MRI磁体系统的计算机辅助设计研究
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摘要
随着现代医学影像技术的快速发展，利用人体组织器官的影像进行临床诊断已经成为一种必不可少的诊疗途径，极大地促进了医疗水平的提高。磁共振成像就是其中一种新兴的医学影像方法，主要利用核磁共振原理对处于均匀静磁场中的人体器官进行成像，具有成像参数多、无电离辐射、可任意方向断层等诸多优点，在临床诊断中被广泛应用。在磁共振成像设备中磁体系统是一个重要的组成部分，负责产生均匀静磁场，为磁共振成像提供基础物理环境，其性能的优劣是决定磁共振图像质量的重要因素，也是衡量整个成像系统性能的主要参考指标，因此具有重要的研究意义。磁体系统主要由磁体和匀场部件两部分组成，其中磁体负责在成像空间产生一定强度的静磁场，匀场部件负责对磁场进行补偿校正，使磁场均匀度满足磁共振成像要求。
由于我国的磁共振成像研究工作起步较晚，在这一领域的相关技术比较落后，目前成像系统仍然以设备引进为主。为了提高我国磁共振成像设备的国产化能力，本文结合我国磁共振成像设备的实际发展需要，针对性价比较高的永磁型磁体系统设计方法进行了深入研究，内容涉及到永磁体磁路设计、磁体结构优化以及成像磁场的无源匀场计算和有源匀场设计等，主要在以下几个方面进行了有特色和创新性的研究工作：
(1)磁体磁路设计
磁共振成像永磁体的结构需要根据系统对成像磁场的性能要求进行设计，本文将磁路计算理论与有限元分析方法相结合，利用磁体仿真模型对磁路结构及成像磁场进行分析研究，确保了磁场性能参数满足设计要求。另外针对永磁体成像磁场均匀性差的缺点，设计了匀场环部件用以减弱磁场的边缘效应，增强了成像磁场均匀性，同时改进了磁体匀场板结构，提高了磁场的磁感应强度。
(2)磁体结构优化
针对永磁体重量大的缺点，本文利用计算机仿真分析方法对磁体中的磁场分布特征进行了详细研究，根据精确的磁场分布数据对磁体结构进行优化，去除了磁体支架材料中的冗余部分，大幅减轻了磁体重量，同时对磁体匀场环的尺寸结构进行了优化设计，进一步提高了成像磁场的均匀性。
(3)无源匀场计算
由于永磁体的初始磁场均匀性通常无法满足磁共振成像要求，一般采用无源匀场方法对磁场进行校正。本文提出了一种利用磁场分布特征对匀场磁片进行综合规划的方法，根据成像磁场样本数据分析磁场分布特征，利用电磁理论建立磁片校正磁场的数学模型，以此为基础提出了极值定位方法对磁片匀场轨道半径进行规划，同时提出了逐次逼近算法进行匀场计算，减少了磁场的重复校正。该算法能够准确算出匀场磁片的校正位置及使用数量，有效地指导无源匀场工作。
(4)匀场线圈设计
当外界温度发生变化时成像磁场的均匀性会随之改变，通常采用有源匀场方法对磁场进行再校正。本文针对永磁体的组成结构提出了一种平板式匀场线圈设计方法，通过研究成像磁场的球谐函数以及通电圆弧在自由空间产生磁场的泰勒展开级数，导出了x、y、xz、xz~2、x~2-y~2等低阶径向线圈的设计方法，同时针对轴向线圈提出了一种新的混合式设计方案，能够通过控制线圈中的电流组合实现对不同轴向磁场分量的获取，简化了轴向线圈的设计过程，提高了校正磁场的准确性。
(5)匀场电流计算
对匀场线圈施加适当的电流后才能产生校正磁场消除成像磁场的非均匀分量，本文在匀场电流间接计算方法的基础上，利用磁共振成像原理中磁场均匀性与系统FID信号强度成正比的关系提出了一种电流迭代优化算法，将系统FID信号强度的变化作为反馈信息，能够快速计算出匀场电流对成像磁场进行补偿校正，平均耗时不到三分钟，满足磁共振成像系统对有源匀场的实时性要求。
With rapid development of modern medical imaging technology, it has been a necessary clinical diagnosis approach to get symptom information from human organic imaging. MRI (Magnetic Resonance Imaging) is a medical imaging method which has lots of advantages such as multi-parameter imaging, no ionizing radiation, arbitrary direction imaging and so on. MRI is based on NMR (Nuclear Magnetic Resonance) theory, so static magnetic field is required for imaging. In MRI system, magnet system is the main part that generates imaging magnetic field. Its performance is an important factor to affect image quality. Therefore, the research on methods for magnet system has important theoretical and applied values. Magnet system is mainly composed of magnet and shimming component. Magnet's responsibility is to generate homogeneous magnetic field in imaging work space. Shimming component is used to improve magnetic field homogeneity to satisfy MRI requirement.
Because the research work on MRI theory is started late in our country, we have lagged behind in this technology field. At present, most MRI equipments need to be imported from foreign companies. To improve the capability of MRI system made in China, this thesis has a deep research on design methods for permanent magnet system. The investigation contents include four parts: magnetic circuit design of permanent magnet, magnet configuration optimization, passive shimming calculation and active shimming design. Primary innovation works are described as the following:
(1) Magnetic circuit design of permanent magnet
In MRI system, permanent magnet's configuration is decided by imaging magnetic field performance parameters. In this thesis, we combined magnetic circuit design theory and FEA (Finite Element Analysis), and had a detailed research on permanent magnet design method. Using simulation analysis method, we studied the magnetic field distribution characteristic of permanent magnet. It made sure that magnetic circuit configuration and magnet performance achieved the MRI requirement. In addition, we designed shim-loop and trapezoid shim-board to improve the magnetic field homogeneity and magnetic induction intensity.
(2) Permanent magnet configuration optimization
To lighten the weight of permanent magnet, this thesis presented a simulation method to study and improve the optimization technique of magnet configuration. With finite element method, we analyzed the magnetic field distribution of permanent magnet. Based on the magnetic field distribution characteristic, the redundancy parts of magnetic circuit configuration were removed accurately. The magnet weight was lightened significantly. Moreover, we optimized the shim-loop configuration. This made the imaging magnetic field more homogeneous.
(3) Passive shimming calculation
Generally, the original magnetic field homogeneity of permanent magnet can't satisfy MRI requirement. Passive shimming is usually used to correct the magnetic field. In this thesis, we presented a novel algorithm to calculate the position and number of shimming ferromagnetic elements. With magnetic induction intensity samples, we analyzed imaging magnetic field distribution characteristic. Based on electromagnetic theory, we established the correcting magnetic field mathematical model, and proposed a localizing method to program the shim radius. To reduce the quantities of the ferromagnetic material and avoid repeated correction, successive approximation was presented to calculate the number of shimming ferromagnetic elements. The experiment results show that this method can effectively guide passive shimming work and obtain highly homogeneous magnetic field.
(4) Active shimming design
The magnetic field homogeneity will descend when temperature changes. Generally, active shimming is used to re-correct magnetic field. We proposed a set of tabulate shim coils to improve the imaging field homogeneity for permanent magnet. By studying the magnetic spherical harmonics and its Taylor expansion, the proposal of designing lower-order radial coils was derived including x, y, xz, xz~2, x~2-y~2 etc. In addition, a novel composite design was presented for the axial coils. By controlling the shim currents of axial coils, different axial magnetic fields can be obtained. This method simplified the design process of axial coils, and made the axial correcting magnetic field more precise.
(5) Shim current calculation
When shim coils are excited by appropriate current, they can generate correcting magnetic field to compensate the imaging magnetic field unhomogeneity. Based on the indirect calculation method, we proposed a novel iterative optimization algorithm to calculate shim current. Because there is a direct proportion relationship between FID (Free Induction Decay) signal and magnetic field homogeneity, we use the intensity of FID signal as feedback information to the calculation. Shim current can be worked out in less than 3 minutes. This satisfied the requirement that active shimming should be performed in real time.
