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Compact, Cylindrical, Distributed-Current, Transverse-Gradient Coils for Use in MRI
JOURNAL OF MAGNETIC RESONANCE, ARTICLE NO.
Series B 110, 158–163 (1996)
0025
Compact, Cylindrical, Distributed-Current, Transverse-Gradient Coils for Use in MRI D. G. HUGHES,* R. TESHIMA,* Q. LIU,*
AND
P. S. ALLEN†
Departments of *Physics and †Biomedical Engineering, University of Alberta, Edmonton, Alberta, T6G 2J1, Canada Received July 17, 1995
Conventional cylindrical, distributed-current, transverse-gradient coils used in MRI occupy the entire surface of the cylinder. Since the two transverse-gradient coils, x and y, must be rigidly supported and electrically isolated from one another, they occupy a significant amount of space in the radial direction, with a consequent reduction in the utilization efficiency of the magnet bore volume. In this work, transverse-gradient coil designs which allow both x and y coils to be located on the same cylindrical surface are studied. One design in particular is shown to generate a gradient which is as linear as that generated by a conventional distributed-current gradient coil over a sphere whose radius is 60% of the radius of the cylinder. q 1996 Academic Press, Inc.
INTRODUCTION
A most important advance in the design of gradient coils for use in magnetic resonance imaging and spectroscopy was the development by Turner (1, 2) of the target field approach for distributed-current cylindrical coils. Such coils can generate longitudinal or transverse gradients of excellent linearity over much larger volumes than earlier types of gradient coils. Moreover, they can be designed to minimize either the inductance or the power dissipation for a given coil efficiency (gradient strength per unit current) (2, 3). Since, for the present distributed-current designs, the current distribution in each gradient coil occupies essentially the whole surface of the cylinder and since each coil must be rigidly supported and electrically isolated from adjacent coils, a complete set of one longitudinal and two transverse coils occupies a significant amount of space in the radial direction. Such space is at a premium, not only for existing small-bore, high-field animal magnets but also for the magnets being developed as part of the trend to higher fields for studies on humans, since at any given field strength, smallerbore magnets are cheaper to build, cheaper and easier to site, and cheaper to operate. For example, a 100 cm bore 3 T magnet costs about 30% more than an 80 cm bore 3 T magnet. Also, the operating costs of a 100 cm bore magnet are about 25% higher than for an 80 cm bore magnet pos-
1064-1866/96 $12.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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sessing the same field. The utilization efficiency of the bore volume is therefore a significant economic issue. The purpose of the present paper is to report transversegradient coil designs which allow both (x and y) transversegradient coils to be located on the same cylindrical surface, thereby saving space while at the same time facilitating construction. THEORETICAL CONSIDERATIONS
Although it will always be possible to design active shielding (4, 5) that will largely eliminate eddy-current fields (6) caused by switched currents in such coils, for the sake of brevity in introducing the new design concept, we shall only consider unshielded gradient coils. A distributed current flowing in a thin cylindrical shell of radius a, whose axis is taken to be the z direction, will generate a z component of magnetic field at a point (r, f, z) inside the cylinder given by (1, 2, 5, 7) `
Bz (r, f, z) Å 0 ( m0a/2p ) ∑ m Å0`
1 e
`
ÉkÉdkJ fm (k)
0`
e Im (ÉkÉr)K *m (ÉkÉa),
imf ikz
[1]
where Im (z) and Km (z) are modified Bessel functions of the first and second kind, the prime indicating a derivative with respect to the argument. Also, J fm (k) is the Fourier transform of the azimuthal component of the current density, Jf(f, z), given by J fm (k) Å (1/2p )
*
p
df e 0imf
0p
*
`
dz e 0ikz Jf(f, z). [2]
0`
The Fourier transform of the longitudinal component of the current density, Jz ( f, z), is similarly given by J zm (k) Å (1/2p )
*
p
0p
df e 0imf
*
`
dz e 0ikz Jz ( f, z). [3]
0`
It follows from the equation of continuity, div J Å 0, that
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J zm (k) Å 0 (m/ka)J fm (k).
[4]
equations can be replaced by cosines. Equations [1], [7], and [8] then become
The inductance of the coil is given by 2
L Å 0 ( m0a /I ) `
∑
1
m Å0`
*
`
n Å1
dkÉJ (k)É I *m (ÉkÉa)K *m (ÉkÉa), [5] m f
∑ m Å1,3,5. . .
