Structural and Multidisciplinary Optimization

, Volume 49, Issue 3, pp 523–535

Design multiple-layer gradient coils using least-squares finite element method

  • Feng Jia
  • Zhenyu Liu
  • Maxim Zaitsev
  • Jürgen Hennig
  • Jan G. Korvink
INDUSTRIAL APPLICATION

Abstract

The design of gradient coils for magnetic resonance imaging is an optimization task in which a specified distribution of the magnetic field inside a region of interest is generated by choosing an optimal distribution of a current density geometrically restricted to specified non-intersecting design surfaces, thereby defining the preferred coil conductor shapes. Instead of boundary integral type methods, which are widely used to design coils, this paper proposes an optimization method for designing multiple layer gradient coils based on a finite element discretization. The topology of the gradient coil is expressed by a scalar stream function. The distribution of the magnetic field inside the computational domain is calculated using the least-squares finite element method. The first-order sensitivity of the objective function is calculated using an adjoint equation method. The numerical operations needed, in order to obtain an effective optimization procedure, are discussed in detail. In order to illustrate the benefit of the proposed optimization method, example gradient coils located on multiple surfaces are computed and characterised.

Keywords

Magnetic resonance imaging Gradient coil design Stream function Least-squares finite element method Adjoint method 

References

  1. Adamiak K, Rutt B, Dabrowski W (1992) Design of gradient coils for magnetic-resonance-imaging. IEEE Trans Magn 28(5):2403–2405CrossRefGoogle Scholar
  2. Bergström R (2002) Adaptive finite element methods for div-curl problems. PhD thesis, Chalmers University of TechnologyGoogle Scholar
  3. Bochev PB, Gunzburger MD (2009) Least-squares finite element methods. Springer, New YorkMATHGoogle Scholar
  4. Bowtell R, Robyr P (1998) Multilayer gradient coil design. J Magn Reson 131:286–294CrossRefGoogle Scholar
  5. Carlson JW, Derby KA, Hawryszko KC, Weideman M (1992) Design and evaluation of shielded gradient coils. Magn Reson Med 26(2):191–206CrossRefGoogle Scholar
  6. Chen Y, Davis TA, Hager WW, Rajamanickam S (2008) Algorithm 887:cholmod, supernodal sparse cholesky factorization and update/downdate. ACM Trans Math Softw 35:3CrossRefMathSciNetGoogle Scholar
  7. Chronik BA, Rutt BK (1998) Constrained length minimum inductance gradient coil design. Magn Reson Med 39(2):270–278CrossRefGoogle Scholar
  8. Davis TA, Hager WW (2005) Row modifications of a sparse cholesky factorization. SIAM J Matrix Anal Appl 26:621–639CrossRefMATHMathSciNetGoogle Scholar
  9. Du YP, Parker DL (1998) Optimal design of gradient coils in mr imaging:optimizing coil performance versus minimizing cost functions. Magn Reson Med 40(3):500–503CrossRefGoogle Scholar
  10. Ern A, Guermond JL (2004) Theory and practice of finite elements. Springer-Verlag, New YorkCrossRefMATHGoogle Scholar
  11. Forbes LK, Crozier S (2004) Novel target-field method for designing shielded biplanar shim and gradient coils. IEEE Trans Magn 40:1929–1938CrossRefGoogle Scholar
  12. Forbes LK, Brideson MA, Crozier S (2005) A target-field method to design circular biplanar coils for asymmetric shim and gradient fields. IEEE Trans Magn 41:2134–2144CrossRefGoogle Scholar
  13. Gill PE, Murray W, Wright MH (1982) Practical optimization. Academic Press Inc, LondonGoogle Scholar
  14. Gross PW, Kotiuga PR (2004) Electromagnetic theory and computation:a topological approach. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  15. Hidalgo-Tobon S (2010) Theory of gradient coil design methods for magnetic resonance imaging. Concepts Magn Reson A 36A(4):223–242CrossRefGoogle Scholar
  16. Jackson JD (1998) Classical electrodynamics, 3rd edn. Wiley, New YorkGoogle Scholar
  17. Jia F, Liu Z, Korvink JG (2011) A novel coil design method for manufacturable configurations at optimal performance In: Proceedings ISMRM, vol 19, p 3780Google Scholar
  18. Jiang B (1998) The least squares finite element method: theory and applications in computational fluid dynamics and electromagnetics. Springer-Verlag, Berlin HeidelbergCrossRefMATHGoogle Scholar
