Mathematical optimization in intensity modulated radiation therapy

4OR

Volumn 6, Issue 3, 2008, Pages 199-262

  Ehrgott, Matthias ^a   Güler, Çiǧdem ^b   Hamacher, Horst W ^b   Shao, Lizhen ^a  

^a UNIVERSITY OF AUCKLAND   (New Zealand)

^b Rheinland Pfälzische Technische Universität Kaiserslautern Landau   (Germany)

Author keywords

Intensity modulated radiation therapy; Mathematical models; Medical physics; Optimization

Indexed keywords

EID: 50849095488     PISSN: 16194500     EISSN: 16142411     Source Type: Journal    
DOI: 10.1007/s10288-008-0083-7     Document Type: Article

References (216)

1. Agmon S (1954) The relaxation method for linear inequalities. Can J Math 6: 382-392
2. 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
3. Ahuja R, Magnanti T, Orlin J (1993) Network flows: theory, algorithms and applications. Prentice-Hall, Englewood Cliffs
4. Alber M, Nüsslin F (2001) A representation of an NTCP function for local complication mechanisms. Phys Med Biol 46: 439-447
5. Alber M, Reemtsen R (2007) Intensity modulated radiation therapy planning by use of a barrier-penalty multiplier method. Optim Meth Softw 22: 391-411
6. 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
7. 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
8. 3, Lambrecht, Germany, pp 11-29
9. Baatar D, Hamacher H, Ehrgott M, Woeginger G (2005) Decomposition of integer matrices and multileaf collimator sequencing. Discr Appl Math 152: 6-34
10. 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
11. 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)
12. 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
13. 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
14. 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
15. Bednarz G, Michalski D, Anne P, Valicenti R (2004) Inverse treatment planning using volume-based objective functions. Phys Med Biol 49: 2503-2514
16. 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
17. Billups S, Kennedy J (2003) Minimum-support solutions for radiotherapy planning. Ann Oper Res 119: 229-245
18. Boland N, Hamacher H, Lenzen F (2004) Minimizing beam-on time in cancer radiation treatment using multileaf collimators. Networks 43(4): 226-240
19. Bortfeld T (1999) Optimized planning using physical objectives and constraints. Semin Radiat Oncol 9: 20-34
20. 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
21. 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
22. 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
23. 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
24. Boyer AL, Yu CX (1999) Intensity-modulated radiation therapy with dynamic multileaf collimators. Semin Radiat Oncol 9(1): 48-59
25. Brahme A (1988) Optimization of stationary and moving beam radiation therapy techniques. Radiother Oncol 12: 129-140
26. Brahme A (2001) Individualizing cancer treatment: biological optimization models in treatment and planning. Int J Radiat Oncol Biol Phys 49: 327-337
27. Brahme A, Agren AK (1987) Optimal dose distribution for eradication of heterogeneous tumours. Acta Oncol 26: 77-85
28. Burkard R (2002) Open problem session. In: Oberwolfach conference on combinatorial optimization, 24-29 November 2002
29. Carlsson F, Forsgren A (2006) Iterative regularization in intensity-modulated radiation therapy optimization. Med Phys 33: 225-234
30. 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
31. 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
32. 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
33. 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
34. 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
35. 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
36. 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
37. 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
38. 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)
39. 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
40. 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
41. 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
42. 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
43. Collins M, Kempe D, Saia J, Young M (2007) Nonnegative integral subset representations of integer sets. Inf Process Lett 101(3): 129-133
44. Convery D, Rosenbloom M (1992) The generation of intensity-modulated fields for conformal radiotherapy by dynamic collimation. Phys Med Biol 37(6): 1359-1374
45. 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
46. Cotrutz C, Xing L (2002) Using voxel-dependent importance factors for interactive DVH-based dose optimisation. Phys Med Biol 47: 1659-1669
47. Cotrutz C, Xing L (2003) Segment-based dose optimisation using a genetic algorithm. Phys Med Biol 48: 2987-2998
48. 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
49. Craft D (2007) Local beam angle optimization with linear programming and gradient search. Phys Med Biol 52: 127-135
50. Craft D, Halabi T, Bortfeld T (2005) Exploration of tradeoffs in intensity-modulated radiotherapy. Phys Med Biol 50: 5857-5868
51. Craft D, Halabi T, Shih H, Bortfeld T (2006) Approximating convex Pareto surfaces in multiobjective radiotherapy planning. Med Phys 33: 3399-3407
52. 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
53. 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
