基于中心磁场参照的MRI永磁主磁体匀场方法研究
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摘要
本文的研究受到教育部高等学校博士学科点专项科研基金项目“开放式磁共振成像系统永磁型磁体匀场技术研究(基金号:20050142003)”的资助。
开放式磁共振成像(Magnetic Resonance Imaging,MRI)系统是目前医用磁共振成像设备的主要发展方向之一。MRI系统的成像质量与其主磁体的匀场技术有着密切的联系。对于开放式永磁MRI装置来说,无源匀场更是当前研究的热点、难点。针对这一现状,本文就开放式永磁型MRI系统主磁体的无源匀场问题从以下几个方面进行了深入的研究。
无源匀场主要靠贴补小磁片产生的磁场来抵消主磁体在空间成像区域产生磁场的不均匀谐波分量。因此需要确定单个小磁片在空间区域产生的附加磁场。本文采用了数值分析、解析求解以及实地测量三种方法,通过比较、验证,最终确定了单个小磁片在空间产生的磁场值,为无源匀场工作的研究奠定了基础。
本文在研究整数线性规划优化理论的基础上,深入研究了实际的人工匀场过程,并吸收了相应的匀场经验,在此基础上提出了基于整数线性规划的中心匀场方法。该方法将成像球面采样点的磁场值逼近成像球域的中心磁场值,通过建立无源匀场数学模型,采用分枝定界法对磁体极面上所需垫补磁片的位置和数目进行合理规划,最终达到提高成像区域磁场均匀度的目的。
本文提出采用数值分析方法计算极面上匀场磁片阵列在空间产生的磁场值,获得磁场后进行软件编程实现匀场过程,然后以此为基础反复计算匀场。结合匀场的粗调与细调过程,最后得到了在极面上先外圈后里圈的理论匀场方案。为了验证匀场算法与匀场方案的正确性,与MRI设备生产企业合作,进行了实地匀场实验。通过对测量与计算结果的比较,证实了本文提出的基于中心磁场值的匀场算法是有效的。
The research of this thesis is supported by the Doctoral Program Foundation of Chinese Higher Education Institutions, Ministry of Education of China—Study of the shimming technique of the permanent magnet for open magnetic resonance imaging system(under Grant 20050142003).
The system of open MRI magnet is one of the significant development aspects of medical MRI device at the present time. Its image quality is closely related with the shimming technique. The passive shimming with permanent MRI magnet is still a hot and difficult investigation problem. In this thesis, the research on shimming problem for open permanent MRI magnet has been presented in detail. The main work is as follows.
The way of passive shimming is to shim the magnet passively, using pieces of the permanent magnetic material placed at specified locations on the pole surfaces to bring an additional magnetic field which corrects the initial field. Therefore, the numeric analysis method, analytic solving and practical measuring are presented to determine the additional magnetic field created by a single shimming sheet in this paper, which lays a foundation on the passive shimming process.
An improved method for allowing an MRI magnet to meet its demand to the field homogeneity has been developed. The method is denoted as central shimming based on the theory of integer linear programming, which involves procedure of practical shimming and experience from shimming engineers. The method proposed in the paper uses the value of central magnetic field in the imaging region to approximately correct the magnetic field of sampling points. After building the passive shimming model, the branch and bound method is adopted to optimize the shim locations and the number of shims at each allowable location which will bring the magnetic field uniformities to an acceptable level.
Firstly, the numerical analysis method is proposed to compute the distribution of the magnetic field in the imaging region, which created by the shim arrays on the pole surface. Secondly, the related software is programmed to fulfill shimming steps. Thirdly, shimming is repeated based on the magnetic field corrected in the second step. The theoretical shimming scheme is obtained through combining process of the coarse and fine shimming, which is completed from outer rings to inner rings on the pole surface. In order to check the precision of the shimming algorithm proposed in this paper, shimming experiment of the queen-post permanent magnet is completed in the enterprise that produces the magnet used for MRI. According to the results comparison of measuring and computing, the validity of the method is testified finally.
