Gradient Coil Design Research Articles

Equivalent Magnetization Current Method Applied to the Design of Gradient Coils for Magnetic Resonance Imaging

A new method is described for the design of gradient coils for magnetic resonance imaging systems. The method is based on the known equivalence between a magnetized volume surrounded by a conducting surface and its equivalent representation by a surface current density. The curl of a vertical magnetization vector of a magnetized thin volume is equivalent to a surface current density whose stream line defines a coil current pattern. This concept is applied to the design of gradient coils of arbitrary shape. The thin magnetized volume is discretized in small triangular elements. By calculating the contribution of each magnetized block at target points a field source matrix is obtained. The equivalent magnetization current concept is applied to obtain the equivalent coil impedance, force and torques. A quadratic programming optimization algorithm is used to obtain the stream-magnetization-thickness function value at each node such that coils of optimal performance are obtained. This method can be used for gradient coils wound on arbitrary surface shapes and can be applied to hybrid current/iron solutions. A variety of examples are shown to demonstrate the versatility of the method. A novel partially shielded 3-D biplanar gradient coil for open MRI magnets is presented.

Local bi‐planar gradient array design using conformal mapping and simulated annealing

AbstractMany magnetic resonance imaging applications require high spatial and temporal resolution. The improved gradient performance required to achieve high spatial and temporal resolution may be achieved by using local gradient coils such as planar gradient inserts. The planar gradient set provides higher gradient performance because it is placed inside of the imaging bore of the magnet (within the body gradients) in close proximity to the imaging region. Although the wire patterns for planar gradients can be designed using two dimensional stream functions and simulated annealing, optimization of the two dimensional stream functions can be much more computationally intensive and time consuming than optimizing the one dimensional stream functions required for cylindrical gradients. To address this problem, we have developed a simple and rapid method for the design of planar gradient inserts to produce a high strength local gradient field and a reasonably uniform imaging region. By using conformal mapping, the two dimensional problem can be simplified to a faster and more easily calculated one dimensional problem. The mapping transforms the magnetic field and wire patterns in the cylindrical system into a magnetic field and wire patterns in the bi‐planar geometry providing a tool for bi‐planar gradient coil design using a one dimensional stream function. © 2009 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 35B: 23–31, 2009

Reverse‐engineering of gradient coil designs based on experimentally measured magnetic fields and approximate knowledge of coil geometry‐application in exposure evaluations

AbstractFor many MRI installations, the coil pattern that generates pulsed magnetic fields produced by gradient coils is not provided by the manufacturer. This has implications for accurate assessments of MRI worker exposures, which is currently an important topic of research. To correctly model the level of exposure, a full three‐dimensional distribution of the magnetic field in the locality of the magnet end is required, which can be difficult to obtain by experimental measurements. This research presents one possible approach, in which the prediction of a current distribution that generates an approximately identical magnetic field profile is constrained by a small number of experimentally measured magnetic field sample points within a plane outside the gradient set. The presented methodology may take into consideration other important descriptors such as field uniformity in the imaging volume, gradient coil geometry/dimension, driving current, central field strength, etc. To exemplify the application of the proposed approach, the current density and matching magnetic field distributions of x‐ and z‐axis gradient coils are approximated without preknowledge of the gradient coil patterns for a MRI system, followed by exposure evaluations of a tissue‐equivalent numerical worker model in the vicinity of said gradients. © 2009 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 35B: 32–43, 2009

3-D Gradient Coil Design—Initial Theoretical Framework

An analytic inverse method is presented for the theoretical design of 3-D transverse gradient coils. Existing gradient coil design methods require the basic geometry of the coil to be predetermined before optimization. Typically, coil windings are constrained to lie on cylindrical, planar, spherical, or conical surfaces. In this paper, a fully 3-D region in the solution space is explored and the precise geometry of the gradient coils is obtained as part of the optimization process. Primary interest lies in minimizing the field error between induced and target gradient fields within a spherical target region. This is achieved using regularization, in which the field error is minimized along with the total coil power, to obtain a 3-D current density solution within the coil volume. A novel priority streamline technique is used to create 3-D coil windings that approximate this current density, and a secondary optimization is performed to obtain appropriate coil currents. The 3-D coil windings display an interesting general geometric form involving sets of closed loops plus spiral-type coils, and a number of examples are presented and discussed. The corresponding induced magnetic field is found to be highly linear within the region of interest, and a shielding constraint may be implemented to minimize the field outside the coil volume.