*
`
k 2 dk
0
1 [K *m (ka)/I *m (ka)]Im (krn )Im (krl )
2
0`
1 cos mfl cos mfn cos kzl cos kzn ,
where I is the current entering the coil, and the power dissipation P is given by `
P Å 0 ( ra/t) ∑ mÅ0`
*
N
` m f
2
2
dkÉJ (k)É [1 / (m/ka) ], [6]
0`
∑ ln Im (krn )cos mfn cos kzn ,
[10]
n Å1
and
N
`
Bz (rl , fl , zl ) Å 0 ( m0 I/8p ) ∑ ln { ∑ 2
n Å1
m Å0`
*
` 2
k dk
0`
1 [K *m (ÉkÉa)/I *m (ÉkÉa)]Im (ÉkÉrn ) 1 Im (ÉkÉrl )exp[im( fl 0 fn )] 1 exp[ik(zl 0 zn )]},
[7]
where Bz (rl , fl , zl ) is the target field at the point (rl , fl , zl ), l takes the values 1, 2, . . . , N, and N is the total number of target field points. By solving the N simultaneous equations in [7] for the Lagrange multipliers ln , J fm (k) can be found using the expression J fm (k) Å {ÉkÉI/[4paI *m (ÉkÉa)]} N
∑ ln exp( 0imfn )exp( 0ikzn )Im (ÉkÉrn ), [8] n Å1
from which J zm (k) can be found using Eq. [4]. The components Jf(f, z) and Jz ( f, z) of the current density that will generate the desired field at the target points can then be found by inverse Fourier transformation of J fm (k) and J zm (k). The theory so far presented is general and can be used to obtain the distributed current that will, in principle, create any desired field profile. In order to generate an x gradient whose field is zero at the center of the cylinder, Bz must be invariant with respect to the transformations f r 0 f and z r 0z. Moreover, Bz must change sign with the transformation f r f / p. It follows that terms involving even values of m must vanish and the exponentials in the preceding
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[9]
J fm (k) Å [kI/4paI *m (ka)] 1
where r is the resistivity and t is the thickness of the material composing the coil. The inductance, the power dissipation, or a suitably weighted sum of the two can be minimized using the targetfield approach, yielding, in the case of minimum inductance,
1
`
Bz (rl , fl , zl ) Å 0 ( m0 I/2p 2 ) ∑ ln
2
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`
Bz (r, f, z) Å 0 (2m0a/ p )
∑ m Å1,3,5. . .
*
`
kdkJ fm (k)
0
1 cos mf cos kz Im (kr)K *m (ka).
[11]
In order to design transverse-gradient coils such that x and y coils of similar design can occupy the same cylindrical surface, the current in each coil must obviously be restricted in the azimuthal direction. The simplest, though as is shown below not the optimal, such arrangement is one in which the x coil occupies the regions 0 p /4 õ f õ p /4 and 3p /4 õ f õ 5p /4, with the y coil occupying the intervening regions. To accommodate the restriction in the azimuthal direction, Jf(f, z) must be separable, i.e., of the form a( f ) b(z), so that J fm (k) can be expressed as J fm (k) Å am f (k),
[12]
where f (k) is the Fourier transform of b(z), and am is the amplitude of the cos mf Fourier component of a( f ). The amplitudes am can be readily calculated once a( f ) has been selected. We shall evaluate f (k), which we emphasize is the same for all values of m, using Eqs. [9] and [10]. Since the major contribution to the integral in Eq. [11] comes from values of ka õ 1, and since x Å r cos f and I1 (z) á z/2 when z õ 1, it follows from Eq. [11] that the m Å 1 term alone will generate an approximately linear gradient. Indeed, it is found that the gradient linearity is slightly worse if the summation is carried out over all values of m. We shall therefore follow Turner (1, 2) in taking only the m Å 1 term when determining f (k). APPLICATIONS
A simple x-gradient coil, which we refer to as a ‘‘singleparabola coil,’’ will have the desired spatial property if it is described by Jf(f, z) Å a( f ) b(z), where
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a( f ) Å 1 0 (16f 2 / p 2 ) 0 p /4 õ f õ p /4 Å0
elsewhere between 0 p /2 and p /2.
[13]
Thus, a( f ) is a parabolic function, centered at f Å 0 and falling to zero at f Å {p /4 radians. The name single-parabola coil refers to the number of parabolic distributions in the range ÉfÉ õ p /2. There is, however, a second, inverted parabola in the region p /2 õ ÉfÉ õ p, since a( f ) must, because of symmetry, satisfy the condition a( f / p ) Å 0 a( f ).
[14]
The values of am from m Å 1 to 19, when normalized so that a1 is unity, are 1, 0.579, 0.109, 00.088, 00.039, 0.034, 0.020, 00.018, 00.012, 0.011.
[15]
The function b(z) depends upon the target-field points selected. If the field generated by the coil is specified to be an ideal x gradient at the target points x/a Å 0.25, 0.5, y/a Å 0, z/a Å 0, 0.25, 0.5, then b(z) is found using Eqs. [9] and [10] to be as shown in Fig. 1a. (It is unnecessary to include target points with negative values of x/a and z/a because of the symmetry properties invoked in deriving Eqs. [9] – [11].) The corresponding locations of the cuts that delineate the current paths in a single-parabola coil are shown in Fig. 1b. For purposes of illustration, the total number of cuts was chosen to be 11. In practice, that number is governed by the desired coil efficiency: the larger the desired coil efficiency, the larger the number of cuts required. The pattern shown in Fig. 1b was obtained after apodization of f (k) (2, 3), by multiplying it by the Gaussian function exp[ 00.02 (ka) 2 ]. Unfortunately, the gradient generated by the single-parabola coil is quite nonlinear, the deviation from linearity, defined as (actual field 0 desired field)/desired field, along the x axis being 0.2% at x/a á 0.06, 1% at x/a á 0.14, and 5% at x/a á 0.31. For comparison purposes, these values are listed in Table 1. The single-parabola coil is, therefore, quite unsuitable for MRI applications because of the poor linearity of the gradient it generates. The inductance and power dissipation of such a coil are found, using Eqs. [5] and [6], to be 4.80 ( p 2a 5 / m0 ) (G/I) 2 and 26.5 ( r /t) ( pa 2G/ m0 ) 2 , respectively, where G is the gradient strength. The inductance and power dissipation of a conventional minimum-inductance transverse-gradient coil (2, 3), where the current occupies the entire sur-
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FIG. 1. (a) The azimuthal component of the current density, Jf , plotted in arbitrary units as a function of z/a. The location of cuts that delineate the current paths for the (b) single-, (c) double-, and (d) quadruple-parabola coil. Only one quadrant of the entire coil is shown in each case. The direction of current flow is in the same sense in each section, as indicated by the arrows.
