Introduction
Magnetic resonance imaging relies on the use of magnetic field gradients – that is a magnetic field, Bz(r) which varies linearly with position, r, such that Bz(r) = G.r – to spatially encode the NMR signal. Such gradients are generated by passing currents through specially arranged coils of wire, placed on a former that surrounds the imaging subject. Three separate coils are needed in order to produce a linear variation of the z-component of the magnetic field along each of the three Cartesian directions (x, y and z). The performance of the gradient coils and the amplifiers that are used to drive them dictates the kind of gradient waveforms that can be used in an MR scanner, particularly controlling the maximum gradient strength and rate of change of gradient with time that can employed. Since the use of strong and rapidly switched gradients generally improves image quality and the speed of image acquisition, considerable benefits can result from optimising gradient coil performance.There are a number of inter-related parameters that characterise the performance of a gradient coil.
Uniformity One of the most important of these is the uniformity of the field gradient that the coil produces. Since the imaging process is based on the assumption that the field varies linearly with position, deviation from linearity leads to image distortion. While some degree of distortion can be corrected by post-processing of the image data, in designing a gradient coil it is desirable to achieve a high degree of uniformity over the imaging volume. The gradient uniformity can be characterised in terms of the local deviation of the field from the desired value:
$$E_B(\textbf{r})=\frac{B_z(\textbf{r})-\textbf{G}.\textbf{r}}{\textbf{G}.\textbf{r}}.\space\space\space [1]$$
It is particularly important to make sure that the field gradient does not tend to zero within the volume from which the RF coil can detect a signal, as this can cause artefacts which cannot be corrected by post-processing .
Efficiency Another important parameter is the gradient coil efficiency, η, which is the gradient strength produced by current, I,
$$\eta=\frac{G}{I}.\space\space\space [2]$$
Inductance and Resistance Ideally, η should be as large possible, but its magnitude is however strongly linked to two other parameters that play an important role in gradient coil performance: the inductance, L, and the resistance, R, of the gradient coil. The inductance value is crucial because it limits the rate of change of current in the coil and thus dictates the maximum possible rate of change of gradient per unit time,, that can be achieved. Using a gradient amplifier capable of generating a maximum voltage, V, gives
$$\frac{dG}{dt}=\frac{\eta V}{L}, \space\space\space[3]$$
so that the rise-time, τ, that it takes to ramp up a gradient to amplitude, G, is $$$\frac{GL}{\eta V}$$$. Gradient coils with low inductance and amplifiers capable of producing high voltages are therefore needed to achieve short rise-times. Reduced rise-times mean that less time is wasted in ramping up gradients in MR sequences, potentially yielding faster image acquisition and improved signal to noise ratio. The inductance can be made small by using a small number of turns, n, in the gradient coil as $$$L\propto n^2$$$, but since $$$\eta \propto n$$$, this entails a reduction of gradient coil efficiency. A useful parameter in characterising the performance of gradient coil is the ratio $$$\frac{\eta^2}{L}$$$, which is independent of n. For example, the power, $$$P_A$$$, that a gradient coil amplifier needs to be able to generate to produce a maximum gradient, G, with a rise-time, t, is given by:
$$P_A=\frac{G^2 L}{\eta^2 \tau},\space\space\space [4]$$
indicating that maximising the value of $$$\frac{\eta^2}{L}$$$ reduces the necessary amplifier power. Importantly, it can be shown that $$$\frac{\eta^2}{L}$$$ depends on $$$a^{-5}$$$ where a is the characteristic size of the coil. It is therefore useful to limit the gradient coil size to the minimum that will accommodate the sample and RF coil (1), while providing a region of gradient uniformity of adequate size. The interplay of η, L and gradient uniformity is generally at the heart of the process of designing gradient coils for human MR systems. The gradient coil resistance, R, is also important, particularly for the smaller coils used in NMR microscopy, since it dictates the amount of power ($$$P = I^2 R$$$) that is dissipated in the gradient coil. Excessive coil heating can lead to large temperature rises in the coil former, and eventually coil failure, and also may make the inside of the gradient coil undesirably hot. It is therefore important to limit the coil resistance.
