SCG算法优化调强放射治疗计划子野权重研究
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摘要
放射治疗是医治癌症的三大主要手段之一，据估计，约有60％-70％的肿瘤病人需要进行放射治疗，40％的癌症治愈病人是由放射治疗治愈的，具有十分重大的意义。从经典的三维适形放射治疗技术发展到现在的调强放射治疗技术(intensity-modulated radiotherapy，IMRT)，是放射肿瘤学史上的一次重大变革。但鉴于实际问题的复杂性，IMRT的优势还远没有在临床应用中完全发挥出来。IMRT治疗计划作为调强放疗的核心和基础，其中尚有许多急需解决的问题。
随着IMRT研究的深入和逆向治疗计划的发展，放射治疗中如何自动选择射野参数引起了广泛关注。在传统放射治疗中，射野参数的选择是通过反复试错(try and error)的方法实现的，治疗计划的优劣往往依赖于计划设计者的经验，由于IMRT采用逆向计划，需要选择的参数很多，通过试错的方式根本无法完成，必须使用优化方法来提高治疗计划设计水平。在过去的十多年中，人们对IMRT射野参数优化的优化方法进行了大量研究，最常用的包括：线性规划法、均方优化法、梯度方法、有约束模拟退火法和遗传算法等。
梯度算法是目前商用调强放疗(IMRT)计划系统中最常用的算法之一。M(?)ller提出的SCG(scaled conjugate gradient，缩放共轭梯度法)算法很好地解决了梯度算法中的线性搜索过程，进一步提高了梯度算法的性能。X.D.Zhang等人在Philips的Pinnacle计划系统上证实，缩放共轭梯度法的速度是常规共轭梯度法的
Radiotherapy, one of the three main treatments for tumor, is treat with about 60%-70% patients suffering from cancer and cures 40% of them. It is a historic advancement that the classical three-dimensional (3D) conformal radiotherapy (3DCRT) evolved into the intensity-modulated radiotherapy (IMRT). However, the advantages of IMRT have not been fully utilized yet, due to the complicated clinical conditions. There are still many problems in the IMRT planning, which is one of the basics of IMRT application, should be solved.
With the development of IMRT and inverse planning , the automatic determination of suitable beam parameters in external beam radiotherapy has gained wide interests. In the conventional radiotherapy, the beam parameters are usually obtained by trial and error. The planning outcomes usually depend on the personal experience of planners. But there are large-scale parameters to be determined in the inverse planning of IMRT. It can hardly be carried out by trial and error. During last ten years, many methods have been proposed to optimize treatment plans such as linear programming, quadratic programming, gradient algorithms, constrained simulated annealing and genetic algorithms.
随着IMRT研究的深入和逆向治疗计划的发展，放射治疗中如何自动选择射野参数引起了广泛关注。在传统放射治疗中，射野参数的选择是通过反复试错(try and error)的方法实现的，治疗计划的优劣往往依赖于计划设计者的经验，由于IMRT采用逆向计划，需要选择的参数很多，通过试错的方式根本无法完成，必须使用优化方法来提高治疗计划设计水平。在过去的十多年中，人们对IMRT射野参数优化的优化方法进行了大量研究，最常用的包括：线性规划法、均方优化法、梯度方法、有约束模拟退火法和遗传算法等。
梯度算法是目前商用调强放疗(IMRT)计划系统中最常用的算法之一。M(?)ller提出的SCG(scaled conjugate gradient，缩放共轭梯度法)算法很好地解决了梯度算法中的线性搜索过程，进一步提高了梯度算法的性能。X.D.Zhang等人在Philips的Pinnacle计划系统上证实，缩放共轭梯度法的速度是常规共轭梯度法的
Radiotherapy, one of the three main treatments for tumor, is treat with about 60%-70% patients suffering from cancer and cures 40% of them. It is a historic advancement that the classical three-dimensional (3D) conformal radiotherapy (3DCRT) evolved into the intensity-modulated radiotherapy (IMRT). However, the advantages of IMRT have not been fully utilized yet, due to the complicated clinical conditions. There are still many problems in the IMRT planning, which is one of the basics of IMRT application, should be solved.
