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IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4,JULY 2004 1929 Novel Target-Field Method for Designing Shielded Biplanar Shim and Gradient Coils Larry K. Forbes and Stuart Crozier, Member, IEEE Abstract—A new method is presented here for the systematic design of biplanar shielded shim and gradient coils, for use in mag- netic resonance imaging (MRI) and other applications. The desired target field interior to the coil is specified in advance, and a winding pattern is then designed to produce a field that matches the target as closely as possible. Both gradient and shim coils can be designed by this approach, and the target region can be located asymmet- rically within the coil. The interior target field may be matched at two or more interior locations, to improve accuracy. When shields are present, the winding patterns are designed so that the fields exterior to the biplanar coil are made as small as possible. The method is illustrated here by the design of some transverse gra- dient and shim coils. Index Terms—Biot–Savart law, biplanar coils, integral equation, magnetic resonance imaging, target field, winding patterns. I. INTRODUCTION T HIS paper presents a method for designing biplanar coils that produce a desired stationary magnetic field that is specified in advance. A biplanar coil consists simply of wind- ings arranged on two parallel plates, and the desired field is pro- duced in some region between them. Shields can also be present, and these consist simply of another pair of plates parallel to the primary plates, placed farther out from the region of interest. The shields serve the dual purposes of isolating the primary coil from stray external magnetic fields, and of minimizing exterior fields generated by the coil. The primary purpose intended for the coils presented here in- volves their use in magnetic resonance imaging (MRI) equip- ment. In that application, biplanar coils present an opportunity to carry out high-resolution medical imaging in a manner that nevertheless reduces the claustrophobia experienced by many patients (see Fishbain et al. [1]) and also allows access by med- ical specialists. However, the applicability of the present tech- nique is by no means restricted to MRI technology, and is avail- able in any equipment for which parallel plates might be used to generate a desired magnetic field. Thus, it is envisaged that the method presented here might also find application in the design of wiggler magnets in synchrotron technology, as discussed by Cover et al. [2], for example. In MRI applications, which are the primary focus of the tech- nique presented here, a patient is placed within a strong and Manuscript received January 14, 2004; revised March 24, 2004. L. K. Forbes is with the School of Mathematics and Physics at the University of Tasmania, Hobart 7001, Australia (e-mail: [email protected]). S. Crozier is with the School of Information Technology and Electrical Engineering, University of Queensland, Queensland 4072, Australia (e-mail: [email protected]). Digital Object Identifier 10.1109/TMAG.2004.828934 homogeneous static magnetic field. This causes the otherwise randomly oriented magnetic moments of the protons, in water molecules within the body, to precess around the direction of the applied field. The part of the body in the homogeneous re- gion of the magnet is then irradiated with radio-frequency (RF) energy, causing some of the protons to change their spin orien- tation. When the RF energy source is removed, the protons in the sample return to their original configuration, inducing mea- surable signal in a receiver coil tuned to the frequency of pre- cession. This is the magnetic resonance (MR) signal. Most im- portantly, the frequency at which protons absorb the RF signal depends on the background magnetic field. In MRI applications, the strong magnetic field is perturbed slightly by the presence of the patient’s body. To correct for this effect, shim coils are used to adjust the magnetic field so as to generate the best possible final image. The field within the spec- ified target volume is usually represented in terms of spherical harmonics, and the impurities in the field are then represented in terms of the coefficients of an expansion in these harmonics. Shim coils are therefore designed to correct a perturbed mag- netic field by producing a particular spherical harmonic that can be added to the background magnetic field, so as to cancel the effect of a certain harmonic caused by an impurity. There may be many such coils in an MRI device, each correcting for a par- ticular spherical harmonic in the impurity. The design task for gradient and shim coils is therefore to determine the winding pattern on the coil such that the desired magnetic field will be produced in a designated region within the coil. In MRI applications, the coil is usually wound on a cylindrical former; this has certain advantages in terms of the quality of the image that is finally produced. A detailed discus- sion of conventional cylindrical coils in MRI is given in the book by Jin [3]. Possibly the best-known method for designing gra- dient and shim windings for cylindrical coils is the “target-field” approach due to Turner [4], [5]. This technique specifies the de- sired “target” field inside the cylinder in advance, and then em- ploys Fourier transform methods to calculate the current density on the surface of the coil, so as to generate the target magnetic field. The ill-conditioned nature normally expected from such an inverse problem is overcome by the Fourier transform tech- nique, which essentially assumes the coil formers are notionally infinite in length. In practice, however, this assumption can usu- ally be circumvented with an appropriate choice of current-den- sity function, which in turn sometimes requires the use of certain smoothing functions in the Fourier space. A similar method for designing coils has been advanced by Forbes et al. [6] and Forbes and Crozier [7]. Their approach is intended to account for the true (finite) length of the coil ex- 0018-9464/04$20.00 © 2004 IEEE

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Page 1: Novel Target-Field Method for Designing Shielded Biplanar Shim and Gradient Coils

IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004 1929

Novel Target-Field Method for Designing ShieldedBiplanar Shim and Gradient Coils

Larry K. Forbes and Stuart Crozier, Member, IEEE

Abstract—A new method is presented here for the systematicdesign of biplanar shielded shim and gradient coils, for use in mag-netic resonance imaging (MRI) and other applications. The desiredtarget field interior to the coil is specified in advance, and a windingpattern is then designed to produce a field that matches the targetas closely as possible. Both gradient and shim coils can be designedby this approach, and the target region can be located asymmet-rically within the coil. The interior target field may be matched attwo or more interior locations, to improve accuracy. When shieldsare present, the winding patterns are designed so that the fieldsexterior to the biplanar coil are made as small as possible. Themethod is illustrated here by the design of some transverse gra-dient and shim coils.