face of the cylinder, a coil we refer to as a Turner coil, are 2.65 ( p 2a 5 / m0 ) (G/I) 2 and 9.08 ( r /t) ( pa 2G/ m0 ) 2 , respectively. For comparison, these values are listed in Table 2. As expected, the inductance and power dissipation of the single-parabola coil are somewhat larger than those of a
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TABLE 1 Values of x/a, at which the Deviation of the Gradient from Linearity on the x Axis is 0.2, 1, and 5%, for the Various Coils Considered in the Text
a( f ) consists of four parabolic functions centered at f Å {0.295 and {7p /24 radians, such that a( f ) Å [( p /6) 0 f]( f 0 0.06676)
0.06676 õ ÉfÉ õ ( p /6)
x/a where deviation is Coil
0.2%
1%
5%
Single-parabola Double-parabola Quadruple-parabolaa
0.06 0.22 0.56
0.14 0.34 0.64
0.31 0.49 0.74
Å 3.5258[( p /3) 0 f]( f 0 p /4)
( p /4) õ ÉfÉ õ ( p /3) Å0
elsewhere between 0 p /2 and p /2,
a
The deviations from linearity of the gradient generated by the ‘‘quadruple-parabola’’ coil and the Turner Coil differ by less than 0.5% over a sphere of radius 0.6a.
Turner coil of the same efficiency, radius, and conductor resistivity and thickness. The reason for the poor linearity of the gradient generated by the single-parabola coil can be understood by reference to Eq. [11], the expression for Bz . Since the main contribution to the integral comes from values of ka õ 1, J 3f (k)cos 3f I3 (kr) Ç a3r 3 cos 3f, and the observed deviation from gradient linearity is almost entirely due to the difference between this third-order term and the desired r cos f behavior. It is logical, therefore, to consider alternative gradient coil designs which cause higher-order terms in the Fourier expansion of a( f ) to go to zero. Consider, for example, a ‘‘double-parabola coil’’ where a( f ) consists of two parabolic functions centered at f Å {p /12 radians such that a( f ) Å [( p /4) 0 f][ f 0 ( p /12)]
( p /12) õ ÉfÉ õ ( p /4) Å0
elsewhere between 0 p /2 and p /2.
[16]
In this case, a3 is zero from symmetry, and the values of am from m Å 1 to 19 are 1, 0, 00.844, 00.707, 0, 0.385, 0.232, 0, 00.006, 0.055.
[17]
If the same target-field points as before are chosen, so that b(z) is as shown in Fig. 1a, the cuts delineating current paths of the double-parabola coil are as shown in Fig. 1c. The gradient is much more linear (see Table 1) than that generated by the single-parabola coil, but is still inadequate for most MRI applications because of the influence of the a5 term. Consider, therefore, a ‘‘quadrupole-parabola coil’’ where
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[18]
and where the peak height of the parabolas centered at {7p /24 radians is 1.158 times larger than the peak height of the parabolas centered at {0.295 radians. In this case, a3 and a5 are both zero and the values of am from m Å 1 to 19 are 1, 0, 0, 0.177, 00.578, 00.677, 0.086, 0.078, 00.259, 0.046.
[19]
It can be seen that a( f ) now extends from 0 p /3 to p /3 instead of from 0 p /4 to p /4. However, since gaps in the current distribution exist between 0 p /6 and 0 p /4 and between p /6 and p /4, an identical coil design, rotated through p /2 so as to generate a y gradient, can still be accommodated on the same cylindrical surface. The cuts delineating current paths of the quadruple-parabola coil, evaluated using the same target field points as before, are shown in Fig. 1d. For clarity, the number of cuts in the narrower sections where p /4 õ ÉfÉ õ p /3 is arbitrarily chosen to be seven. The deviation from gradient linearity of the quadrupleparabola coil on the axis is less than 1% out to x/a á 0.64 (see Table 1). The deviation from linearity is 2.5% for small values of x off axis at y/a Å 0 and z/a Å 0.5. However, this characteristic is shared by the Turner coil and stems from the choice of target-field points. If the field is specified at x/a Å 0.25, 0.5, y/a Å 0, z/a Å 0, 0.25, 0.5, 0.75, 1, the deviations from linearity at y/a Å 0 and z/a Å 0.5 are less than 0.1%. However, this results in a longer coil. Over a TABLE 2 Calculated Values of the Inductance and Power Dissipation for the Various Coils Coil
Inductancea
Power dissipationb
Single-parabola Double-parabola Quadruple-parabola Turner
4.80 19.9 22.9 2.65
26.5 318 627 9.08
a b
In units of (p2a 5/mo)(G/I)2. In units of (r/t)(pa 2G/mo)2.