With the development of IMRT and inverse planning , the automatic determination of suitable beam parameters in external beam radiotherapy has gained wide interests. In the conventional radiotherapy, the beam parameters are usually obtained by trial and error. The planning outcomes usually depend on the personal experience of planners. But there are large-scale parameters to be determined in the inverse planning of IMRT. It can hardly be carried out by trial and error. During last ten years, many methods have been proposed to optimize treatment plans such as linear programming, quadratic programming, gradient algorithms, constrained simulated annealing and genetic algorithms.
引文
1. Fletcher, R. Practical Methods of Optimization, John Wiley & Sons.1975.
2. Gill, P.E., W. Murray, M.H. Wright. Practical Optimization, Academic Press, inc. 1980.
3. Hestenes, M. Conjugate Direction Methods in Optimization, Springer Verlag, New York. 1980.
4. Powell, M.. Restart Procedures for the Conjugate Gradient Method, in: Mathematical Programming, 1977, 241-254.
5. M. F. Moller, "A scaled conjugate-gradient algorithm for fast supervised learning, " Neural Networks 1993, 6:525-533.
6. Fletcher, R. Practical Methods of Optimization, John Wiley & Sons. 1975.
7. Johansson, E.M., F.U. Dowla, D.M. Goodman, Backpropagation Learning for Multi-Layer Feed-Forward Neural Networks Using the Conjugate Gradient Method, Lawrence Livermore National Laboratory, Preprint UCRL-JC-104850. 1990.
8. Bahr G K, Kereiakes J G et al. The method of linear programming applied to radiation therapy treatment planning. Radiology 1968, 91:686-693.
9. X. D. Zhang, H. Liu, X. C. Wang et al., Speed and convergence properties of gradient algorithms for optimization of IMRT Med. Phys. 31.5., May 2004.
10. Webb S. Optimization of conformal radiotherapy dose distribution by simulated annealing. Phys.Med.Biol. 1989, 34:1349-1370.
11．胡逸民,肿瘤放射物理学,北京：原子能出版社,1999．
12. BoyerAL, Zhu Y, Wang L, et al. Fast Fourier transform convolution calculation of X-ray isodose distribution in homogeneous media. Med Physl989:16(2):248-253.
13. M.J.Day, E.GA.Arid. The equivalent field method for dose determination in rectangular fields. Br.J.Radiol.Suppl., 1983, 17:105-114.
14. Morris T, Bengt F.B. Equivalent squares of irregular photon fields. Med. Phys. 1992，20(4)：1229-32．
15．戴建荣，胡逸民，杨勇．多叶准直器射野处方剂量的快速计算，中华放射肿瘤学杂志，2000，9(2)：131-134．
16．金浩宇，吕庆文，周凌宏．用解卷积方法提取笔射束核的实现．中国生物医学工程学报，2004，23：382-385．
17. Chui C. S. Mohan R. Extraction of pencil beam kernels by the deconvolution method. Med. Phys., 1988, 15(2):138-144.
18. T. Bortfeld et al., "Physical vs biological objectives for treatment plan optimization, "Radiother. Oncol. 1996, 40:185-185.
19. L. Jones, P. Hoban "A comparison of physically and radiobiologically based optimization for IMRT, "Med. Phys. 2002, 29(7):1447-1455.
20. C. Thieke, T. Bortfeld and A. Niemierko et al From physical dose constraints to equivalent uniform dose constraints in inverse radiotherapy planning Med. Phys. 2003, 30(9):2332-2339.
21. Agren A., Brahme A., Turesson I., Optimization of un-complicated control for head and neck tumors. Int. J. Radiat Oncol Biol Phys., 1990, Vol. 19, p1077-1085.
22. Mohan R., Mageras G. S., et al., Clinically relevant optimization of 3D conformal treatments. Med. Phys., 1992, 19:933-944.
23. A. Niemierko, "Reporting and analyzing dose distributions: A concept of equivalent uniform dose, "Med. Phys. 1997, 24:103-110.