  19. Jin JM (2002) The finite element method in electromagnetics, 2nd edn. Wiley, New YorkMATHGoogle Scholar
  20. Karypis G, Kumar V (1998) Multilevel k-way partitioning scheme for irregular graphs. J Parallel Distrib Comput 48:96–129CrossRefGoogle Scholar
  21. Lauterbur PC (1973) Image formation by induced local interactions: examples of employing nuclear magnetic resonance. Nature 242:190–191CrossRefGoogle Scholar
  22. Leggett J, Crozier S, Blackband S, Beck B, Bowtell RW (2003) Multilayer transverse gradient coil design. Concepts Magn Reson B 16:38–46CrossRefGoogle Scholar
  23. Lemdiasov RA, Ludwig R (2005) A stream function method for gradient coil design. Concepts Magn Reson B 26B(1):67–80CrossRefGoogle Scholar
  24. Liu W, Zu D, Tang X, Guo H (2007) Target-field method for mri biplanar gradient coil design. J Phys D Appl Phys 40:4418– 4424CrossRefGoogle Scholar
  25. Lopez HS, Poole M, Crozier S (2009) An improved equivalent magnetization current method applied to the design of local breast gradient coils. J Magn Reson 199:48–55CrossRefGoogle Scholar
  26. Mansfield P, Chapman B (1986) Active magnetic screening of gradient coils in nmr imaging. J Magn Reson 66(3):573–576Google Scholar
  27. Marin L, Power H, Bowtell RW, Sanchez CC, Becker AA, Glover P, Jones A (2008) Boundary element method for an inverse problem in magnetic resonance imaging gradient coils. CMES-Comp Model Eng Sci 23:149–173MATHGoogle Scholar
  28. Marinus VT, Jacques BA (2010) Magnetic resonance imaging theory and practice, 3rd edn. Springer, Berlin HeidelbergGoogle Scholar
  29. Nocedal J, Wright SJ (1999) Numerical optimization. Springer Science + Business Media IncGoogle Scholar
  30. Olesen LH, Okkels F, Bruus H (2006) A high-level programming-language implementation of topology optimization applied to steady-state navier-stokes flow. Int J Numer Meth Eng 65:975–1001CrossRefMATHMathSciNetGoogle Scholar
  31. Peeren GN (2003) Stream function approach for determining optimal surface currents. J Comput Phys 191:305–321CrossRefMATHMathSciNetGoogle Scholar
  32. Pissanetzky S (1992) Minimum energy mri gradient coils of general geometry. Meas Sci Technol 3:567–673CrossRefGoogle Scholar
  33. Poole M, Bowtell R (2007) Novel gradient coils designed using a boundary element method. Concepts Magn Reson B 31B:162–175CrossRefGoogle Scholar
  34. Poole M, Weiss P, Lopez HS, Ng M, Crozier S (2010) Minimax current density coil design. J Phys D Appl Phys 43:095001CrossRefGoogle Scholar
  35. Roemer PB, Hickey JS (1988) Self-shielded gradient coils for nuclear magnetic resonance imaging. US 4737716 AGoogle Scholar
  36. Shi F, Ludwig R (1998) Magnetic resonance imaging gradient coil design by combining optimization techniques with the finite element method. IEEE Trans Magn 34:671–683CrossRefGoogle Scholar
  37. Shvartsman S, Steckner MC (2007) Discrete design method of transverse gradient coils for mri. Concepts Magn Reson Part B Magn Reson Eng 31B(2):95–115CrossRefGoogle Scholar
  38. Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization:a survey on procedures dealing with checkerboards, mesh-dependencies and local minimal. Struct Optim 16:68–75CrossRefGoogle Scholar
  39. Turner R (1986) A target field approach to optimal coil design. J Phys D Appl Phys 19:147–151CrossRefGoogle Scholar
  40. Turner R (1988) Minimum inductance coils. J Phys E Sci Instrum 21:948–952CrossRefGoogle Scholar
  41. Turner R (1993) Gradient coil design: a review of methods. Magn Reson Imaging 11:903–920CrossRefGoogle Scholar
  42. Ungersma SE, Xu H, Chronik BA, Scott GC, Macovski A, Conolly SM (2004) Shim design using a linear programming algorithm. Magn Reson Med 52:619627CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Feng Jia
    • 1
  • Zhenyu Liu
    • 2
  • Maxim Zaitsev
    • 1
  • Jürgen Hennig
    • 1
  • Jan G. Korvink
    • 3
    • 4
  1. 1.Department of Radiology, Medical PhysicsUniversity Hospital FreiburgFreiburgGermany
  2. 2.State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics (CIOMP)Chinese Academy of SciencesChangchunChina
  3. 3.Department of Microsystems Engineering (IMTEK)University of FreiburgFreiburgGermany
  4. 4.Freiburg Institute for Advanced Studies (FRIAS)University of FreiburgFreiburgGermany

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