超椭圆柱面梯度线圈设计

王亮; 曹英晖; 贾峰; 刘震宇

doi:10.7498/aps.63.238301

摘要

超椭圆柱设计表面能够减小线圈与目标的距离, 提高空间利用率, 扩大成像区域的有效范围. 提出利用流函数法及柱面的可展性在超椭圆柱面上设计核磁共振成像系统中的梯度线圈. 根据Biot-Savart定律建立磁场强度与流函数的表达式, 采用最小二乘法和Tikhonov正则化方法构造了双目标设计函数. 利用柱面的可展性提高了基于分片离散流函数计算电磁场的数值精度, 通过L-曲线方法实现了正则参数的合理选取. 通过引入适当的流函数边界约束条件, 把梯度线圈的优化问题转化为适定线性方程组的直接求解问题. 通过数值算例验证了超椭圆柱面展开求解方法的正确性. 优化结果显示, 在满足线性度误差小于5 %的设计约束下, 该方法在设计超椭圆柱面线圈驱动电流分布的同时有效控制了梯度线圈的能耗.

关键词:

Abstract

A super-elliptical cylinder surface can shorten the distance between coils and target, enhance the space utilization, and enlarge the homogeneous imaging volumes. This paper proposes a method to design magnetic resonance imaging (MRI) gradient coils using the stream function and the developable property of the super-elliptical cylindrical surface. Based on the Biot-Savart law, the relationship between the magnetic flux density and stream function is established firstly, and the objective is chosen in the least-square form with the additional Tikhonov regularization term. Numerical accuracy of the magnetic flux density in the region of interest is maintained through transforming the cylindrical surface to the corresponding flat surface, and the value of regularization coefficient of the dissipated powers is chosen automatically by using the L-curve method. Via imposing specified boundary conditions to the stream function on the developed surface, the optimization of gradient coils is gained by directly solving well-posed linear algebraic equations. Numerical examples illustrate the feasibility of the proposed design method. The designed coils on the super-elliptical cylindrical surface show that the electric current and the dissipated powers are adequately optimized under the condition that the linear gradient deviation is less than 5%.

Keywords:

作者及机构信息

1.
中国科学院长春光学精密机械与物理研究所, 长春 130033;

2.
中国科学院大学, 北京 100039;

3.
弗莱堡大学医学院放射与医学物理系, 弗莱堡 79106

基金项目: 国家自然科学基金(批准号:51275504)、吉林省科技发展计划(批准号:20140519007JH)和欧盟研究署‘RANGEMRI'启动项目(批准号:282345)资助的课题.

Authors and contacts

1.
Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China;

2.
University of Chinese Academy of Sciences, Beijing 100039, China;

3.
Department of Diagnostic Radiology, Medical Physics, University Medical Center Freiburg, Freiburg 79106, Germany

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51275504), the Science and Technology Development Plan of Jilin Province, China (Grant No. 20140519007JH), and the European Research Council Starting Grant ‘RANGEMRI' (Grant No. 282345).