24. A. Niemierko, "A generalized concept of equivalent uniform dose", Med. Phys. 1999, 26:1100.
25. B. Choi and J. O. Deasy, "The generalized equivalent uniform dose function as a basis for intensity-modulated treatment planning, " Phys. Med. Biol. 2002, 47:3579-3589.
26. Jones L. C. and Hoban P. W., Treatment plan comparison using equivalent uniform biologically effective dose (EUBED). Phys. Med. Biol. 2000, 45:159-170.
27. Bortfeld T, Stein J, Preise Kr, Clinically relevant intensity modulation optimization using physical criteria, In: Leavitt DD, Starkschall G, 12th International Conference on the Use of Computers in Radiation Therapy. Madison, WI: Medical Physics Publishing; 1997, 1-4.
28. Austin-Seynour MM, Cheb GTY, et al. Dose volume histogram anaysis of liver radiation tolerance. Int J Radiat Oncol Biol Phys, 1986, 12: 31-35.
29. Lawrence Ts, Tesser RJ, Ten Haken RK. An application of dose volume histograms to treatment of intrahepatic malignancies with radiation therapy. Int J Radiat Oncol Biol Phys, 1990, 19:1041-1047
30. Drzymala RE, Mohan R, Brewster L, et al. Dose-volume histograms. Int J Radiat Oncol Biol Phys, 1991, 21: 71-78.
31. Haken RKT, Kesder M. Use of DVHs, TCPs, and NTCPs in Treatment Planning. In S. Sternick ES(ed). The theory and practice of intensity modulated radiation therapy. Chapter H. Advanced Medical Publishing, 1997.
32. Q. W. Wu and R. Mohan, "Multiple local minima in IMRT optimization based on dose-volume criteria," Med. Phys. 29, 1514-1527 (2002).
33. J. Llacer et al., "Absence of multiple local minima effects in intensity modulated optimization with dose-volume constraints," Phys. Med. Biol. 48, 183-210 (2003).
34. C. G. Rowbottom and S. Webb, "Configuration space analysis of common cost functions in radiotherapy beam-weight optimization algorithms, " Phys. Med. Biol. 2002, 47:65-77.
35. M. Alber et al., "On the degeneracy of the IMRT optimization problem, " Med. Phys. 2002, 29:2584-2589.
36. R. Jeraj, C. A. Wu, and T. R. Mackie, "Optimizer convergence and local minima errors and their clinical importance, " Phys. Med. Biol. 2003, 48:2809-2827.
37．杨瑞杰，戴建荣，胡逸民．调强放射治疗的计划优化，中国医疗器械信息，2005．11(2)：12-16
38. S. Webb: "Inverse Planning with Constraints to. Generate smoothed intensity-modulated beams", Phys. Med. Biol. 1998.43:2785-2794.
39. Webb S. The future of photon external-beam radiotherapy: the dream and the reality. Phys Med 2001.17:207-15.
40. M. Alber, F. Nusslin: "Intensity Modulated Photon. Beams Subject to a Minimal Surface Constraint",. Phys. Med. Biol. 2000.45:N49-N52.
41. Spirou SV, Fournier-Bidoz N, Yang J, Chui CS, Ling CC, Smoothing intensity-modulated beam profiles to improve the efficiency of delivery.Med Phys. 2001.28(10):2105-12.
1 Q. W. Wu and R. Mohan, "Algorithms and functionality of an intensity modulated radiotherapy optimization system", Med. Phys. 27, 701-711 (2000)
2 Q. W. Wu et al, "Optimization of intensity-modulated radiotherapy plans based on the equivalent uniform dose", Int. J. Radiat. Oncol. Biol. Phys. 52, 224-235 (2002)
3 T. Bortfeld, "Optimized planning using physical objectives and constraints", Semin Radiat. Oncol. 9, 20 (1999)
4 D. Hristov et al, "On the implementation of dose-volume objectives in gradient algorithms for inverse treatment planning", Med. Phys. 29, 848-856 (2002)
5 G. Starkschall, A. Pollack, and C. W. Stevens, "Treatment planning using a dose-volume feasibility search algorithm", Int. J. Radiat. Oncol. Biol. Phys. 49,1419-1427 (2001)
6 Q. Hou et al., "An optimization algorithm for intensity modulated radiotherapy-The simulated dynamics with dose-volume constraints," Med. Phys. 30, 61-68 (2003).
