, Eklöf A, Brahme A (1988) Shaping of arbitrary dose distributions by dynamic multileaf collimation. Phys Med Biol 33(11): 1291–1300

    Article  Google Scholar 

  • Källman P, Ågren A, Brahme A (1992) Tumor and normal tissue responses to fractionated non uniform dose delivery. Int J Radiat Biol 62(2): 249–262

    Article  Google Scholar 

  • Kamath S, Sahni S, Li J, Palta J, Ranka S (2003) Leaf sequencing algorithms for segmented multileaf collimation. Phys Med Biol 48(3): 307–324

    Article  Google Scholar 

  • Kamath S, Sahni S, Palta J, Ranka S (2004a) Algorithms for optimal sequencing of dynamic multileaf collimators. Phys Med Biol 49: 33–54

    Article  Google Scholar 

  • Kamath S, Sahni S, Ranka S, Li J, Palta J (2004b) A comparison of step-and-shoot leaf sequencing algorithms that eliminate tongue-and-groove effects. Phys Med Biol 49: 3137–3143

    Article  Google Scholar 

  • Kamath S, Sahni S, Palta J, Ranka S, Li J (2004c) Optimal leaf sequencing with elimination of tongue-and-groove underdosage. Phys Med Biol 49: N7–N19

    Article  Google Scholar 

  • Kennedy JM (2000) Minimum support solutions for radiotherapy treatment planning. Master’s Thesis, Department of Mathematics, University of Colorado at Denver, Denver

  • Khan FM (2003) The physics of radiation therapy. Lippincott Williams & Wilkins, Philapelphia

    Google Scholar 

  • Kolmonen P, Tervo J, Lahtinen P (1998) Use of the Cimmino algorithm and continuous approximation for the dose deposition kernel in the inverse problem of radiation treatment planning. Phys Med Biol 43: 2539–2554

    Article  Google Scholar 

  • Küfer K-H, Hamacher H (2000) A multicriteria optimization approach for inverse radiotherapy planning. In: Schlegel W, Bortfeld T(eds) XIIIth International conference on the use of computers in radiation therapy, Heidelberg, Germany. Springer, Berlin, pp 26–28

    Google Scholar 

  • Küfer K-H, Scherrer A, Monz M, Alonso F, Trinkaus H, Bortfeld T, Thieke C (2003) Intensity-modulated radiotherapy—a large scale multi-criteria programming problem. OR Spectr 25: 223–249

    Article  Google Scholar 

  • Kutcher GJ, Burman C, Brewster L, Goitein M, Mohan R (1991) Histogram reduction method for calculating complication probabilities for three-dimensional treatment planning evalutions. Int J Radiat Oncol Biol Phys 21: 137–146

    Google Scholar 

  • Lahanas M, Schreibmann E, Milickovic N, Baltas D (2003) Intensity modulated beam radiation therapy dose optimization with multi-objective evolutionary algorithms. In: Fonseca C, Fleming P, Zitzler E, Deb K, Thiele L(eds) Evolutionary multi-criterion optimization. 2nd international conference, EMO 2003, Faro, Portugal, 8–11 April 2003, Proceedings, volume 2632 of Lecture Notes in Computer Science. Springer, Berlin, pp 648–661

    Google Scholar 

  • Lahanas M, Schreibmann E, Baltas D (2003b) Multiobjective inverse planning for intensity modulated radiotherapy with constraint-free gradient-based optimization algorithms. Phys Med Biol 48: 2843–2871

    Article  Google Scholar 

  • Langer M (1987) Optimization of beam weights under dose–volume restrictions. Int J Radiat Oncol Biol Phys 13: 1255–1260

    Google Scholar 

  • Langer M, Morrill S (1996) A comparison of mixed integer programming and fast simulated annealing for optimized beam weights in radiation therapy. Med Phys 23: 957–964

    Article  Google Scholar 

  • 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: 887–893

    Google Scholar 

  • Langer M, Brown R, Morill S, Lane R, Lee O (1996) A generic genetic algorithm for generating beam weights. Med Phys 23: 965–971

    Article  Google Scholar 

  • Langer M, Thai V, Papiez L (2001) Improved leaf sequencing reduces segments of monitor units needed to deliver IMRT using MLC. Med Phys 28: 2450–2458