由于我国的磁共振成像研究工作起步较晚，在这一领域的相关技术比较落后，目前成像系统仍然以设备引进为主。为了提高我国磁共振成像设备的国产化能力，本文结合我国磁共振成像设备的实际发展需要，针对性价比较高的永磁型磁体系统设计方法进行了深入研究，内容涉及到永磁体磁路设计、磁体结构优化以及成像磁场的无源匀场计算和有源匀场设计等，主要在以下几个方面进行了有特色和创新性的研究工作：
(1)磁体磁路设计
磁共振成像永磁体的结构需要根据系统对成像磁场的性能要求进行设计，本文将磁路计算理论与有限元分析方法相结合，利用磁体仿真模型对磁路结构及成像磁场进行分析研究，确保了磁场性能参数满足设计要求。另外针对永磁体成像磁场均匀性差的缺点，设计了匀场环部件用以减弱磁场的边缘效应，增强了成像磁场均匀性，同时改进了磁体匀场板结构，提高了磁场的磁感应强度。
(2)磁体结构优化
针对永磁体重量大的缺点，本文利用计算机仿真分析方法对磁体中的磁场分布特征进行了详细研究，根据精确的磁场分布数据对磁体结构进行优化，去除了磁体支架材料中的冗余部分，大幅减轻了磁体重量，同时对磁体匀场环的尺寸结构进行了优化设计，进一步提高了成像磁场的均匀性。
(3)无源匀场计算
由于永磁体的初始磁场均匀性通常无法满足磁共振成像要求，一般采用无源匀场方法对磁场进行校正。本文提出了一种利用磁场分布特征对匀场磁片进行综合规划的方法，根据成像磁场样本数据分析磁场分布特征，利用电磁理论建立磁片校正磁场的数学模型，以此为基础提出了极值定位方法对磁片匀场轨道半径进行规划，同时提出了逐次逼近算法进行匀场计算，减少了磁场的重复校正。该算法能够准确算出匀场磁片的校正位置及使用数量，有效地指导无源匀场工作。
(4)匀场线圈设计
当外界温度发生变化时成像磁场的均匀性会随之改变，通常采用有源匀场方法对磁场进行再校正。本文针对永磁体的组成结构提出了一种平板式匀场线圈设计方法，通过研究成像磁场的球谐函数以及通电圆弧在自由空间产生磁场的泰勒展开级数，导出了x、y、xz、xz~2、x~2-y~2等低阶径向线圈的设计方法，同时针对轴向线圈提出了一种新的混合式设计方案，能够通过控制线圈中的电流组合实现对不同轴向磁场分量的获取，简化了轴向线圈的设计过程，提高了校正磁场的准确性。
(5)匀场电流计算
对匀场线圈施加适当的电流后才能产生校正磁场消除成像磁场的非均匀分量，本文在匀场电流间接计算方法的基础上，利用磁共振成像原理中磁场均匀性与系统FID信号强度成正比的关系提出了一种电流迭代优化算法，将系统FID信号强度的变化作为反馈信息，能够快速计算出匀场电流对成像磁场进行补偿校正，平均耗时不到三分钟，满足磁共振成像系统对有源匀场的实时性要求。
With rapid development of modern medical imaging technology, it has been a necessary clinical diagnosis approach to get symptom information from human organic imaging. MRI (Magnetic Resonance Imaging) is a medical imaging method which has lots of advantages such as multi-parameter imaging, no ionizing radiation, arbitrary direction imaging and so on. MRI is based on NMR (Nuclear Magnetic Resonance) theory, so static magnetic field is required for imaging. In MRI system, magnet system is the main part that generates imaging magnetic field. Its performance is an important factor to affect image quality. Therefore, the research on methods for magnet system has important theoretical and applied values. Magnet system is mainly composed of magnet and shimming component. Magnet's responsibility is to generate homogeneous magnetic field in imaging work space. Shimming component is used to improve magnetic field homogeneity to satisfy MRI requirement.
Because the research work on MRI theory is started late in our country, we have lagged behind in this technology field. At present, most MRI equipments need to be imported from foreign companies. To improve the capability of MRI system made in China, this thesis has a deep research on design methods for permanent magnet system. The investigation contents include four parts: magnetic circuit design of permanent magnet, magnet configuration optimization, passive shimming calculation and active shimming design. Primary innovation works are described as the following:
(1) Magnetic circuit design of permanent magnet
In MRI system, permanent magnet's configuration is decided by imaging magnetic field performance parameters. In this thesis, we combined magnetic circuit design theory and FEA (Finite Element Analysis), and had a detailed research on permanent magnet design method. Using simulation analysis method, we studied the magnetic field distribution characteristic of permanent magnet. It made sure that magnetic circuit configuration and magnet performance achieved the MRI requirement. In addition, we designed shim-loop and trapezoid shim-board to improve the magnetic field homogeneity and magnetic induction intensity.
(2) Permanent magnet configuration optimization
To lighten the weight of permanent magnet, this thesis presented a simulation method to study and improve the optimization technique of magnet configuration. With finite element method, we analyzed the magnetic field distribution of permanent magnet. Based on the magnetic field distribution characteristic, the redundancy parts of magnetic circuit configuration were removed accurately. The magnet weight was lightened significantly. Moreover, we optimized the shim-loop configuration. This made the imaging magnetic field more homogeneous.
(3) Passive shimming calculation
Generally, the original magnetic field homogeneity of permanent magnet can't satisfy MRI requirement. Passive shimming is usually used to correct the magnetic field. In this thesis, we presented a novel algorithm to calculate the position and number of shimming ferromagnetic elements. With magnetic induction intensity samples, we analyzed imaging magnetic field distribution characteristic. Based on electromagnetic theory, we established the correcting magnetic field mathematical model, and proposed a localizing method to program the shim radius. To reduce the quantities of the ferromagnetic material and avoid repeated correction, successive approximation was presented to calculate the number of shimming ferromagnetic elements. The experiment results show that this method can effectively guide passive shimming work and obtain highly homogeneous magnetic field.
(4) Active shimming design
The magnetic field homogeneity will descend when temperature changes. Generally, active shimming is used to re-correct magnetic field. We proposed a set of tabulate shim coils to improve the imaging field homogeneity for permanent magnet. By studying the magnetic spherical harmonics and its Taylor expansion, the proposal of designing lower-order radial coils was derived including x, y, xz, xz~2, x~2-y~2 etc. In addition, a novel composite design was presented for the axial coils. By controlling the shim currents of axial coils, different axial magnetic fields can be obtained. This method simplified the design process of axial coils, and made the axial correcting magnetic field more precise.
(5) Shim current calculation
When shim coils are excited by appropriate current, they can generate correcting magnetic field to compensate the imaging magnetic field unhomogeneity. Based on the indirect calculation method, we proposed a novel iterative optimization algorithm to calculate shim current. Because there is a direct proportion relationship between FID (Free Induction Decay) signal and magnetic field homogeneity, we use the intensity of FID signal as feedback information to the calculation. Shim current can be worked out in less than 3 minutes. This satisfied the requirement that active shimming should be performed in real time.
引文
[1] 赵喜平．磁共振成像．北京科学出版社，2004．
[2] S.Kakugawa, N.Hino, A.Komura, M.Kitamura, et al. Shielding stray magnetic fields of open high field MRI magnets. IEEE transactions on applied superconductivity, Vol.14, No.2, Jun.2004.
[3] A.Podol'skii. Desgin procedure for permanent magnet assemblies with uniform magnetic fields for MRI devices. IEEE transactions on magnetics, Vol.36, No.2, pp.484-490, Mar.2000.
[4] Donald G．Mitchell．MRI原理．四川科学技术出版社，2004．
[5] Stuart Crozier, David M.Doddrell. Compact MRI magnet design by stochastic optimization. Journal of magnetic resonance, 127, pp.233-237, 1997.
