The performance of gradient systems in magnetic resonance imaging (MRI) can have a large impact on image quality, patient comfort, and system cost. The creation of large magnetic field gradients and the rapid transitions between the magnetic field gradients required for high-performance MRI and improved image quality in some applications can result in large local magnetic field (B-field) transitions that in turn can cause noticeable and sometimes painful nerve stimulation in some subjects (1–5).
In the design of gradient coils there is always a trade-off between gradient efficiency (defined as gradient strength per unit driving current), slew rate, region of coverage, and uniformity (6). Within certain limits, improvements in any one performance measure can be obtained only at the expense of one or more of the other measures. The relationships between these different gradient design factors are not linear, and nonoptimal designs are easily obtained (6).
High gradient amplitude is needed for diffusion imaging and high spatial resolution imaging in general. High gradient slew rate is important for shorter echo times (TEs) in nearly all imaging applications. Gradient field uniformity, which is important to avoid image distortions, limits the field of view (FOV) from which images can be acquired in many current MRI scanners to a small region of about 40 cm in diameter. One can improve gradient performance by increasing the field amplitude and slew rate, without increasing nerve stimulation, by reducing the available gradient FOV (GFOV). However, as gradients become physically smaller in size, the uniformity over an FOV of fixed size is generally worse. Reducing gradient performance to attain an increased GFOV is not desirable.
There are many applications in which it would be useful to acquire images from FOVs that are larger than those available on current MRI systems. For some applications, such as vascular runoff, tumor staging, and whole-body screening, it would be of value to be able to image the entire body simultaneously, although this would require significant changes in existing magnet designs. However, the ability to extend the GFOV in existing magnets while maintaining high gradient performance for rapid image acquisition and reducing the potential for nerve stimulation would constitute a significant improvement in these applications over current imaging systems.
Vascular runoff studies are performed in conjunction with an intravenous contrast agent to visualize the arterial system and diagnose vascular disease from the aorta to the toes (a distance of >1 m in most subjects). In the ideal case, the full region would be imaged dynamically at relatively high spatial and temporal resolution so that the timing of the filling of small arterial branches could be observed. Such large-FOV imaging is not possible in current MRI systems. The full FOV must be divided into several small regions that are then imaged either separately in conjunction with a contrast injection for each region, or sequentially following a single contrast bolus injection by moving the table rapidly between regions (7–13).
For all techniques there are trade-offs between the time spent at any imaging station, resulting in improved spatial and/or temporal resolution, and the spatial and temporal resolution obtainable from the full FOV desired. Because of nerve-stimulation effects, there are also trade-offs between the gradient performance parameters achievable and the FOV over which the gradients can be applied.
In this paper we present the concept of multiple-region gradient arrays as a potential method to achieve extended FOV imaging and improve gradient performance while reducing the potential for nerve stimulation. The concept of spatially oscillating gradients was first described by Oppelt (14), but no attempt was made to investigate the design or potential performance properties of such gradient systems. In this work we implement the stream function approach to design both longitudinal and transverse gradient arrays, and use simulated annealing to obtain a family of near-optimal solutions. By examining the properties of the resulting solutions, we can select a design that yields the most desirable trade-off of gradient performance factors. We explicitly compute the coil parameters (inductance, efficiency, and gradient homogeneity) as well as the magnitude of the B-field. The latter is important because it is rapid change in the magnitude of the B-field that induces nerve stimulation.
Multiple-Region Gradient Arrays: Trade-Off Between Performance and FOV
Nerve stimulation can be caused by rapidly changing magnetic fields, such as those that occur in high-performance magnetic field gradients for high-performance MRI. The sensitivity to nerve stimulation depends on the rate and magnitude of the field transition. Although the sensitivity to nerve stimulation varies among individuals, Chronik and Rutt (3, 4) demonstrated that the threshold of the gradient change for nerve stimulation can be expressed as a function of a minimum gradient amplitude, ΔGmin, and a minimum slew rate, SRmin: ΔGstim(τ)=SRmin · τ+ΔGmin. ΔGmin and SRmin are subject-dependent and vary with the size and imaging extent (GFOV) of specific gradient designs. Despite the large variability among individuals, experiments demonstrate that individual thresholds follow this basic relationship and that the threshold limits, ΔGmin and SRmin, increase (stimulation decreases) as the GFOV decreases.