54. Dai J, Zhu Y (2003) Conversion of dose-volume constraints to dose limits. Phys Med Biol 48: 3927-3941
55. 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
56. 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
57. 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
58. De Werra D, Costa M, Picouleau C, Ries B (2008) On the use of graphs in discrete tomography. 4OR 6: 101-123
59. Deasy JO (1997) Multiple local minima in radiotherapy optimization problems with dose-volume constraints. Med Phys 24(7): 1157-1161
60. 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
61. 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
62. 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
63. Ehrgott M (2005) Multicriteria optimization, 2nd edn. Springer, Berlin
64. 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
65. Ehrgott M, Johnston R (2003) Optimisation of beam directions in intensity modulated radiation therapy planning. OR Spectr 25(2): 251-264
66. 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
67. Ehrgott M, Holder A, Reese J (2008) Beam selection in radiotherapy design. Lin Algebra Appl 428: 1272-1312
68. Emami B, Lyman J, Brown A (1991) Tolerance of normal tissue to therapeutic irradiation. Int J Radiat Oncol Biol Phys 21: 109-122
69. Engel K (2005) A new algorithm for optimal MLC field segmentation. Discr Appl Math 152: 35-51
70. Engel K, Tabbert E (2005) Fast simultaneous angle, wedge, and beam intensity optimization in inverse radiotherapy planning. Optim Eng 6: 393-419
71. 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
72. 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
73. 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
74. Ezzell GA (1996) Genetic and geometric optimization of three-dimensional radiation therapy. Med Phys 23: 293-305
75. Ferris M, Voelker M (2004) Fractionation in radiation treatment planning. Math Program Ser B 101: 387-413
76. 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
77. Gersho A, Gray R (1991) Vector quantization and signal compression. Kluwer, Boston
78. Gong Y (2006) Integer programming methods for beam selection in radiotherapy treatment planning. Master's Thesis, Department of Engineering Science, The University of Auckland
79. 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
80. Haas O, Burnham K, Mills J (1998) Optimization of beam orientation in radiotherapy using planar geometry. Phys Med Biol 43: 2179-2193
81. Halabi T, Craft D, T B (2006) Dose-volume objectives in multi-criteria optimization. Phys Med Biol 51: 3809-3818
82. Hamacher H, Küfer K-H (2002) Inverse radiation therapy planning-a multiple objective optimization approach. Discr Appl Math 118(1-2): 145-161
83. Hodes L (1974) Semiautomatic optimization of external beam radiation treatment planning. Radiology 110: 191-196
84. Holder A (2003) Designing radiotherapy plans with elastic constraints and interior point methods. Health Care Manage Sci 6: 5-16
85. 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
86. Holder A (2006) Partitioning multiple objective optimal solutions with applications in radiotherapy design. Optim Eng 7: 501-526
87. Holder A, Llagostera D (2008) Optimal treatments for photodynamic therapy. 4OR 6: 167-182
88. 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
89. Holmes T, Mackie TR (1994) A filtered backprojection dose calculation method for inverse treatment planning. Med Phys 21: 303-313
90. 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
91. Hristov D, Fallone B (1997) An active set algorithm for treatment planning optimization. Med Phys 24: 91-106
92. Hristov D, Fallone B (1998) A continuous penalty function method for inverse treatment planning. Med Phys 25(2): 208-223
93. 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
94. Jeleń U, Söhn M, Alber M (2005) A finite pencil beam for IMRT dose optimization. Med Phys 50: 1747-1766
95. Jeraj R, Wu C, Mackie T (2003) Optimizer convergence and local minima errors and their clinical importance. Phys Med Biol 48: 2809-2827
96. 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
97. Kalinowski T (2005) A duality based algorithm for multileaf collimator field segmentation with interleaf collision constraint. Discr Appl Math 152: 52-88
98. Kalinowski T (2008) Reducing the tongue-and-groove underdosage in mlc shape matrix decomposition. Algor Oper Res 3: 165-174
99. 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
100. 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
101. 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
102. Kamath S, Sahni S, Palta J, Ranka S (2004a) Algorithms for optimal sequencing of dynamic multileaf collimators. Phys Med Biol 49: 33-54
103. 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
104. 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
105. Kennedy JM (2000) Minimum support solutions for radiotherapy treatment planning. Master's Thesis, Department of Mathematics, University of Colorado at Denver, Denver
106. Khan FM (2003) The physics of radiation therapy. Lippincott Williams & Wilkins, Philapelphia
107. 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
108. 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
109. 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