[6] Anna Salvati, Iseult Lynch, Carin Malmborg, Daniel Topgaard. Chemical shift imaging of molecular transport in colloidal system:visualization and quantification of diffusion processes. Journal of colloid and interface science, 308, pp.542-550, 2007.
[7] Darren H.Brouwer, John A.Ripmeester. Symmetry-based recoupling of proton chemical shift anisotropies in ultrahigh-field solid-state NMR. Journal of magnetic resonance, 185, pp.173-178, 2007.
[8] 冯晓源．MRI技术发展十年回顾．上海生物医学工程，Vol．27，No．2，pp．119-123，2006．
[9] D.A. Lowther, W.Mai. A comparison of MRI magnet design using a hopfield network and optimized material distribution method. IEEE transactions on magnetics, Vol.34, No.5, pp.2885-2888, Sep. 1998.
[10] Koji Miyata, Ken Ohashi. Analysis of magnetic characteristics of permanent magnet assembly for MRI devices taking account of hysteresis and eddy current. IEEE transactions on magnetics, Vol.34, No.5, pp.3556-3559, Sep. 1998.
[11] G.Morrow. Progress in MRI magnets. IEEE transations on applied superconductivity, Vol. 10, No. 1, pp.744-751, Mar.2000.
[12] A.Trequattrini, G.Coscia, S.Pittaluga. Double-cavity open permanent magnet for dedicated MRI. IEEE transactions on applied superconductivity, Vol. 10, No. 1, pp.756-758, Mar.2000.
[13] T.Miyamoto, H.Sakurai, H.Takabayashi, M.Aoki. A developmen of a permanent magnet assembly for MRI devices using Nd-Fe-B material. IEEE transactions on magnetics, Vol.25, No.5, pp.3907-3909, Sept. 1989.
[14] Na Lu, Baoci Shan, Jianyang Xu, et al. Improved temporal clustering analysis method applied to whole-brain data in acupuncture fMRI study. Magnetic resonance imagine, 2007.
[15] 黄继英，梁星原．磁共振成像原理．陕西科学技术出版社，1998．
[16] Wolfgang, H.G. Mueller. Optimization of asymmetric MRI magnets. IEEE transactions on applied superconductivity, Vol.10, No.1, pp.752-755, Mar.2000.
[17] Jin J M. Electromagnetic analysis and design in magnetic resonance imaging. New York CRC Press,1995.
[18] Robert L. Sterigerwald, William F. Wirth. High-power, high-performance switching amplifier driving magnetic resonance imaging gradient coils. Power Electronics Specialists Conference, Vol.2, pp.643-648, Jun.2000.
[19] Tang Xin, Zu Donglin, Bao Shanglian. A new length constraint method for MRI gradient coil design. Chinese journal of medical imaging technology, Vo.20, No.2, pp.290-293, 2004.
[20] Chu Xu, Jiang xiaohua, Jiang jianguo. An inverse method to design RF coils with arbitrary conductor patterns for MRI systems. Proceedings of the CSEE, Vol.25, No. 13, pp. 139-142, Jul.2005.
[21] Larry K.Forbes, Michael A.Brideson, Stuart Crozier. A target field method to design circular biplanar coils for asymmetric shim and gradient fields. IEEE transactions on magnetics, Vol.41, No.6, pp.2134-2144, Jun.2005.
[22] 张锦，张晋兴．MRI的近期进展．中国医疗器械杂志，Vol．26，No．5，pp．361-362，2002．
[23] Arndt Glowinski, Jurgen Kursch, Gerhard Adam, et al. Device visualization for interventional MRI using local magnetic fields:basic theory and its application to catheter visualization. IEEE transactions on medical imaging, Vol.17, No.5, pp.786-793, Oct. 1998.
[24] Norio Takahashi, Takeshi Kayano. Effect of minor loop on magnetic characteristics of permanent magnet type of MRI. IEEE transactions on magnetics, Vol.35, No.3, pp. 1893-1896, May. 1999.
[25] P.Rub, G.Maret. A new 18-T resistive magnet with radial bores. IEEE transactions on magnetics, Vol.30, No.4, pp.2158-2161, Jul. 1994.
[26] Y.M. Eyssa, W.Denis, Markiewicz, P.Pernambuco Wise. Plastic stress analysis of pulse and resistive magnets. IEEE transactions on magnetics, Vol.32, No.4, pp.2526-2529, Jul. 1996.
[27] S.Prestemon, P.gilmore, B.J. Gao, M.D. Bird. Design optimization of multi-coil resistive magnets. IEEE transactions on magnetics, Vol.32, No.4, pp.2550-2553, Jul. 1996.
[28] B.J. Gao, M.D. Bird, Y.M. Eyssa, H.J. Schneider-Muntau. Material and cooling requirements for poly-bitter resistive magnets and hybrid inserts generating continuous fields up to 50T. IEEE transactions on magnetics, Vol.30, No.4, pp.2196-2199, Jul. 1994.
[29] Vincenzo Cavaliere, Marco Cioffi, Alessandro Formisano, Raffaele Martone. Improvement of MRI magnet design through sensitivity analysis. IEEE transactions on applied superconductivity, Vol. 12, No. 1, pp. 1413-1416, Mar.2002.
[30] S.Noguchi, A.Ishiyama. Optimal design method for MRI superconducting magnets with ferromagnetic shied. IEEE transactions on magnetics, Vol.33, No.2, pp.1904-1907, Mar. 1997.
[31] Yuri Lvovsky, Peter Jarvis. Superconducting systems for MRI present solutions and new trends. IEEE transactions on applied superconductivity, Vol.15, No.2, pp.1317-1325, Jun.2005.
[32] Jing Fang, M.S. Chow, K.C. Chan, K.K. Wong, et al. Design of superconducting MRI surface coil by using method of moment. IEEE transactions on applied superconductivity, Vol. 12, No.2, pp. 1823-1827, Jun.2002.
[33] Anatoly Podol'skii. Development of permanent magnet assembly for MRI Devices. IEEE transations on magnetics, Vol.34, No. 1, pp.248-254, Jan. 1998.
[34] Chun Li, Michael Devine. Efficiency of permanent magnet assemblies for MRI devices. IEEE transactions on magnetics, Vol.41, No. 10, pp.3835-3837, Oct.2005.
[35] 王洪,林庆德．永磁型MRI技术探讨．中国医疗仪器杂志,Vol．5,No．2,pp．72-74,2001．
[36] A.Bonito Oliva, M.N. Biltcliffe, M.Cox, et al. Preliminary results of final test of the GHMFL 40T hybrid magnet. IEEE transactions on applied superconductivity, Vol. 15, No.2, pp. 1311-1316, Jun.2005.
[37] R.D. Schlueter, S.Marks, C.Loper, K.Halbach. Temperature compensation in hybrid magnets with application to the fermilab stacker and recycler ring dipole design. IEEE transactions on magnetics, Vol.32, No.4, pp.2203-2205, Jul. 1996.
[38] E.S. Bobrov, W.F.B. Punchard. A general method of design of axial and radial shim coils for NMR and MRI magnets. IEEE transactions on magnetics, Vol.24, No.l, pp.533-536, Jan. 1988.
[39] Shigeru Kakugawa, Noriaki Hino, Akiyoshi Komura, Masashi Kitamura. A study on optimal coil configurations in split-type superconducting MRI magnet. IEEE transactions on applied superconductivity, Vol.9, No.2, pp.366-369, Jun. 1999.
[40] H.W. Weijers, Y.S. Hascicek, S.W. Van Sciver. Critical current, quench and stability of a BSCCO/Ag high field shim coil. IEEE transactions on applied superconductivity, Vol.5, No.2, pp.499-452, Jun. 1995.
[41] S.Crozier, H.Zhao, L.K. Forbes, et al. Asymmetric MRI systmes:shim and RF coil designs. 2001 proceedings of the 23rd annual EMBS international conference, October 25-28, Istanbul, Turkey, pp.2323-2325.