3D Gradient coil design – Toroidal surfaces

Gradient coil design typically involves optimisation of current densities or coil windings on familiar cylindrical, planar, spherical or conical surfaces. In this paper, an analytic inverse method is presented for the theoretical design of toroidal transverse gradient coils. This novel geometry is based on previous work involving a 3D current density solution, in which the precise geometry of the gradient coils was obtained as part of the optimisation process. Regularisation is used to solve for the toroidal current densities, whereby the field error is minimised in conjunction with the total power of the coil. The method is applied to the design of unshielded and shielded, whole-body and head coil gradient systems. Preliminary coil windings displaying high gradient homogeneity, low inductance, high efficiency and good force balancing are displayed and discussed. Potential benefits associated with this morphology include self-shielding gradient sets, greater access to cooling mechanisms, a reduction in acoustic noise due to force-balancing, a lessening of patient claustrophobia and greater patient access for clinicians.

Deformation-Space Method for the Design of Biplanar Transverse Gradient Coils in Open MRI Systems

We propose an efficient real-space algorithm for the design of biplanar transverse gradient coils for use in open magnetic resonance imaging (MRI) systems. In our method, each wire arc is represented by a closed contour (Limacon). Using parametric equations, we deform/reshape an ensemble of closed contours in a simple manner, controllable by just a few parameters. These parameters are used to define system rearrangements in the design procedure. We use an iterative optimization procedure to adjust the control parameters in order to minimize cost functions such as gradient homogeneity and inductance. Here, we compare the coil pattern designed by our deformation-space method with a pattern designed by the conventional stream function approach, and we discuss the merit of the new method.

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 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.

Controlled E-field gradient coils for MRI

Peripheral neural stimulation is a major problem in current gradient coil designs. Induced current problems in patients relate directly to gradient strength and modulation frequency. Present designs of gradient coil tend to limit ultra-high-speed imaging methods such as echo-planar imaging (EPI) and echo-volumar imaging (EVI) because of the effect of induced currents in the patient which produce neural stimulation resulting in tingling sensations and involuntary muscle twitch. Neural stimulation could also trigger epileptic fits and/or cardiac fibrillation. For reduction of induced currents, an important aspect is the coil geometry. It is desirable to design the gradient coil in such a way as to prevent closed loop circulating currents within the body. Preliminary results using a four-sector gradient coil with a rectangular geometry, operating in a low mutual coupling mode and using passive E-field control, indicate significant reduction of the E-field within the subject volume of the coil, leading to a reduction of induced currents in a patient. Such a reduction allows safer operation using higher magnetic gradient strengths together with faster scans. Currently very fast scans are prohibited by virtue of the neural stimulation effects produced in present standard coil geometries.

Computing Lorentz forces generated by gradient coils in an open MRI system

Magnetic Resonance Imaging requires the use of gradient coils to perturb the main homogeneous static magnetic field. Sequential perturbations in each of the orthogonal cartesian directions results in the magnetic field becoming spatially encoded. The fast switching of the gradient coils produces large Lorentz forces which act on the coils. These unwanted forces dissipate as acoustic noise at sound pressure levels upward of 130dB. We present a method for designing open magnetic resonance imaging coils that produce reduced Lorentz forces whilst still producing highly accurate gradient fields. We show that when gradient fields of approximately 1% are considered, a linearised regularisation solves for the Fourier series coefficients. References R. Turner. A target field approach to optimal coil design. J. Phys. D: Appl. Phys., 19, 1986, pages L147--L151. doi:10.1088/0022-3727/19/8/001 M. A. Brideson, L. K. Forbes and S. Crozier. Determining complicated winding patterns for shim coils using stream functions and the target-field method. Concepts in Magnetic Resonance, 14(1), 2002, pages 9--18. doi:10.1002/cmr.10000 P. Mansfield, P. M. Glover and J. Beaumont. Sound generation in gradient coil structures for mri. Magnetic Resonance in Medicine, 39(4), 1998, 539--550. doi:10.1002/mrm.1910390406 P. Mansfield, B. Haywood and R. Coxon. Active Acoustic Control in Gradient Coils for mri. Magnetic Resonance in Medicine, 46(4), 2001, 807--818. doi:10.1002/mrm.1261 M. A. Brideson, L. K. Forbes and S. Crozier. Winding patterns for actively shielded shim coils with asymmetric target-fields. Measurement Science and Technlogy, 14, 2003, pages 484--493. doi:10.1088/0957-0233/14/4/312 L. K. Forbes, M. A. Brideson, S. Crozier and P. T. While, An analytical approach to the design of quiet cylindrical asymmetric gradient coils in mri. Concepts in Magnetic Resonance Part B. 31B(4), 2007, 218--236. doi:10.1002/cmr.b.20095.