110. 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
111. 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
112. 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
113. Langer M (1987) Optimization of beam weights under dose-volume restrictions. Int J Radiat Oncol Biol Phys 13: 1255-1260
114. 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
115. 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
116. Langer M, Brown R, Morill S, Lane R, Lee O (1996) A generic genetic algorithm for generating beam weights. Med Phys 23: 965-971
117. 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
118. Lee E, Fox T, Crocker I (2003) Integer programming applied to intensity-modulated radiation therapy treatment planning. Ann Oper Res 119: 165-181
119. 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
120. Legras J, Legras B, Lambert J (1982) Software for linear and non-linear optimization in external radiotherapy. Comput Programs Biomed 15: 233-242
121. 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
122. 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
123. Liu DC, Nocedal J (1989) On the limited memory BFGS method for large scale optimization. Math Program 45: 503-528
124. Llacer G (1997) Inverse radiation treatment planning using the dynamically penalized likelihood method. Med Phys 24(11): 1751-1764
125. 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
126. Lloyd S (1982) Least squares quantization in PCM. IEEE Trans Inf Theory 28: 127-135
127. Löf J (2000) Development of a general framework for optimization of radiation therapy. PhD Thesis, Department of Medical Radiation Physics, Karolinska Institute, Stockholm, Sweden
128. Luan S, Saia J, Young M (2007) Approximation algorithms for minimizing segments in radiation therapy. Inf Process Lett 101: 239-244
129. 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
130. 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
131. 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
132. 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
133. Mageras GS, Mohan R (1993) Application of fast simulated annealing to optimization of conformal radiation treatments. Med Phys 20(3): 1-3
134. Mak V (2007) Iterative variable aggregation and disaggregation in IP: an application. Oper Res Lett 35: 36-44
135. McDonald SC, Rubin P (1977) Optimization of external beam radiation therapy. Int J Radiat Oncol Biol Phys 2: 307-317
136. Meedt G, Alber M, Nüsslin F (2003) Non-coplanar beam direction optimization for intensity-modulated radiotherapy. Phys Med Biol 48: 2999-3019
137. 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
138. Messac A, Ismail-Yahaya A, Mattson CA (2003) The normalized constraint method for generating the Pareto frontier. Struct Multidisciplin Optim 25: 86-98
139. 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
140. 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
141. Morrill S, Lane R, Wong J, Rosen I (1991a) Dose-volume considerations with linear programming optimization. Med Phys 18(6): 1201-1210
142. Morrill S, Lane R, Jacobson G, Rosen I (1991b) Treatment planning optimization using constrained simulated annealing. Phys Med Biol 36(10): 1341-1361
143. 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
144. Morrill SM, Lane RG, Rosen II (1990b) Constrained simulated annealing for optimized radiation therapy treatment planning. Comput Meth Programs Biomed 33: 135-144
145. Motzkin TS, Schoenberg IJ (1954) The relaxation method for linear inequalities. Can J Math 6: 393-404
146. 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
147. Niemierko A (1997) Reporting and analysing dose distributions: a concept of equivalent uniform dose. Med Phys 24: 103-110
148. Niemierko A (1999) A generalized concept of equivalent uniform dose. Med Phys 26: 1100
149. 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
150. Nizin P, Kania A, Ayyangar K (2001) Basic concepts of CORVUS dose model. Med Dosim 26(1): 65-69
151. Nußbaum M (2006) Min Cardinality C1-Decomposition of Integer Matrices. Master's Thesis, Department of Mathematics, Technical University of Kaiserslautern
152. Ólafsson A, Wright S (2006) Efficient schemes for robust IMRT treatment planning. Phys Med Biol 51: 5621-5642
153. 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
154. 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
155. 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
156. 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
157. Que W (1999) Comparison of algorithms for multileaf collimator field segmentation. Med Phys 26: 2390-2396
158. 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
159. Redpath AT, Vickery BL, Wright DH (1976) A new technique for radiotherapy planning using quadratic programming. Phys Med Biol 21: 781-91
160. 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
161. Romeijn H, Dempsey J, Li J (2004) A unifying framework for multi-criteria fluence map optimization models. Phys Med Biol 49: 1991-2013
162. 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
163. Rosen I, Lane R, Morrill S, Belli J (1991) Treatment planning optimisation using linear programming. Med Phys 18(2): 141-152