    Article  Google Scholar 

  • Lee E, Fox T, Crocker I (2003) Integer programming applied to intensity-modulated radiation therapy treatment planning. Ann Oper Res 119: 165–181

    Article  Google Scholar 

  • Lee S, Cho P, Marks R, Oh S (1997) Conformal radiotherapy computation by the method of alternating projections onto convex sets. Phys Med Biol 42: 1065–1086

    Article  Google Scholar 

  • Legras J, Legras B, Lambert J (1982) Software for linear and non-linear optimization in external radiotherapy. Comput Programs Biomed 15: 233–242

    Article  Google Scholar 

  • Lim G, Ferris M, Wright S, Shepard D, Earl M (2007) An optimization framework for conformal radiation treatment planning. INFORMS J Comput 19: 366–380

    Article  Google Scholar 

  • Lim G, Choi J, Mohan R (2008) Iterative solution methods for beam angle and fluence map optimization in intensity modulated radiation therapy planning. OR Spectr 30: 289–309

    Article  Google Scholar 

  • Liu DC, Nocedal J (1989) On the limited memory BFGS method for large scale optimization. Math Program 45: 503–528

    Article  Google Scholar 

  • Llacer G (1997) Inverse radiation treatment planning using the dynamically penalized likelihood method. Med Phys 24(11): 1751–1764

    Article  Google Scholar 

  • Llacer J, Deasy J, Bortfeld T, Solberg T, Promberger C (2003) Absence of multiple local minima effects in intensity modulated optimization with dose–volume constraints. Phys Med Biol 48: 183–210

    Article  Google Scholar 

  • Lloyd S (1982) Least squares quantization in PCM. IEEE Trans Inf Theory 28: 127–135

    Article  Google Scholar 

  • Löf J (2000) Development of a general framework for optimization of radiation therapy. PhD Thesis, Department of Medical Radiation Physics, Karolinska Institute, Stockholm, Sweden

  • Luan S, Saia J, Young M (2007) Approximation algorithms for minimizing segments in radiation therapy. Inf Process Lett 101: 239–244

    Article  Google Scholar 

  • Lyman JT, Wolbrast AB (1989) Optimization of radiation therapy IV: a dose–volume histogram reduction algorithm. Int J Radiat Oncol Biol Phys 17: 433–436

    Google Scholar 

  • Ma L, Boyer AL, Xing L, Ma C-M (1998) An optimized leaf-setting algorithm for beam intensity modulation using dynamic multileaf collimators. Phys Med Biol 43: 1629–1643

    Article  Google Scholar 

  • Ma L, Boyer AL, Ma C-M, Xing L (1999) Synchronizing dynamic multileaf collimators for producing two-dimensional intensity-modulated fields with minimum beam delivery time. Int J Radiat Oncol Biol Phys 44(5): 1147–1154

    Google Scholar 

  • Mackie T, Holmes T, Swerdloff S, Reckwerdt P, Deasy J, Yang J, Paliwal B, Kinsella T (1993) Tomotherapy: a new concept for the delivery of dynamic conformal radiotherapy. Med Phys 20: 1709–1719

    Article  Google Scholar 

  • Mageras GS, Mohan R (1993) Application of fast simulated annealing to optimization of conformal radiation treatments. Med Phys 20(3): 1–3

    Article  Google Scholar 

  • Mak V (2007) Iterative variable aggregation and disaggregation in IP: an application. Oper Res Lett 35: 36–44

    Article  Google Scholar 

  • McDonald SC, Rubin P (1977) Optimization of external beam radiation therapy. Int J Radiat Oncol Biol Phys 2: 307–317

    Google Scholar 

  • Meedt G, Alber M, Nüsslin F (2003) Non-coplanar beam direction optimization for intensity-modulated radiotherapy. Phys Med Biol 48: 2999–3019

    Article  Google Scholar 

  • Merritt M, Zhang Y (2002) A successive linear programming approach to IMRT optimization problem. Technical report, Department of Computational and Applied Mathematics, Rice University. http://www.caam.rice.edu/~zhang/reports/tr0216.pdf

  • Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized constraint method for generating the Pareto frontier. Struct Multidisciplin Optim 25: 86–98

    Article  Google Scholar 

  • Michalski D, Xiao Y, Censor Y, Galvin J (2004) The dose–volume constraint satisfaction problem for inverse treatment planning with field segments. Phys Med Biol 49: 601–616

    Article  Google Scholar 

  • Morrill S, Rosen I, Lane R, Belli J (1990a) The influence of dose constraint point placement on optimized radiation therapy treatment planning. Int J Radiat Oncol Biol Phys 19: 129–141

    Google Scholar 

  • Morrill S, Lane R, Wong J, Rosen I (1991a) Dose–volume considerations with linear programming optimization. Med Phys 18(6): 1201–1210

    Article  Google Scholar 

  • Morrill S, Lane R, Jacobson G, Rosen I (1991b) Treatment planning optimization using constrained simulated annealing. Phys Med Biol 36(10): 1341–1361

    Article  Google Scholar 

  • Morrill S, Lam K, Lane R, Langer M, Rosen I (1995) Very fast simulated annealing in radiation therapy treatment plan optimization. Int J Radiat Oncol Biol Phys 31: 179–188

    Google Scholar 

  • Morrill SM, Lane RG, Rosen II (1990b) Constrained simulated annealing for optimized radiation therapy treatment planning. Comput Meth Programs Biomed 33: 135–144

    Article  Google Scholar 

  • Motzkin TS, Schoenberg IJ (1954) The relaxation method for linear inequalities. Can J Math 6: 393–404

    Google Scholar 

  • Niemierko A (1992) Random search algorithm (RONSC) for optimization of radiation therapy with both physical and biological end points and constraints. Int J Radiat Oncol Biol Phys 33: 89–98

    Google Scholar 

  • Niemierko A (1997) Reporting and analysing dose distributions: a concept of equivalent uniform dose. Med Phys 24: 103–110

    Article  Google Scholar 

  • Niemierko A (1999) A generalized concept of equivalent uniform dose. Med Phys 26: 1100

    Google Scholar 

  • Niemierko A, Goitein M (1991) Calculation of normal tissue complication probability and dose–volume histogram reduction schemes for tissues with a critical element architecture. Radiother Oncol 20: 166–176

    Article  Google Scholar 

  • Nizin P, Kania A, Ayyangar K (2001) Basic concepts of CORVUS dose model. Med Dosim 26(1): 65–69

    Article  Google Scholar 

  • Nußbaum M (2006) Min Cardinality C1-Decomposition of Integer Matrices. Master’s Thesis, Department of Mathematics, Technical University of Kaiserslautern

  • Ólafsson A, Wright S (2006) Efficient schemes for robust IMRT treatment planning. Phys Med Biol 51: 5621–5642

    Article  Google Scholar 

  • Peñagarícano JA, Papanikolaou N, Wu C, Yan Y (2005) An assessment of biologically-based optimization (BORT) in the IMRT era. Med Dosim 30(1): 12–19

    Article  Google Scholar 

  • Powlis W, Altschuler M, Censor Y, Buhle J (1989) Semi-automatic radiotherapy treatment planning with a mathemathical model to satisfy treatment goals. Int J Radiat Oncol Biol Phys 16: 271–276

    Google Scholar 

  • Preciado-Walters F, Rardin R, Langer M, Thai V (2004) A coupled column generation, mixed integer approach to optimal planning of intensity modulated radiation therapy for cancer. Math Program Ser B 101: 319–338

    Article  Google Scholar 

  • Pugachev A, Xing L (2001) Pseudo beam’s-eye-view as applied to beam orientation selection in intensity-modulated radiation therapy. Int J Radiat Oncol Biol Phys 51(5): 1361–1370

    Google Scholar 

  • Que W (1999) Comparison of algorithms for multileaf collimator field segmentation. Med Phys 26: 2390–2396

    Article  Google Scholar 

  • Que W, Kung J, Dai J (2004) Tongue-and-groove effect in intensity modulated radiotherapy with static multileaf collimator fields. Phys Med Biol 49: 399–405

    Article  Google Scholar 

  • Redpath AT, Vickery BL, Wright DH (1976) A new technique for radiotherapy planning using quadratic programming. Phys Med Biol 21: 781–91