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sphere of radius 0.6a, which encompasses the volume of interest in most MRI and MRS systems, the deviations from linearity of the quadrupole-parabola coil and the Turner coil differ by less than 0.5%. This is due to the fact that not only are a3 and a5 both zero, but a7 is quite small. The price to be paid for saving radial space by confining both transverse-coil sets to the same radius, and for maintaining excellent linearity at the same time, is a limitation on the gradient switching speed due to a large increase in inductance, and the necessity of efficient cooling because of a large increase in power dissipation. Table 2 quantifies these increases for each of the coil systems evaluated. Replacing the parabolas by other functions such as cosines or triangles has relatively little effect on gradient linearity, inductance, and power dissipation. The summation over m to determine the inductance using Eq. [5] converges extremely slowly for the quadrupole-parabola coil. For example, values of the inductance obtained by summing to m Å 39 and 99 are, respectively, 96.8 and 99.6% of the value 22.91 ( p 2a 5 / m0 )(G/I) 2 found by summing over all m. In order to confirm the above theory, a quadrupole-parabola coil of radius 0.196 m was constructed, the current distribution being approximated by discrete wires as described by Turner (1–3). The inner and outer parabolic regions consisted of six and seven turns of 14 AWG wire, respectively, giving a theoretical coil efficiency, G/I, of 4.3 1 10 05 T/mrA and an inductance of 96 mH. The measured coil efficiency and inductance are 4.5 1 10 05 T/mrA and 138 mH, respectively. Since the inductance of the leads is estimated to be no more than 3 mH, we attribute the discrepancy between the theoretical and experimental values of the inductance to the use of thin wire instead of a continuous current distribution. DISCUSSION
We have shown that it is possible to design distributedcurrent transverse-gradient coils which occupy only half of the surface area of a cylinder, thereby allowing both orthogonal sets to be constructed on the same surface. This minimizes the radial space occupied by the coils and allows both coils to be fabricated at the same time. Of the three types considered, the quadruple-parabola coil is by far the most attractive, because of the excellent linearity of the gradient it generates. The space saving advantage of the quadrupleparabola coil over the Turner coil must, however, be weighed against its limitations on switching speed. The ultrafast switching capability of the Turner design is reduced to that to which one became accustomed for discrete wired Golay coils, namely rise and fall times )1 ms. If high-speed gradient switching is not essential, the quadruple-parabola coil is well suited for use in MRI and MRS systems.
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A further advantage of the quadruple-parabola coil is the ease with which efficient water cooling can be incorporated. An incidental feature of its design is the concentration of current, and therefore Joule heating, at the edges of the parabolic components. Therefore, not only can the gap between f Å 00.0667 and 0.0667 radians (i.e., from 03.87 to 3.87 ), where there is no current, be used for water cooling, but each of the parabolas could all be made slightly, say 1.57, narrower, without significantly affecting gradient linearity, to allow spaces on either side for water cooling. Such an arrangement would have the advantage that the water cooling would be located immediately adjacent to the regions of most intense Joule heating. Since the quadruple-parabola coil and the Turner coil generate Bz fields which are essentially identical over a sphere of radius 0.6a, it seems, at first sight, surprising that the inductances of the two coils should differ by so much. However, as is shown in the Appendix, for a given coil efficiency in generating a distant field gradient, a small circular coil will have a larger inductance than a large circular coil because of the greater number of turns required. While the individual coils in our double-parabola coil and quadruple-parabola coil are far from circular, it would seem that the large values of their inductance are a manifestation of the use of small coils to generate the magnetic field gradient. APPENDIX
The magnetic field on the axis of a circular coil consisting of N turns of radius a is given by Bz Å m0NIa 2 /[2(a 2 / z 2 ) 3 / 2 ],
(A.1)
where I is the current flowing in the coil and z is the distance of the field point from the center of the coil. It follows that if z @ a, then N Å 0 [(dBz /dz)/I](2z 4 /3m0a 2 ).
(A.2)
Thus, treating N and a as variables, N is inversely proportional to a 2 for a given coil efficiency, [(dBz /dz)/I], at a distant field point. However, since the self-inductance of the coil is given by (8) L Å m0aN 2[ln(8a/b) 0 (7/4)],
(A.3)
where b is the radius of the wire, it follows that L Ç a 03[ln(8a/b) 0 (7/4)].
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163
TRANSVERSE-GRADIENT COILS FOR MRI
This shows that the self-inductance of a coil with a given coil efficiency at a distant field point increases rapidly with decreasing radius, because of the larger number of turns required.
ACKNOWLEDGMENT
This work was supported by a Collaborative Project Grant from the Natural Science and Engineering Research Council of Canada.
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REFERENCES 1. 2. 3. 4. 5.
R. Turner, J. Phys. D: Appl. Phys. 19, L147 (1986). R. Turner, J. Phys. E: Sci. Instrum. 21, 948 (1988). R. Turner, Magn. Reson. Imaging 11, 903 (1993). P. Mansfield and B. Chapman, J. Magn. Reson. 66, 573 (1986). R. Turner and R. M. Bowley, J. Phys. E: Sci. Instrum. 19, 876 (1986). 6. Q. Liu, D. G. Hughes, and P. S. Allen, Magn. Reson. Med. 31, 73 (1994). 7. Q. Liu, M.Sc. Thesis, University of Alberta, 1991. 8. W. T. Scott, ‘‘The Physics of Electricity and Magnetism,’’ p. 322, Wiley, New York, 1959.
