基于PET/CT的IMRT数学模型及算法研究
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摘要
利用适形调强(IMRT)医疗方法对肿瘤放射治疗方案进行优化,可以使放射线向肿瘤的活性组织集中而使得在有效杀伤癌细胞的前提下,人体的功能器官和组织受到有效保护,是近10多年来放射医学发展较快且应用普及的一项重要技术。近几年来,由于PET/CT检验技术的引入,人们不但能利用螺旋CT的肿瘤成像技术得到清晰的肿瘤几何轮廓,而且可以利用PET检验提供的肿瘤的生物信息,得到肿瘤细胞的活性指标和扩散信息,为放射治疗提供更准确地病灶信息。但PET生物信息图像给出的是肿瘤的生物信息,如何将这些信息以几何的形式表现出来并应用于适形调强医疗方案的优化是目前研究的热点问题,涉及到优化算法、图形、图像处理等一系列问题的研究,目前已有许多进展。
在本文中,根据PET/CT的适形调强算法的特点和算法研究开发的需要,我们对以下几个问题进行了研究：
(1)由于PET图像的清晰程度比较差,在图像的处理中需要引入图像的识别算法。我们基于离散数据,给出了图像识别模型及其相应的数值计算方法,算法可以应用于一般的n维空问上任意多面体的识别问题,且计算效率较高。
(2)在对适形调强医疗方案的算法研究中,我们分析了对调强矩阵分解为放射子野组合的优化方法,基于两个基本的目标函数,导出了达到最优的条件并给出了证明。
(3)对于基于PET/CT的放疗的临床靶区的确定,我们利用微分方程的热传导模型建立了一种基于肿瘤生物信息的临床靶区的确定方法。
Intensity modulated radiation therapy (IMRT), a widely used optimizing approach for the radiation therapy in cancer treatment, is an important technology which has been developing rapidly in the last 10 years. In recent years, with the invention and development of PET/CT equipment, both geometric and biological information of cancer can be obtained through PET/CT diagnosis, which provide us more information about cancer such as the active index and diffusion trend of the tumor cells which show more accurate information for radiotherapy. However, PET shows only the biological information of the tumor with is of poor visibility. The current hot issues are how to demonstrate these information in the form of geometry.
(1) Since PET image is of poor definition, image identification has to be introduced in the information treatment. Based on the discrete data of the image, an information identification model and a relative algorithm are introduced. The numerical algorithm is efficient, and can be used in N dimensional problems.
(2) In the study of the numerical treatment of IMRT medical plans, the optimization combination of shape matrices to form a aperture is analyzed. The optimal conditions are derived and proved basing on two basic objective functions.
(3) A method for defining clinical target volumn according to biological information is given by the revised heat conduction model of differential equations for defining clinical target volumn according to PET/CT radiotherapy.
在本文中,根据PET/CT的适形调强算法的特点和算法研究开发的需要,我们对以下几个问题进行了研究：
(1)由于PET图像的清晰程度比较差,在图像的处理中需要引入图像的识别算法。我们基于离散数据,给出了图像识别模型及其相应的数值计算方法,算法可以应用于一般的n维空问上任意多面体的识别问题,且计算效率较高。
(2)在对适形调强医疗方案的算法研究中,我们分析了对调强矩阵分解为放射子野组合的优化方法,基于两个基本的目标函数,导出了达到最优的条件并给出了证明。
(3)对于基于PET/CT的放疗的临床靶区的确定,我们利用微分方程的热传导模型建立了一种基于肿瘤生物信息的临床靶区的确定方法。
Intensity modulated radiation therapy (IMRT), a widely used optimizing approach for the radiation therapy in cancer treatment, is an important technology which has been developing rapidly in the last 10 years. In recent years, with the invention and development of PET/CT equipment, both geometric and biological information of cancer can be obtained through PET/CT diagnosis, which provide us more information about cancer such as the active index and diffusion trend of the tumor cells which show more accurate information for radiotherapy. However, PET shows only the biological information of the tumor with is of poor visibility. The current hot issues are how to demonstrate these information in the form of geometry.