    Article  Google Scholar 

  • Romeijn H, Ahuja R, Dempsey J, Kumar A, Li J (2003) A novel linear programming approach to fluence map optimization for intensity modulated radiation therapy treatment planning. Phys Med Biol 48: 3521–3542

    Article  Google Scholar 

  • Romeijn H, Dempsey J, Li J (2004) A unifying framework for multi-criteria fluence map optimization models. Phys Med Biol 49: 1991–2013

    Article  Google Scholar 

  • Romeijn H, Ahuja R, Dempsey J, Kumar A (2006) A new linear programming approach to radiation therapy treatment planning problems. Oper Res 54(2): 201–216

    Article  Google Scholar 

  • Rosen I, Lane R, Morrill S, Belli J (1991) Treatment planning optimisation using linear programming. Med Phys 18(2): 141–152

    Article  Google Scholar 

  • Rowbottom C, Webb S (2002) Configuration space analysis of common cost functions in radiotherapy beam-weight optimization algorithms. Phys Med Biol 47: 65–77

    Article  Google Scholar 

  • Rowbottom C, Khoo V, Webb S (2001) Simultaneous optimization of beam orientations and beam weights in conformal radiotherapy. Med Phys 28(8): 1696–1702

    Article  Google Scholar 

  • Schlegel W, Mahr A (2002) 3D-conformal radiation therapy: a multimedia introduction to methods and techniques. Springer, Heidelberg, Berlin

    Google Scholar 

  • Shao L, Ehrgott M (2007) Finding representative nondominated points in multiobjective linear programming. In: Proceedings of the IEEE symposium on computational intelligence in multi-criteria decision-making, 1–5 April 2007, Honolulu, pp 245–252. IEEE Service Center, Piscataway

  • Shao L, Ehrgott M (2008a) Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning. Math Methods Oper Res doi:10.1007/s00186-008-0220-2

  • Shao L, Ehrgott M (2008b) Approximating the nondominated set of an MOLP by approximately solving its dual problem. Math Methods Oper Res. doi:10.1007/s00186-007-0194-5

  • Shepard D, Ferris M, Olivera G, Mackie T (1999) Optimizing the delivery of radiation therapy to cancer patients. SIAM Rev 41(4): 721–744

    Article  Google Scholar 

  • Siochi R (1999) Minimizing static intensity modulation delivery time using an intensity solid paradigm. Int J Radiat Oncol Biol Phys 43: 671–689

    Google Scholar 

  • Sir M, Pollock S, Epelman M, Lam K, Haken R (2006) Ideal spatial radiotherapy dose distributions subject to positional uncertainties. Phys Med Biol 51: 6329–6347

    Article  Google Scholar 

  • Söderström S, Brahme A (1992) Selection of beam orientations in radiation therapy using entropy and fourier transform measures. Phys Med Biol 37(4): 911–924

    Article  Google Scholar 

  • Sonderman D, Abrahamson P (1985) Radiotherapy treatment design using mathematical programming models. Oper Res 33(4): 705–725

    Article  Google Scholar 

  • Spirou SV, Chui CS (1994) Generation of arbitrary intensity profiles by dynamic jaws or multileaf collimators. Med Phys 27(7): 1031–1041

    Article  Google Scholar 

  • Spirou SV, Chui C-S (1998) A gradient inverse planning algorithm with dose–volume constraints. Med Phys 25(3): 321–333

    Article  Google Scholar 

  • Starkschall G (1984) A constrained least-squares optimization method for external beam treatment planning. Med Phys 11(5): 659–665

    Article  Google Scholar 

  • Starkschall G, Pollack A, Stevens CW (2001) Treatment planning using a dose–volume feasibility search algorithm. Int J Radiat Oncol Biol Phys 49(5): 1419–1427

    Google Scholar 

  • Stavrev P, Hristov D, Warkentin B, Fallone BG (2003) Inverse treatment planning by physically constrained minimization of a biological objective function. Med Phys 30: 2948–2958

    Article  Google Scholar 

  • Stein J, Bortfeld T, Dörschel B, Schlegel W (1994) Dynamic X-ray compensation for conformal radiotherapy by means of multileaf collimation. Radiother Oncol 32: 163–173