AP: Mag Res
Series B 110, 158–163 (1996)
0025
Compact, Cylindrical, Distributed-Current, Transverse-Gradient Coils for Use in MRI D. G. HUGHES,* R. TESHIMA,* Q. LIU,*
AND
P. S. ALLEN†
Departments of *Physics and †Biomedical Engineering, University of Alberta, Edmonton, Alberta, T6G 2J1, Canada Received July 17, 1995
Conventional cylindrical, distributed-current, transverse-gradient coils used in MRI occupy the entire surface of the cylinder. Since the two transverse-gradient coils, x and y, must be rigidly supported and electrically isolated from one another, they occupy a significant amount of space in the radial direction, with a consequent reduction in the utilization efficiency of the magnet bore volume. In this work, transverse-gradient coil designs which allow both x and y coils to be located on the same cylindrical surface are studied. One design in particular is shown to generate a gradient which is as linear as that generated by a conventional distributed-current gradient coil over a sphere whose radius is 60% of the radius of the cylinder. q 1996 Academic Press, Inc.
INTRODUCTION
A most important advance in the design of gradient coils for use in magnetic resonance imaging and spectroscopy was the development by Turner (1, 2) of the target field approach for distributed-current cylindrical coils. Such coils can generate longitudinal or transverse gradients of excellent linearity over much larger volumes than earlier types of gradient coils. Moreover, they can be designed to minimize either the inductance or the power dissipation for a given coil efficiency (gradient strength per unit current) (2, 3). Since, for the present distributed-current designs, the current distribution in each gradient coil occupies essentially the whole surface of the cylinder and since each coil must be rigidly supported and electrically isolated from adjacent coils, a complete set of one longitudinal and two transverse coils occupies a significant amount of space in the radial direction. Such space is at a premium, not only for existing small-bore, high-field animal magnets but also for the magnets being developed as part of the trend to higher fields for studies on humans, since at any given field strength, smallerbore magnets are cheaper to build, cheaper and easier to site, and cheaper to operate. For example, a 100 cm bore 3 T magnet costs about 30% more than an 80 cm bore 3 T magnet. Also, the operating costs of a 100 cm bore magnet are about 25% higher than for an 80 cm bore magnet pos-
1064-1866/96 $12.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
m4679$7318
01-16-96 07:54:01
sessing the same field. The utilization efficiency of the bore volume is therefore a significant economic issue. The purpose of the present paper is to report transversegradient coil designs which allow both (x and y) transversegradient coils to be located on the same cylindrical surface, thereby saving space while at the same time facilitating construction. THEORETICAL CONSIDERATIONS
Although it will always be possible to design active shielding (4, 5) that will largely eliminate eddy-current fields (6) caused by switched currents in such coils, for the sake of brevity in introducing the new design concept, we shall only consider unshielded gradient coils. A distributed current flowing in a thin cylindrical shell of radius a, whose axis is taken to be the z direction, will generate a z component of magnetic field at a point (r, f, z) inside the cylinder given by (1, 2, 5, 7) `
Bz (r, f, z) Å 0 ( m0a/2p ) ∑ m Å0`
1 e
`
ÉkÉdkJ fm (k)
0`
e Im (ÉkÉr)K *m (ÉkÉa),
imf ikz
[1]
where Im (z) and Km (z) are modified Bessel functions of the first and second kind, the prime indicating a derivative with respect to the argument. Also, J fm (k) is the Fourier transform of the azimuthal component of the current density, Jf(f, z), given by J fm (k) Å (1/2p )
*
p
df e 0imf
0p
*
`
dz e 0ikz Jf(f, z). [2]
0`
The Fourier transform of the longitudinal component of the current density, Jz ( f, z), is similarly given by J zm (k) Å (1/2p )
*
p
0p
df e 0imf
*
`
dz e 0ikz Jz ( f, z). [3]
0`
It follows from the equation of continuity, div J Å 0, that
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TRANSVERSE-GRADIENT COILS FOR MRI
J zm (k) Å 0 (m/ka)J fm (k).
[4]
equations can be replaced by cosines. Equations [1], [7], and [8] then become
The inductance of the coil is given by 2
L Å 0 ( m0a /I ) `
∑
1
m Å0`
*
`
n Å1
dkÉJ (k)É I *m (ÉkÉa)K *m (ÉkÉa), [5] m f
∑ m Å1,3,5. . .