Magnetic Screening Whenever the current in a gradient coil changes, the magnetic fields produced both inside and outside the coil change with time. In a conventional MR scanner, there is a considerable amount of conducting material, such as the cryostat and heat shields of the superconducting magnet that is located outside, but quite close to the gradient coil. A temporally varying magnetic field induces eddy currents in this material. The resulting currents decay in an uncontrolled manner over a time-scale that may be as long as seconds. These eddy currents generate time-varying magnetic fields inside the sample, which can significantly degrade the quality of spectra and images. Although this effect can be ameliorated to some extent by pre-emphasis whereby the gradient current waveform is modified so that the combination of the fields produced by the coil and eddy currents best approximates the desired gradient waveform, it is better to prevent the formation of eddy currents in the first place. This can be achieved by active magnetic screening (2, 3), in which the gradient coil is composed of two co-axial, cylindrical coils, that together generate the desired field gradient inside the inner or primary coil, and a field that is as small as possible outside the outer or screen coil.
Other Parameters A variety of other parameters are also important in designing a gradient coil. These include the “buildability” of the coil – there is no point in designing a great coil if it is impossible to construct. In addition the effect of the Lorentz forces experienced by the coil windings when they carry current in the scanner’s strong magnet field are also important. In particular it should be ensured that the net force and torque experienced by the coil are zero, so as to avoid excessive motion of the coil former. Such motion could potentially lead to dangerous mechanical failure of the coil, and also may cause excessive acoustic noise – a particular problem since gradient waveforms vary at audio frequencies. Finally for human studies it is important to consider the propensity of the gradient coil to cause peripheral nerve stimulation (PNS) in the subject being studied. Nerve stimulation arises because the temporally varying magnetic fields produced inside the conducting tissue of the body generate electric fields that can be large enough to stimulate peripheral nerves. Fortunately the threshold for peripheral nerve stimulation (PNS) of approximately ~ 6 Vm-1 is significantly lower than that for stimulation of the heart. Consequently, regulatory limits on gradient switching rates have been generally set so as to avoid PNS, generally providing a significant safety margin for cardiac stimulation.
Most gradient coils that are used in MRI consist of wire arrangements on the surface of a cylindrical former, and it is this type of coil that will initially be considered here.
Coils with Discrete Windings The simplest cylindrical gradient coils consist of arrangements of elemental wire units, such as circular loops and saddles, which are positioned so as to produce as pure a gradient as possible at the coil centre. This is achieved by expanding the field produced by each unit as a series of spherical harmonics about the coil centre. By studying the form of these expansions, arrangements that null out a number of unwanted spherical harmonics can be found, leaving the desired field gradient as the lowest order and dominant harmonic (4). The more units that are included in the coil arrangement, the greater the number of unwanted spherical harmonics that can be zeroed – leading to the generation of a uniform field gradient over a larger region of space. Simple application of this process leads to the Maxwell coil, which generates a z-gradient, and the Golay coil (5), which can be used to produce an x- or y-gradient.