Index Terms—Biot–Savart law, biplanar coils, integral equation,magnetic resonance imaging, target field, winding patterns.

I. INTRODUCTION

THIS paper presents a method for designing biplanar coilsthat produce a desired stationary magnetic field that is

specified in advance. A biplanar coil consists simply of wind-ings arranged on two parallel plates, and the desired field is pro-duced in some region between them. Shields can also be present,and these consist simply of another pair of plates parallel to theprimary plates, placed farther out from the region of interest.The shields serve the dual purposes of isolating the primary coilfrom stray external magnetic fields, and of minimizing exteriorfields generated by the coil.

The primary purpose intended for the coils presented here in-volves their use in magnetic resonance imaging (MRI) equip-ment. In that application, biplanar coils present an opportunityto carry out high-resolution medical imaging in a manner thatnevertheless reduces the claustrophobia experienced by manypatients (see Fishbain et al. [1]) and also allows access by med-ical specialists. However, the applicability of the present tech-nique is by no means restricted to MRI technology, and is avail-able in any equipment for which parallel plates might be used togenerate a desired magnetic field. Thus, it is envisaged that themethod presented here might also find application in the designof wiggler magnets in synchrotron technology, as discussed byCover et al. [2], for example.

In MRI applications, which are the primary focus of the tech-nique presented here, a patient is placed within a strong and

Manuscript received January 14, 2004; revised March 24, 2004.L. K. Forbes is with the School of Mathematics and Physics at the University

of Tasmania, Hobart 7001, Australia (e-mail: [email protected]).S. Crozier is with the School of Information Technology and Electrical

Engineering, University of Queensland, Queensland 4072, Australia (e-mail:[email protected]).

Digital Object Identifier 10.1109/TMAG.2004.828934

homogeneous static magnetic field. This causes the otherwiserandomly oriented magnetic moments of the protons, in watermolecules within the body, to precess around the direction ofthe applied field. The part of the body in the homogeneous re-gion of the magnet is then irradiated with radio-frequency (RF)energy, causing some of the protons to change their spin orien-tation. When the RF energy source is removed, the protons inthe sample return to their original configuration, inducing mea-surable signal in a receiver coil tuned to the frequency of pre-cession. This is the magnetic resonance (MR) signal. Most im-portantly, the frequency at which protons absorb the RF signaldepends on the background magnetic field.

In MRI applications, the strong magnetic field is perturbedslightly by the presence of the patient’s body. To correct for thiseffect, shim coils are used to adjust the magnetic field so as togenerate the best possible final image. The field within the spec-ified target volume is usually represented in terms of sphericalharmonics, and the impurities in the field are then representedin terms of the coefficients of an expansion in these harmonics.Shim coils are therefore designed to correct a perturbed mag-netic field by producing a particular spherical harmonic that canbe added to the background magnetic field, so as to cancel theeffect of a certain harmonic caused by an impurity. There maybe many such coils in an MRI device, each correcting for a par-ticular spherical harmonic in the impurity.

The design task for gradient and shim coils is therefore todetermine the winding pattern on the coil such that the desiredmagnetic field will be produced in a designated region withinthe coil. In MRI applications, the coil is usually wound on acylindrical former; this has certain advantages in terms of thequality of the image that is finally produced. A detailed discus-sion of conventional cylindrical coils in MRI is given in the bookby Jin [3]. Possibly the best-known method for designing gra-dient and shim windings for cylindrical coils is the “target-field”approach due to Turner [4], [5]. This technique specifies the de-sired “target” field inside the cylinder in advance, and then em-ploys Fourier transform methods to calculate the current densityon the surface of the coil, so as to generate the target magneticfield. The ill-conditioned nature normally expected from suchan inverse problem is overcome by the Fourier transform tech-nique, which essentially assumes the coil formers are notionallyinfinite in length. In practice, however, this assumption can usu-ally be circumvented with an appropriate choice of current-den-sity function, which in turn sometimes requires the use of certainsmoothing functions in the Fourier space.

A similar method for designing coils has been advanced byForbes et al. [6] and Forbes and Crozier [7]. Their approach isintended to account for the true (finite) length of the coil ex-

0018-9464/04$20.00 © 2004 IEEE

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1930 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

plicitly, but likewise involves approximations based on the useof Fourier series. Nevertheless, it is capable of designing coils,for target fields located asymmetrically with respect to the coillength, in a very systematic fashion.

Coils of finite length can also be designed directly usingthe approach of Crozier and Doddrell [8]. They used theBiot–Savart law to calculate the magnetic field resultingdirectly from a collection of wires wound on a former. Theinverse problem of arranging the wires to produce a desiredtarget magnetic field inside the coil was solved using “simu-lated annealing,” which is a stochastic optimization strategy.The method is extremely robust and can accommodate manytypes of constraints easily, simply by adding them to thepenalty function. On the other hand, it is likely that complicatedmagnetic fields (with tesseral components, for example) wouldbe very difficult to design by this method, particularly in viewof the number of numerical iterations required in the simulatedannealing technique.