[42] R.Gupta, M.Anerella, J.Cozzolino, B.Erickson, et al. Tuning shims for high field quality in superconducting magnets. IEEE transactions on magnetics, Vol.32, No.4, pp.2069-2074, Jul. 1996.
[43] D.I. Hoult, D. Lee. Shimming a superconducting nuclear magnetic resonance imaging magnet with steel. Science instrument, Vol.56, No.1, 1985.
[44] M.Cavenago, M.F. Moisio. Iron shimming of superconductive rotating dipoles. IEEE transations on magnetics, Vol.30, No.4, pp. 1923-1926, 1994.
[45] Hoult D, Deslauries R. Accurate shim coil design and magnet profiling by a power minimization matrix method. Journal of magnetic resonace. Vol. 108, No.1, pp.9-20, 1994.
[46] Larry K.Forbes, Stuart Crozier. Novel target field method for designing shielded biplanar shim and gradient coils. IEEE transactions magnetics, Vol.40, No.4, pp.1929-1938, Jul.2004.
[47] http://www.moh.gov.cn/open/statistics/digest06/y96.htm http://www.moh.gov.cn/open/statistics/digest05/s14.htm http://www.moh.gov.cn/open/statistics/digest04/s12.htm
[48] J.E.C Williams. Superconducting magnets for MRI. IEEE transactions on nuclear science, Vol. NS-31, No.2, pp.1303-1308, 1986.
[49] B.Dorri, M.E. Vermilyea. Passive shimming of MR magnets: algorithm, hardware, and results. IEEE transactions on applied superconductivity, Vol.3, No. 1, pp. 254-257, 1993.
[50] Stuart Crozier, Larry K. Forbes, Michael Brideson. Ellipsoidal harmonic (Lame) MRI shims. IEEE transactions on applied superconductivity, Vol. 12, No.4, pp. 1880-1885, Dec.2002.
[51] Rolf Gruetter, Chris Boesch. Fast, Noniterative Shimming of Spatially Localized Signals in Vivo Analysis of the Magnetic Field along Axes. Journal of Magnetic Resonance, Vol.96, pp.323-334, 1992.
[52] 叶慧．中国研制出世界上最高场强的医用MRI永磁磁体．中国医疗器械信息，Vol．12，NO．4，Dec．2006．
[53] Nathan Ida, Joao P.A. Bastos. Electromagnetics and Calculation of Fields. New York Springer, 1999.
[54] 张玉民，戚伯云．电磁学．中国科学技术大学出版社，1997．
[55] 冯定五．永磁铠装磁路设计．磁性材料及器件，Vol．28，No．2，pp．37-40，Apr．1997．
[56] Susanne C. Brenner, L. Ridgeway scott. The mathematical theory of finite element methods, New York, Spring-Verlag, 2002.
[57] Jianming Jin. The finite element method in electromagnetic. New York: Wiley, 2002.
[58] 孙敏，孙亲锡，叶齐政．工程电磁场基础．北京：科学出版社，2001．
[59] 盛剑霓．电磁场数值分析．北京科学出版社，1984．
[60] Graeme fairweather. Finite element Galerkin methods for differential equations. New York, M. Dekker, 1978.
[61] 金建铭．电磁场有限元方法．西安电子科技大学出版社，1998．
[62] Herbert, A.Leupold. Approaches to permanent magnet circuit design. IEEE transactions on magnetic, Vol.29, No.6, pp.2341-2346, Nov. 1993.
[63] 王以真．实用磁路设计．天津科学技术出版社，1992．
[64] 冯定五，孙仲元，陈荩，彭世英．一种永磁磁路设计的新方法．金属矿山，Vol．12，pp．58-60，Dec．1994．
[65] 赵韩，王勇，李露．材料线性特性对铁心磁路磁场计算的影响分析．磁性材料及器件，Vol．36，No．6，pp．24-26，Dec．2005．
[66] Yi Mingjun, Gu Chenglin. Analysis of magnetic circuit on ALA rotor motor. Proceedings of the CSEE, Vol.20, No.9, pp.5-8, Sep.2000.
[67] 郑永成，王洋，何建国，黄文．基于磁场分割的磁导计算与磁路设计．机械与电子，No．7，pp．10-12，2006．
[68] Xiang Honggang, Chen Degui, Li Xingwen, et al. Construction of equivalent magnetic circuit for electromagnet based on 3D magnetic field. Joumal of xi'an jiao tong university, Vo.37, No.8, pp.807-810, Aug.2003.
[69] 林其壬，赵佑民．磁路设计原理．机械工业出版社，1987．
[70] 高联辉．磁路和铁磁器件．高等教育出版社，1982．
[71] Qing Zhao, Zhongliang An, Zhemin Liu, Renyuan Tang. Analysis of flux leakage coefficient of permanent magnet synchronous motors with U-shaped magnets rotor. IEEE transactions on magnetics, Vol.35, Issue 1, Part 2, pp.543-547, Jan.1999.
[72] 吴刚，张育林，刘昆等．永磁电磁轴承的磁路设计方法．轴承，No．3，pp．4-7，2004．
[73] 杨红保，刘黎明．真空密封传动装置的磁路设计．真空，No．4，pp．58-60，Jul．2003．
[74] 李金，张华良．磁流变液阻尼器的磁路设计方法．机械加工与自动化，No．8，2003．
[75] 易敬曾．磁场计算与磁路设计．成都电讯工程学院出版社，1987．
[76] Yingying Yao, Youtong Fang, Chang Seop Koh, Guangzheng Ni. A new design method for completely open architecture permanent magnet for MRI. IEEE transactions on magnetics, Vol.41, No.5, pp. 1504-1507, May.2005.
[77] Donghun Kim, Byungsung Kim, Joonho Lee, et al. 3D optimal shape design of ferromagnetic pole in MRI magnet of open permanent magnet type. IEEE transactions on applied superconductivity, Vol. 12, No. 1, pp. 1467-1470, Mar.2002.
[78] C. Salustri, Y. Yang, G.H. Glover. Simple but reliable solutions for spiral MRI gradient design. Journal of magnetic resonance, 140, pp.347-350, 1999.
[79] Takao, Takahashi. Numerical analysis of eddy current problems involving saddle-shaped coils in superconducting MRI magnets. IEEE transactions on magnetics, Vol.27, No.5, pp. 3996-3999, Sep. 1991.
[80] Yongchuan Lai, Xiaohua Jiang, Guanghui Shen. A method to reduce eddy currents within iron pole plates of 0.3T NdFeB MRI magnet. IEEE transactions on applied supercoductivity, Vol.12, No.1, pp.373-739, Mar.2002.
[81] K. Sivasubramaniam, M. Xu, X. Huang, W. Barber, et al. Transient perturbation to permanent magnetic field by gradient pulses in MRI mangets. IEEE transactions on applied superconductivity, Vol.16, No.2, pp. 1550-1553, Jun.2006.
[82] H. Sanchez, F. Liu, A. Trakic, S. Crozier. A magnetization mapping approach for passive shim design in MRI. Proceedings of the 28th IEEE EMBS annual international conference, New York City, USA, Aug30-Sept3, 2006.
[83] Y.Zhai, L. Vu-Quoc. Analysis of power magnetic components with nonlinear static hysteresis: finite element formulation. IEEE transactions on magnetics, Vol.41, No.7, Jul.2005.
[84] R. H. Vander Heiden, J. R. Brauer, J. J. ruehl, G. A. Zimmerlee. Utilizing permanent magnets in nonlinear three-dimensional finite element models. IEEE transactions on magnetic, Vol.24, No.6, pp.2931-2933, Nov. 1988.
[85] Themistoklis Kefalas, George Kalokiris, Antonios Kladas, John Tegopoulos. Design of skewed mounted permanent magnet synchronous generators based on 2D and 3D finite element techniques. Journal of materials processing technology, 161, pp.288-293, 2005.