*
`
k 2 dk
0
1 [K *m (ka)/I *m (ka)]Im (krn )Im (krl )
2
0`
1 cos mfl cos mfn cos kzl cos kzn ,
where I is the current entering the coil, and the power dissipation P is given by `
P Å 0 ( ra/t) ∑ mÅ0`
*
N
` m f
2
2
dkÉJ (k)É [1 / (m/ka) ], [6]
0`
∑ ln Im (krn )cos mfn cos kzn ,
[10]
n Å1
and
N
`
Bz (rl , fl , zl ) Å 0 ( m0 I/8p ) ∑ ln { ∑ 2
n Å1
m Å0`
*
` 2
k dk
0`
1 [K *m (ÉkÉa)/I *m (ÉkÉa)]Im (ÉkÉrn ) 1 Im (ÉkÉrl )exp[im( fl 0 fn )] 1 exp[ik(zl 0 zn )]},
[7]
where Bz (rl , fl , zl ) is the target field at the point (rl , fl , zl ), l takes the values 1, 2, . . . , N, and N is the total number of target field points. By solving the N simultaneous equations in [7] for the Lagrange multipliers ln , J fm (k) can be found using the expression J fm (k) Å {ÉkÉI/[4paI *m (ÉkÉa)]} N
∑ ln exp( 0imfn )exp( 0ikzn )Im (ÉkÉrn ), [8] n Å1
from which J zm (k) can be found using Eq. [4]. The components Jf(f, z) and Jz ( f, z) of the current density that will generate the desired field at the target points can then be found by inverse Fourier transformation of J fm (k) and J zm (k). The theory so far presented is general and can be used to obtain the distributed current that will, in principle, create any desired field profile. In order to generate an x gradient whose field is zero at the center of the cylinder, Bz must be invariant with respect to the transformations f r 0 f and z r 0z. Moreover, Bz must change sign with the transformation f r f / p. It follows that terms involving even values of m must vanish and the exponentials in the preceding
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[9]
J fm (k) Å [kI/4paI *m (ka)] 1
where r is the resistivity and t is the thickness of the material composing the coil. The inductance, the power dissipation, or a suitably weighted sum of the two can be minimized using the targetfield approach, yielding, in the case of minimum inductance,
1
`
Bz (rl , fl , zl ) Å 0 ( m0 I/2p 2 ) ∑ ln
2
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`
Bz (r, f, z) Å 0 (2m0a/ p )
∑ m Å1,3,5. . .
*
`
kdkJ fm (k)
0
1 cos mf cos kz Im (kr)K *m (ka).
[11]
In order to design transverse-gradient coils such that x and y coils of similar design can occupy the same cylindrical surface, the current in each coil must obviously be restricted in the azimuthal direction. The simplest, though as is shown below not the optimal, such arrangement is one in which the x coil occupies the regions 0 p /4 õ f õ p /4 and 3p /4 õ f õ 5p /4, with the y coil occupying the intervening regions. To accommodate the restriction in the azimuthal direction, Jf(f, z) must be separable, i.e., of the form a( f ) b(z), so that J fm (k) can be expressed as J fm (k) Å am f (k),
[12]
where f (k) is the Fourier transform of b(z), and am is the amplitude of the cos mf Fourier component of a( f ). The amplitudes am can be readily calculated once a( f ) has been selected. We shall evaluate f (k), which we emphasize is the same for all values of m, using Eqs. [9] and [10]. Since the major contribution to the integral in Eq. [11] comes from values of ka õ 1, and since x Å r cos f and I1 (z) á z/2 when z õ 1, it follows from Eq. [11] that the m Å 1 term alone will generate an approximately linear gradient. Indeed, it is found that the gradient linearity is slightly worse if the summation is carried out over all values of m. We shall therefore follow Turner (1, 2) in taking only the m Å 1 term when determining f (k). APPLICATIONS
A simple x-gradient coil, which we refer to as a ‘‘singleparabola coil,’’ will have the desired spatial property if it is described by Jf(f, z) Å a( f ) b(z), where
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a( f ) Å 1 0 (16f 2 / p 2 ) 0 p /4 õ f õ p /4 Å0
elsewhere between 0 p /2 and p /2.
[13]
Thus, a( f ) is a parabolic function, centered at f Å 0 and falling to zero at f Å {p /4 radians. The name single-parabola coil refers to the number of parabolic distributions in the range ÉfÉ õ p /2. There is, however, a second, inverted parabola in the region p /2 õ ÉfÉ õ p, since a( f ) must, because of symmetry, satisfy the condition a( f / p ) Å 0 a( f ).
[14]
The values of am from m Å 1 to 19, when normalized so that a1 is unity, are 1, 0.579, 0.109, 00.088, 00.039, 0.034, 0.020, 00.018, 00.012, 0.011.
[15]
The function b(z) depends upon the target-field points selected. If the field generated by the coil is specified to be an ideal x gradient at the target points x/a Å 0.25, 0.5, y/a Å 0, z/a Å 0, 0.25, 0.5, then b(z) is found using Eqs. [9] and [10] to be as shown in Fig. 1a. (It is unnecessary to include target points with negative values of x/a and z/a because of the symmetry properties invoked in deriving Eqs. [9] – [11].) The corresponding locations of the cuts that delineate the current paths in a single-parabola coil are shown in Fig. 1b. For purposes of illustration, the total number of cuts was chosen to be 11. In practice, that number is governed by the desired coil efficiency: the larger the desired coil efficiency, the larger the number of cuts required. The pattern shown in Fig. 1b was obtained after apodization of f (k) (2, 3), by multiplying it by the Gaussian function exp[ 00.02 (ka) 2 ]. Unfortunately, the gradient generated by the single-parabola coil is quite nonlinear, the deviation from linearity, defined as (actual field 0 desired field)/desired field, along the x axis being 0.2% at x/a á 0.06, 1% at x/a á 0.14, and 5% at x/a á 0.31. For comparison purposes, these values are listed in Table 1. The single-parabola coil is, therefore, quite unsuitable for MRI applications because of the poor linearity of the gradient it generates. The inductance and power dissipation of such a coil are found, using Eqs. [5] and [6], to be 4.80 ( p 2a 5 / m0 ) (G/I) 2 and 26.5 ( r /t) ( pa 2G/ m0 ) 2 , respectively, where G is the gradient strength. The inductance and power dissipation of a conventional minimum-inductance transverse-gradient coil (2, 3), where the current occupies the entire sur-
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FIG. 1. (a) The azimuthal component of the current density, Jf , plotted in arbitrary units as a function of z/a. The location of cuts that delineate the current paths for the (b) single-, (c) double-, and (d) quadruple-parabola coil. Only one quadrant of the entire coil is shown in each case. The direction of current flow is in the same sense in each section, as indicated by the arrows.