In a series of three recent papers, a new method has been pre-sented for designing conventional cylindrical coils in such a waythat the exact finite-length geometry of the coil is accountedfor, without approximation. This technique appears in Forbesand Crozier [9]–[12]. In this approach, the Biot–Savart law isused in a form appropriate to a current sheet distributed over thecylindrical surface of the coil, and an inverse problem is solvedin which the resulting magnetic field is specified in advance (asa desired target field), and the required current density on thecoil is found by solving an integral equation. As expected, thegoverning equations are so ill-conditioned as to be incapable ofyielding a solution in the usual sense; however, this difficultyis overcome using a regularization approach similar to the fa-mous Tikhonov method: see Delves and Mohamed [13, p. 307].This approach works extremely well in practice and has beenused to design a range of different cylindrical shim coils, withasymmetrically located target fields. Once the current-densitysheet on the cylinder has been determined by this technique, astreamfunction method is immediately available for designingthe complicated winding patterns automatically. For further de-tails, see the tutorial article by Brideson et al. [14].

A similar approach has been presented very recently by Greenet al. [15], and has been adapted to the design of “uniplanar”coils. A uniplanar coil consists simply of windings located ona single plane, and is intended for producing a desired gradientfield in a small volume adjacent to the coil. A strong motivationfor this work is again evidently the desire to create truly openMRI systems, as discussed above. Petropoulos [16], [17] hasalso presented a method for designing uniplanar coils, based onthe use of Fourier transforms. Mathematically, this techniqueassumes that the plane of the coil is of infinite extent, but usesa smoothing technique (“apodization”) to confine the current toa region of acceptable size.

Biplanar coils consist of windings placed on parallel planes,and the magnetic field of interest is created in the space betweenthem. They also offer the possibility of more open MRI systems.Some designs have been presented by Martens et al. [18] forinsertable biplanar gradient coils. These authors also assumedplates of infinite extent, so that a solution based on Fourier trans-forms was again available, and they computed some winding

Fig. 1. Diagram illustrating the biplanar coil. The primary coil is located onthe pair of planes x = �a, and the shield coil consists of windings on theparallel planes x = �b. The desired target field is specified on the two sets ofinterior planes x = �c and x = �c and there is an exterior zero target fieldimposed on the plane x = �c . The coordinate system is indicated, in whichthe z axis lies along the center of the coil.

patterns for symmetric gradient coils. This type of approachwas extended by Crozier et al. [19] to allow for the presenceof shields exterior to the primary biplanar coil. A similar tech-nique has been used by Petropoulos [20], so that the presence ofan external secondary winding set of coils may also be incorpo-rated. In that work, the thrust forces on each coil set due to thepresence of the other was minimized.

In the present paper, the design philosophy of Forbes andCrozier [11], [12] is applied to the design of biplanar target coils.Depending on the type of field required, the windings on the op-posing parallel plates of the biplanar coil may be either counter-wound or else wound in phase, and shields may also be present.The required target field may be placed at an arbitrary asym-metric location within the coil, and the finite size of each coil isaccounted for explicitly in the method.

II. GOVERNING EQUATIONS

Fig. 1 outlines the geometry of the coils, the shields, and thetarget locations for the biplanar coil system to be modeled here.The primary coils are located on the parallel planes .These coils are taken here to be rectangular in shape, and tooccupy the region , . The shieldingcoils, also shown in Fig. 1, lie on the two planes . Again,these are assumed to have rectangular geometry, and to occupythe region , . Here,the two constants and are dimensionless scaling ratios(relative to the size of the primary coils).

The target zones are also indicated in Fig. 1. We choose tospecify the target field on two sets of inner planes, followingForbes and Crozier [11], [12], since this provides the opportu-nity for greater accuracy in matching the desired target field. Thetwo sets of inner target zones are located on the planes ,

with . Each of these target zonesis specified on the rectangular region ,

, in which the dimensionless numbers andsatisfy the constraints .

There is also an outer target zone, located on the parallelplanes , as shown in the diagram in Fig. 1. On thisregion, the imposed target field is zero, representing the effects

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FORBES AND CROZIER: NOVEL TARGET-FIELD METHOD FOR DESIGNING SHIELDED BIPLANAR SHIM AND GRADIENT COILS 1931

of the shields. These target zones are again taken to be rect-angular in shape, and to lie over the area ,

. The constants and are again dimension-less scaling ratios.

The symbol A/m will be used to denote the mag-netic field vector at a field point either inside oroutside the coil. On either the primary or the shield coil, thereis a current-density sheet A/m at the source point

on the primary coil, or at a sourcepoint on the shields. The magnetic field pro-duced by the current sheets in the coil at any field point is thengiven by the generalized Biot–Savart law

(2.1)

(see, e.g., [21, p. 144, eqn. 5.48]). The extra factor of two inthis equation accounts for the fact that the current flows as twosheets on each surface of the coil. It is assumed that no magneticmaterial is present.

The current density vector has a component in the directionand another in the direction, and so it can be represented inthe form

(2.2)

in which the two vectors and are the usual unit vectors inthe and directions, respectively. A similar notation is usedfor the current density on the shields. The magneticfield induced by these current densities is likewise expressed inCartesian form

(2.3)The two components of the current density in (2.2) are relatedthrough the steady-state continuity equation

(2.4)

on both of the primary coil planes. A similar equation holds forthe current-density components on the shielding planes.

Equations (2.2) and (2.3) may be substituted into theBiot–Savart law (2.1) to yield expressions for the three compo-nents , , and of the magnetic field. The calculationis straightforward, but the final equations are lengthy andso all three will not be written out in full here. Instead, thispaper will illustrate the general technique by reference only tothe transverse component, and coils will be designed tomatch target fields specified on this component of the magneticfield. Nevertheless, it should be emphasized that the methodpresented here is capable of being applied equally to any of thethree magnetic field components , , or , or indeed to

some linear combination of all three. The first component ofthe magnetic field is given from (2.1) by

(2.5)

Similar expressions are also obtained for the other two com-ponents of the magnetic field vector, and in fact the methodspresented in this paper have also been used to design coils forgenerating target fields, although this will not be discussedfurther here.