参考文献

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[2]	Wang H Z, Zhang X L, Wu J 2008 Experiment Study of Magnetic Resonance Imaging (Beijing: Science Press) pp3-29 (in Chinese) [汪红志, 张学龙, 武杰2008核磁共振成像技术试验教程(北京: 科学出版社)第3–29页]
[3]	Turner R 1986 J. Phys. D: Appl. Phys. 19 L147
[4]	Alsop D C, Connick T J 1996 Magn. Reson. Med. 35 875
[5]	Liu Z Y, Jia F, Hennig J, Korvink J G 2012 IEEE Trans. Magn. 48 1179
[6]	Liu W T, Zu D L, Tang X 2010 Chin. Phys. B 19 018701
[7]	Poole M, Bowtell R 2007 Concepts Magn. Reson. B: Magn. Reson. Eng. 31B 162
[8]	Peeren G N 2003 J. Comput. Phys. 191 305
[9]	Tomasi D 2001 Magn. Reson. Med. 45 505
[10]	Jia F, Liu Z Y, Zaitsev M, Hennig J, Korvink J 2014 Struct. Multidis. Optim. 49 523
[11]	Pissanetzky S 1992 Meas. Sci. Technol. 3 667
[12]	Liu H Y 1998 IEEE Trans. Magn. 34 2162
[13]	Tomasi D, Caparelli E C, Panepucci H, Foerster B 1999 J. Magn. Reson. 140 325
[14]	Liu W T, Zu D L, Tang X, Guo H 2007 J. Phys. D: Appl. Phys. 40 4418
[15]	Zhu M H, Xia L, Liu F, Crozier S 2008 IEEE Trans. Magn. 44 2035
[16]	Feng Z K, Hu G L, Xu Y, Zhu G, Zhou F, Dai Y M, Wang Q L 2013 Acta Phys. Sin. 62 230701 (in Chinese) [冯忠奎, 胡格丽, 许莹, 朱光, 周峰, 戴银明, 王秋良 2013 物理学报 62 230701]
[17]	Forbes L K, Brideson M A, Crozier S 2005 IEEE Trans. Magn. 41 2134
[18]	Wang L Q, Wang W M 2014 Chin. Phys. B 23 028703
[19]	Moon S M, Goodrich K C, Hadley J R, Parker D L 2009 Concepts Magn. Reson. B: Magn. Reson. Eng. 35B 23
[20]	Moon S M, Goodrich K C, Hadley J R, Kim S, Zeng G L, Morrell G R, McAlpine M A, Chronik B A, Parker D L 2011 Magn. Reson. Med. 65 863
[21]	Brideson M A, Forbes L K, Crozier S 2002 Concepts Magn. Reson. 14 9
[22]	Barr A H 1981 IEEE Comput. Graph. Appl. 1 11
[23]	Franklin W R, Barr A H 1981 IEEE Comput. Graph. Appl. 1 41
[24]	Pilu M, Fisher R 1995 Proceedings of the British Machine Vision Conference Birmingham, British, September 11-14, 1995 p257
[25]	Jackson J D 1998 Classical Electrodynamics (New York: Wiley) pp174-224
[26]	Ungersma S E, Xu H, Chronik B A, Scott G C, Macovski A, Conolly S M 2004 Magn. Reson. Med. 52 619
[27]	Shi F, Ludwig R 1998 IEEE Trans. Magn. 34 671
[28]	Cheng Y P, Zhang K Y, Xu Z 1989 The Theory of Matrices (Xi'an: Northwestern Polytechnical University Press) pp360-370 (in Chinese) [程云鹏, 张凯院, 徐仲1989矩阵论(西安: 西北工业大学出版社)第360–370页]
[29]	Hu G L, Ni Z P, Wang Q L 2014 Acta Phys. Sin. 63 018301 (in Chinese) [胡格丽, 倪志鹏, 王秋良 2014 物理学报 63 018301]
[30]	Xu W L, Zhang J C, Li X, Xu B Q, Tao G S 2013 Chin. Phys. B 22 010203
[31]	Li Q Y, Wang N C, Yi D Y 2008 Numerical Analysis (Beijing: TsingHua University Press) pp202-208 (in Chinese) [李庆阳, 王能超, 易大义2008数值分析(北京: 清华大学出版社)第202–208页]
[32]	Hansen P C, Nagy J G, O'Leary D P 2006 Deblurring Images: Matrices, Spectra, and Filtering (Philadelphia: Society for Industrial and Applied Mathematics) pp71-100
[33]	Hansen P C 1994 Numer. Algorithms. 6 1