Until recently the physical extent of each gradient system was chosen at the time of construction and could not be changed after installation. However, with the recent development of modular, dual, and twin gradient systems, gradient performance can be reduced to image a large FOV, and increased to image a small FOV (15–18). Such modular gradient systems allow a dynamic trade-off between the FOV and gradient performance while they avoid peripheral nerve stimulation. In all cases, large-FOV imaging cannot be performed without a reduction in gradient performance.
An alternative method to attain large-FOV imaging that would not require a reduction in gradient performance would be to create gradient systems with regions of spatially alternating gradient polarity (14). The proposed design uses repetitions of the longitudinal and transverse gradients to create multiple nonadjacent imaging regions. Imaging is performed in the regions where the applied field gradients are approximately linear.
z-Gradient Array Design
Just as two alternating current distributions (Fig. 1a) form the Maxwell pair gradient that is used to image a single region, three alternating current distributions (as shown in Fig. 1b) can be used to form a Maxwell triplet, a z-gradient coil with two linear gradient regions of opposite polarity separated by a transition region where the gradient changes polarity. Such a Maxwell triplet might be used to perform image acquisition over two noncontiguous regions of linear field variation and thereby improve imaging efficiency.
The concept of the Maxwell triplet can be generalized to an array consisting of an arbitrary number of alternating dipole elements designed to give a spatially oscillating magnetic field. The field between each dipole pair is very similar and only slightly reduced from that obtained from a Maxwell pair. In each dipole the number and position of wire loops are chosen to optimize the trade-off between gradient performance and uniformity.
Overlapping Maxwell Arrays
The three-loop Maxwell array, or triplet, is of particular interest because it is the simplest multiple-region gradient array. Although imaging can be performed in the two nearly linear regions, it cannot be performed in the nonlinear (rounded) central part. As shown in Fig. 2, the central part can be imaged with the use of a Maxwell pair that is centered with the Maxwell triplet such that the linear region of the Maxwell pair fills in the imaging gap of the triplet. We note that the image acquisition technique would have to acquire image data alternately from the two coil arrays. Instead of the Maxwell pair, the overlapping gradient could be a second Maxwell triplet with two imaging regions or a Maxwell quadruplet with three imaging regions, shifted such that their imaging regions interleave with the two regions of the Maxwell triplet.
Transverse-Gradient Array Design
The design of a transverse gradient array is more complicated than that of the Maxwell array, but the principle is the same. The gradient in Bz in the transverse direction is formed by currents that create dipoles perpendicular to the cylinder surface. The conventional transverse gradient consists of four sets of wire loops placed on the cylindrical former in a “fingerprint” pattern. Each dipole represents one quadrant of the fingerprint gradient wire windings (16, 18–20). The performance of transverse gradients is also limited by nerve-stimulation effects. To maintain gradient performance while extending the FOV without increasing nerve stimulation, we propose an array design that is conceptually similar to the Maxwell triplet.
Relations Used for Gradient Design
In this work stream functions (21) are used to specify the current distributions on cylindrical surfaces for both longitudinal and transverse gradient coils. From these current distributions, the magnetic field as a function of position within the cylindrical volume and the coil inductance are computed. Following the example of Tomasi (20), imaging regions are specified and simulated annealing is used to adjust the stream function to maximize the gradient efficiency and minimize the inductance and the deviation of the gradient from linear within the specified regions. Although this use of stream functions results in a monolayer design for cylindrical gradient coils, it provides designs that are adequate for proof of the gradient array concept. Designs without and with shields were considered.
The continuity equation, ∇ · =0, allows the currents on the cylinder surface to be defined in terms of the curl of a stream function, S, which for convenience is defined as a radial vector: S=(S(φ, z), 0, 0) (20). The magnetic vector potential is defined in terms of the surface currents, and the magnetic fields are computed from the curl of the vector potential.
For the z-gradient coil, the stream function has no angular dependence: S(φ, z)=hz(z). For the case of no shield, the z-component of the magnetic field and the inductance can be written as (22):
respectively, where ρ, the radius of measurement, is less than a, the coil radius, I and K are the modified Bessel functions, and jφm(k), jzm(k) are the components of the Fourier transform of the current density on the cylinder.