[86] Zhou Yong, Tang Zesheng. Partitioning 3D finite element meshes for volume rendering. Chinese Journal of computers, Vol. 18, No.5, pp.339-351, May.1995.
[87] Sanjia Xu, Xinqing Sheng. Automatic division technique for 3-D finite element. ACTA electronica sinica, Vol.22, No.6, Jun. 1994.
[88] 周寿增，董清飞．超强永磁体．北京：冶金工业出版社，2004．
[89] Juan Parra-Robels, Albert R.Cross, Giles E.Santyr. Passive shimming of the fringe field of a superconducting magnet for ultra-low field hyperpolarized noble gas MRI. Journal of magnetic resonace, 174, pp. 116-124, 2005.
[90] Z.X.Feng, X.H.Jiang, S.Han. IEEE transactions on magnetics on magnetics, Vol.28, No.1, pp.641-643, Jan. 1992.
[91] Joseph H. Battocletti, Husain A.Kamal, Thomas J.Myers, Thomas A. Knox. Systematic passive shimming of a permanent magnet for p-31 NMR spectroscopy of bone mineral. IEEE transactions on magnetics, Vol.29, No.3, pp.2139-2151, 1993.
[92] 夏平畴．永磁机构．北京：工业大学出版社，2000．
[93] Horst Alzer, Song-Liang Qiu. Monotonicity theorms and inequalities for the complete ellipc integrals. Journal of computational and applied mathematics, 172, pp.289-312, 2004.
[94] Haseeb Kazi, Edward Neuman. Inqualities and bounds for elliptic integrals. Journal of Approximation theory, pp. 1-15, 2006.
[95] Thome R J, Tarrh J M. MHD and fusion magnets field and force design concepts. New York John Wiley and Sons Inc., 1982.
[96] Che Wenhua, Xia Pingchou, Zhang Yiming. Generation of a slice with homogeneous magnetic field outside of the used magnet. Proceedings of the CSEE, Vol.23, No.4, pp.71-76, 2003.
[97] Yao Yingying, Xie Dexin. A hybrid optimal analysis for the design of passive shimming elements in high homogeneous permanent magnets. Computation in Electromagnetics, pp.8-11, 2002.
[98] A.Belov, V.Bushuev, M.Emelianov et al. Passive shimming of the superconducting magnet for MRI. IEEE transactions on applied superconductivity, Vol.5, No.2, pp. 679-681, 1995.
[99] Manfred Kaltenbacher, Stefan Reitzinger. Nonlinear three dimensional magnetic field computations using lagrange finite element functions and algebraic multigrid. IEEE transactions on magnetics, Vol.38, No.3, pp.1489-1495, May.2002.
[100] Liu Guoyue, Xu Xuejing. On applying mirror image method to resolve steady magnetic field. Journal of southwest institute of technology, Vol. 13, No.l, pp.69-53, Mar. 1998.
[101] M.Fujita. The coil design of the superconducting MRI magnet. IEEE transactions on magnetics, Vol.24, No.6, pp.2907-2909, Nov.1988.
[102] 储岳森．有源匀场的优化算法．高电压技术，Vol．22，No．2，pp．15-18，Jun．1996．
[103] William H. Hayt, Jr., John A. Buck. Engineering electromagnetics. Boston McGraw-Hill, 2001.
[104] Samuel Seely, Alexander D. Poularikas. Electromagnetic: classical and modem theory and applications. New York: M. Dekker, 1979.
[105] Nannapaneni Narayana Rao. Elements of engineering electromagnetics. Upper Saddle River, N.J.: Pretice Hall, 2000.
[106] Qian Jingen, Ni Guangzheng, Cheng Weiying. The single scalar magnetic potential method applied in numerical analysis on magnetostatic field. Proceedings of the CSEE, Vol.15, No.4, 228-233, 1995.
[107] Andrew E.Marble, Igor V.Mastikhin, Brace G.Colpitts. A constant gradient unidateral magnet for near surface MRI profiling. Journal of magnetic resonance, 183, pp.228-234, 2006.
[108] Rastislav Vadovic. Magnetic field correction using magnetized shims. IEEE transactions on magnetics, Vol.25, No.4, pp.3133-3139, 1989.
[109] Bai Ye, Wang Qiuliang, Yu Yunjia, Xia Pingchou. An approach to spherical coils design based on the target field approach. Proceedings of the CSEE, Vol.24, No.6, pp.132-136, 2004.
[110] Vjeran Vrankovic, David C.George. Smooth shimming of pole profiles. IEEE transactions on applied superconductivity, Vol. 10, No. 1, pp. 1372-1375, Mar.2000.
[111] Jan Chladek, Pavel Konzbul, Pavel Osmera et al. Evolutionary and genetic optimization of NMR gradient and shim coils. IEEE Transactions on Magnetics, Vol.36, No.4, pp.1102-1105, 2000.
[112] P.Konzbul, K.Sveda, A.Srnka. Superconducting shim system with high purity magneticfield for NMR and MRI solenoid magnet MIDI-200. IEEE Transactions on Magnetics, Vol.32, No.4, pp.2643-2646, 1996.
[113] V.S. Vladimirov. Equations of mathematical physics. Moscow: Mir, 1984.
[114] Chu Yuesen, Huang Fang. The design and field calculation of MRI magnet shimming coils. Journal of shanghai jiaotong university, Vol.30, No. 12, pp. 103-109, 1996.
[115] Pavel Konzbul, Karel Sveda, Ales Srnka. Design of matrix shim coils system for nuclear magnetic resonance. IEEE Transactions on Magnetics, Vol.36, No.4, pp. 1732-1735, 2000.
[116] Zhao Haixiang. Influence of magnetic field of dry-type air core reactor on spatial closed loop. Power System Technology, Vol.24, No.2, pp. 17-19, 2000.
[117] Jun Shen, Douglas L. Rothman. Fast automatic adjustment of on-axis shims for high resolution NMR. Joumal of Magnetic Resonance, 127, pp.229-232, 1997.
[118] Jeff Wilkins, Steve Miller. The use of adaptive algorithms for obtaining optimal electrical shimming in magnetic resonance imaging. IEEE Transactions on Biomedical Engineering, Vol.36, No.2, pp.202-210, 1989.
[119] P.Konzbul, K.Sveda. Shim coils for NMR and MILI solenoid magnets. Measurement science and Technology, Vol.6, pp.1116-1123, Aug.1995.
[120] Rolf Gruetter. Automatic, Localized in Vivo Adjustment of All First and Second-Order Shim Coils. Magnetic Resonance in Medicine, Vol.29, pp.804-811, 1993.
[121] Haacke M, Brown RW, Thompson MR, Venkatesan R, Magnetic Resonance Imaging: Physical Principles and Sequence Design. John Wiley and Sons Publication, 1999.
[122] 陈历明，彭承琳，朱学武．0.2T开放式永磁型医用核磁共振成像仪有源匀场系统的研制．生物医学工程杂志，Vol．21，No．2，pp．288-291，2004．
[123] Che Wenhua, Xia Pingchou, Zhang Yiming. Considerations and magnet design on the unilateral magnetic resonance imaging system with homogeneous magnetic field. The second Beijing international conference on physics and engineering of medical imaging, pp.24-28, Oct.2001.
[124] Jae Seop Ryu, Yingying Yao, Chang Seop Koh, Young Jun Shin. 3D optimal shape design of pole piece in permanent magnet MRI using parameterized nonlinear design sensitivity analysis. IEEE transactions on magnetics, Vol.42, No.4, pp. 1351-1354, Apr.2006.
[2] S.Kakugawa, N.Hino, A.Komura, M.Kitamura, et al. Shielding stray magnetic fields of open high field MRI magnets. IEEE transactions on applied superconductivity, Vol.14, No.2, Jun.2004.