Depending upon the type of field desired, the biplanar coilsare either counterwound or else wound in phase. If the chosentarget field has odd symmetry in , then the windings on op-posing planes must be counterwound, so that

and

for odd fields (2.6a)

Alternatively, if the target field has even symmetry in , thenthe opposing planes of the primary and shield coils are woundin phase. This is expressed by the mathematical relationships

and

for even fields (2.6b)

Once the particular target field has been chosen, either ex-pression (2.6a) for an odd target field or (2.6b) for an even fieldis substituted into the relation (2.5). This results in an integralequation for designing the current densities on the coil, when thedesired magnetic field component is specified in advance.

III. REPRESENTATION OF THE SOLUTION

The continuity equation (2.4) on the primary coils permitsa streamfunction to be defined immediately, bymeans of the relations

on (3.1a)

It can be seen at once that (3.1a) satisfies the continuity condi-tion (2.4) identically. An equation similar to (2.4) also holds on

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1932 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

the shield coils, and likewise leads to the definition of a secondstreamfunction from the equations

on (3.1b)

These streamfunctions are dimensionally equivalent to cur-rents, and Brideson et al. [14] have shown that contours of thestreamfunctions immediately give the required winding patternson the coils.

It may be shown that the two current density components andthe streamfunction on the primary coils can be represented toany desired accuracy by the mathematical expressions

on (3.2)

Here, the sets of coefficients are to be determined, andthe integers and may be chosen as large as is requiredfor purposes of numerical accuracy. The three expressions in(3.2) satisfy the continuity equation (2.4) and the two relationsin (3.1a). In addition, it can be seen that onand on , as is required on physical grounds.The streamfunction vanishes on all four boundaries of therectangular primary coil.

Similarly, the two components of the current density and thestreamfunction on the shielding coils can be found from theanalogous formulas

on

(3.3)

Again, the sets of coefficients are to be determined.

IV. NUMERICAL SOLUTION—ILLUSTRATION FOR

ANTISYMMETRIC TARGET FIELDS

As has been previously discussed, in connection with (2.6),the method of this paper has been applied to the design of coilsthat produce magnetic fields that are either symmetric or anti-symmetric with respect to . This requires biplanar coils thatare either wound in phase or counterwound, respectively, wheneach involves the same geometry as indicated in Fig. 1. Bothtypes of coil are of interest, but for brevity only counterwoundcoils and their resulting antisymmetric fields will be discussedhere.

For antisymmetric fields, the relationship between the currentdensities on the left and right planes of the biplanar coils (andtheir shields) is that given by (2.6a). When this expression andthe formulas (3.2) and (3.3) are substituted into the integral rela-tion (2.5), it is possible to derive a formal relationship betweenthe magnetic field component and the sets of unknown coeffi-cients and . Equation (2.5) becomes

(4.1)

The functions in this expression are defined to be

(4.2)

Equation (4.1) is of little practical value in its present form,however, since it is both overdetermined and ill-conditioned.This is a well-known difficulty with inverse problems of thistype, and is documented extensively in the literature: see, forexample, Delves and Mohamed [13]. (For even target fields,where the biplanar coils are wound in phase, the only changeneeded in the above is to replace the minus sign before the lastterm of (4.2) with a plus sign).

As in Forbes and Crozier [11], [12], the coefficients in (4.1)are determined using least-squares minimization and a regular-ization strategy. The desired target field on the inner two sets oftarget planes and shown in Fig. 1 is writtenas

for

Page 5: Novel Target-Field Method for Designing Shielded Biplanar Shim and Gradient Coils

FORBES AND CROZIER: NOVEL TARGET-FIELD METHOD FOR DESIGNING SHIELDED BIPLANAR SHIM AND GRADIENT COILS 1933

There is also a third target field on the outer set of planes at, namely

A total error function is now defined, and takes the form

(4.3)

In this expression, the first three terms on the right-hand side(RHS) represent the squared error in the satisfaction of the gov-erning equation (4.1) on the three sets of target planes shown inFig. 1. These terms may be written as

(4.4)

The symbol beneath the integrals in (4.4) refers to the targetzone on each plane , , as indicated in Fig. 1and described above.

The remaining two terms on the RHS of (4.3) are penaltyterms in the regularization process, and these may be chosenlargely at the discretion of the designer. Thus, the two con-stants and are regularization parameters; they must bechosen to be small enough to ensure that the expression (4.1) isstill represented to a high degree of accuracy, but neverthelesslarge enough to give a well-conditioned system of equations. Wehave found that values of the order of 10 are appropriate tothis purpose. In addition, the two functions and arepenalty functions. They can be chosen by the designer to corre-spond to physical parameters, such as the inductance of the coilor the power it consumes, and minimized accordingly, as part ofthe system (4.3).