7 T. Bortfeld et al, "Physical vs biological objectives for treatment plan optimization," Radiother. Oncol. 40,185-185 (1996).
8 S. V. Spirou and C. S. Chui, "A gradient inverse planning algorithm with dose-volume constraints," Med. Phys. 25, 321-333 (1998).
9 M. F. Moller, "A scaled conjugate-gradient algorithm for fast supervised learning," Neural Networks 6, 525-533 (1993).
10 J. O. Deasy, "Multiple local minima in radiotherapy optimization problems with dose-volume constraints," Med. Phys. 24,1157-1161 (1997).
11 Q. W. Wu and R. Mohan, "Multiple local minima in IMRT optimization based on dose-volume criteria" Med. Phys. 29,1514-1527 (2002).
12 J. Llacer et al., "Absence of multiple local minima effects in intensity modulated optimization with dose-volume constraints," Phys. Med. Biol. 48,183-210 (2003).
13 C. G. Rowbottom and S. Webb, "Configuration space analysis of common cost functions in radiotherapy beam-weight optimization algorithms," Phys. Med. Biol. 47, 65-77 (2002).
14 M. Alber et al., "On the degeneracy of the IMRT optimization problem," Med. Phys. 29, 2584-2589 (2002).
15 R. Jeraj, C. A. Wu, and T. R. Mackie, "Optimizer convergence and local minima errors and their clinical importance," Phys. Med. Biol. 48, 2809-2827 (2003).
16 C. Wu, R. Jeraj, and T. R. Mackie, "The method of intercepts in parameter, space for the analysis of local minima caused by dose-volume constraints," Phys. Med. Biol. 48, N149-N157 (2003).
17 A. Niemierko, "Reporting and analyzing dose distributions: A concept of equivalent uniform dose," Med. Phys. 24,103-110 (1997).
18 Q. W. Wu et al, "A fast dose calculation method based on table lookup for IMRT optimization," Phys. Med. Biol. 48, N159-N166 (2003).
19 L. Jones, P. Hoban "A comparison of physically and radiobiologically based optimization for IMRT," Med. Phys. 29(7), 1447-1455 July 2002
20 C. Thieke, T. Bortfeld and A. Niemierko et al From physical dose constraints to equivalent uniform dose constraints in inverse radiotherapy planning Med. Phys. 30 .9., September 2003 2332-2339
21 A. Pugachev, J. G. Li, A. L. Boyer, et al "Role of beam orientation optimization in intensity-modulated radiation therapy," Int. J. Radiat. Oncol., Biol., Phys. 50, 551-560 (2001).
22 A. Niemierko, "A generalized concept of equivalent uniform dose" Med. Phys. 26, 1100 (1999).
23 S. Nill, "Development and application of a multi-modality inverse treatment planning system" Ph. D. thesis, Ruprecht-Karls-Universitaet Heidelberg, Germany, 2001.
24 C. Scholz, S. Nill, U. Oelfke, and T. Bortfeld, "Application of an advanced photon dose algodthrn for inverse IMRT planning," Med. Phys. 29, 1335 (2002).
25 C. Thieke, S. Nill, U. Oelfke, and T. Bortfeld, "Acceleration of intensitymodulated dose calculation by importance sampling of the calculation matrices," Med. Phys. 29, 676-681 (2002).
26 B. Choi and J. O. Deasy, "The generalized equivalent uniform dose function as a basis for intensity-modulated treatment planning," Phys. Meal. Biol. 47, 3579-3589 (2002).
27 A. Pugachev and L. Xing, "Incorporating prior knowledge into beam orientation optimization in IMRT," Int. J. Radiat. Oncol., Biol., Phys. 54, 1565-1574 (2002).