(1) Since PET image is of poor definition, image identification has to be introduced in the information treatment. Based on the discrete data of the image, an information identification model and a relative algorithm are introduced. The numerical algorithm is efficient, and can be used in N dimensional problems.
(2) In the study of the numerical treatment of IMRT medical plans, the optimization combination of shape matrices to form a aperture is analyzed. The optimal conditions are derived and proved basing on two basic objective functions.
(3) A method for defining clinical target volumn according to biological information is given by the revised heat conduction model of differential equations for defining clinical target volumn according to PET/CT radiotherapy.
引文
1 Takahashi S, Conformation radiotherapy:rotation techniques as applied to radiotherapy and radiotherapy of cancer, Acta Radiologica,1965
2 Bjarngard, Kijewski Computer-controlled radiation therapy machine for pelvic and para-aortic nodal areas, Int J Radiat Oncol Biol Phys,1981,61-70
3 Bortfeld T Optimized planning using physical objectives and constraints. Semin Radiat Oncol, (1999)19(1):20-34
4 Romeijn HE, Ahuja RK, Dempsey JF, Kumar A, Li JG (2003) A novel linear programming approach to luence map optimization for intensity modulated radiation therapy treatment planning. Phys Med iol 48(21):3521-3542
5 Munro TR, Gilbert CW(1961) The relation between tumour lethal doses and the radiosensitivity of tumour cells. Br J Radiol 34:246-251
6 Withers HR, McBride WH (1998) Biologic basis of radiation therapy. In:Perez CA, Brady LW (eds)Principles and practice of radiotherapy. Lippincott-Raven, Philadelphia, pp 79-118, chap 2
7 Niemierko A (1997) Reporting and analyzing dose distributions:a concept of equivalent uniform dose. Med Phys 24(1):103-110
8 Bortfeld T, Craft D, Dempsey JF, Halabi T, Romeijn HE (2008b) Evaluating target cold spots by use of tail EUDs. Phys Med Biol 71 (3):880-889
9 Hoffmann AL, Siem AYD, den Hertog D, Kaanders JHAM, Huizenga H (2006) Derivative-free generation and interpolation of convex Pareto optimal IMRT plans. Phys Med Biol 51(24):6349-6369
10 Niemierko A (1999) A generalized concept of equivalent uniform dose. Med Phys 26(6):1100
11 Romeijn HE, Ahuja RK, Dempsey JF, Kumar A (2005) A column generation approach to radiation therapy treatment planning using aperture modulation. SIAM J Optim 15 (3):838-862
12 Carlsson F (2008) Combining segment generation with direct step-and-shoot optimization in intensitymodulated radiation therapy. Med Phys 35(9):3828-3838
13金志超,陆健,等.基于偏最小二乘的原发性肝癌细胞遗传学异常区域识别方法[J].中国卫生统计,2009,6(26)：237-240
14 Brahme A. Biologically optimized 3-dimensional in vivo predictive assay-based radiation therapy using positron emission tomographcomputerized tomography imaging [J]. Acta Oncol,2003,42 (2):123-136.
15 Bradley JD, Thorstad WL, Mutic S, et al. Impact of FDG-PET on radiation therapy volume delineation in non-small cell lung cancer[J]. Int J Radiat Oncol Biol Phys,2004, 59(1):78-86.
16 Mah K, Caldwell C, Ung Y, et al. The impact of (18)FDG-PET on target and critical organs in CT-based treatment planning of patients with poorly defined non-small cell lung carcinoma:a prospective study[J]. Int J Radiat Oncol Biol Phys,2002,52(2):339-350.
17 Ciernik IF, Dizendor E, Baumert BG, et al. Radiation treatment planning with an integrated positron emission and computer tomography(PET-CT):a feasibility study[J]. Int J Radiat Oncol Biol Phys,2003,57(3):853-863.
18 Spaepen K, Stroobants S, Dupont P, et al. Prognostic value of positron emission tomography with Fluorine-18 Fluorodeoxyglucose ([18F]-FDG) after first-line chemotherapy in non-Hodgkin's lymphoma:[18F]FDG-PET a valid alternative to conventional diagnosticmethods[J]. J Clin Oncol,2001,19(2):414-419.