Similarly, for the transverse gradient coils the simplest stream function has a simple harmonic dependence on φ given by: S(φ,z)=hx(z)cosφ. In the unshielded case, the field (at the angle for which Bz(ρ,z) is maximum) and inductance become:
To assess the relative potential for nerve stimulation, we calculate the full magnetic field including the components, Bρ and Bϕ. For the z-gradient coil:
For the transverse gradient:
From Eqs. , , and  it is evident that the magnitude of the magnetic field is much greater for φ=0 than for φ=π/2, and for simplicity we consider the field in the plane defined by φ=0 where Bφ=0.
Finally, for shielded gradients the passive shielding current density (as described in Ref.22) is used and the above equations are modified appropriately.
Our purpose in using simulated annealing was not to obtain a single solution that is optimal for some figure of merit, but to obtain a large number of near-optimal solutions by sampling the solution space in the region close to that optimal solution. In this manner the set of possible solutions can be reviewed, and trade-offs between the solutions driven by the chosen figure of merit and other potentially important factors can be evaluated (23).
For the longitudinal gradient array, the figure of merit FOM used was:
η is the gradient per unit current, and V is the volume of the GFOV. For convenience we compute these factors in the cross section of the GFOV in the x/z plane, and define FOVz and FOVx as the length and diameter of the GFOV, respectively. For the transverse gradient array, a large variability was observed in the total magnetic field external to the imaging volume. To create solutions that constrained this variability, it became useful to explore solutions that also minimized the total field external to the imaging regions of interest (ROIs), but potentially within the bodies of large subjects. In this case we used:
where Bmax(ρ) is the maximum of the total magnetic field on a cylinder of radius, ρ. Although the potential for nerve stimulation is difficult to assess, it is assumed that this potential increases as the gradient magnetic field increases as long as other parameters, such as the imaging volume and gradient slew rate, do not change. In general, the magnetic field created by the gradients increases toward the gradient windings. To obtain a qualitative comparison of the potential for nerve stimulation, we chose to calculate the maximum of the total magnetic field at a distance of 5/6 of the winding radius, Bmax5/6. This field is outside the imaging volume, given that the radius of the imaging volume was selected to be 3/4 of the winding radius but may still lie within the subject being imaged.
MATERIALS AND METHODS
The gradient system designs created in this work were obtained by using simulated annealing to explore the space of possible stream functions. As in Ref.20, the stream function, h(z), was divided into monotonically changing sections. Instead of polynomial and exponential representations for h(z), each monotonically changing section was modeled by a set of monotonically changing piecewise linear segments (Fig. 3). We found that these monotonically changing piecewise linear segments were easy to define in terms of realizations of sets of random variables, and at the same time provided a more general representation of the ends of the transverse gradients. Because all gradients are symmetric or antisymmetric in z, it was only necessary to determine h(z) for z > 0 and apply the appropriate symmetry to obtain the full h(z).
As shown in Fig. 3, the set of random numbers from the simulated annealing process is used in such a manner that the resulting segment heights and slopes are both monotonically increasing. The segment widths are predefined to monotonically decrease. For the transverse gradient coil, the segments were defined to be equally spaced and the heights in each section to change monotonically, but no restraint was placed on the slopes. More general forms, or forms with slope continuity, could have been used; however, the forms evaluated were sufficient for the purposes of this demonstration.
To compare gradient performance parameters between standard gradients and corresponding two-region gradients, we divided the cylindrical imaging volume, GFOV (defined by length FOVz and diameter FOVx) into three equal subvolumes of length FOVz/3. The single-region gradients were designed and optimized to image the entire cylindrical volume. The corresponding two-region gradients were designed and optimized to image the outer two of the three subvolumes.