开放式磁共振成像(Magnetic Resonance Imaging,MRI)系统是目前医用磁共振成像设备的主要发展方向之一。MRI系统的成像质量与其主磁体的匀场技术有着密切的联系。对于开放式永磁MRI装置来说,无源匀场更是当前研究的热点、难点。针对这一现状,本文就开放式永磁型MRI系统主磁体的无源匀场问题从以下几个方面进行了深入的研究。
无源匀场主要靠贴补小磁片产生的磁场来抵消主磁体在空间成像区域产生磁场的不均匀谐波分量。因此需要确定单个小磁片在空间区域产生的附加磁场。本文采用了数值分析、解析求解以及实地测量三种方法,通过比较、验证,最终确定了单个小磁片在空间产生的磁场值,为无源匀场工作的研究奠定了基础。
本文在研究整数线性规划优化理论的基础上,深入研究了实际的人工匀场过程,并吸收了相应的匀场经验,在此基础上提出了基于整数线性规划的中心匀场方法。该方法将成像球面采样点的磁场值逼近成像球域的中心磁场值,通过建立无源匀场数学模型,采用分枝定界法对磁体极面上所需垫补磁片的位置和数目进行合理规划,最终达到提高成像区域磁场均匀度的目的。
本文提出采用数值分析方法计算极面上匀场磁片阵列在空间产生的磁场值,获得磁场后进行软件编程实现匀场过程,然后以此为基础反复计算匀场。结合匀场的粗调与细调过程,最后得到了在极面上先外圈后里圈的理论匀场方案。为了验证匀场算法与匀场方案的正确性,与MRI设备生产企业合作,进行了实地匀场实验。通过对测量与计算结果的比较,证实了本文提出的基于中心磁场值的匀场算法是有效的。
The research of this thesis is supported by the Doctoral Program Foundation of Chinese Higher Education Institutions, Ministry of Education of China—Study of the shimming technique of the permanent magnet for open magnetic resonance imaging system(under Grant 20050142003).
The system of open MRI magnet is one of the significant development aspects of medical MRI device at the present time. Its image quality is closely related with the shimming technique. The passive shimming with permanent MRI magnet is still a hot and difficult investigation problem. In this thesis, the research on shimming problem for open permanent MRI magnet has been presented in detail. The main work is as follows.
The way of passive shimming is to shim the magnet passively, using pieces of the permanent magnetic material placed at specified locations on the pole surfaces to bring an additional magnetic field which corrects the initial field. Therefore, the numeric analysis method, analytic solving and practical measuring are presented to determine the additional magnetic field created by a single shimming sheet in this paper, which lays a foundation on the passive shimming process.
An improved method for allowing an MRI magnet to meet its demand to the field homogeneity has been developed. The method is denoted as central shimming based on the theory of integer linear programming, which involves procedure of practical shimming and experience from shimming engineers. The method proposed in the paper uses the value of central magnetic field in the imaging region to approximately correct the magnetic field of sampling points. After building the passive shimming model, the branch and bound method is adopted to optimize the shim locations and the number of shims at each allowable location which will bring the magnetic field uniformities to an acceptable level.
Firstly, the numerical analysis method is proposed to compute the distribution of the magnetic field in the imaging region, which created by the shim arrays on the pole surface. Secondly, the related software is programmed to fulfill shimming steps. Thirdly, shimming is repeated based on the magnetic field corrected in the second step. The theoretical shimming scheme is obtained through combining process of the coarse and fine shimming, which is completed from outer rings to inner rings on the pole surface. In order to check the precision of the shimming algorithm proposed in this paper, shimming experiment of the queen-post permanent magnet is completed in the enterprise that produces the magnet used for MRI. According to the results comparison of measuring and computing, the validity of the method is testified finally.
引文
[1]史泰华.核磁共振成像仪磁场装置的混合结构研究.上海大学学报,2001,7(3):231-234.
[2]武海澄,刘正敏,周荷琴.MRI永磁体设计方法研究.中国医疗器械杂志,2006,30(3):176-179.
[3]夏平畴.永磁机构.北京工业大学出版社,2000.
[4]张奇,黄刚,刘曼芳.如何做好核磁共振安装时的无源匀场工作.医疗装备,1999,4:21.
[5]蒲钔.磁共振成像系统主磁体技术的工程研究.核工业西南物理研究,1998.
[6]H.Sakurai,M.Aoki,T.Miyamoto.Improvement of the Field Homogeneity with a Permanent Magnet Assembly for MRI.IEEE Trans.Magn.,1991,6(6):524-530.
[7]Abele M,G,H.Rusinek.Open C-Shaped Permanent Magnets Assembly for MRI Device.IEEE Trans.Magn.,1995,31(6):3740-3742.