28 M. Alber and F. Nusslin, "An objective function for radiation treatment optimization based on local biological measures," Phys. Med. Biol. 44, 479-493 (1999).
29 A. Brahme, "Optimized radiation therapy based on radiobiological objectives," Semin Radiat. Oncol. 9(1), 35-47 (1999).
30 A. Brahme, "Development of radiation therapy optimization," Acta Oncol. 39(5), 579-595 (2000).
31 V. Moiseenko, J. Battista, and J. Van Dyk, "Normal tissue complication probabilities: dependence on choice of biological model and dose-volume histogram reduction scheme," Int. J. Radiat. Oncol., Biol., Phys. 46(4), 983-993 (2000).
32 X. D. Zhang, H. Liu, X. C. Wang et al., Speed and convergence properties of gradient algorithms for optimization of IMRT Med. Phys. 31. 5., May 2004
33 高磊，吕庆文，李树祥 一种调强放射治疗计划系统的研究 医疗卫生装备2001年第5期 24-26
34 周凌宏，王志远，吕庆文等 CT模拟定位技术研究医疗装备2003第1期Vol16，No11，8-10
35 李宝生，于金明，王立英等 调强放射治疗计划 中国肿瘤2001年第10卷第8期 461-463
36 蒋国梁 三维适形放疗和调强放疗 肿瘤 2003，Vol．23，No．4
2. Gill, P.E., W. Murray, M.H. Wright. Practical Optimization, Academic Press, inc. 1980.
3. Hestenes, M. Conjugate Direction Methods in Optimization, Springer Verlag, New York. 1980.
4. Powell, M.. Restart Procedures for the Conjugate Gradient Method, in: Mathematical Programming, 1977, 241-254.
5. M. F. Moller, "A scaled conjugate-gradient algorithm for fast supervised learning, " Neural Networks 1993, 6:525-533.
6. Fletcher, R. Practical Methods of Optimization, John Wiley & Sons. 1975.
7. Johansson, E.M., F.U. Dowla, D.M. Goodman, Backpropagation Learning for Multi-Layer Feed-Forward Neural Networks Using the Conjugate Gradient Method, Lawrence Livermore National Laboratory, Preprint UCRL-JC-104850. 1990.
8. Bahr G K, Kereiakes J G et al. The method of linear programming applied to radiation therapy treatment planning. Radiology 1968, 91:686-693.
9. X. D. Zhang, H. Liu, X. C. Wang et al., Speed and convergence properties of gradient algorithms for optimization of IMRT Med. Phys. 31.5., May 2004.
10. Webb S. Optimization of conformal radiotherapy dose distribution by simulated annealing. Phys.Med.Biol. 1989, 34:1349-1370.
11．胡逸民,肿瘤放射物理学,北京：原子能出版社,1999．
12. BoyerAL, Zhu Y, Wang L, et al. Fast Fourier transform convolution calculation of X-ray isodose distribution in homogeneous media. Med Physl989:16(2):248-253.
13. M.J.Day, E.GA.Arid. The equivalent field method for dose determination in rectangular fields. Br.J.Radiol.Suppl., 1983, 17:105-114.
14. Morris T, Bengt F.B. Equivalent squares of irregular photon fields. Med. Phys. 1992，20(4)：1229-32．
15．戴建荣，胡逸民，杨勇．多叶准直器射野处方剂量的快速计算，中华放射肿瘤学杂志，2000，9(2)：131-134．
16．金浩宇，吕庆文，周凌宏．用解卷积方法提取笔射束核的实现．中国生物医学工程学报，2004，23：382-385．
17. Chui C. S. Mohan R. Extraction of pencil beam kernels by the deconvolution method. Med. Phys., 1988, 15(2):138-144.
18. T. Bortfeld et al., "Physical vs biological objectives for treatment plan optimization, "Radiother. Oncol. 1996, 40:185-185.
19. L. Jones, P. Hoban "A comparison of physically and radiobiologically based optimization for IMRT, "Med. Phys. 2002, 29(7):1447-1455.