Simulated annealing was used to create families of stream functions and corresponding gradient wiring patterns for a variety of imaging volumes. From the range of results, the stream function that gave a desirable compromise in performance parameters was selected. The gradient wiring dimensions, the desired imaging volumes, and the performance parameters resulting from the selected stream functions are summarized in Table 1. Using a conductor cylinder diameter (2a) of 60 cm, we designed the gradients to cover cylindrical imaging volumes with FOVz/FOVx = 90 cm/45 cm, 60 cm/45 cm, 51 cm/45 cm, 51 cm/22 cm, and 51 cm/21 cm (with shield). The latter designs could potentially be useful for small animals and proof of principle. FOVz and FOVx are respectively the length and diameter of the cylinder over which the resulting homogeneity is measured. For the single-region coil, the homogeneity (defined as the root mean square (RMS) deviation of the predicted field from the desired field) is computed over the entire cylinder volume. For the double-region coil, the homogeneity is computed over the desired smaller volume (section length = FOVz/3). The coil length, w, and radius, a, are the dimensions of the cylinder over which the coil windings are placed. As shown in the table, in every case the efficiency of the double-region coil is higher than that of the single-region coil for the same inductance.
All gradients are designed assuming inductance, L = 600 μH.
In Fig. 4 scattergrams are plotted of the FOM factors obtained during the simulated annealing process for the longitudinal gradient coils for FOVz = 90 cm and FOVx = 45 cm. In the figure the RMS deviation (Fig. 4a) and Bmax5/6 (Fig. 4b) are plotted as a function of the relative efficiency, ηr (defined as the efficiency divided by the square root of the inductance). Defined in this manner, ηr is independent of the scaling of the stream function (the number of wires per unit change in h(z)). In general, the ηr of the double-region gradient is substantially higher than that of the single-region gradient. Thus, for the two clusters, the results for the single-region gradient are to the left and those of the double-region gradient are to the right in this plot. The plot in Fig. 4b illustrates how, even for the single-region gradient, this local maximum magnetic field can be substantially reduced by allowing greater inhomogeneity within the imaging volume. However, picking an operating point where the inhomogeneity is minimized, the plots illustrate that the local maximum magnetic field for the two-region gradient is much lower (less than half) than that of the single-region gradient. If the same level of inhomogeneity is used as the operating point, the efficiency is increased while the local maximum magnetic field for the double-region gradient is only slightly more than half that of the single-region gradient.
The stream functions for the operating points indicated in Fig. 4 are plotted in Fig. 5. The gradient field plots are shown in Fig. 6. The rectangles show the cross section of the cylindrical imaging volume for the single-region coil and the two imaging volumes of the double-region coil. The contour lines represent changes in the gradient field that are 5% of the gradient field variation across the full GFOV. In Fig. 7 we plot the gradient fields for single- and double-region coils of 60-cm length (only slightly larger than the 50-cm length of at least one commercially available MRI scanner). For all coils, the winding diameter is 60 cm and the imaging volume diameter is 45 cm.
The simulated annealing results for the 60-cm long, 45-cm diameter transverse gradient coil designs plotted in Fig. 8 illustrate the increased difficulty of designing a double-region transverse gradient. The operating points selected yield a substantial increase in inhomogeneity for the double-region coil over the single-region coil, while the local maximum magnetic field for the double-region coil is larger than half that of the single-region coil. The stream functions for the selected operating points are shown in Fig. 9. The gradient field plots for the 90-cm-long, 45-cm-diameter transverse gradient are shown in Fig. 10 and those for the 60-cm-long, 45-cm-diameter transverse gradient are shown in Fig. 11.
The unshielded gradients demonstrate the basic concepts of this paper. However, for completeness we have added shielded designs using the methods of Turner and Bowley (22). The stream functions for a double-region shielded gradient and the resulting gradient field for FOVz = 51 cm and FOVx = 21 cm are shown in Fig. 12.
The purpose of this study was to demonstrate the potential value of using gradients with spatially oscillating magnetic fields to facilitate imaging extended FOVs with high gradient performance, while minimizing the total magnetic fields and the resulting potential for nerve stimulation. In this work we only calculated and considered the magnetic fields in air generated by the various current distributions. While this provides an indication of the potential for nerve stimulation depends directly on the intensity, distribution, and timing of electric fields generated within a given subject. These fields in turn depend on the applied magnetic fields and the distributions of electric properties (conductivity, permittivity, and permeability) within the subject, and are thus subject-dependent.