[8]焦玮玮,董增仁,孙威,刘卫核.磁共振永磁体无源匀场的神经网络方法.中国科学院研究生院学报,2006,23(2):149-154.
[9]周涛.新型磁共振磁体的研究:(博士学位论文).上海:上海交通大学,2000.
[10]Weston A.Anderson.Electrical Current Shimming for Correcting Magnetic Fields.The Review of Scientific Instruments,1961,241-250.
[11]储岳森.有源匀场的优化算法.高电压技术,1996,22(2):15-17.
[12]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.生物医学工程学杂志,2004,21(2):288-291.
[13]储岳森,黄放.磁共振成像中磁体的匀场线圈设计与磁场计算.上海交通大学学报,1996,30(12):103-109.
[14]Vadovic.R.Magnetic Field Correction Using Magnetized Shims.IEEE Trans.Magn.,1989,25(4):35.
[15]高国灿.开放式C型磁共振成像系统永磁体匀场研究:(硕士学位论文).沈阳:东北大学,2003.
[16]包尚联.现代医学影像物理学.北京:北京大学医学出版社,2004.
[17]Jeff Wilkins and Steve Miller.The Use of Adaptive Algorithms for Optimal Electrical Shimming in Magnetic Resonance Imaging(MRI).IEEE Trans.Magn.,1989,36(2):202-210.
[18]V.E Eregin.Formation of Homogeneous Field in a Shielded Superconducting Solenoid for MRI.IEEE Trans.Magn.,1992,28(1):675-677.
[19]A.Belov,V.Bushuev,M.Emelianov.Passive Shimming of the Superconducting Magnets for MRI.IEEE Trans.on Applied Superconductivity,1995,5(2):679-681.
[20]J.E.C.Williams,E.S.Bobrov,Y.Iwasa.NMR Magnet Technology at MIT.1EEE Trans.Magn.,1992,28(1):627-630.
[21]Forbes,L.K.and S.Crozier.A novel target-field method for finite-length magnetic resonance shim coils:Part 2.Tesseral shims,J.Phys.D,Appl.Phys.,2002,35:839-849.
[22]Forbes,L.K.and S.Crozier,A novel target-field method for magnetic resonance shim coils:Part 3.Shielded zonal and tesseral coils,J.Phys.D,Appl.Phys.,2003,36:68-80.
[23]Brideson,M.A.,L.K.Forbes,and S.Crozier,Determining complicated winding patterns for shim coils using stream functions and the target field method,Concepts Magn.Reson.,2002,14:9-18.
[24]B.Dorri,M.E.Vermilyea.Passive Shimming of MR Magnets:Algorithm,Hardware,and Results.IEEE Trans.on Applied Superconductivity,1993,3(1):254-257.
[25]B.Dorri,M.E.Vermilyea.Method of Optimizing Passive Shim Placement in Magnetic Resonance Magnets.US Patent 5,045,794,1991.
[26]D.I.Hoult,D.Lee.Shimming a superconducting nuclear-magnetic-resonance imaging magnet with steel.Rev.Sct.lnstrum.1985,56(1):131-135.
[27]Xiaohua Jiang,Guanghui Shen,Yongchuan Lai,and Jinfeng Tian.Development of an Open 0.3T NdFeB MRI Magnet.IEEE Trans.on applied superconductivity,2004,14(2):1621-1623.
[28]Zhe jin,Xin Tang,Feng Qi,Donglin Zu,and Weimin wang.A new method for shimming a magnetic field in NMR system.PIERS Online,2007,3(6):870-873.
[29]Liming Hong and Donglin Zu.Shimming permanent magnet of MRI scanner.PIERS online,2007,3(6):859-864.
[30]赵微,唐晓英,胡国军等.一种永磁磁共振成像磁体的被动匀场方法.北京生物医学工程,2006,25(5):493-497.
[31]王恩龙.一种特殊永磁型磁体的匀场技术研究:(硕士学位论文).沈阳:沈阳工业大学,2007.
[32]苏金明,阮沈勇,王永利.MATLAB工程数学.北京:电子工业出版社,2005.
[33]甘应爱,田丰,李维铮等.运筹学(修订版).北京:清华大学出版社,1995.
[34]雍龙泉.求解线性规划的几种方法.江西科学 2007,25(2):202-205.
[35]Kojina M,Megiddo Mizuno S.A primal-dual algorithms for linear programming.Math Prog,1993,(61):263-280.