Boundary Element Method for an Inverse Problem in Magnetic Resonance Imaging Gradient Coils

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 for use in magnetic resonance imaging (MRI) and is formulated using its corresponding integral repre- sentation according to potential theory. A novel boundary element method (BEM) which employs linear interpolationon quadratic surfaces and also satisfies the continuity equation for the current density, i.e. a divergence-free BEM, is presented. Since thediscretised BEM systemis ill-posedand hence the associated least-squares solution may be inaccurate and/or physically meaningless, the Tikhonov regularization method is employed in order to retrieve accurate and physically correct

An analytical approach to the design of quiet cylindrical asymmetric gradient coils in MRI

AbstractWe consider the design of asymmetric gradient coils in a conventional cylindrical bore magnetic resonance imaging system. The gradient coils are switched on and off during the scan, and are therefore subject to Lorentz forces that cause them to buckle during the switching process. This in turn creates a pressure wave within the coil, and that gives rise to acoustic noise. We present a simplified linearized model for the deflection of the coil due to electromagnetic forces, which is amenable to solution using analytical methods. Closed‐form solutions for the coil deflection and the pressure pulse and noise level within the coil are obtained. These are used to design new coil winding patterns so as to reduce the acoustic noise. Sample results are shown both for unshielded and shielded gradient coils. Extensions of this model are indicated, although it is suggested that the advantages of the present closed‐form solutions might then not be available, and fully numerical solutions may be needed instead. © 2007 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 31B: 218–236, 2007

Calculating the movement of MRI coils, and minimizing their noise

The design of gradient coils within Magnetic Resonance Imaging equipment is considered. These coils produce linear magnetic fields in each of the three orthogonal directions in physical space. In addition, they are turned on and off repeatedly to enhance the clarity of the image, but this produces a great deal of noise within the coil, as its shape distorts under the influence of Lorentz forces. We present a method for calculating the movement of the coil in the background magnetic field, and estimating the consequent noise levels. This involves solving for the current density in the coil coupled with equations for its elastic deformation, along with acoustic equations for the pressure in the surrounding air. Winding patterns are designed to minimize the noise produced by the Magnetic Resonance Imaging coil. References L. M. Delves and J. L. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, Cambridge, 1985. L. K. Forbes and S. Crozier. A novel target-field method for finite-length magnetic resonance shim coils: Part 1: Zonal shims. J. Phys. D: Appl. Phys., 34:3447--3455, 2001. doi:10.1088/0022-3727/34/24/305 L. K. Forbes and S. Crozier. A novel target-field method for finite-length magnetic resonance shim coils: Part 2: Tesseral shims. J. Phys. D: Appl. Phys., 35:839--849, 2002. doi:10.1088/0022-3727/35/9/303 L. K. Forbes 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., 36:68--80, 2003. doi:10.1088/0022-3727/36/2/302 E. M. Haacke, R. W. Brown, M. R. Thompson and R. Venkatesan. Magnetic Resonance Imaging: Physical principles and sequence design. Wiley, New York, 1999. J. Jin. Electromagnetic Analysis and Design in Magnetic Resonance Imaging. Biomedical Engineering Series, CRC Press, Boca Raton, 1999. P. Mansfield, B. Haywood and R. Coxon. Active acoustic control in gradient coils for MRI. Magn. Reson. Med., 46:807--818, 2001. C. K. Mechefske, G. Yao, W. Li, C. Gazdzinski and B. K. Rutt. Modal analysis and acoustic noise characterization of a 4T MRI gradient coil insert. Concepts Magn. Reson. B, 22:37--49, 2004. doi:10.1002/cmr.b.20013 W. Shao and C. K. Mechefske. Acoustic analysis of a gradient coil winding in an MRI scanner. Concepts Magn. Reson. B, 24:15--27, 2005. doi:10.1002/cmr.b.20023 W. Shao and C. K. Mechefske. Analysis of the sound field in finite length infinite baffled cylindrical ducts with vibrating walls of finite impedance. J. Acoust. Soc. Am., 117:1728--1736, 2005. A. P. Boresi and K. P. Chong. Elasticity in engineering mechanics. Elsevier, New York 1987. R. Turner. A target field approach to optimal coil design. J. Phys D: Appl. Phys., 19:147--151, 1986. M. T. Vlaadingerbroeck and J. A. den Boer. Magnetic Resonance Imaging: Theory and practice, 3rd edition. Springer Verlag, Berlin, 2003. M. A. Brideson, L. K. Forbes and S. Crozier. Determining complicated winding patterns for shim coils using stream functions and the target-field method. Concepts Magn. Reson., 14:9--18, 2002. doi:10.1002/cmr.10000 B. L. W. Chapman and P. Mansfield. A quiet gradient-coil set employing optimized, force-shielded, distributed coil designs. J. Magn. Reson. B, 107:152--157, 1995. S. Crozier and D. M. Doddrell. Gradient-coil design by simulated annealing. J. Magn. Reson. A, 103:354--357, 1993.