    Article  Google Scholar 

  • Stein J, Mohan R, Wang X, Bortfeld T, Wu Q, Preiser K, Ling C, Schlegel W (1997) Number and orientations of beams in intensity-modulated radiation treatments. Med Phys 24(2): 149–160

    Article  Google Scholar 

  • Svensson R, Källman P, Brahme A (1994) An analytical solution for the dynamic control of multileaf collimators. Phys Med Biol 39: 37–61

    Article  Google Scholar 

  • Taşkin Z, Smith J, Romeijn H, Dempsey J (2007) Optimal multileaf collimator leaf sequencing in IMRT treatment planning. Technical report, Department of Industrial and Systems Enginnering, University of Florida

  • Thieke C (2003) Multicriteria optimisation in inverse radiotherapy planning. PhD Thesis, Ruprecht–Karls–Universität Heidelberg, Germany

  • Thieke C, Bortfeld T, Küfer K-H (2002) Characterization of dose distributions through the max and mean dose concept. Acta Oncol 41: 158–161

    Article  Google Scholar 

  • Tucker SL, Thames HD, Taylor JMG (1990) How well is the probability of tumor cure after fractionated irradiation described by poisson statistics?. Radiat Res 124: 273–282

    Article  Google Scholar 

  • Ulmer W, Harder D (1995) A triple Gaussian pencil beam model for photon beam treatment planning. Zeitschrift für medizinische Physik 5: 25–30

    Google Scholar 

  • Van Santvoort J, Heijmen B (1996) Dynamic multileaf collimation without tongue-and-groove underdosage effects. Phys Med Biol 41: 2091–2105

    Article  Google Scholar 

  • Verhaegen F (2003) Monte Carlo modelling of external radiotherapy photon beams. Phys Med Biol 48: 107–164

    Article  Google Scholar 

  • Wake G, Boland N, Jennings L (2007) Mixed integer programming approaches to exact minimization of total treatment time in cancer radiotherapy using multileaf collimators. Comput Oper Res. doi:10.1016/j.cor.2007.10.027

  • Wang C, Dai J, Hu Y (2003) Optimization of beam orientations and beam weights for conformal radiotherapy using mixed integer programming. Phys Med Biol 48: 4065–4076

    Article  Google Scholar 

  • Wang X, Mohan R, Jackson A, Leibel S, Fuks Z, Ling C (1995) Optimization of intensity-modulated 3D conformal treatment plans based on biological indices. Radiother Oncol 37: 140–152

    Article  Google Scholar 

  • Wang X, Spirou S, LoSasso T, Stein J, Chui C-S, Mohan R (1996) Dosimetric verification of intensity-modulated fields. Med Phys 23(3): 317–327

    Article  Google Scholar 

  • Webb S (1989) Optimisation of conformal radiotherapy dose distribution by simulated annealing. Phys Med Biol 34: 1349–1370

    Article  Google Scholar 

  • Webb S (1991) Optimization of conformal radiotherapy dose distributions by simulated annealing: 2. Inclusion of scatter in the technique. Phys Med Biol 36: 1227–1237

    Article  Google Scholar 

  • 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: 1689–1704

    Article  Google Scholar 

  • Webb S (1994a) Optimizing the planning of intensity-modulated radiotherapy. Phys Med Biol 39: 2229–2246

    Article  Google Scholar 

  • Webb S (1994b) Optimum parameters in a model for tumor control probability including interpatient heterogeneity. Phys Med Biol 39: 1895–1914

    Article  Google Scholar 

  • Webb S (2001) Intensity-modulated radiation therapy (Ser Med Phys) Institute of Physics Publishing, Bristol, Philadelphia

  • Webb S, Nahum AE (1993) A model for calculating tumor control probability in radiotherapy including the effects of inhomogeneous distributions of dose and clonogenic cell density. Phys Med Biol 38: 653–666

    Article  Google Scholar 

  • Webb S, Bortfeld T, Stein J, Convery D (1997) The effect of stair-step leaf transmission on the "tongue-and-groove problem" in dynamic radiotherapy with a multileaf collimator. Phys Med Biol 42: 595–602