We examined several different gradient aspect ratios to determine the nature of the improvement that might be expected. Observations from the results are summarized in Table 1. First we note that all transverse gradients have substantially higher total magnetic fields than the corresponding z-gradients for the same GFOV. The ratio appears to vary between a factor of 1.6–2.75 for the single-region gradients, and to nearly 3.9 for the double-region gradient. A review of the table indicates that the transverse gradient is least improved when the aspect ratio (FOVz/FOVx) is small. Similarly, the ability of the double-region gradients to reduce the maximum of the double-region magnetic field (Bmax5/6) is greatest for imaging volumes with large aspect ratios.
Most of the results presented were obtained with unshielded gradients; however, we performed comparison simulations to design shielded gradients and obtained essentially the same results, which indicates that similar results would be obtained regardless of whether the gradients are shielded. We performed many design experiments that are not reported here, but the results reported are typical of the results obtained. In every case, improved performance is obtained from the conversion to gradient arrays, and the amount of improvement is always greatest when the aspect ratio, FOVz/FOVx, is greatest. However, there is also some effect based on the total length of the gradient cylinder that is available for receiving the windings. The improvement also depends on the trade-off made with other performance parameters, such as gradient efficiency and homogeneity.
Although the gradient-array concept arose because of the need for extended FOVs in human imaging, the small-bore gradients may be useful for imaging arrays of small animals. Whereas small-bore MRI scanners have the disadvantage that only one animal can be imaged at a time, the gradient-array concept could be used to design high-performance gradients that are capable of imaging multiple animals or groups of animals simultaneously.
The techniques used in this study are relatively simple, and may not be considered most optimal. The use of piecewise linear stream functions results in piecewise constant current distributions, and one could reduce such discontinuities in current density by using more segments. It is likely that some form of polynomial representation, similar to that used by Tomasi (2), could be used to obtain smoothly changing current distributions and hence more homogeneous imaging regions. In general, the use of a functional form with a small number of coefficients will limit the range of stream functions that can be attained. It is likely that more optimal solutions will require more general functional forms. However, it is not likely that more complicated stream functions would change the observations that the gradient-array concept can reduce the maximum magnitude of the gradient magnetic field while improving gradient performance.
Finally, we note that the practical realization of these results will depend upon how well the wire placement patterns can emulate the specified current densities. It is expected that deviations from the predicted results will be small and not of consequence in terms of the general observations made here. For example, the deviations of the magnetic field should be minor for the fields internal and external to the gradient coils. The screening efficiency will not be 100% in a real implementation, but there is no reason to expect that this efficiency will be worse for multiple-region gradients compared to single-region gradients.
We have not addressed the many technical issues that must be resolved before multiple-region gradients can be used in MRI scanners. Such technical issues include 1) decoupling the offset gradient arrays from each other, 2) designing single or multiple RF transmitters and coil arrays to selectively excite the desired regions with minimal saturation of spins in non-imaged regions, 3) designing multiple RF receiver arrays to obtain signals from each region while minimizing signals received from neighboring regions, 4) investigating the feasibility of using homogeneous regions on existing magnets or developing magnets with longer homogeneous regions, 5) designing pulse sequences that utilize the full capabilities of the multiple-region gradient arrays, and 6) determining the mechanical forces that occur on the array. These technical and safety issues pose significant challenges to the design of gradient arrays. However, the decoupling of two-gradient systems has been addressed in the design of twin gradients, which are commercially available. Also, scanners with up to 32 RF receivers are available, and multiple transmitter systems are being investigated for transmit sensitivity encoding (SENSE). These and other new advances will help mitigate the gradient-array imaging problems mentioned above.
We have demonstrated that multiple-region gradient arrays can be designed to achieve multiple-region imaging with increased relative efficiency and reduced magnetic field excursions compared to single-region gradients that cover comparable imaging volumes. The improvement in gradient performance is greatest when the aspect ratio (FOVz/FOVx) is large, but improvement can also be achieved in designs for shorter systems, such as might be found in existing imaging magnets. The potential for improved small-animal imaging is also very real.
The authors appreciate helpful discussions with Blaine Chronik and Brian Rutt of the University of Western Ontario and the Robarts Research Institute, and Craig Goodrich and other members of the Utah Center for Advanced Imaging Research at the University of Utah.