Three-Dimensional Gradient Coil Structures for Magnetic Resonance Imaging Designed Using Fuzzy Membership Functions

The design of high-performance gradient coils is essential for modern magnetic resonance imaging (MRI) applications. This work presents a new and alternative design methodology that explores three-dimensional (3-D) solution space by combining fuzzy membership functions to model and reshape the coil structure, given the magnetic field and electrical and mechanical constraints. The approach includes a stream function generator, a 3-D coil structure generator, and the evaluation of an objective function. An unconventional stream function for asymmetric transverse gradient coils is defined in terms of fuzzy sets. The method was applied to design a short, unshielded asymmetric gradient coil for breast imaging. The resulting dome-shape coil has superior gradient performance compared to a standard fingerprint coil. New quadrupolar gradient coil designs for breast imaging can also be obtained with the proposed method. The paper concludes with a study of the influence of a number of design parameters on the coil performance.

Target-field method for MRI biplanar gradient coil design

Based on the target-field method of solving Fredholm integral equations of the first kind, a new approach is presented in the paper for designing gradient coils that can be used in a permanent-magnet magnetic resonance imaging (MRI) system with biplanar poles. To restrict the current distribution on the coil plane within a finite radius, the current density is pre-expanded into Fourier series by orthogonal basis functions. By setting the target-field points and Bz values over the imaging region of interest, corresponding integral equations are derived from the Biot–Savart law to calculate the current densities. They form a matrix equation, in which the unknown elements of the column vector are the Fourier coefficients for the unknown current density. As long as these target-field points are well chosen, the Fourier coefficients can be solved by inverse matrix calculation instead of the regularization method for Fredholm integral equations of the first kind. Then the current density is discretized using the stream-function method to generate the winding patterns. To verify the feasibility of this approach, the gradient magnetic field generated by the current density is calculated via the Biot–Savart law. Optimized parameters are obtained through computer simulations for some shielded and unshielded transverse gradient coils. The performance of this approach has been demonstrated as well.

Novel gradient coils designed using a boundary element method

AbstractBoundary element methods offer a powerful approach for designing gradient coils, allowing the generation of coils wounds on arbitrarily shaped surface so as to produce any form of field variation that is consistent with Maxwell's equations. These methods are based on meshing the current carrying surface into an array of boundary elements. In this work, we have extended boundary element methods that have previously been used used for coil design and integrated a powerful mesh generating program so as to produce coils with totally arbitrary geometry. Four examples are used to illustrate how the modified method provides a single versatile coil design protocol. These relate to the design of: i) shielded head gradients with highly asymmetric surface geometry that give the highest possible gradient field strengths; ii) very short, shielded gradient coils to allow improved access to the subject; iii) bi‐planar coils generating highly asymmetric, stepped magnetic fields for use in fast imaging by multiple acquisition with micro‐B0 arrays (MAMBA); iv) an insertable set of head gradient coils with shoulder cut‐outs. © 2007 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 31B: 162–175, 2007

Longitudinal gradient coil optimization in the presence of transient eddy currents