[36]唐焕文,秦学志.实用最优化方法.大连:大连理工大学出版社,2000.
[37]马仲藩.线性规划最新进展.北京:科学出版社,1994.
[38]Bazaraa M S,Shetty C M.Nonlinear Programming Theory and Algorithms.New York:Wiley,1979.
[39]申卯兴,宁振民,郝彩丽.线性规划单纯形法的改进与教学.中国企业运筹学,146-151.
[40]薛家庆.最优化原理与方法(修订版).北京:冶金工业出版社,2003.
[41]郭永发,宛士春,陶凤玲.邻域整点搜索法求解标准型纯整数规划.武汉大学学报,2004,37(5):13-17.
[42]刘寿春,胡雁玲,洪文.整数规划模型研究.皖西学院学报,2004,20(2):6-8.
[43]陈静,周晓云,于伟伟.一类整数规划问题的Lagrange求解方法.金陵科技学院学报,2007,23(2):13-17.
[44]李玉娥,胡海波,付大庆.整数规划的一个解法.大庆高等专科学校学报,1996,16(4):14-17.
[45]朱承学,李崧,肖鸣宇.解整数规划问题的目标收敛法.广西科学,2005,12(1):14-17.
[46]高培旺,封全喜.切割定界与整数分枝结合求解整数线性规划.数学的实践与认识,2004,34(4):109-114.
[47]宛士春,郭永发.对求解整数规划方法的新探索.青海大学学报,2003,21(6):63-67.
[48]徐大申,邱启荣,何凤霞,彭武安.求解整数规划方法新探.华北电力大学学报,2004,31(5):110-112.
[49]郭永发,宛士春,陈农,等.邻域整点搜索法求解整数规划.数学的实践与认识,2006,36(11):100-104.
[50]张天鹤.整数线性规划问题解法研究.湖州职业技术学院学报,2005,4:74-76.
[51]许万蓉.线性规划.北京:北京理工大学出版社,1990.
[52]Dantzig G B.Linear Programming and Extension.Princeton,NJ:Princeton University press,1963.
[53]Gomory R.E.Outline of an algorithm for integer solutions to linear problem.Bulletin of the American Mathematical Society,1958,64(5):275-278.
[54]苟格.整数规划中的割平面法与分枝定界法比较.达县师范高等专科学校学报,2005,15(2):18-21
[55]谢德馨,王恩龙,张艳丽.基于动态规划的永磁MRI磁体的无源匀场.沈阳工业大学学报,2007,29(4):396-399.
[56]倪光正.工程电磁场原理.北京:高等教育出版社,2004.
[57]Konzbul P,Sveda K,Srnka A.Superconducting shim system with high purity magnetic field for NMR and MRI solenoid magnet MIDI-200.IEEE Trans.Magn.,1996,32(4):2643-2646.
[58]Jan Chladek,Pavel Konzbul,Pavel Olmera,et al.Evolutionary and genetic optimization of NMR gradient and shim coils.IEEE Trans.Magn.,2000,36(4):1102-1105.
[59]武海澄,周荷琴,刘正敏.有源匀场电流优化算法研究.北京生物医学工程,2007,26(5):466-469.
[60]Hoult,D.I.and R.Deslauriers.Accurate shim-coil design and magnet-field profiling by a power-minimization-matrix method.J.Magn,Reson.Ser,A,1994,108:9-20.
[61]Forbes,L.K.and S.Crozier,Asymmetric zonal shim coils for magnetic resonance applications.Med.Phys.,2001,28:1644-1651.
[62]赵微,刘志文,刘勇等.开放式低场永磁MRI磁体的匀场方法研究.电工电能新技术,2007,26(3):29-32.
[63]武海澄.MRI磁体系统的计算机辅助设计研究:(博士学位论文).合肥:中国科学技术大学,2007.
[64]张艳丽,谢德馨,王恩龙等.开放式永磁机构无源匀场方法.电工技术学报,2007,22(6):1-5.
[2]武海澄,刘正敏,周荷琴.MRI永磁体设计方法研究.中国医疗器械杂志,2006,30(3):176-179.
[3]夏平畴.永磁机构.北京工业大学出版社,2000.
[4]张奇,黄刚,刘曼芳.如何做好核磁共振安装时的无源匀场工作.医疗装备,1999,4:21.
[5]蒲钔.磁共振成像系统主磁体技术的工程研究.核工业西南物理研究,1998.