In this paper, as in Forbes and Crozier [11], [12], we chooseto make use of slightly more abstract penalty functionsand . Here, we choose the total squares of the curvaturesof the streamfunctions on the primary and shield coils, for thereason that minimizing these quantities is equivalent to makingthe winding patterns on each coil as smooth as possible. Coilswith this property are expected to be more amenable to practicalmanufacture. Thus, the penalty function on the primary is takento be

(4.5)

The penalty function for the shield coils is defined similarly,with replaced by in (4.5). The expressions (3.2)and (3.3) for the two streamfunctions are inserted into (4.5), and

after some algebra, the two curvature penalty functions take thefinal forms

(4.6)

The positive-definite error function in (4.3) is now mini-mized, by requiring that

and

(4.7)The derivatives of the quantities in (4.4) and (4.6) are calcu-

lated explicitly, and the system (4.7) then leads to a set of linearalgebraic equations for the unknown coefficients and .This system can be represented as

(4.8)

Here, the sets of coefficients and so on, and the RHSterms and are known in terms of integrals over eachtarget zone. They are lengthy expressions and so will not bepresented here, in the interests of brevity. The system (4.8) maybe represented in block-matrix form as

(4.9)

in which the terms , and so on, in the coefficient matrixare matrices, and the remaining quantities are allvectors of length . The matrix system (4.9) can be solvedby standard software, since it is rendered well-conditioned byappropriate choice of parameters and in (4.3).

In order to use the design algorithm described in this paper, itis therefore necessary to evaluate the elements , and so on,that appear in the large matrix on the left-handside of (4.9). Each of these terms involves integrals over the ap-propriate target regions, and these integrals are evaluated usingtrapezoidal-rule quadrature. In addition, each of these integralscontains the functions defined in (4.2) in the integrand;these functions must also be evaluated by numerical quadrature.The solution of (4.9) for the coefficients and is there-fore a reasonably demanding numerical task, although results ofgood accuracy can generally be obtained with as few asand coefficients.

V. RESULTS AND EXAMPLE DESIGNS

The methods described here have been used to design variousbiplanar shielded coils, to produce symmetrical and asymmet-rically located fields of practical interest. As discussed in Sec-

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1934 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

tion IV, the technique will be illustrated here to design targetfield components (the component normal to the biplanarcoil surfaces), although it has also been applied successfully tothe design of coils for generating target fields.

As indicated in Section II, the interior target zones are locatedon the two pairs of planes , and positionedover the rectangular regions ,on each plane. The dimensionless numbers and satisfy theconstraints . Following Forbes and Crozier[7], it will prove convenient to locate the target fields asymmet-rically with respect to the coordinate by defining the new vari-able

(5.1)

This nondimensional coordinate is centered with respect to thetarget field , and it has the advantage that it allowsthe usual formulas for spherical harmonic target magnetic fieldsto be used naturally over the target region. A list of these fieldsmay be found in the paper by Roméo and Hoult [22], for bothzonal and tesseral target fields.

A. (Constant Field) Biplanar Coil

In this section, the technique of this paper will be used to de-sign shielded and unshielded biplanar coils that are intended toproduce symmetrically located (constant) fields within theprimary biplanar coil. For this coil, the target fields are simplyconstants, and so is simply a constant on each of the threepairs of target planes. We therefore set

(5.2)

Here, the constant transverse magnetic field within the primarycoils has been assumed to take the value A/m. Thetarget field at is set to zero, reflecting the intendedfunction of the shields. Since the field is even in the co-ordinate (it is constant), the primary coil windings on the planeat are in phase with those on , and the same istrue for the shields on the planes . (This even target fieldrequires the last term in (4.2) to take a plus sign).

To begin, the case of an unshielded symmetric coil is firstconsidered. The numerical method is the same as presented inSection IV, except that all the terms involving the Fourier coef-ficients on the shields are simply removed. Consequently,the matrix system (4.9) is replaced with the smaller block system

, which may be solved for the coefficientson the primary coil. Once these coefficients have been deter-mined, the streamfunction is then evaluated using(3.2). The appropriate winding patterns to create the desired coilare then obtained immediately, simply by drawing contours of

at constant increments, using standard software.Fig. 2 shows the winding pattern for the unshielded primary,

obtained by this method. In this calculation, the plate is takento be a square m and the primary biplanar coilis located on the planes with m. The twointerior target zones in (5.2) were specified on the sets of planes

Fig. 2. Winding pattern for a T (constant field) unshielded symmetric coil.The winding pattern was obtained by contouring the computed streamfunction.The parameters used in this calculation are L = B = 1 m, a = 0:5 m, andH = 1 A/m. The target zones are defined by the parameters � = 0:5,p = �0:5, q = 0:5, c = 0:35 m, c = 0:2 m.

defined by the conditions and m, and thetarget regions , are chosento be square and symmetrically located, with and

, . (For an unshielded coil, the zero targetfield condition on in (5.2) is omitted). The windingpattern in Fig. 2 is strongly affected by the square geometry ofthe plates, and approximately circular loops exist in each of thefour corners, to compensate for their presence. An interestingsquare-shaped winding region is also present in the center ofthe pattern, with extra winding lobes near each side.

When shields are introduced, the winding pattern changessubstantially, because the field is altered by the presence ofthe shield. Fig. 3(a) shows the effect on the primary coil ofplacing shields at m with zero target fields imposed at

m. These shields are also square and are the same sizeas the primary coils . The target region is also ofthe same size . All the other parameters pertainingto the primary coil retain the same values as for Fig. 2, so thatthis is still a symmetrically located target field. The winding pat-tern in Fig. 3(a) has now developed an eight-fold pattern in thecenter of the coil. The four regions involving looped windingsnear each of the corners are still present, as before, but with theimportant difference that the direction of current in these cornerwindings is now reversed. Thus, there is positive current on thewindings in the center of the primary coil, but negative currentin each corner.