    Article  Google Scholar 

  • Wu C, Jeraj R, Mackie T (2003a) The method of intercepts in parameter space for the analysis of local minima caused by dose–volume constraints. Phys Med Biol 48: N149–N157

    Article  Google Scholar 

  • Wu Q et al (2003b) Intensity-modulated radiotherapy optimization with gEUD-guided dose–volume objectives. Phys Med Biol 48: 279–291

    Article  Google Scholar 

  • Wu Q, Mohan R (2000) Algorithms and functionality of an intensity modulated radiotherapy optimization system. Med Phys 27(4): 701–711

    Article  Google Scholar 

  • Wu Q, Mohan R (2002) Multiple local minima in IMRT optimization-based on dose–volume criteria. Med Phys 29: 1514–1527

    Article  Google Scholar 

  • Wu Q, Mohan R, Niemierko A, Schmidt-Ulirich R (2002) Optimization of intensity-modulated radiotherapy plans based on the equivalent uniform dose. Int J Radiat Oncol Biol Phys 52(1): 224–235

    Google Scholar 

  • Wu X, Zhu Y (2001) An optimization method for importance factors and beam weights based on genetic algorithms for radiotherapy treatment planning. Phys Med Biol 46: 1085–1099

    Article  Google Scholar 

  • Xia P, Verhey L (1998) Multileaf collimator leaf sequencing algorithm for intensity modulated beams with multiple segments. Med Phys 25(8): 1424–1434

    Article  Google Scholar 

  • Xiao Y, Censor Y, Michalski D, Galvin J (2003) The least-intensity feasible solution for aperture-based inverse planning in radiation therapy. Ann Oper Res 119: 183–203

    Article  Google Scholar 

  • Xing L, Chen GTY (1996) Iterative algorithms for inverse treatment planning. Phys Med Biol 41: 2107–2123

    Article  Google Scholar 

  • Xing L, Hamilton R, Spelbring D, Pelizzari C, Chen G, Boyer A (1998) Fast iterative algorithms for three-dimensional inverse treatment planning. Med Phys 25: 1845–1849

    Article  Google Scholar 

  • Xing L, Li JG, Donaldson S, Le QT, Boyer AL (1999) Optimization of importance factors in inverse planning. Phys Med Biol 44: 2525–2536

    Article  Google Scholar 

  • Yu CX (1995) Intensity-modulated arc therapy with dynamic multileaf collimation: an alternative to tomotherapy. Phys Med Biol 40: 1435–1449

    Article  Google Scholar 

  • Yu CX, Symons M, Du M, Martinez A, Wong J (1995) A method for implementing dynamic photon beam intensity modulation using independent jaws and a multileaf collimator. Phys Med Biol 40: 769–787

    Article  Google Scholar 

  • Yu Y (1997) Multi-objective decision theory for computational optimisation in radiation therapy. Med Phys 24(9): 1445–1454

    Article  Google Scholar 

  • Zaider M, Minerbo GN (2000) Tumor control probability: A formulation applicable to any temporal protocol of dose delivery. Phys Med Biol 45: 279–293

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Science, The University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand

    Matthias Ehrgott & Lizhen Shao

  2. Department of Mathematics, Technical University of Kaiserslautern, Postfach 3049, 67653, Kaiserslautern, Germany

    Çiğdem Güler & Horst W. Hamacher

Authors
  1. Matthias Ehrgott
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  2. Çiğdem Güler
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  3. Horst W. Hamacher
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  4. Lizhen Shao
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Correspondence to Matthias Ehrgott.

Additional information

H.W. Hamacher and Ç. Güler have been partially supported by the Deutsche Forschungsgemeinschaft (DFG), Grant HA 1737/7 “Algorithmik großer und komplexer Netzwerke”, the Julius-von-Haast Award of New Zealand’s MORST, and by the Rhineland-Palatinate cluster of excellence “Dependable adaptive systems and mathematical modeling”. L. Shao has been supported by The University of Auckland, Grant 23364/8214.

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Ehrgott, M., Güler, Ç., Hamacher, H.W. et al. Mathematical optimization in intensity modulated radiation therapy. 4OR 6, 199–262 (2008). https://doi.org/10.1007/s10288-008-0083-7

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