[3] A.Podol'skii. Desgin procedure for permanent magnet assemblies with uniform magnetic fields for MRI devices. IEEE transactions on magnetics, Vol.36, No.2, pp.484-490, Mar.2000.
[4] Donald G．Mitchell．MRI原理．四川科学技术出版社，2004．
[5] Stuart Crozier, David M.Doddrell. Compact MRI magnet design by stochastic optimization. Journal of magnetic resonance, 127, pp.233-237, 1997.
[6] Anna Salvati, Iseult Lynch, Carin Malmborg, Daniel Topgaard. Chemical shift imaging of molecular transport in colloidal system:visualization and quantification of diffusion processes. Journal of colloid and interface science, 308, pp.542-550, 2007.
[7] Darren H.Brouwer, John A.Ripmeester. Symmetry-based recoupling of proton chemical shift anisotropies in ultrahigh-field solid-state NMR. Journal of magnetic resonance, 185, pp.173-178, 2007.
[8] 冯晓源．MRI技术发展十年回顾．上海生物医学工程，Vol．27，No．2，pp．119-123，2006．
[9] D.A. Lowther, W.Mai. A comparison of MRI magnet design using a hopfield network and optimized material distribution method. IEEE transactions on magnetics, Vol.34, No.5, pp.2885-2888, Sep. 1998.
[10] Koji Miyata, Ken Ohashi. Analysis of magnetic characteristics of permanent magnet assembly for MRI devices taking account of hysteresis and eddy current. IEEE transactions on magnetics, Vol.34, No.5, pp.3556-3559, Sep. 1998.
[11] G.Morrow. Progress in MRI magnets. IEEE transations on applied superconductivity, Vol. 10, No. 1, pp.744-751, Mar.2000.
[12] A.Trequattrini, G.Coscia, S.Pittaluga. Double-cavity open permanent magnet for dedicated MRI. IEEE transactions on applied superconductivity, Vol. 10, No. 1, pp.756-758, Mar.2000.
[13] T.Miyamoto, H.Sakurai, H.Takabayashi, M.Aoki. A developmen of a permanent magnet assembly for MRI devices using Nd-Fe-B material. IEEE transactions on magnetics, Vol.25, No.5, pp.3907-3909, Sept. 1989.
[14] Na Lu, Baoci Shan, Jianyang Xu, et al. Improved temporal clustering analysis method applied to whole-brain data in acupuncture fMRI study. Magnetic resonance imagine, 2007.
[15] 黄继英，梁星原．磁共振成像原理．陕西科学技术出版社，1998．
[16] Wolfgang, H.G. Mueller. Optimization of asymmetric MRI magnets. IEEE transactions on applied superconductivity, Vol.10, No.1, pp.752-755, Mar.2000.
[17] Jin J M. Electromagnetic analysis and design in magnetic resonance imaging. New York CRC Press,1995.
[18] Robert L. Sterigerwald, William F. Wirth. High-power, high-performance switching amplifier driving magnetic resonance imaging gradient coils. Power Electronics Specialists Conference, Vol.2, pp.643-648, Jun.2000.
[19] Tang Xin, Zu Donglin, Bao Shanglian. A new length constraint method for MRI gradient coil design. Chinese journal of medical imaging technology, Vo.20, No.2, pp.290-293, 2004.
[20] Chu Xu, Jiang xiaohua, Jiang jianguo. An inverse method to design RF coils with arbitrary conductor patterns for MRI systems. Proceedings of the CSEE, Vol.25, No. 13, pp. 139-142, Jul.2005.
[21] Larry K.Forbes, Michael A.Brideson, Stuart Crozier. A target field method to design circular biplanar coils for asymmetric shim and gradient fields. IEEE transactions on magnetics, Vol.41, No.6, pp.2134-2144, Jun.2005.
[22] 张锦，张晋兴．MRI的近期进展．中国医疗器械杂志，Vol．26，No．5，pp．361-362，2002．
[23] Arndt Glowinski, Jurgen Kursch, Gerhard Adam, et al. Device visualization for interventional MRI using local magnetic fields:basic theory and its application to catheter visualization. IEEE transactions on medical imaging, Vol.17, No.5, pp.786-793, Oct. 1998.
[24] Norio Takahashi, Takeshi Kayano. Effect of minor loop on magnetic characteristics of permanent magnet type of MRI. IEEE transactions on magnetics, Vol.35, No.3, pp. 1893-1896, May. 1999.
[25] P.Rub, G.Maret. A new 18-T resistive magnet with radial bores. IEEE transactions on magnetics, Vol.30, No.4, pp.2158-2161, Jul. 1994.
[26] Y.M. Eyssa, W.Denis, Markiewicz, P.Pernambuco Wise. Plastic stress analysis of pulse and resistive magnets. IEEE transactions on magnetics, Vol.32, No.4, pp.2526-2529, Jul. 1996.
[27] S.Prestemon, P.gilmore, B.J. Gao, M.D. Bird. Design optimization of multi-coil resistive magnets. IEEE transactions on magnetics, Vol.32, No.4, pp.2550-2553, Jul. 1996.
[28] B.J. Gao, M.D. Bird, Y.M. Eyssa, H.J. Schneider-Muntau. Material and cooling requirements for poly-bitter resistive magnets and hybrid inserts generating continuous fields up to 50T. IEEE transactions on magnetics, Vol.30, No.4, pp.2196-2199, Jul. 1994.
[29] Vincenzo Cavaliere, Marco Cioffi, Alessandro Formisano, Raffaele Martone. Improvement of MRI magnet design through sensitivity analysis. IEEE transactions on applied superconductivity, Vol. 12, No. 1, pp. 1413-1416, Mar.2002.
[30] S.Noguchi, A.Ishiyama. Optimal design method for MRI superconducting magnets with ferromagnetic shied. IEEE transactions on magnetics, Vol.33, No.2, pp.1904-1907, Mar. 1997.
[31] Yuri Lvovsky, Peter Jarvis. Superconducting systems for MRI present solutions and new trends. IEEE transactions on applied superconductivity, Vol.15, No.2, pp.1317-1325, Jun.2005.
[32] Jing Fang, M.S. Chow, K.C. Chan, K.K. Wong, et al. Design of superconducting MRI surface coil by using method of moment. IEEE transactions on applied superconductivity, Vol. 12, No.2, pp. 1823-1827, Jun.2002.
[33] Anatoly Podol'skii. Development of permanent magnet assembly for MRI Devices. IEEE transations on magnetics, Vol.34, No. 1, pp.248-254, Jan. 1998.
[34] Chun Li, Michael Devine. Efficiency of permanent magnet assemblies for MRI devices. IEEE transactions on magnetics, Vol.41, No. 10, pp.3835-3837, Oct.2005.
[35] 王洪,林庆德．永磁型MRI技术探讨．中国医疗仪器杂志,Vol．5,No．2,pp．72-74,2001．
[36] A.Bonito Oliva, M.N. Biltcliffe, M.Cox, et al. Preliminary results of final test of the GHMFL 40T hybrid magnet. IEEE transactions on applied superconductivity, Vol. 15, No.2, pp. 1311-1316, Jun.2005.
[37] R.D. Schlueter, S.Marks, C.Loper, K.Halbach. Temperature compensation in hybrid magnets with application to the fermilab stacker and recycler ring dipole design. IEEE transactions on magnetics, Vol.32, No.4, pp.2203-2205, Jul. 1996.
[38] E.S. Bobrov, W.F.B. Punchard. A general method of design of axial and radial shim coils for NMR and MRI magnets. IEEE transactions on magnetics, Vol.24, No.l, pp.533-536, Jan. 1988.
[39] Shigeru Kakugawa, Noriaki Hino, Akiyoshi Komura, Masashi Kitamura. A study on optimal coil configurations in split-type superconducting MRI magnet. IEEE transactions on applied superconductivity, Vol.9, No.2, pp.366-369, Jun. 1999.