[6]H.Sakurai,M.Aoki,T.Miyamoto.Improvement of the Field Homogeneity with a Permanent Magnet Assembly for MRI.IEEE Trans.Magn.,1991,6(6):524-530.
[7]Abele M,G,H.Rusinek.Open C-Shaped Permanent Magnets Assembly for MRI Device.IEEE Trans.Magn.,1995,31(6):3740-3742.
[8]焦玮玮,董增仁,孙威,刘卫核.磁共振永磁体无源匀场的神经网络方法.中国科学院研究生院学报,2006,23(2):149-154.
[9]周涛.新型磁共振磁体的研究:(博士学位论文).上海:上海交通大学,2000.
[10]Weston A.Anderson.Electrical Current Shimming for Correcting Magnetic Fields.The Review of Scientific Instruments,1961,241-250.
[11]储岳森.有源匀场的优化算法.高电压技术,1996,22(2):15-17.
[12]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.生物医学工程学杂志,2004,21(2):288-291.
[13]储岳森,黄放.磁共振成像中磁体的匀场线圈设计与磁场计算.上海交通大学学报,1996,30(12):103-109.
[14]Vadovic.R.Magnetic Field Correction Using Magnetized Shims.IEEE Trans.Magn.,1989,25(4):35.
[15]高国灿.开放式C型磁共振成像系统永磁体匀场研究:(硕士学位论文).沈阳:东北大学,2003.
[16]包尚联.现代医学影像物理学.北京:北京大学医学出版社,2004.
[17]Jeff Wilkins and Steve Miller.The Use of Adaptive Algorithms for Optimal Electrical Shimming in Magnetic Resonance Imaging(MRI).IEEE Trans.Magn.,1989,36(2):202-210.
[18]V.E Eregin.Formation of Homogeneous Field in a Shielded Superconducting Solenoid for MRI.IEEE Trans.Magn.,1992,28(1):675-677.
[19]A.Belov,V.Bushuev,M.Emelianov.Passive Shimming of the Superconducting Magnets for MRI.IEEE Trans.on Applied Superconductivity,1995,5(2):679-681.
[20]J.E.C.Williams,E.S.Bobrov,Y.Iwasa.NMR Magnet Technology at MIT.1EEE Trans.Magn.,1992,28(1):627-630.
[21]Forbes,L.K.and S.Crozier.A novel target-field method for finite-length magnetic resonance shim coils:Part 2.Tesseral shims,J.Phys.D,Appl.Phys.,2002,35:839-849.
[22]Forbes,L.K.and S.Crozier,A novel target-field method for magnetic resonance shim coils:Part 3.Shielded zonal and tesseral coils,J.Phys.D,Appl.Phys.,2003,36:68-80.
[23]Brideson,M.A.,L.K.Forbes,and S.Crozier,Determining complicated winding patterns for shim coils using stream functions and the target field method,Concepts Magn.Reson.,2002,14:9-18.
[24]B.Dorri,M.E.Vermilyea.Passive Shimming of MR Magnets:Algorithm,Hardware,and Results.IEEE Trans.on Applied Superconductivity,1993,3(1):254-257.
[25]B.Dorri,M.E.Vermilyea.Method of Optimizing Passive Shim Placement in Magnetic Resonance Magnets.US Patent 5,045,794,1991.
[26]D.I.Hoult,D.Lee.Shimming a superconducting nuclear-magnetic-resonance imaging magnet with steel.Rev.Sct.lnstrum.1985,56(1):131-135.
[27]Xiaohua Jiang,Guanghui Shen,Yongchuan Lai,and Jinfeng Tian.Development of an Open 0.3T NdFeB MRI Magnet.IEEE Trans.on applied superconductivity,2004,14(2):1621-1623.
[28]Zhe jin,Xin Tang,Feng Qi,Donglin Zu,and Weimin wang.A new method for shimming a magnetic field in NMR system.PIERS Online,2007,3(6):870-873.
[29]Liming Hong and Donglin Zu.Shimming permanent magnet of MRI scanner.PIERS online,2007,3(6):859-864.
[30]赵微,唐晓英,胡国军等.一种永磁磁共振成像磁体的被动匀场方法.北京生物医学工程,2006,25(5):493-497.
[31]王恩龙.一种特殊永磁型磁体的匀场技术研究:(硕士学位论文).沈阳:沈阳工业大学,2007.
