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The design of gradient coils for a magnetic resonance imaging (MRI) system is a multiple objective optimization problem, which usually needs to deal with a couple of conflicting design objectives, such as the stored magnetic energy, power consumption, and target linear gradient distribution. These design requirements usually conflict with each other, and there is no unique optimal solution which is capable of minimizing all objectives simultaneously. Therefore, the design of gradient coils needs to be optimized reasonably with the tradeoff among different design objectives. Based on the developable property of the super-elliptical cylindrical surface and the stream function design method, the multiple objective optimization problem is analyzed by using the Pareto optimization method in this paper. The effect of proposed approach is illustrated by using the stream function method and three aforementioned coil design objectives are analyzed. The influences of the stored magnetic energy and power consumption target on linearity of gradient coil and the configuration of coils are analyzed respectively. The suitable sizes of gradient coils are discussed by analyzing the change of the stored magnetic energy. A weighted sum method is employed to produce the optimal Pareto solutions, in which the multiple objective problem reduces into a single objective function through a weighted sum of all objectives. The quantitative relationship of each design requirement is analyzed in the Pareto solution space, where Pareto optimal solutions can be intuitively found by dealing efficiently with the tradeoff among different coil properties. Numerical examples of super-elliptical gradient coil solutions are provided to demonstrate the effectiveness and versatility of the proposed method to design super-elliptical gradient coils with different coil requirements. The optimization results show that there are multiple available solutions in the convex Pareto solution space under the constraints that the linear gradient deviation is less than 5% and the magnetic stored energy and power dissipated are both no more than user-preset values. In the case that the values of summed objective functions are the same, the proposed method can intuitively see the performance of each individual target, thereby conducting to realizing the final design of gradient coils under the different design requirements. With the proposed approach, coil designers can have a reasonable overview of gradient coil design about the achievable performances of some specific properties and the competing or compatible relationships among coils properties. Therefore, a suitable design of the gradient coils for a given requirement of MRI application can be chosen reasonably.
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Keywords:
- gradient coils /
- stream function /
- super-elliptical cylindrical surface /
- Pareto optimization
[1] Zu D L 2004 Magnetic Resonance Imaging (Beijing: Higher Education Press) pp53-82 (in Chinese) [俎栋林 2004 核磁共振成像学(北京: 高等教育出版社)第5382页]
[2] Wang L, Cao Y H, Jia F, Liu Z Y 2014 Acta Phys. Sin. 63 238301 (in Chinese) [王亮, 曹英晖, 贾峰, 刘震宇 2014 63 238301]
[3] Turner R 1986 J. Phys. D: Appl. Phys. 19 147
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[6] Liu W T, Zu D L, Tang X 2010 Chin. Phys. B 19 018701
[7] Forbes L K, Brideson M A, Crozier S 2005 IEEE Trans. Magn. 41 2134
[8] Liu W T, Zu D L, Tang X, Guo H 2007 J. Phys. D: Appl. Phys. 40 4418
[9] Li X, Xie D X, Wang J M 2009 IEEE Trans. Magn. 45 1804