[40] H.W. Weijers, Y.S. Hascicek, S.W. Van Sciver. Critical current, quench and stability of a BSCCO/Ag high field shim coil. IEEE transactions on applied superconductivity, Vol.5, No.2, pp.499-452, Jun. 1995.
[41] S.Crozier, H.Zhao, L.K. Forbes, et al. Asymmetric MRI systmes:shim and RF coil designs. 2001 proceedings of the 23rd annual EMBS international conference, October 25-28, Istanbul, Turkey, pp.2323-2325.
[42] R.Gupta, M.Anerella, J.Cozzolino, B.Erickson, et al. Tuning shims for high field quality in superconducting magnets. IEEE transactions on magnetics, Vol.32, No.4, pp.2069-2074, Jul. 1996.
[43] D.I. Hoult, D. Lee. Shimming a superconducting nuclear magnetic resonance imaging magnet with steel. Science instrument, Vol.56, No.1, 1985.
[44] M.Cavenago, M.F. Moisio. Iron shimming of superconductive rotating dipoles. IEEE transations on magnetics, Vol.30, No.4, pp. 1923-1926, 1994.
[45] Hoult D, Deslauries R. Accurate shim coil design and magnet profiling by a power minimization matrix method. Journal of magnetic resonace. Vol. 108, No.1, pp.9-20, 1994.
[46] Larry K.Forbes, Stuart Crozier. Novel target field method for designing shielded biplanar shim and gradient coils. IEEE transactions magnetics, Vol.40, No.4, pp.1929-1938, Jul.2004.
[47] http://www.moh.gov.cn/open/statistics/digest06/y96.htm http://www.moh.gov.cn/open/statistics/digest05/s14.htm http://www.moh.gov.cn/open/statistics/digest04/s12.htm
[48] J.E.C Williams. Superconducting magnets for MRI. IEEE transactions on nuclear science, Vol. NS-31, No.2, pp.1303-1308, 1986.
[49] B.Dorri, M.E. Vermilyea. Passive shimming of MR magnets: algorithm, hardware, and results. IEEE transactions on applied superconductivity, Vol.3, No. 1, pp. 254-257, 1993.
[50] Stuart Crozier, Larry K. Forbes, Michael Brideson. Ellipsoidal harmonic (Lame) MRI shims. IEEE transactions on applied superconductivity, Vol. 12, No.4, pp. 1880-1885, Dec.2002.
[51] Rolf Gruetter, Chris Boesch. Fast, Noniterative Shimming of Spatially Localized Signals in Vivo Analysis of the Magnetic Field along Axes. Journal of Magnetic Resonance, Vol.96, pp.323-334, 1992.
[52] 叶慧．中国研制出世界上最高场强的医用MRI永磁磁体．中国医疗器械信息，Vol．12，NO．4，Dec．2006．
[53] Nathan Ida, Joao P.A. Bastos. Electromagnetics and Calculation of Fields. New York Springer, 1999.
[54] 张玉民，戚伯云．电磁学．中国科学技术大学出版社，1997．
[55] 冯定五．永磁铠装磁路设计．磁性材料及器件，Vol．28，No．2，pp．37-40，Apr．1997．
[56] Susanne C. Brenner, L. Ridgeway scott. The mathematical theory of finite element methods, New York, Spring-Verlag, 2002.
[57] Jianming Jin. The finite element method in electromagnetic. New York: Wiley, 2002.
[58] 孙敏，孙亲锡，叶齐政．工程电磁场基础．北京：科学出版社，2001．
[59] 盛剑霓．电磁场数值分析．北京科学出版社，1984．
[60] Graeme fairweather. Finite element Galerkin methods for differential equations. New York, M. Dekker, 1978.
[61] 金建铭．电磁场有限元方法．西安电子科技大学出版社，1998．
[62] Herbert, A.Leupold. Approaches to permanent magnet circuit design. IEEE transactions on magnetic, Vol.29, No.6, pp.2341-2346, Nov. 1993.
[63] 王以真．实用磁路设计．天津科学技术出版社，1992．
[64] 冯定五，孙仲元，陈荩，彭世英．一种永磁磁路设计的新方法．金属矿山，Vol．12，pp．58-60，Dec．1994．
[65] 赵韩，王勇，李露．材料线性特性对铁心磁路磁场计算的影响分析．磁性材料及器件，Vol．36，No．6，pp．24-26，Dec．2005．
[66] Yi Mingjun, Gu Chenglin. Analysis of magnetic circuit on ALA rotor motor. Proceedings of the CSEE, Vol.20, No.9, pp.5-8, Sep.2000.
[67] 郑永成，王洋，何建国，黄文．基于磁场分割的磁导计算与磁路设计．机械与电子，No．7，pp．10-12，2006．
[68] Xiang Honggang, Chen Degui, Li Xingwen, et al. Construction of equivalent magnetic circuit for electromagnet based on 3D magnetic field. Joumal of xi'an jiao tong university, Vo.37, No.8, pp.807-810, Aug.2003.
[69] 林其壬，赵佑民．磁路设计原理．机械工业出版社，1987．
[70] 高联辉．磁路和铁磁器件．高等教育出版社，1982．
[71] Qing Zhao, Zhongliang An, Zhemin Liu, Renyuan Tang. Analysis of flux leakage coefficient of permanent magnet synchronous motors with U-shaped magnets rotor. IEEE transactions on magnetics, Vol.35, Issue 1, Part 2, pp.543-547, Jan.1999.
[72] 吴刚，张育林，刘昆等．永磁电磁轴承的磁路设计方法．轴承，No．3，pp．4-7，2004．
[73] 杨红保，刘黎明．真空密封传动装置的磁路设计．真空，No．4，pp．58-60，Jul．2003．
[74] 李金，张华良．磁流变液阻尼器的磁路设计方法．机械加工与自动化，No．8，2003．
[75] 易敬曾．磁场计算与磁路设计．成都电讯工程学院出版社，1987．
[76] Yingying Yao, Youtong Fang, Chang Seop Koh, Guangzheng Ni. A new design method for completely open architecture permanent magnet for MRI. IEEE transactions on magnetics, Vol.41, No.5, pp. 1504-1507, May.2005.
[77] Donghun Kim, Byungsung Kim, Joonho Lee, et al. 3D optimal shape design of ferromagnetic pole in MRI magnet of open permanent magnet type. IEEE transactions on applied superconductivity, Vol. 12, No. 1, pp. 1467-1470, Mar.2002.
[78] C. Salustri, Y. Yang, G.H. Glover. Simple but reliable solutions for spiral MRI gradient design. Journal of magnetic resonance, 140, pp.347-350, 1999.
[79] Takao, Takahashi. Numerical analysis of eddy current problems involving saddle-shaped coils in superconducting MRI magnets. IEEE transactions on magnetics, Vol.27, No.5, pp. 3996-3999, Sep. 1991.
[80] Yongchuan Lai, Xiaohua Jiang, Guanghui Shen. A method to reduce eddy currents within iron pole plates of 0.3T NdFeB MRI magnet. IEEE transactions on applied supercoductivity, Vol.12, No.1, pp.373-739, Mar.2002.
[81] K. Sivasubramaniam, M. Xu, X. Huang, W. Barber, et al. Transient perturbation to permanent magnetic field by gradient pulses in MRI mangets. IEEE transactions on applied superconductivity, Vol.16, No.2, pp. 1550-1553, Jun.2006.
[82] H. Sanchez, F. Liu, A. Trakic, S. Crozier. A magnetization mapping approach for passive shim design in MRI. Proceedings of the 28th IEEE EMBS annual international conference, New York City, USA, Aug30-Sept3, 2006.
[83] Y.Zhai, L. Vu-Quoc. Analysis of power magnetic components with nonlinear static hysteresis: finite element formulation. IEEE transactions on magnetics, Vol.41, No.7, Jul.2005.
[84] R. H. Vander Heiden, J. R. Brauer, J. J. ruehl, G. A. Zimmerlee. Utilizing permanent magnets in nonlinear three-dimensional finite element models. IEEE transactions on magnetic, Vol.24, No.6, pp.2931-2933, Nov. 1988.
