Numerical solution for an inverse MRI problem using a regularised boundary element method

We investigate the reconstruction of a divergence-free surface current distribution from knowledge of the magnetic flux density in a prescribed region of interest in the framework of static electromagnetism. This inverse problem is motivated by the design of gradient coils used in magnetic resonance...

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Published in:	Engineering analysis with boundary elements Vol. 32; no. 8; pp. 658 - 675
Main Authors:	Marin, Liviu, Power, Henry, Bowtell, Richard W., Cobos Sanchez, Clemente, Becker, Adib A., Glover, Paul, Jones, Arthur
Format:	Journal Article
Language:	English
Published:	Elsevier Ltd 01.08.2008
Subjects:	Boundary element method Coil design Coils Current density Density Design engineering Divergence-free boundary element method (BEM) Electromagnetism Inverse problem Magnetic resonance imaging (MRI) Mathematical analysis Mathematical models Regularisation Streams Magnetic resonance imaging (MRI) Electromagnetism Coil design Regularisation Divergence-free boundary element method (BEM) Inverse problem
ISSN:	0955-7997, 1873-197X
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Abstract	We investigate the reconstruction of a divergence-free surface current distribution from knowledge of the magnetic flux density in a prescribed region of interest in the framework of static electromagnetism. This inverse problem is motivated by the design of gradient coils used in magnetic resonance imaging (MRI) and is formulated using its corresponding integral representation according to potential theory. A constant boundary element method (BEM) which satisfies the continuity equation for the current density, i.e. divergence-free BEM, and was originally proposed by Lemdiasov and Ludwig [A stream function method for gradient coil design. Concepts Magn Reson B Magn Reson Eng 2005;26B:67–80], is presented based on geometrical arguments with respect to the linear (flat) triangular boundary elements employed in order to emphasise its possible extension to further higher-order divergence-free interpolations. Since the discretised BEM system is ill-posed and hence the associated least-squares solution may be inaccurate and/or physically meaningless, the Tikhonov regularisation method is employed in order to retrieve accurate and physically correct solutions. A rigorous numerical approximation for the calculation the magnetic energy, which reduces the errors induced by employing the approach of Lemdiasov and Ludwig [A stream function method for gradient coil design. Concepts Magn Reson B Magn Reson Eng 2005;26B:67–80], is also proposed.
AbstractList	We investigate the reconstruction of a divergence-free surface current distribution from knowledge of the magnetic flux density in a prescribed region of interest in the framework of static electromagnetism. This inverse problem is motivated by the design of gradient coils used in magnetic resonance imaging (MRI) and is formulated using its corresponding integral representation according to potential theory. A constant boundary element method (BEM) which satisfies the continuity equation for the current density, i.e. divergence-free BEM, and was originally proposed by Lemdiasov and Ludwig [A stream function method for gradient coil design. Concepts Magn Reson B Magn Reson Eng 2005; 26B:67-80], is presented based on geometrical arguments with respect to the linear (flat) triangular boundary elements employed in order to emphasise its possible extension to further higher-order divergence-free interpolations. Since the discretised BEM system is ill-posed and hence the associated least-squares solution may be inaccurate and /or physically meaningless, the Tikhonov regularisation method is employed in order to retrieve accurate and physically correct solutions. A rigorous numerical approximation for the calculation the magnetic energy, which reduces the errors induced by employing the approach of Lemdiasov and Ludwig [A stream function method for gradient coil design. Concepts Magn Reson B Magn Reson Eng 2005; 26B:67-80], is also proposed.
Author	Marin, Liviu Power, Henry Becker, Adib A. Jones, Arthur Glover, Paul Bowtell, Richard W. Cobos Sanchez, Clemente
Author_xml	– sequence: 1 givenname: Liviu surname: Marin fullname: Marin, Liviu email: liviu.marin@nottingham.ac.uk organization: School of Mechanical, Materials and Manufacturing Engineering, The University of Nottingham, University Park, Nottingham NG7 2RD, UK – sequence: 2 givenname: Henry surname: Power fullname: Power, Henry organization: School of Mechanical, Materials and Manufacturing Engineering, The University of Nottingham, University Park, Nottingham NG7 2RD, UK – sequence: 3 givenname: Richard W. surname: Bowtell fullname: Bowtell, Richard W. organization: Sir Peter Mansfield Magnetic Resonance Centre, School of Physics and Astronomy, The University of Nottingham, Nottingham Park, Nottingham NG7 2RD, UK – sequence: 4 givenname: Clemente surname: Cobos Sanchez fullname: Cobos Sanchez, Clemente organization: Sir Peter Mansfield Magnetic Resonance Centre, School of Physics and Astronomy, The University of Nottingham, Nottingham Park, Nottingham NG7 2RD, UK – sequence: 5 givenname: Adib A. surname: Becker fullname: Becker, Adib A. organization: School of Mechanical, Materials and Manufacturing Engineering, The University of Nottingham, University Park, Nottingham NG7 2RD, UK – sequence: 6 givenname: Paul surname: Glover fullname: Glover, Paul organization: Sir Peter Mansfield Magnetic Resonance Centre, School of Physics and Astronomy, The University of Nottingham, Nottingham Park, Nottingham NG7 2RD, UK – sequence: 7 givenname: Arthur surname: Jones fullname: Jones, Arthur organization: School of Mechanical, Materials and Manufacturing Engineering, The University of Nottingham, University Park, Nottingham NG7 2RD, UK
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SubjectTerms	Boundary element method Coil design Coils Current density Density Design engineering Divergence-free boundary element method (BEM) Electromagnetism Inverse problem Magnetic resonance imaging (MRI) Mathematical analysis Mathematical models Regularisation Streams
Title	Numerical solution for an inverse MRI problem using a regularised boundary element method
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