19 D. Dink, S. Orcun, M. P. Langer, J. F. Pekny, G. V. Reklaitis, R. L. Rardin, Importance of sensitivity analysis in intensity modulated radiation therapy (IMRT). EuroInforms Presentation 2003.
20 H. E. Romeijn, R. K. Ahuja, A COLUMN GENERATION APPROACH TO RADIATION THERAPY TREATMENT PLANNING USING APERTURE MODULATION, SIAM J. OPTIM._c 2005 Society for Industrial and Applied Mathematics, Vol.15, No.3, pp.838-862
20 Philipp Suss(?) and Karl-Heinz Kufer, Balancing control and simplicity:A variable aggregation method in intensity modulated radiation therapy planning, Linear Algebra Appl.2008 March 1; 428(5-6):1388-1405.
22 K. Engel, A new algorithm for optimal multileaf collimator field segmentation, University Rostock, Germany, March 2003.
23 M. Langer, V. Thai, and L. Papiez, Improved leaf sequencing reduces segments or monitor units needed to deliver IMRT using multileaf collimators, Medical Physics, 28 (12),2001.
24 P. Xia, L. J. Verhey, Multileaf collimator leaf sequencing algorithm for intensity modulated beams with multiple static segments, Medical Physics,25 (8),1998.
25 H. Edwin Romeijn, James F. Dempsey, Intensity modulated radiation therapy treatment plan Optimization, Sociedad de Estadistica e Investigacion Operativa 2008.
26 Agazaryan N, Solberg TD, Segmental and dynamic intensity-modulated radiotherapy delivery techniques, Med Phys 30 (7):1758-1765
27 Bednarz G, Michalski D, Houser C, Huq MS, Xiao Y, Anne PR, Galvin JM (2002), The use of mixedinteger programming for inverse treatment planning with pre-defined field segments. Phys Med Biol,47(13):2235-2245.
28 Boland N, Hamacher HW, Lenzen F (2004), Minimizing beam-on time in cancer radiation treatment using multileaf collimators, Networks 43(4):226-240
29 Bortfeld T (1999), Optimized planning using physical objectives and constraints, Semin Radiat Oncol,19(1):20-34
30 Carlsson F (2008), Combining segment generation with direct step-and-shoot optimization in intensitymodulated radiation therapy, Med Phys 35 (9):3828-3838
31 Choi B, Deasy JO (2002), The generalized equivalent uniform dose function as a basis for intensitymodulated treatment planning, Phys Med Biol 47:3579-3589
32 Dai J, Zhu Y (2001) Minimizing the number of segments in a delivery sequence for intensity-modulated radiation therapy with a multileaf collimator, Med Phys 28(10):2113-2120
33 Engel K (2005), A new algorithm for optimal multileaf collimator field segmentation. Discrete Appl Math 152(1):35-51
34 Fraass B, Balter J, Ten Haken R,McShan D (2003), Margins, errors and plan optimization. Radiother Oncol 68(Suppl.1):S34
35 Kalinowski T (2004), The algorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentation. Technical report, Fachbereich Mathematik, Universitat Rostock, Germany, August 2004
36 Kamath S, Sahni S, Li J, Palta J, Ranka S (2003), Leaf sequencing algorithms for segmented multileafcollimation, Phys Med Biol 48(3):307-324
37 Kamath S, Sahni S, Palta J, Ranka S (2004a), Algorithms for optimal sequencing of dynamic multileafcollimators, Phys Med Biol 49(1):33-54
38 Langer M, Thai V, Papiez L (2001), Improved leaf sequencing reduces segments or monitor units needed to deliver IMRT using multileaf collimators. Med Phys 28(12):2450-2458
39 Lee EK, Fox T, Crocker I (2003) Integer programming applied to intensity-modulated radiation treatment planning, Ann Operat Res 119:165-181
40 D. Baatar, H. W. Hamacher, New LP model for multileaf collimators in radiation therapy, Contribution to the Conference ORP3, University of Kaiserslautern,2003.
41 T. R. Bortfeld, D. L. Kahler, T.J Waldron and A. L. Boyer, "X-ray field compensation with multileaf collimators. " International Journal of Radiation Oncology Biology 28 (1994), pp.723-730.