face of the cylinder, a coil we refer to as a Turner coil, are 2.65 ( p 2a 5 / m0 ) (G/I) 2 and 9.08 ( r /t) ( pa 2G/ m0 ) 2 , respectively. For comparison, these values are listed in Table 2. As expected, the inductance and power dissipation of the single-parabola coil are somewhat larger than those of a
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TABLE 1 Values of x/a, at which the Deviation of the Gradient from Linearity on the x Axis is 0.2, 1, and 5%, for the Various Coils Considered in the Text
a( f ) consists of four parabolic functions centered at f Å {0.295 and {7p /24 radians, such that a( f ) Å [( p /6) 0 f]( f 0 0.06676)
0.06676 õ ÉfÉ õ ( p /6)
x/a where deviation is Coil
0.2%
1%
5%
Single-parabola Double-parabola Quadruple-parabolaa
0.06 0.22 0.56
0.14 0.34 0.64
0.31 0.49 0.74
Å 3.5258[( p /3) 0 f]( f 0 p /4)
( p /4) õ ÉfÉ õ ( p /3) Å0
elsewhere between 0 p /2 and p /2,
a
The deviations from linearity of the gradient generated by the ‘‘quadruple-parabola’’ coil and the Turner Coil differ by less than 0.5% over a sphere of radius 0.6a.
Turner coil of the same efficiency, radius, and conductor resistivity and thickness. The reason for the poor linearity of the gradient generated by the single-parabola coil can be understood by reference to Eq. [11], the expression for Bz . Since the main contribution to the integral comes from values of ka õ 1, J 3f (k)cos 3f I3 (kr) Ç a3r 3 cos 3f, and the observed deviation from gradient linearity is almost entirely due to the difference between this third-order term and the desired r cos f behavior. It is logical, therefore, to consider alternative gradient coil designs which cause higher-order terms in the Fourier expansion of a( f ) to go to zero. Consider, for example, a ‘‘double-parabola coil’’ where a( f ) consists of two parabolic functions centered at f Å {p /12 radians such that a( f ) Å [( p /4) 0 f][ f 0 ( p /12)]
( p /12) õ ÉfÉ õ ( p /4) Å0
elsewhere between 0 p /2 and p /2.
[16]
In this case, a3 is zero from symmetry, and the values of am from m Å 1 to 19 are 1, 0, 00.844, 00.707, 0, 0.385, 0.232, 0, 00.006, 0.055.
[17]
If the same target-field points as before are chosen, so that b(z) is as shown in Fig. 1a, the cuts delineating current paths of the double-parabola coil are as shown in Fig. 1c. The gradient is much more linear (see Table 1) than that generated by the single-parabola coil, but is still inadequate for most MRI applications because of the influence of the a5 term. Consider, therefore, a ‘‘quadrupole-parabola coil’’ where
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[18]
and where the peak height of the parabolas centered at {7p /24 radians is 1.158 times larger than the peak height of the parabolas centered at {0.295 radians. In this case, a3 and a5 are both zero and the values of am from m Å 1 to 19 are 1, 0, 0, 0.177, 00.578, 00.677, 0.086, 0.078, 00.259, 0.046.
[19]
It can be seen that a( f ) now extends from 0 p /3 to p /3 instead of from 0 p /4 to p /4. However, since gaps in the current distribution exist between 0 p /6 and 0 p /4 and between p /6 and p /4, an identical coil design, rotated through p /2 so as to generate a y gradient, can still be accommodated on the same cylindrical surface. The cuts delineating current paths of the quadruple-parabola coil, evaluated using the same target field points as before, are shown in Fig. 1d. For clarity, the number of cuts in the narrower sections where p /4 õ ÉfÉ õ p /3 is arbitrarily chosen to be seven. The deviation from gradient linearity of the quadrupleparabola coil on the axis is less than 1% out to x/a á 0.64 (see Table 1). The deviation from linearity is 2.5% for small values of x off axis at y/a Å 0 and z/a Å 0.5. However, this characteristic is shared by the Turner coil and stems from the choice of target-field points. If the field is specified at x/a Å 0.25, 0.5, y/a Å 0, z/a Å 0, 0.25, 0.5, 0.75, 1, the deviations from linearity at y/a Å 0 and z/a Å 0.5 are less than 0.1%. However, this results in a longer coil. Over a TABLE 2 Calculated Values of the Inductance and Power Dissipation for the Various Coils Coil
Inductancea
Power dissipationb
Single-parabola Double-parabola Quadruple-parabola Turner
4.80 19.9 22.9 2.65
26.5 318 627 9.08
a b
In units of (p2a 5/mo)(G/I)2. In units of (r/t)(pa 2G/mo)2.