164. Rowbottom C, Webb S (2002) Configuration space analysis of common cost functions in radiotherapy beam-weight optimization algorithms. Phys Med Biol 47: 65-77
165. Rowbottom C, Khoo V, Webb S (2001) Simultaneous optimization of beam orientations and beam weights in conformal radiotherapy. Med Phys 28(8): 1696-1702
166. Schlegel W, Mahr A (2002) 3D-conformal radiation therapy: a multimedia introduction to methods and techniques. Springer, Heidelberg, Berlin
167. 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
168. 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
169. 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
170. Shepard D, Ferris M, Olivera G, Mackie T (1999) Optimizing the delivery of radiation therapy to cancer patients. SIAM Rev 41(4): 721-744
171. Siochi R (1999) Minimizing static intensity modulation delivery time using an intensity solid paradigm. Int J Radiat Oncol Biol Phys 43: 671-689
172. 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
173. 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
174. Sonderman D, Abrahamson P (1985) Radiotherapy treatment design using mathematical programming models. Oper Res 33(4): 705-725
175. Spirou SV, Chui CS (1994) Generation of arbitrary intensity profiles by dynamic jaws or multileaf collimators. Med Phys 27(7): 1031-1041
176. Spirou SV, Chui C-S (1998) A gradient inverse planning algorithm with dose-volume constraints. Med Phys 25(3): 321-333
177. Starkschall G (1984) A constrained least-squares optimization method for external beam treatment planning. Med Phys 11(5): 659-665
178. 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
179. 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
180. 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
181. 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
182. Svensson R, Källman P, Brahme A (1994) An analytical solution for the dynamic control of multileaf collimators. Phys Med Biol 39: 37-61
183. 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
184. Thieke C (2003) Multicriteria optimisation in inverse radiotherapy planning. PhD Thesis, Ruprecht-Karls-Universität Heidelberg, Germany
185. Thieke C, Bortfeld T, Küfer K-H (2002) Characterization of dose distributions through the max and mean dose concept. Acta Oncol 41: 158-161
186. 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
187. Ulmer W, Harder D (1995) A triple Gaussian pencil beam model for photon beam treatment planning. Zeitschrift für medizinische Physik 5: 25-30
188. Van Santvoort J, Heijmen B (1996) Dynamic multileaf collimation without tongue-and-groove underdosage effects. Phys Med Biol 41: 2091-2105
189. Verhaegen F (2003) Monte Carlo modelling of external radiotherapy photon beams. Phys Med Biol 48: 107-164
190. 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
191. 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
192. 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
193. 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
194. Webb S (1989) Optimisation of conformal radiotherapy dose distribution by simulated annealing. Phys Med Biol 34: 1349-1370
195. 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
196. 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
197. Webb S (1994a) Optimizing the planning of intensity-modulated radiotherapy. Phys Med Biol 39: 2229-2246
198. Webb S (1994b) Optimum parameters in a model for tumor control probability including interpatient heterogeneity. Phys Med Biol 39: 1895-1914
199. Webb S (2001) Intensity-modulated radiation therapy (Ser Med Phys) Institute of Physics Publishing, Bristol, Philadelphia
200. 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
201. 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
202. 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
203. Wu Q et al (2003b) Intensity-modulated radiotherapy optimization with gEUD-guided dose-volume objectives. Phys Med Biol 48: 279-291
204. Wu Q, Mohan R (2000) Algorithms and functionality of an intensity modulated radiotherapy optimization system. Med Phys 27(4): 701-711
205. Wu Q, Mohan R (2002) Multiple local minima in IMRT optimization-based on dose-volume criteria. Med Phys 29: 1514-1527
206. 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
207. 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
208. Xia P, Verhey L (1998) Multileaf collimator leaf sequencing algorithm for intensity modulated beams with multiple segments. Med Phys 25(8): 1424-1434
209. 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
210. Xing L, Chen GTY (1996) Iterative algorithms for inverse treatment planning. Phys Med Biol 41: 2107-2123
211. 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
212. Xing L, Li JG, Donaldson S, Le QT, Boyer AL (1999) Optimization of importance factors in inverse planning. Phys Med Biol 44: 2525-2536
213. Yu CX (1995) Intensity-modulated arc therapy with dynamic multileaf collimation: an alternative to tomotherapy. Phys Med Biol 40: 1435-1449
214. 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
215. Yu Y (1997) Multi-objective decision theory for computational optimisation in radiation therapy. Med Phys 24(9): 1445-1454
216. Zaider M, Minerbo GN (2000) Tumor control probability: A formulation applicable to any temporal protocol of dose delivery. Phys Med Biol 45: 279-293