The switching of magnetic field gradient coils in magnetic resonance imaging (MRI) inevitably induces transient eddy currents in conducting system components, such as the cryostat vessel. These secondary currents degrade the spatial and temporal performance of the gradient coils, and compensation methods are commonly employed to correct for these distortions. This theoretical study shows that by incorporating the eddy currents into the coil optimization process, it is possible to modify a gradient coil design so that the fields created by the coil and the eddy currents combine together to generate a spatially homogeneous gradient that follows the input pulse. Shielded and unshielded longitudinal gradient coils are used to exemplify this novel approach. To assist in the evaluation of transient eddy currents induced within a realistic cryostat vessel, a low-frequency finite-difference time-domain (FDTD) method using the total-field scattered-field (TFSF) scheme was performed. The simulations demonstrate the effectiveness of the proposed method for optimizing longitudinal gradient fields while taking into account the spatial and temporal behavior of the eddy currents.

A hybrid optimization method for biplanar transverse gradient coil design

The optimization of transverse gradient coils is one of the fundamental problems in designing magnetic resonance imaging gradient systems. A new approach is presented in this paper to optimize the transverse gradient coils' performance. First, in the traditional spherical harmonic target field method, high order coefficients, which are commonly ignored, are used in the first stage of the optimization process to give better homogeneity. Then, some cosine terms are introduced into the series expansion of stream function. These new terms provide simulated annealing optimization with new freedoms. Comparison between the traditional method and the optimized method shows that the inhomogeneity in the region of interest can be reduced from 5.03% to 1.39%, the coil efficiency increased from 3.83 to 6.31 mT m−1 A−1 and the minimum distance of these discrete coils raised from 1.54 to 3.16 mm.

Gradient coil with active endcap shielding

AbstractShielded gradient coil assemblies produce eddy currents in surrounding metal structures. Such eddy currents can degrade image quality and produce acoustic noise. These effects are magnified for high‐field MRI scanners because of increased Lorentz forces. Conventional actively shielded gradient assemblies consist of primary and secondary coils in the shape of cylindrical shells surrounding the imaging volume. One of the principal regions of field leakage is at the ends of the gradient structure, and these fields are responsible for substantial eddy current generation. We present here a new shielded gradient coil design with an endcap current distribution having both azimuthal and radial components. Our active endcap design significantly reduces the fringe field at the cryostat inner bore. A peak fringe field reduction near the end of the gradient coil of 42% is obtained for the Z gradient relative to a traditional design without active endcaps. The reduction of peak fringe field near the end of the coil for transverse gradients (X and Y) is approximately 40%. We discuss the degree to which the suppression of peak fringe fields corresponds to a reduction in the acoustic noise generated near the end of the warm bore. © 2007 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 31B: 12–23, 2007

Toward designing asymmetric head gradient coils for high‐resolution imaging

AbstractThe aim of the research described here is to design head gradient coils using approaches previously developed by the authors. The wave equation method for designing gradient coils is used to design a transverse gradient coil winding on a cylindrical/spherical former. The results are compared with asymmetric cylindrical designs and other well‐known head gradient coil designs to establish the quality of the gradient coil winding. Comparisons of field, inductance, torque, and other quality factors are made before conclusions are drawn about the adequacy of such a winding pattern to perform high‐resolution imaging of the head. We show that it is possible to generate a robust winding pattern that has good efficiency and produces a sufficiently large imaging volume. © 2007 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 31B: 1–11, 2007

A Novel Target Field Method for Designing Uniplanar Self-shield Gradient Coils of Fully Open MRI Device

A new approach to the design of uniplanar self-shield gradient coils for fully open magnetic resonance imaging system is proposed to build a specified gradient magnetic field in a target region. In this new design methodology, the surface current density of the coils is represented by a two-dimensional Fourier series expansion. A cost function is constructed in terms of the summation of squared error and penalty functions. Minimizing the cost function, we obtain the continuous current density distribution for the uniplanar gradient coils. Applying the stream function technique to the resulting continuous current distribution, discrete current patterns can be obtained. Employing the Biot-Savart integral equation to the discrete current loops, the gradient magnetic field has been re-evaluated in order to validate the method proposed. Using this approach, uniplanar self-shield x-gradient coils which are capable of creating a gradient field of 40 mT/m have been designed. The results indicate that with this method a magnetic field with high level of linearity and a satisfied screening effect are obtained.