Coils with Distributed Windings In both the Golay and Maxwell coils, the gradient is uniform only within a small fraction of the volume enclosed by the coil and these coils also have quite low values of $$$\frac{\eta^2}{L}$$$ . Better performance can be achieved by using coils consisting of windings that are distributed over the whole cylindrical surface, so as to approximate a continuous current distribution. Such coils can produce a linear field variation over a larger fraction of the coil volume and tend to have higher values of $$$\frac{\eta^2}{L}$$$ because there is less flux concentration close to the coil windings than in coils composed of discrete units. Distributed coils can be designed using analytic methods which are based on representing the distributed coil windings by a current distribution. This current distribution can be written using cylindrical polar co-ordinates as
$$\textbf{J}(\phi,z)=J_\phi(\phi,z)\widehat{\phi}+J_z(\phi,z)\widehat{z} \space\space\space[5]$$
on the surface of a cylinder of radius, a. Using the Green’s function of the vector potential (5), the z-component of the magnetic field produced inside the cylinder can be described by:
$$B_z(\rho,\phi,z)=-\frac{\mu_0}{2\pi}\sum_{m=-\infty}^{m=\infty}\int_{-\infty}^{\infty} dk\space e^{im\phi}e^{ikz}J_{\phi}^{m}(k)\space ka\space I_m(k\rho)K'_m(ka) \space\space\space [6] $$
where $$$I_m(x)$$$ and $$$K_m(x)$$$ are the modified Bessel functions and $$$J_{\phi}^{m}(k)$$$ is the Fourier transform of the azimuthal component of the current distribution:
$$J_{\phi}^{m}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}d\phi \int_{-\infty}^{\infty} dz\space e^{-im\phi}e^{-ikz}J_{\phi}(\phi,z)\space\space\space [7] $$
A similar Fourier transform, $$$J_{z}^{m}(k)$$$ , of the z-component of the current distribution can be defined and as a result of the fact that the divergence of the current must be zero everywhere on the cylinder surface, the simple relationship,
$$J_{\phi}^{m}(k)=-\frac{ka}{m}J_{z}^{m}(k)\space\space\space [8] $$
also holds.
Target Field Approach By defining a similar Fourier transform of the magnetic field variation at radius c < a ($$$B_{z}^{m}(c,k)=FT[B_{z}(c,\phi,z)]$$$) Eq. [6] can be inverted to give:
$$ J_{\phi}^{m}(k)=-\frac{B_{z}^{m}(c,k)}{\mu_0kaI_m(kc)K'_m(ka)} \space\space\space [9] $$
This equation forms the basis of the Target Field approach to gradient coil design (7), in which a target field variation is first specified over a cylindrical surface inside the coil cylinder. The target field is Fourier transformed and substituted into Eq. [9] to yield the current distribution needed to produce the target field.
To complete the process of designing the coil it is necessary to identify conductor paths which when carrying current will best represent the current distribution. This is best achieved by using the stream function, $$$S(\phi,z)$$$, which describes the integral of the current distribution and is related to $$$\textbf{J}(\phi,z)$$$ via
$$\textbf{J}(\phi,z)=\triangledown S(\phi,z)\times \widehat{\textbf{n}}\space\space\space [10]$$
where $$$\widehat{\textbf{n}}$$$ is the unit vector that is locally normal to the cylinder surface.
Appropriate paths for wires or cuts in a conducting surface are given by the contours of this stream function. This process can also be applied to the current distributions calculated by any of the coil design methods described below.
Minimum Inductance or Power Relatively simple expressions for the inductance (or stored energy) (8) and resistance (or power dissipation) of the coil can also be written down:
$$L=-\frac{\mu_0a^2}{I^2}\sum_{m=-\infty}^{m=\infty}\int_{-\infty}^{\infty} dk\space \left| J_{\phi}^{m}(k)\right|^2 \space I'_m(ka)K'_m(ka) \space\space\space [11] $$
$$R=\frac{a}{\sigma tI^2}\sum_{m=-\infty}^{m=\infty}\int_{-\infty}^{\infty} dk\space \left| J_{\phi}^{m}(k)\right|^2 \space \left( 1+\frac{m^2}{k^2 a^2}\right) \space\space\space [12]$$
Here I is the current in the coil, while t and σ are the thickness and conductivity of the conducting material in which the current distribution flows.