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sphere of radius 0.6a, which encompasses the volume of interest in most MRI and MRS systems, the deviations from linearity of the quadrupole-parabola coil and the Turner coil differ by less than 0.5%. This is due to the fact that not only are a3 and a5 both zero, but a7 is quite small. The price to be paid for saving radial space by confining both transverse-coil sets to the same radius, and for maintaining excellent linearity at the same time, is a limitation on the gradient switching speed due to a large increase in inductance, and the necessity of efficient cooling because of a large increase in power dissipation. Table 2 quantifies these increases for each of the coil systems evaluated. Replacing the parabolas by other functions such as cosines or triangles has relatively little effect on gradient linearity, inductance, and power dissipation. The summation over m to determine the inductance using Eq. [5] converges extremely slowly for the quadrupole-parabola coil. For example, values of the inductance obtained by summing to m Å 39 and 99 are, respectively, 96.8 and 99.6% of the value 22.91 ( p 2a 5 / m0 )(G/I) 2 found by summing over all m. In order to confirm the above theory, a quadrupole-parabola coil of radius 0.196 m was constructed, the current distribution being approximated by discrete wires as described by Turner (1–3). The inner and outer parabolic regions consisted of six and seven turns of 14 AWG wire, respectively, giving a theoretical coil efficiency, G/I, of 4.3 1 10 05 T/mrA and an inductance of 96 mH. The measured coil efficiency and inductance are 4.5 1 10 05 T/mrA and 138 mH, respectively. Since the inductance of the leads is estimated to be no more than 3 mH, we attribute the discrepancy between the theoretical and experimental values of the inductance to the use of thin wire instead of a continuous current distribution. DISCUSSION
We have shown that it is possible to design distributedcurrent transverse-gradient coils which occupy only half of the surface area of a cylinder, thereby allowing both orthogonal sets to be constructed on the same surface. This minimizes the radial space occupied by the coils and allows both coils to be fabricated at the same time. Of the three types considered, the quadruple-parabola coil is by far the most attractive, because of the excellent linearity of the gradient it generates. The space saving advantage of the quadrupleparabola coil over the Turner coil must, however, be weighed against its limitations on switching speed. The ultrafast switching capability of the Turner design is reduced to that to which one became accustomed for discrete wired Golay coils, namely rise and fall times )1 ms. If high-speed gradient switching is not essential, the quadruple-parabola coil is well suited for use in MRI and MRS systems.
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A further advantage of the quadruple-parabola coil is the ease with which efficient water cooling can be incorporated. An incidental feature of its design is the concentration of current, and therefore Joule heating, at the edges of the parabolic components. Therefore, not only can the gap between f Å 00.0667 and 0.0667 radians (i.e., from 03.87 to 3.87 ), where there is no current, be used for water cooling, but each of the parabolas could all be made slightly, say 1.57, narrower, without significantly affecting gradient linearity, to allow spaces on either side for water cooling. Such an arrangement would have the advantage that the water cooling would be located immediately adjacent to the regions of most intense Joule heating. Since the quadruple-parabola coil and the Turner coil generate Bz fields which are essentially identical over a sphere of radius 0.6a, it seems, at first sight, surprising that the inductances of the two coils should differ by so much. However, as is shown in the Appendix, for a given coil efficiency in generating a distant field gradient, a small circular coil will have a larger inductance than a large circular coil because of the greater number of turns required. While the individual coils in our double-parabola coil and quadruple-parabola coil are far from circular, it would seem that the large values of their inductance are a manifestation of the use of small coils to generate the magnetic field gradient. APPENDIX
The magnetic field on the axis of a circular coil consisting of N turns of radius a is given by Bz Å m0NIa 2 /[2(a 2 / z 2 ) 3 / 2 ],
(A.1)
where I is the current flowing in the coil and z is the distance of the field point from the center of the coil. It follows that if z @ a, then N Å 0 [(dBz /dz)/I](2z 4 /3m0a 2 ).
(A.2)
Thus, treating N and a as variables, N is inversely proportional to a 2 for a given coil efficiency, [(dBz /dz)/I], at a distant field point. However, since the self-inductance of the coil is given by (8) L Å m0aN 2[ln(8a/b) 0 (7/4)],
(A.3)
where b is the radius of the wire, it follows that L Ç a 03[ln(8a/b) 0 (7/4)].
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This shows that the self-inductance of a coil with a given coil efficiency at a distant field point increases rapidly with decreasing radius, because of the larger number of turns required.
ACKNOWLEDGMENT
This work was supported by a Collaborative Project Grant from the Natural Science and Engineering Research Council of Canada.
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REFERENCES 1. 2. 3. 4. 5.
R. Turner, J. Phys. D: Appl. Phys. 19, L147 (1986). R. Turner, J. Phys. E: Sci. Instrum. 21, 948 (1988). R. Turner, Magn. Reson. Imaging 11, 903 (1993). P. Mansfield and B. Chapman, J. Magn. Reson. 66, 573 (1986). R. Turner and R. M. Bowley, J. Phys. E: Sci. Instrum. 19, 876 (1986). 6. Q. Liu, D. G. Hughes, and P. S. Allen, Magn. Reson. Med. 31, 73 (1994). 7. Q. Liu, M.Sc. Thesis, University of Alberta, 1991. 8. W. T. Scott, ‘‘The Physics of Electricity and Magnetism,’’ p. 322, Wiley, New York, 1959.
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