20. C. Thieke, T. Bortfeld and A. Niemierko et al From physical dose constraints to equivalent uniform dose constraints in inverse radiotherapy planning Med. Phys. 2003, 30(9):2332-2339.
21. Agren A., Brahme A., Turesson I., Optimization of un-complicated control for head and neck tumors. Int. J. Radiat Oncol Biol Phys., 1990, Vol. 19, p1077-1085.
22. Mohan R., Mageras G. S., et al., Clinically relevant optimization of 3D conformal treatments. Med. Phys., 1992, 19:933-944.
23. A. Niemierko, "Reporting and analyzing dose distributions: A concept of equivalent uniform dose, "Med. Phys. 1997, 24:103-110.
24. A. Niemierko, "A generalized concept of equivalent uniform dose", Med. Phys. 1999, 26:1100.
25. B. Choi and J. O. Deasy, "The generalized equivalent uniform dose function as a basis for intensity-modulated treatment planning, " Phys. Med. Biol. 2002, 47:3579-3589.
26. Jones L. C. and Hoban P. W., Treatment plan comparison using equivalent uniform biologically effective dose (EUBED). Phys. Med. Biol. 2000, 45:159-170.
27. Bortfeld T, Stein J, Preise Kr, Clinically relevant intensity modulation optimization using physical criteria, In: Leavitt DD, Starkschall G, 12th International Conference on the Use of Computers in Radiation Therapy. Madison, WI: Medical Physics Publishing; 1997, 1-4.
28. Austin-Seynour MM, Cheb GTY, et al. Dose volume histogram anaysis of liver radiation tolerance. Int J Radiat Oncol Biol Phys, 1986, 12: 31-35.
29. Lawrence Ts, Tesser RJ, Ten Haken RK. An application of dose volume histograms to treatment of intrahepatic malignancies with radiation therapy. Int J Radiat Oncol Biol Phys, 1990, 19:1041-1047
30. Drzymala RE, Mohan R, Brewster L, et al. Dose-volume histograms. Int J Radiat Oncol Biol Phys, 1991, 21: 71-78.
31. Haken RKT, Kesder M. Use of DVHs, TCPs, and NTCPs in Treatment Planning. In S. Sternick ES(ed). The theory and practice of intensity modulated radiation therapy. Chapter H. Advanced Medical Publishing, 1997.
32. Q. W. Wu and R. Mohan, "Multiple local minima in IMRT optimization based on dose-volume criteria," Med. Phys. 29, 1514-1527 (2002).
33. J. Llacer et al., "Absence of multiple local minima effects in intensity modulated optimization with dose-volume constraints," Phys. Med. Biol. 48, 183-210 (2003).
34. C. G. Rowbottom and S. Webb, "Configuration space analysis of common cost functions in radiotherapy beam-weight optimization algorithms, " Phys. Med. Biol. 2002, 47:65-77.
35. M. Alber et al., "On the degeneracy of the IMRT optimization problem, " Med. Phys. 2002, 29:2584-2589.
36. R. Jeraj, C. A. Wu, and T. R. Mackie, "Optimizer convergence and local minima errors and their clinical importance, " Phys. Med. Biol. 2003, 48:2809-2827.
37．杨瑞杰，戴建荣，胡逸民．调强放射治疗的计划优化，中国医疗器械信息，2005．11(2)：12-16
38. S. Webb: "Inverse Planning with Constraints to. Generate smoothed intensity-modulated beams", Phys. Med. Biol. 1998.43:2785-2794.
39. Webb S. The future of photon external-beam radiotherapy: the dream and the reality. Phys Med 2001.17:207-15.
40. M. Alber, F. Nusslin: "Intensity Modulated Photon. Beams Subject to a Minimal Surface Constraint",. Phys. Med. Biol. 2000.45:N49-N52.
41. Spirou SV, Fournier-Bidoz N, Yang J, Chui CS, Ling CC, Smoothing intensity-modulated beam profiles to improve the efficiency of delivery.Med Phys. 2001.28(10):2105-12.