The winding pattern for the shield in this case is shown inFig. 3(b). Here, the current has the opposite polarity to thaton the primary, so as to cancel the exterior field as nearlyas possible. Thus, the large central windings on the shieldin Fig. 3(b) possess negative current, while a small positivecurrent is present on each of the four small corner windings.

The effectiveness of this coil can be determined by investi-gating the magnetic field it produces, with particular referenceto the component. Accordingly, this field component hasbeen computed from the Fourier coefficients, using (4.1). Forease of viewing, the field on the center planeis shown in Fig. 4. Here, contours have been drawn at equal in-crements at the 5% level. The influence of the primary coils at

m and the shield coils at m is evidentfrom the figure. In addition, the three target zones are indicated

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FORBES AND CROZIER: NOVEL TARGET-FIELD METHOD FOR DESIGNING SHIELDED BIPLANAR SHIM AND GRADIENT COILS 1935

Fig. 3. Winding patterns for a T (constant field) symmetric coil, for (a) theprimary coil, and (b) the shield. The parameters used in this calculation are L =B = 1 m, a = 0:5, b = 0:75 m, and H = 1 A/m. The target zonesare defined by the parameters � = 0:5, p = �0:5, q = 0:5, c = 0:35,c = 0:2 m, and � = = 1, r = r = 1, c = 0:9 m.

Fig. 4. Contours for the H component of the magnetic field produced by theshielded T symmetric coil of Fig. 3, on the center plane y = 0. The dashedlines indicate the three target regions.

with dashed lines on the diagram. The absence of field contourswithin the two interior planes indicates that the aim of producinga constant strength field in this region has largely been achieved,at least to the 5% level, although there is a small distortion at thetop and the bottom of this target zone. Likewise, the absence offield lines beyond the exterior target planes at m onthe diagram indicates that the shields have been very successfulin suppressing lateral fields beyond the coil region, although it isevident that there is considerable field leakage above and belowthe coil in the figure.

A more detailed look at the field produced by this coil is af-forded by Fig. 5, which shows the field on a portion of

Fig. 5. Comparison of the H field on a portion of the center line (x axis),obtained with the unshielded biplanar coil of Fig. 2 (drawn with a dot-dashedline) and the shielded coil of Fig. 3 (solid line). The vertical dashed lines on thefigure indicate the locations of the three pairs of target zones.

the center line (the axis). Here, the fields produced by boththe unshielded coil of Fig. 2 and the shielded coil of Figs. 3and 4 are shown, indicated with a dot-dashed line and a solidline, respectively. It is evident that the field from the shieldedcoil is superior to the unshielded case, on a number of dif-ferent criteria. First, the shielding has dramatically reduced themagnetic field exterior to the coil, as intended. For example, at

m, the unshielded magnetic field component is ap-proximately A/m and the shielded component isabout A/m. Thus, the shielding gives a reductionin the exterior field to about 4% of the unshielded field, evenquite close to the coil; a similar factor is likewise experiencedby the other field components. In addition, the field inside theprimary coil is closer to the (constant) target field in the shieldedcase.

The methods of this paper have also been used to design coilsfor which the target field is positioned asymmetrically with re-spect to the biplanar plates. The winding pattern for an asym-metrically located constant field is displayed in Fig. 6. Here, theparameters are the same as for Fig. 3, except that nowand , so that the target field is very asymmetrically lo-cated along the axis. The primary coil is shown in Fig. 6(a),where it is clear that the target zone has been shifted toward thebottom of the picture. Positive current flows in the large wind-ings in the bottom section of the diagram, but there is negativecurrent in each of the three smaller windings at the top of thepicture.

The shield windings for this asymmetric (constant field)coil are presented in Fig. 6(b). The direction of the current isreversed in these windings, so as to cancel the magnetic fieldbeyond the shields. Thus, there is negative current in the largewindings that cover most of the coil, although a small positivecurrent flows in the two small loops in the top two corners ofthe shield pattern in Fig. 6(b).

B. ( -Gradient) Biplanar Coil

As a further illustration of the use of this technique, shieldedbiplanar coils will be designed in this section, both for sym-metric and asymmetrically located fields. The transversemagnetic field component for this case has the general form

with an arbitrary constant (see [22]).

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1936 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

Fig. 6. Winding patterns for a T asymmetric constant field coil, for (a) theprimary coil, and (b) the shield. The parameters used in this calculation are L =B = 1 m, a = 0:5, b = 0:75 m, and H = 1 A/m. The target zonesare defined by the parameters � = 0:5, p = �0:7, q = 0:1, c = 0:35,c = 0:2 m, and � = = 1, r = r = 1, c = 0:9 m.

It follows that the target fields on the three pairs of target planesused in this paper must be given by the equations

(5.3)

Here, the constant is now the maximum magnetic fieldstrength (in the component) on the plane of the primarycoil. Its value is set to 1 A/m here, for illustrative purposes. Asthis target field is now antisymmetric in , it follows that thecoils on the opposing planes of the biplanar coil (and also theshields) are counterwound. This is the case discussed explicitlyin Section IV.

The primary winding for a shielded coil in shown inFig. 7(a). In this case, the target zone is located symmetrically,in the central half of the coil ( , , ).The primary coil is square m and the shield isthe same size . The other parameters are as inFig. 3. The winding pattern in Fig. 7(a) possesses a nearly cir-cular central portion surrounded by an eight-fold pattern. Thereare again small reverse windings in each of the four corners, inwhich the current is negative.