[10] Tomasi D 2001 Magn. Reson. Med. 45 505
[11] Peeren G N 2003 J. Comput. Phys. 191 305
[12] Lemdiasov R A, Ludwig R 2005 Concepts Magn. Reson. B: Magn. Reson. Eng. 26B 67
[13] Liu Z Y, Jia F, Hennig J, Korvink J G 2012 IEEE Trans. Magn. 48 1179
[14] Wang Q L 2013 Practical Design of Magnetostatic Structure Using Numerical Simulation (Singapore: John Wiley Sons) pp39-142
[15] Hu G L, Ni Z P, Wang Q L 2012 IEEE Trans. Appl. Supercond. 22 4900604
[16] Zhu X C, Wang Q L, Wang H S 2016 Adv. Technol. Electral. Eng. Energ. 35 43 (in Chinese) [朱旭晨, 王秋良, 王厚生 2016 电工电能技术 35 43]
[17] Li X, Xia L, Chen W F, Liu F, Crozier S, Xie D X 2011 J. Magn. Reson. 208 148
[18] Hu Y, Wang Q L, Li Y, Zhu X C, Niu C Q 2016 Acta Phys. Sin. 65 218301 (in Chinese) [胡洋, 王秋良, 李毅, 朱旭晨, 牛超群 2016 65 218301]
[19] Turner R 1993 Magn. Reson. Imag. 11 903
[20] Abduljalil A M, Aletras A H, Robilaille P M L 1994 Magn. Reson. Med. 31 450
[21] Alsop D C, Connick T J 1996 Magn. Reson. Med. 35 875
[22] Pissanetzky S 1992 Meas. Sci. Technol. 3 667
[23] Bowtell R, Robyr P 1998 J. Magn. Reson. 131 286
[24] Wang L Q, Wang W M 2014 Chin. Phys. B 23 028703
[25] Sanchez C C, Pantoja M F, Poole M, Bretones A R 2012 IEEE Trans. Magn. 48 1967
[26] Marler R T, Arora J S 2004 Struct. Multid. Optim. 26 369
[27] Marler R T, Arora J S 2005 Eng. Optim. 37 551
[28] Xie D X, Sun X W, Bai B D, Yang S Y 2008 IEEE Trans. Magn. 44 1006
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[1] Zu D L 2004 Magnetic Resonance Imaging (Beijing: Higher Education Press) pp53-82 (in Chinese) [俎栋林 2004 核磁共振成像学(北京: 高等教育出版社)第5382页]
[2] Wang L, Cao Y H, Jia F, Liu Z Y 2014 Acta Phys. Sin. 63 238301 (in Chinese) [王亮, 曹英晖, 贾峰, 刘震宇 2014 63 238301]
[3] Turner R 1986 J. Phys. D: Appl. Phys. 19 147
[4] Turner R 1988 J. Phys. E: Sci. Instrum. 21 948
[5] Forbes L K, Crozier S 2002 J. Phys. D: Appl. Phys. 35 839
[6] Liu W T, Zu D L, Tang X 2010 Chin. Phys. B 19 018701
[7] Forbes L K, Brideson M A, Crozier S 2005 IEEE Trans. Magn. 41 2134
[8] Liu W T, Zu D L, Tang X, Guo H 2007 J. Phys. D: Appl. Phys. 40 4418
[9] Li X, Xie D X, Wang J M 2009 IEEE Trans. Magn. 45 1804
[10] Tomasi D 2001 Magn. Reson. Med. 45 505
[11] Peeren G N 2003 J. Comput. Phys. 191 305
[12] Lemdiasov R A, Ludwig R 2005 Concepts Magn. Reson. B: Magn. Reson. Eng. 26B 67
[13] Liu Z Y, Jia F, Hennig J, Korvink J G 2012 IEEE Trans. Magn. 48 1179
[14] Wang Q L 2013 Practical Design of Magnetostatic Structure Using Numerical Simulation (Singapore: John Wiley Sons) pp39-142
[15] Hu G L, Ni Z P, Wang Q L 2012 IEEE Trans. Appl. Supercond. 22 4900604
[16] Zhu X C, Wang Q L, Wang H S 2016 Adv. Technol. Electral. Eng. Energ. 35 43 (in Chinese) [朱旭晨, 王秋良, 王厚生 2016 电工电能技术 35 43]
[17] Li X, Xia L, Chen W F, Liu F, Crozier S, Xie D X 2011 J. Magn. Reson. 208 148
[18] Hu Y, Wang Q L, Li Y, Zhu X C, Niu C Q 2016 Acta Phys. Sin. 65 218301 (in Chinese) [胡洋, 王秋良, 李毅, 朱旭晨, 牛超群 2016 65 218301]
[19] Turner R 1993 Magn. Reson. Imag. 11 903
[20] Abduljalil A M, Aletras A H, Robilaille P M L 1994 Magn. Reson. Med. 31 450
[21] Alsop D C, Connick T J 1996 Magn. Reson. Med. 35 875
[22] Pissanetzky S 1992 Meas. Sci. Technol. 3 667
[23] Bowtell R, Robyr P 1998 J. Magn. Reson. 131 286
[24] Wang L Q, Wang W M 2014 Chin. Phys. B 23 028703
[25] Sanchez C C, Pantoja M F, Poole M, Bretones A R 2012 IEEE Trans. Magn. 48 1967
[26] Marler R T, Arora J S 2004 Struct. Multid. Optim. 26 369
[27] Marler R T, Arora J S 2005 Eng. Optim. 37 551
[28] Xie D X, Sun X W, Bai B D, Yang S Y 2008 IEEE Trans. Magn. 44 1006
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