1 Q. W. Wu and R. Mohan, "Algorithms and functionality of an intensity modulated radiotherapy optimization system", Med. Phys. 27, 701-711 (2000)
2 Q. W. Wu et al, "Optimization of intensity-modulated radiotherapy plans based on the equivalent uniform dose", Int. J. Radiat. Oncol. Biol. Phys. 52, 224-235 (2002)
3 T. Bortfeld, "Optimized planning using physical objectives and constraints", Semin Radiat. Oncol. 9, 20 (1999)
4 D. Hristov et al, "On the implementation of dose-volume objectives in gradient algorithms for inverse treatment planning", Med. Phys. 29, 848-856 (2002)
5 G. Starkschall, A. Pollack, and C. W. Stevens, "Treatment planning using a dose-volume feasibility search algorithm", Int. J. Radiat. Oncol. Biol. Phys. 49,1419-1427 (2001)
6 Q. Hou et al., "An optimization algorithm for intensity modulated radiotherapy-The simulated dynamics with dose-volume constraints," Med. Phys. 30, 61-68 (2003).
7 T. Bortfeld et al, "Physical vs biological objectives for treatment plan optimization," Radiother. Oncol. 40,185-185 (1996).
8 S. V. Spirou and C. S. Chui, "A gradient inverse planning algorithm with dose-volume constraints," Med. Phys. 25, 321-333 (1998).
9 M. F. Moller, "A scaled conjugate-gradient algorithm for fast supervised learning," Neural Networks 6, 525-533 (1993).
10 J. O. Deasy, "Multiple local minima in radiotherapy optimization problems with dose-volume constraints," Med. Phys. 24,1157-1161 (1997).
11 Q. W. Wu and R. Mohan, "Multiple local minima in IMRT optimization based on dose-volume criteria" Med. Phys. 29,1514-1527 (2002).
12 J. Llacer et al., "Absence of multiple local minima effects in intensity modulated optimization with dose-volume constraints," Phys. Med. Biol. 48,183-210 (2003).
13 C. G. Rowbottom and S. Webb, "Configuration space analysis of common cost functions in radiotherapy beam-weight optimization algorithms," Phys. Med. Biol. 47, 65-77 (2002).
14 M. Alber et al., "On the degeneracy of the IMRT optimization problem," Med. Phys. 29, 2584-2589 (2002).
15 R. Jeraj, C. A. Wu, and T. R. Mackie, "Optimizer convergence and local minima errors and their clinical importance," Phys. Med. Biol. 48, 2809-2827 (2003).
16 C. Wu, R. Jeraj, and T. R. Mackie, "The method of intercepts in parameter, space for the analysis of local minima caused by dose-volume constraints," Phys. Med. Biol. 48, N149-N157 (2003).
17 A. Niemierko, "Reporting and analyzing dose distributions: A concept of equivalent uniform dose," Med. Phys. 24,103-110 (1997).
18 Q. W. Wu et al, "A fast dose calculation method based on table lookup for IMRT optimization," Phys. Med. Biol. 48, N159-N166 (2003).
19 L. Jones, P. Hoban "A comparison of physically and radiobiologically based optimization for IMRT," Med. Phys. 29(7), 1447-1455 July 2002
20 C. Thieke, T. Bortfeld and A. Niemierko et al From physical dose constraints to equivalent uniform dose constraints in inverse radiotherapy planning Med. Phys. 30 .9., September 2003 2332-2339
21 A. Pugachev, J. G. Li, A. L. Boyer, et al "Role of beam orientation optimization in intensity-modulated radiation therapy," Int. J. Radiat. Oncol., Biol., Phys. 50, 551-560 (2001).
22 A. Niemierko, "A generalized concept of equivalent uniform dose" Med. Phys. 26, 1100 (1999).
23 S. Nill, "Development and application of a multi-modality inverse treatment planning system" Ph. D. thesis, Ruprecht-Karls-Universitaet Heidelberg, Germany, 2001.
24 C. Scholz, S. Nill, U. Oelfke, and T. Bortfeld, "Application of an advanced photon dose algodthrn for inverse IMRT planning," Med. Phys. 29, 1335 (2002).
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