The corresponding windings for the shield are presented inFig. 7(b). The current in this coil opposes that in the primary;

Fig. 7. Winding patterns for a T (x-gradient field) symmetric coil, for (a)the primary coil, and (b) the shield. The parameters used in this calculation areL = B = 1 m, a = 0:5, b = 0:75 m and H = 1 A/m. The target zonesare defined by the parameters � = 0:5, p = �0:5, q = 0:5, c = 0:35,c = 0:2 m, and � = = 1, r = r = 1, c = 0:9 m.

Fig. 8. H field on a portion of the center line (x axis), obtained with theshielded coil of Fig. 7. The target field is indicated by the dot-dashed line onthe figure. The vertical dashed lines indicate the locations of the three pairs oftarget zones.

thus, the central windings contain negative current and there aresmall loops in each corner on which the current is positive.

The effectiveness of the coil shown in Fig. 7 is examined inFig. 8. Here, the magnetic field component pro-duced by this coil is shown along the axis. The locations ofthe three pairs of target planes are indicated by vertical dashedlines on the figure, and the target field is shown as a dot-dashedline. Since the field is linear in for this particular harmonic,the target field in Fig. 8 is simply a straight line passing throughthe origin. The magnetic field generated by the coil is drawn inFig. 8 with a heavier solid line. It shows that the target field is

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FORBES AND CROZIER: NOVEL TARGET-FIELD METHOD FOR DESIGNING SHIELDED BIPLANAR SHIM AND GRADIENT COILS 1937

Fig. 9. Winding patterns for aT asymmetric gradient coil, for (a) the primarycoil, and (b) the shield. The parameters used in this calculation are L = B =

1 m, a = 0:5, b = 0:75 m, and H = 1 A/m. The target zones are definedby the parameters � = 0:5, p = �0:7, q = 0:1, c = 0:35, c = 0:2 m,and � = = 1, r = r = 1, c = 0:9 m.

matched very accurately inside the primary coil, and particularlywithin the inner target region (here, m).This is also confirmed by an examination of the field contourson the center plane , similarly to Fig. 4, although this isnot shown here in the interests of brevity.

Coils have also been designed using the techniques outlinedin this paper, for gradient fields located asymmetrically withinrespect to the biplanar plates. This only requires the designer tochange the values of the target-field parameters and in thealgorithm presented above. The winding patterns for such anasymmetrically positioned target field are presented in Fig. 9.Here, the off-set parameters have been chosen to beand , so that this is a very asymmetric coil indeed.The primary windings are presented in Fig. 9(a). The mainbody of the windings is now moved down toward the bottomof the picture, since the target zone is located toward that endof the coil. These windings carry positive current. There isalso a smaller isolated pattern of windings toward the top ofthe figure, and these carry negative current. Finally, anotherdistinct small winding appears in each of the two corners at thetop of the diagram, and the current is once again positive there.Thus, the shielded asymmetrically located target field requiresa primary coil with alternating patterns of current direction inthe windings at the end farthest from the target zone.

The windings on the shield coil are shown for this case inFig. 9(b). The asymmetry of the target field is again reflected inthe location of the windings presented in this diagram. Consis-

Fig. 10. Contours for the H component of the magnetic field produced bythe shielded T asymmetric coil of Fig. 9, on the center plane y = 0. Thedashed lines indicate the three target regions.

tently with the function of the shield, the current in these coilsopposes that in the primary, so that the current has negative signon the windings in Fig. 9(b).

Field contours for the component of the mag-netic field are presented in Fig. 10, again on the center plane

. Field lines are shown at the 5% level, and the threetarget regions are indicated by dashed lines. The asymmetryin the location of the interior target fields is very clear fromthe picture. The coil has been designed with the intention ofachieving a linear gradient field within the target zones insidethe primary plates, and Fig. 10 shows that this has largely beenachieved, except for some irregularity at the top and bottom ofthese target zones. This is evident from the fact that the field con-tours within the target region are evenly spaced straight lines,with the zero-field contour running directly down the middle ofthe diagram.

C. Biplanar Coil

The final illustration of the use of this technique to be pre-sented in this paper concerns the design of shielded biplanarcoils. The method has been used to design both symmetric andasymmetrically located fields, although only the latter case willbe discussed here.

The transverse component for a symmetrically locatedmagnetic field has the general form ,where is again an arbitrary constant (see [22]). For a fieldthat is asymmetrically positioned with respect to the origin, ittherefore follows that the target fields take the forms

(5.4)

Again, the constant is set to the value 1 A/m here, for il-lustrative purposes. This field (5.4) is also antisymmetric in ,as for the case discussed in Section V-B, and so the coils on theopposing planes of the biplanar coil and shield are also coun-terwound. The quantity in (5.4) is the coordinate defined in(5.1).

Fig. 11(a) shows the winding pattern for the primary coil de-signed to produce an asymmetric field within the interior

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1938 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

Fig. 11. Winding patterns for a T asymmetric shim coil, for (a) the primarycoil, and (b) the shield. The parameters used in this calculation are L = B =

1 m, a = 0:5, b = 0:75 m, and H = 1 A/m. The target zones are definedby the parameters � = 0:5, p = �0:7, q = 0:1, c = 0:35, c = 0:2 m,and � = = 1, r = r = 1, c = 0:9 m.

target region. The primary coils and the shields are both as-sumed to be square and of the same size m , as inprevious examples, and the target field is located very asymmet-rically toward the bottom of the picture, with off-set parameters

and . The streamfunction in (3.1a) hasbeen contoured to produce the windings in Fig. 11(a), and thesemay be seen to have a fascinating and complicated form, withalternating patterns of current in neighboring sections of wire.Thus, the current is negative in the loops of wire at the bottomof the figure, positive in the middle sections and negative againin the portion right at the top of the diagram.