42 K. Engel, "A new algorithm for optimal multileaf collimator field segmentation. " University Rostock, Germany, March 2003.
43 T. Bortfeld, et. al. "Current IMRT optimization algorithms:principles, potential and limitations. " Massachusetts General Hospital, Harvard Medical School, Presentation 2000.
2 Bjarngard, Kijewski Computer-controlled radiation therapy machine for pelvic and para-aortic nodal areas, Int J Radiat Oncol Biol Phys,1981,61-70
3 Bortfeld T Optimized planning using physical objectives and constraints. Semin Radiat Oncol, (1999)19(1):20-34
4 Romeijn HE, Ahuja RK, Dempsey JF, Kumar A, Li JG (2003) A novel linear programming approach to luence map optimization for intensity modulated radiation therapy treatment planning. Phys Med iol 48(21):3521-3542
5 Munro TR, Gilbert CW(1961) The relation between tumour lethal doses and the radiosensitivity of tumour cells. Br J Radiol 34:246-251
6 Withers HR, McBride WH (1998) Biologic basis of radiation therapy. In:Perez CA, Brady LW (eds)Principles and practice of radiotherapy. Lippincott-Raven, Philadelphia, pp 79-118, chap 2
7 Niemierko A (1997) Reporting and analyzing dose distributions:a concept of equivalent uniform dose. Med Phys 24(1):103-110
8 Bortfeld T, Craft D, Dempsey JF, Halabi T, Romeijn HE (2008b) Evaluating target cold spots by use of tail EUDs. Phys Med Biol 71 (3):880-889
9 Hoffmann AL, Siem AYD, den Hertog D, Kaanders JHAM, Huizenga H (2006) Derivative-free generation and interpolation of convex Pareto optimal IMRT plans. Phys Med Biol 51(24):6349-6369
10 Niemierko A (1999) A generalized concept of equivalent uniform dose. Med Phys 26(6):1100
11 Romeijn HE, Ahuja RK, Dempsey JF, Kumar A (2005) A column generation approach to radiation therapy treatment planning using aperture modulation. SIAM J Optim 15 (3):838-862
12 Carlsson F (2008) Combining segment generation with direct step-and-shoot optimization in intensitymodulated radiation therapy. Med Phys 35(9):3828-3838
13金志超,陆健,等.基于偏最小二乘的原发性肝癌细胞遗传学异常区域识别方法[J].中国卫生统计,2009,6(26)：237-240
14 Brahme A. Biologically optimized 3-dimensional in vivo predictive assay-based radiation therapy using positron emission tomographcomputerized tomography imaging [J]. Acta Oncol,2003,42 (2):123-136.
15 Bradley JD, Thorstad WL, Mutic S, et al. Impact of FDG-PET on radiation therapy volume delineation in non-small cell lung cancer[J]. Int J Radiat Oncol Biol Phys,2004, 59(1):78-86.
16 Mah K, Caldwell C, Ung Y, et al. The impact of (18)FDG-PET on target and critical organs in CT-based treatment planning of patients with poorly defined non-small cell lung carcinoma:a prospective study[J]. Int J Radiat Oncol Biol Phys,2002,52(2):339-350.
17 Ciernik IF, Dizendor E, Baumert BG, et al. Radiation treatment planning with an integrated positron emission and computer tomography(PET-CT):a feasibility study[J]. Int J Radiat Oncol Biol Phys,2003,57(3):853-863.
18 Spaepen K, Stroobants S, Dupont P, et al. Prognostic value of positron emission tomography with Fluorine-18 Fluorodeoxyglucose ([18F]-FDG) after first-line chemotherapy in non-Hodgkin's lymphoma:[18F]FDG-PET a valid alternative to conventional diagnosticmethods[J]. J Clin Oncol,2001,19(2):414-419.
19 D. Dink, S. Orcun, M. P. Langer, J. F. Pekny, G. V. Reklaitis, R. L. Rardin, Importance of sensitivity analysis in intensity modulated radiation therapy (IMRT). EuroInforms Presentation 2003.