Coils of Restricted Length In the coil design approaches described above, the length of the gradient coil is not so easily constrained. To incorporate the limited length in the definition of the current distribution (9), it can be defined in terms of a Fourier series of finite axial extent (2L), which for a z-gradient coil would for example take the form:
$$J_\phi(\phi,z) =\sum_{n=1}^{N}a_n sin(\frac{\pi z}{L}) \space\space\space \left| z\right|<L$$
$$J_\phi(\phi,z) =0 \space \space\space\space\space \left| z\right|>L \space \space \space [13]$$
To design the coil a mesh of P points defining the region in which a uniform gradient is desired is initially defined and then the coefficients $$$a_n$$$ which minimise a functional,
$$U=\alpha L+\beta R +\sum_{p=1}^PE_B^2(\textbf{r}_p)\space\space\space [14]$$
are then found (where $$$E_B$$$ the field deviation described in Eq. [1]). Adjustment of the weighting coefficients, α and β, allows varying significance to be placed upon reduction of the inductance or resistance. Generally increasing α and β leads to a decrease of the gradient uniformity as well as a reduction in the coil’s inductance and resistance.
1) Lee SK, Mathieu JB, Graziani D, Piel J, Budesheim E, Fiveland E, Hardy CJ, Tan ET, Amm B, Foo TKF, Bernstein MA, Huston J, Shu YH, Schenck JF. (2016) Peripheral Nerve Stimulation Characteristics of an Asymmetric Head-Only Gradient Coil Compatible with a High-Channel-Count Receiver Array. Magnetic Resonance in Medicine 76:1939-50.
2) Mansfield P, Chapman B (1986) Active magnetic screening of gradient coils in NMR imaging. J. Phys. E 15:235-239.
3) Roemer P, Edelstein WA, Hickey J (1986) Self shielded gradient coils. Proceeding of the 5th Annual Meeting of the SMRM, Montreal, P1067.
4) Romeo F, Hoult DI (1984) Magnetic field profiling: analysis and correcting coil design. Magn. Reson. Med. 1:44-65.
5) Golay MJE (1957) Magnetic field control apparatus. U.S. patent 3,515,979.
6) Turner R, Bowley RM (1986) Passive screening of switched magnetic field gradients. J. Phys. E 19:876-879.
7) Turner R (1986) A target field approach to optimal coil design. J. Phys. D 19:L147-151.
8) Turner R (1988) Minimum inductance coils. J. Phys. E 21:948-952.
9) Carlson JW, Derby KA, Hawrysko KC, Weidemann M (1992) Design and evaluation of shielded gradient coils. Magn. Reson. Med. 26: 191-206.
10) Moon CH, Park HW, Lee SY (1999) A design method for minimum-inductance planar magnetic-resonance-imaging gradient coils considering the pole-piece effect. Meas. Sci. Technol. 10: N136-N141.
11) Martens MA, Petropoulos LS, Brown RW, Andrews JH, Morich MA, Patrick JL (1991) Insertable biplanar gradient coil for MR Imaging. Rev. Sci. Instr. 62:2639-2645.
12) Aksel B, Marinelli L, Collick BD, Von Morze C, Bottomley PA, Hardy CJ (2007) Local planar gradients with order-of-magnitude strength and speed advantage. Magn. Reson. Med. 58:134-143.
13) Lemdiasov RA, Ludwig R (2005) A stream function method for gradient coil design. Conc. in MR: B 26:67-80.
14) Poole M, Bowtell R (2007) Novel gradient coils designed using a boundary element method. Conc. in MR: B 31:162-175.
15) Poole M, Bowtell R, Green D, Pittard S, Lucas A, Hawkes R, Carpenter A (2009) Split gradient coils for simultaneous PET/MRI Magn. Reson. Med. 62:1106-111.
16) Tang FF, Liu F, Freschi F, Li Y, Repetto M, Giaccone L, Wang YH, Crozier S. (2016) An improved asymmetric gradient coil design for high-resolution MRI head imaging. Physics in Medicine and Biology 61:8875-89.
17) Handler WB, Harris CT, Scholl TJ, Parker DL, Goodrich KC, Dalrymple B, Van Sass F, Chronik BA. (2014) New head gradient coil design and construction techniques. Journal of Magnetic Resonance Imaging 39:1088-95.
18) Edelstein WA, Hedeen RA, Mallozzi RP, El-Hamamsy SA, Ackermann RA, Havens TJ. (2002) Making MRI Quieter. Magnetic Resonance Imaging 20:155-63.