施引文献

[1]	Xu Y, Liang B L 2009 Medical Imaging Equipment (Beijing: People's Medical Publishing House) pp162-195 (in Chinese) [徐跃, 梁碧玲 2009 医学影像设备学(北京: 人民卫生出版社)第162–195 页]
[2]	Wang H Z, Zhang X L, Wu J 2008 Experiment Study of Magnetic Resonance Imaging (Beijing: Science Press) pp3-29 (in Chinese) [汪红志, 张学龙, 武杰2008核磁共振成像技术试验教程(北京: 科学出版社)第3–29页]
[3]	Turner R 1986 J. Phys. D: Appl. Phys. 19 L147
[4]	Alsop D C, Connick T J 1996 Magn. Reson. Med. 35 875
[5]	Liu Z Y, Jia F, Hennig J, Korvink J G 2012 IEEE Trans. Magn. 48 1179
[6]	Liu W T, Zu D L, Tang X 2010 Chin. Phys. B 19 018701
[7]	Poole M, Bowtell R 2007 Concepts Magn. Reson. B: Magn. Reson. Eng. 31B 162
[8]	Peeren G N 2003 J. Comput. Phys. 191 305
[9]	Tomasi D 2001 Magn. Reson. Med. 45 505
[10]	Jia F, Liu Z Y, Zaitsev M, Hennig J, Korvink J 2014 Struct. Multidis. Optim. 49 523
[11]	Pissanetzky S 1992 Meas. Sci. Technol. 3 667
[12]	Liu H Y 1998 IEEE Trans. Magn. 34 2162
[13]	Tomasi D, Caparelli E C, Panepucci H, Foerster B 1999 J. Magn. Reson. 140 325
[14]	Liu W T, Zu D L, Tang X, Guo H 2007 J. Phys. D: Appl. Phys. 40 4418
[15]	Zhu M H, Xia L, Liu F, Crozier S 2008 IEEE Trans. Magn. 44 2035
[16]	Feng Z K, Hu G L, Xu Y, Zhu G, Zhou F, Dai Y M, Wang Q L 2013 Acta Phys. Sin. 62 230701 (in Chinese) [冯忠奎, 胡格丽, 许莹, 朱光, 周峰, 戴银明, 王秋良 2013 物理学报 62 230701]
[17]	Forbes L K, Brideson M A, Crozier S 2005 IEEE Trans. Magn. 41 2134
[18]	Wang L Q, Wang W M 2014 Chin. Phys. B 23 028703
[19]	Moon S M, Goodrich K C, Hadley J R, Parker D L 2009 Concepts Magn. Reson. B: Magn. Reson. Eng. 35B 23
[20]	Moon S M, Goodrich K C, Hadley J R, Kim S, Zeng G L, Morrell G R, McAlpine M A, Chronik B A, Parker D L 2011 Magn. Reson. Med. 65 863
[21]	Brideson M A, Forbes L K, Crozier S 2002 Concepts Magn. Reson. 14 9
[22]	Barr A H 1981 IEEE Comput. Graph. Appl. 1 11
[23]	Franklin W R, Barr A H 1981 IEEE Comput. Graph. Appl. 1 41
[24]	Pilu M, Fisher R 1995 Proceedings of the British Machine Vision Conference Birmingham, British, September 11-14, 1995 p257
[25]	Jackson J D 1998 Classical Electrodynamics (New York: Wiley) pp174-224
[26]	Ungersma S E, Xu H, Chronik B A, Scott G C, Macovski A, Conolly S M 2004 Magn. Reson. Med. 52 619
[27]	Shi F, Ludwig R 1998 IEEE Trans. Magn. 34 671
[28]	Cheng Y P, Zhang K Y, Xu Z 1989 The Theory of Matrices (Xi'an: Northwestern Polytechnical University Press) pp360-370 (in Chinese) [程云鹏, 张凯院, 徐仲1989矩阵论(西安: 西北工业大学出版社)第360–370页]
[29]	Hu G L, Ni Z P, Wang Q L 2014 Acta Phys. Sin. 63 018301 (in Chinese) [胡格丽, 倪志鹏, 王秋良 2014 物理学报 63 018301]
[30]	Xu W L, Zhang J C, Li X, Xu B Q, Tao G S 2013 Chin. Phys. B 22 010203
[31]	Li Q Y, Wang N C, Yi D Y 2008 Numerical Analysis (Beijing: TsingHua University Press) pp202-208 (in Chinese) [李庆阳, 王能超, 易大义2008数值分析(北京: 清华大学出版社)第202–208页]
[32]	Hansen P C, Nagy J G, O'Leary D P 2006 Deblurring Images: Matrices, Spectra, and Filtering (Philadelphia: Society for Industrial and Applied Mathematics) pp71-100
[33]	Hansen P C 1994 Numer. Algorithms. 6 1

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