20 H. E. Romeijn, R. K. Ahuja, A COLUMN GENERATION APPROACH TO RADIATION THERAPY TREATMENT PLANNING USING APERTURE MODULATION, SIAM J. OPTIM._c 2005 Society for Industrial and Applied Mathematics, Vol.15, No.3, pp.838-862
20 Philipp Suss(?) and Karl-Heinz Kufer, Balancing control and simplicity:A variable aggregation method in intensity modulated radiation therapy planning, Linear Algebra Appl.2008 March 1; 428(5-6):1388-1405.
22 K. Engel, A new algorithm for optimal multileaf collimator field segmentation, University Rostock, Germany, March 2003.
23 M. Langer, V. Thai, and L. Papiez, Improved leaf sequencing reduces segments or monitor units needed to deliver IMRT using multileaf collimators, Medical Physics, 28 (12),2001.
24 P. Xia, L. J. Verhey, Multileaf collimator leaf sequencing algorithm for intensity modulated beams with multiple static segments, Medical Physics,25 (8),1998.
25 H. Edwin Romeijn, James F. Dempsey, Intensity modulated radiation therapy treatment plan Optimization, Sociedad de Estadistica e Investigacion Operativa 2008.
26 Agazaryan N, Solberg TD, Segmental and dynamic intensity-modulated radiotherapy delivery techniques, Med Phys 30 (7):1758-1765
27 Bednarz G, Michalski D, Houser C, Huq MS, Xiao Y, Anne PR, Galvin JM (2002), The use of mixedinteger programming for inverse treatment planning with pre-defined field segments. Phys Med Biol,47(13):2235-2245.
28 Boland N, Hamacher HW, Lenzen F (2004), Minimizing beam-on time in cancer radiation treatment using multileaf collimators, Networks 43(4):226-240
29 Bortfeld T (1999), Optimized planning using physical objectives and constraints, Semin Radiat Oncol,19(1):20-34
30 Carlsson F (2008), Combining segment generation with direct step-and-shoot optimization in intensitymodulated radiation therapy, Med Phys 35 (9):3828-3838
31 Choi B, Deasy JO (2002), The generalized equivalent uniform dose function as a basis for intensitymodulated treatment planning, Phys Med Biol 47:3579-3589
32 Dai J, Zhu Y (2001) Minimizing the number of segments in a delivery sequence for intensity-modulated radiation therapy with a multileaf collimator, Med Phys 28(10):2113-2120
33 Engel K (2005), A new algorithm for optimal multileaf collimator field segmentation. Discrete Appl Math 152(1):35-51
34 Fraass B, Balter J, Ten Haken R,McShan D (2003), Margins, errors and plan optimization. Radiother Oncol 68(Suppl.1):S34
35 Kalinowski T (2004), The algorithmic complexity of the minimization of the number of segments in multileaf collimator field segmentation. Technical report, Fachbereich Mathematik, Universitat Rostock, Germany, August 2004
36 Kamath S, Sahni S, Li J, Palta J, Ranka S (2003), Leaf sequencing algorithms for segmented multileafcollimation, Phys Med Biol 48(3):307-324
37 Kamath S, Sahni S, Palta J, Ranka S (2004a), Algorithms for optimal sequencing of dynamic multileafcollimators, Phys Med Biol 49(1):33-54
38 Langer M, Thai V, Papiez L (2001), Improved leaf sequencing reduces segments or monitor units needed to deliver IMRT using multileaf collimators. Med Phys 28(12):2450-2458
39 Lee EK, Fox T, Crocker I (2003) Integer programming applied to intensity-modulated radiation treatment planning, Ann Operat Res 119:165-181
40 D. Baatar, H. W. Hamacher, New LP model for multileaf collimators in radiation therapy, Contribution to the Conference ORP3, University of Kaiserslautern,2003.
41 T. R. Bortfeld, D. L. Kahler, T.J Waldron and A. L. Boyer, "X-ray field compensation with multileaf collimators. " International Journal of Radiation Oncology Biology 28 (1994), pp.723-730.
42 K. Engel, "A new algorithm for optimal multileaf collimator field segmentation. " University Rostock, Germany, March 2003.
43 T. Bortfeld, et. al. "Current IMRT optimization algorithms:principles, potential and limitations. " Massachusetts General Hospital, Harvard Medical School, Presentation 2000.