[85] Themistoklis Kefalas, George Kalokiris, Antonios Kladas, John Tegopoulos. Design of skewed mounted permanent magnet synchronous generators based on 2D and 3D finite element techniques. Journal of materials processing technology, 161, pp.288-293, 2005.
[86] Zhou Yong, Tang Zesheng. Partitioning 3D finite element meshes for volume rendering. Chinese Journal of computers, Vol. 18, No.5, pp.339-351, May.1995.
[87] Sanjia Xu, Xinqing Sheng. Automatic division technique for 3-D finite element. ACTA electronica sinica, Vol.22, No.6, Jun. 1994.
[88] 周寿增，董清飞．超强永磁体．北京：冶金工业出版社，2004．
[89] Juan Parra-Robels, Albert R.Cross, Giles E.Santyr. Passive shimming of the fringe field of a superconducting magnet for ultra-low field hyperpolarized noble gas MRI. Journal of magnetic resonace, 174, pp. 116-124, 2005.
[90] Z.X.Feng, X.H.Jiang, S.Han. IEEE transactions on magnetics on magnetics, Vol.28, No.1, pp.641-643, Jan. 1992.
[91] Joseph H. Battocletti, Husain A.Kamal, Thomas J.Myers, Thomas A. Knox. Systematic passive shimming of a permanent magnet for p-31 NMR spectroscopy of bone mineral. IEEE transactions on magnetics, Vol.29, No.3, pp.2139-2151, 1993.
[92] 夏平畴．永磁机构．北京：工业大学出版社，2000．
[93] Horst Alzer, Song-Liang Qiu. Monotonicity theorms and inequalities for the complete ellipc integrals. Journal of computational and applied mathematics, 172, pp.289-312, 2004.
[94] Haseeb Kazi, Edward Neuman. Inqualities and bounds for elliptic integrals. Journal of Approximation theory, pp. 1-15, 2006.
[95] Thome R J, Tarrh J M. MHD and fusion magnets field and force design concepts. New York John Wiley and Sons Inc., 1982.
[96] Che Wenhua, Xia Pingchou, Zhang Yiming. Generation of a slice with homogeneous magnetic field outside of the used magnet. Proceedings of the CSEE, Vol.23, No.4, pp.71-76, 2003.
[97] Yao Yingying, Xie Dexin. A hybrid optimal analysis for the design of passive shimming elements in high homogeneous permanent magnets. Computation in Electromagnetics, pp.8-11, 2002.
[98] A.Belov, V.Bushuev, M.Emelianov et al. Passive shimming of the superconducting magnet for MRI. IEEE transactions on applied superconductivity, Vol.5, No.2, pp. 679-681, 1995.
[99] Manfred Kaltenbacher, Stefan Reitzinger. Nonlinear three dimensional magnetic field computations using lagrange finite element functions and algebraic multigrid. IEEE transactions on magnetics, Vol.38, No.3, pp.1489-1495, May.2002.
[100] Liu Guoyue, Xu Xuejing. On applying mirror image method to resolve steady magnetic field. Journal of southwest institute of technology, Vol. 13, No.l, pp.69-53, Mar. 1998.
[101] M.Fujita. The coil design of the superconducting MRI magnet. IEEE transactions on magnetics, Vol.24, No.6, pp.2907-2909, Nov.1988.
[102] 储岳森．有源匀场的优化算法．高电压技术，Vol．22，No．2，pp．15-18，Jun．1996．
[103] William H. Hayt, Jr., John A. Buck. Engineering electromagnetics. Boston McGraw-Hill, 2001.
[104] Samuel Seely, Alexander D. Poularikas. Electromagnetic: classical and modem theory and applications. New York: M. Dekker, 1979.
[105] Nannapaneni Narayana Rao. Elements of engineering electromagnetics. Upper Saddle River, N.J.: Pretice Hall, 2000.
[106] Qian Jingen, Ni Guangzheng, Cheng Weiying. The single scalar magnetic potential method applied in numerical analysis on magnetostatic field. Proceedings of the CSEE, Vol.15, No.4, 228-233, 1995.
[107] Andrew E.Marble, Igor V.Mastikhin, Brace G.Colpitts. A constant gradient unidateral magnet for near surface MRI profiling. Journal of magnetic resonance, 183, pp.228-234, 2006.
[108] Rastislav Vadovic. Magnetic field correction using magnetized shims. IEEE transactions on magnetics, Vol.25, No.4, pp.3133-3139, 1989.
[109] Bai Ye, Wang Qiuliang, Yu Yunjia, Xia Pingchou. An approach to spherical coils design based on the target field approach. Proceedings of the CSEE, Vol.24, No.6, pp.132-136, 2004.
[110] Vjeran Vrankovic, David C.George. Smooth shimming of pole profiles. IEEE transactions on applied superconductivity, Vol. 10, No. 1, pp. 1372-1375, Mar.2000.
[111] Jan Chladek, Pavel Konzbul, Pavel Osmera et al. Evolutionary and genetic optimization of NMR gradient and shim coils. IEEE Transactions on Magnetics, Vol.36, No.4, pp.1102-1105, 2000.
[112] P.Konzbul, K.Sveda, A.Srnka. Superconducting shim system with high purity magneticfield for NMR and MRI solenoid magnet MIDI-200. IEEE Transactions on Magnetics, Vol.32, No.4, pp.2643-2646, 1996.
[113] V.S. Vladimirov. Equations of mathematical physics. Moscow: Mir, 1984.
[114] Chu Yuesen, Huang Fang. The design and field calculation of MRI magnet shimming coils. Journal of shanghai jiaotong university, Vol.30, No. 12, pp. 103-109, 1996.
[115] Pavel Konzbul, Karel Sveda, Ales Srnka. Design of matrix shim coils system for nuclear magnetic resonance. IEEE Transactions on Magnetics, Vol.36, No.4, pp. 1732-1735, 2000.
[116] Zhao Haixiang. Influence of magnetic field of dry-type air core reactor on spatial closed loop. Power System Technology, Vol.24, No.2, pp. 17-19, 2000.
[117] Jun Shen, Douglas L. Rothman. Fast automatic adjustment of on-axis shims for high resolution NMR. Joumal of Magnetic Resonance, 127, pp.229-232, 1997.
[118] Jeff Wilkins, Steve Miller. The use of adaptive algorithms for obtaining optimal electrical shimming in magnetic resonance imaging. IEEE Transactions on Biomedical Engineering, Vol.36, No.2, pp.202-210, 1989.
[119] P.Konzbul, K.Sveda. Shim coils for NMR and MILI solenoid magnets. Measurement science and Technology, Vol.6, pp.1116-1123, Aug.1995.
[120] Rolf Gruetter. Automatic, Localized in Vivo Adjustment of All First and Second-Order Shim Coils. Magnetic Resonance in Medicine, Vol.29, pp.804-811, 1993.
[121] Haacke M, Brown RW, Thompson MR, Venkatesan R, Magnetic Resonance Imaging: Physical Principles and Sequence Design. John Wiley and Sons Publication, 1999.
[122] 陈历明，彭承琳，朱学武．0.2T开放式永磁型医用核磁共振成像仪有源匀场系统的研制．生物医学工程杂志，Vol．21，No．2，pp．288-291，2004．
[123] Che Wenhua, Xia Pingchou, Zhang Yiming. Considerations and magnet design on the unilateral magnetic resonance imaging system with homogeneous magnetic field. The second Beijing international conference on physics and engineering of medical imaging, pp.24-28, Oct.2001.
[124] Jae Seop Ryu, Yingying Yao, Chang Seop Koh, Young Jun Shin. 3D optimal shape design of pole piece in permanent magnet MRI using parameterized nonlinear design sensitivity analysis. IEEE transactions on magnetics, Vol.42, No.4, pp. 1351-1354, Apr.2006.