The corresponding shield windings for this coil are given inFig. 11(b). Again, the current in this plane opposes that on theprimary coils, so that the windings at the bottom of the diagramnow contain positive current, while those at the top possess cur-rent in the opposite (negative) direction.

VI. SUMMARY

In this paper, a new method for designing rectangular biplanarshielded coils has been presented. The intention here is to usethese devices in MRI applications, although it is clearly the casethat very similar techniques may be applied to the design of anysimilar device which makes use of biplanar coils. It is a partic-ular feature of the method described here that the location ofthe target field may be chosen arbitrarily along the axis, sothat it might be positioned either symmetrically or asymmetri-cally within the coil. The technique has been illustrated here by

reference to five particular example designs, involving biplanarcoils for producing constant fields, gradient fields, and higherorder shim fields of practical interest in MRI engineering.

The approach discussed in this paper can be adapted to thedesign of different types of coils to those illustrated here. Inparticular, the technique is capable of generalization, so as toallow the target field to have any desired form at all, whether thisfield consists purely of spherical harmonics or otherwise. In ad-dition, biplanar coils consisting of circular pole pieces may alsobe designed by these techniques, simply by replacing the as-sumed double Fourier series forms (3.2) and (3.3) of the currentdensity components and streamfunctions by their correspondingrepresentations in polar coordinates. Such coil geometries willbe discussed in a future article.

REFERENCES

[1] D. Fishbain, M. Goldberg, and E. Labbe, “Long-term claustrophobiafollowing MRI,” Amer. J. Psych., vol. 145, pp. 1038–1039, 1988.

[2] R. A. Cover, G. Rakowsky, B. L. Bobbs, and P. K. Kennedy, “Undulatordesign for synchrotron radiation sources using simulated annealing,”IEEE J. Quantum Electron., vol. 31, pp. 664–672, Apr. 1995.

[3] J. Jin, Electromagnetic Analysis and Design in Magnetic Resonance En-gineering. Boca Raton, FL: CRC, 1999.

[4] R. Turner, “A target field approach to optimal coil design,” J. Phys. D,Appl. Phys., vol. 19, pp. 147–151, 1986.

[5] , “Electrical Coils,” U.S. Patent 5 289 151, 1994.[6] L. K. Forbes, S. Crozier, and D. M. Doddrell, “Asymmetric Zonal Shim

Coils for Magnetic Resonance,” U.S. Patent 6 377 148, 2002.[7] L. K. Forbes and S. Crozier, “Asymmetric zonal shim coils for magnetic

resonance applications,” Med. Phys., vol. 28, pp. 1644–1651, 2001.[8] S. Crozier and D. M. Doddrell, “Gradient-coil design by simulated an-

nealing,” J. Magn. Reson. A, vol. 103, pp. 354–357, 1993.[9] 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., vol. 34, pp. 3447–3455, 2001.

[10] , “A novel target-field method for finite-length magnetic resonanceshim coils: Part 2. Tesseral shims,” J. Phys. D, Appl. Phys., vol. 35, pp.839–849, 2002.

[11] , “A novel target-field method for magnetic resonance shim coils:Part 3. shielded zonal and tesseral coils,” J. Phys. D, Appl. Phys., vol.36, pp. 68–80, 2003.

[12] , “The Efficient Design of Asymmetric Tesseral Shim Coils forMagnetic Resonance Imaging Applications,” U.S. Patent 6 664 879,2003.

[13] L. M. Delves and J. L. Mohamed, Computational Methods for IntegralEquations. Cambridge, U.K.: Cambridge Univ. Press, 1985.

[14] M. A. Brideson, L. K. Forbes, and S. Crozier, “Determining complicatedwinding patterns for shim coils using streamfunctions and the target-field method,” Concepts Magn. Reson., vol. 14, pp. 9–18, 2002.

[15] D. Green, R. W. Bowtell, and P. G. Morris, “Uniplanar gradient coils forbrain imaging,” in Proc. Int. Soc. Mag. Reson. Med., vol. 10, 2002.

[16] L. S. Petropoulos, “Single Gradient Coil Configuration for MRI Sys-tems With Orthogonal Directed Magnetic Fields,” U.S. Patent 5 977 771,1999.

[17] , “Phased Array Planar Gradient Coil Set for MRI Systems,” U.S.Patent 6 262 576, 2001.

[18] M. A. Martens, L. S. Petropoulos, R. W. Brown, J. H. Andrews, M. A.Morich, and J. L. Patrick, “Insertable biplanar gradient coils for mag-netic resonance imaging,” Rev. Sci. Instrum., vol. 62, pp. 2639–2645,1991.

[19] S. Crozier, S. Dodd, K. Luescher, J. Field, and D. M. Doddrell, “The de-sign of biplanar, shielded, minimum energy, or minimum power pulsedB coils,” Magn. Reson. Mater. Phys., Biol. Med. (MAGMA), vol. 3, pp.49–55, 1995.

[20] L. S. Petropoulos, “Thrust Balanced Bi-Planar Gradient Set for MRIScanners,” U.S. Patent 5 942 898, 1999.

[21] B. I. Bleaney and B. Bleaney, Electricity and Magnetism, 2nded. Oxford, U.K.: Clarendon, 1965.

[22] F. Roméo and D. I. Hoult, “Magnetic field profiling: analysis and cor-recting coil design,” Magn. Reson. Med., vol. 1, pp. 44–65, 1984.