[32]苏金明,阮沈勇,王永利.MATLAB工程数学.北京:电子工业出版社,2005.
[33]甘应爱,田丰,李维铮等.运筹学(修订版).北京:清华大学出版社,1995.
[34]雍龙泉.求解线性规划的几种方法.江西科学 2007,25(2):202-205.
[35]Kojina M,Megiddo Mizuno S.A primal-dual algorithms for linear programming.Math Prog,1993,(61):263-280.
[36]唐焕文,秦学志.实用最优化方法.大连:大连理工大学出版社,2000.
[37]马仲藩.线性规划最新进展.北京:科学出版社,1994.
[38]Bazaraa M S,Shetty C M.Nonlinear Programming Theory and Algorithms.New York:Wiley,1979.
[39]申卯兴,宁振民,郝彩丽.线性规划单纯形法的改进与教学.中国企业运筹学,146-151.
[40]薛家庆.最优化原理与方法(修订版).北京:冶金工业出版社,2003.
[41]郭永发,宛士春,陶凤玲.邻域整点搜索法求解标准型纯整数规划.武汉大学学报,2004,37(5):13-17.
[42]刘寿春,胡雁玲,洪文.整数规划模型研究.皖西学院学报,2004,20(2):6-8.
[43]陈静,周晓云,于伟伟.一类整数规划问题的Lagrange求解方法.金陵科技学院学报,2007,23(2):13-17.
[44]李玉娥,胡海波,付大庆.整数规划的一个解法.大庆高等专科学校学报,1996,16(4):14-17.
[45]朱承学,李崧,肖鸣宇.解整数规划问题的目标收敛法.广西科学,2005,12(1):14-17.
[46]高培旺,封全喜.切割定界与整数分枝结合求解整数线性规划.数学的实践与认识,2004,34(4):109-114.
[47]宛士春,郭永发.对求解整数规划方法的新探索.青海大学学报,2003,21(6):63-67.
[48]徐大申,邱启荣,何凤霞,彭武安.求解整数规划方法新探.华北电力大学学报,2004,31(5):110-112.
[49]郭永发,宛士春,陈农,等.邻域整点搜索法求解整数规划.数学的实践与认识,2006,36(11):100-104.
[50]张天鹤.整数线性规划问题解法研究.湖州职业技术学院学报,2005,4:74-76.
[51]许万蓉.线性规划.北京:北京理工大学出版社,1990.
[52]Dantzig G B.Linear Programming and Extension.Princeton,NJ:Princeton University press,1963.
[53]Gomory R.E.Outline of an algorithm for integer solutions to linear problem.Bulletin of the American Mathematical Society,1958,64(5):275-278.
[54]苟格.整数规划中的割平面法与分枝定界法比较.达县师范高等专科学校学报,2005,15(2):18-21
[55]谢德馨,王恩龙,张艳丽.基于动态规划的永磁MRI磁体的无源匀场.沈阳工业大学学报,2007,29(4):396-399.
[56]倪光正.工程电磁场原理.北京:高等教育出版社,2004.
[57]Konzbul P,Sveda K,Srnka A.Superconducting shim system with high purity magnetic field for NMR and MRI solenoid magnet MIDI-200.IEEE Trans.Magn.,1996,32(4):2643-2646.
[58]Jan Chladek,Pavel Konzbul,Pavel Olmera,et al.Evolutionary and genetic optimization of NMR gradient and shim coils.IEEE Trans.Magn.,2000,36(4):1102-1105.
[59]武海澄,周荷琴,刘正敏.有源匀场电流优化算法研究.北京生物医学工程,2007,26(5):466-469.
[60]Hoult,D.I.and R.Deslauriers.Accurate shim-coil design and magnet-field profiling by a power-minimization-matrix method.J.Magn,Reson.Ser,A,1994,108:9-20.
[61]Forbes,L.K.and S.Crozier,Asymmetric zonal shim coils for magnetic resonance applications.Med.Phys.,2001,28:1644-1651.
[62]赵微,刘志文,刘勇等.开放式低场永磁MRI磁体的匀场方法研究.电工电能新技术,2007,26(3):29-32.
[63]武海澄.MRI磁体系统的计算机辅助设计研究:(博士学位论文).合肥:中国科学技术大学,2007.
[64]张艳丽,谢德馨,王恩龙等.开放式永磁机构无源匀场方法.电工技术学报,2007,22(6):1-5.