As the number of available receiver channels on modern MR systems increases, greater attention will be paid to the design and performance of many-element RF coil arrays. Questions regarding the balance of coil-noise and sample-noise, or the suitability of any particular array design for parallel imaging, promise to take on new significance as the number of elements increases. In the present work a method has been described to evaluate the performance of any particular coil array against a well-defined absolute benchmark. Coil performance is strictly constrained by the behavior of electromagnetic fields within the sample 6-8
. For a chosen imaging task, the best possible SNR achievable for any coil configuration can be computed using a complete set of coil sensitivity basis functions. In this study, we used ultimate intrinsic SNR as a reference to assess the efficiency of a 32-element head receive array.
It should be noted that, although our experiments were performed only at 2.89 T, the basis functions used in this work take full account of the frequency-dependence associated with operation at different field strengths, and that coil performance maps are possible at arbitrary field strength. That said, simple phantom geometries with uniform electrical properties are expected to become an increasingly poor approximation of in vivo SNR behavior with increasing field strength. The characterization of coil performance for arbitrary electrically inhomogeneous objects remains possible with appropriate choices of surface-based field or current basis functions, but such characterization would likely require extensive computational effort. In fact, the calculation of ultimate intrinsic SNR would require a full-wave numerical solution, for example using the Finite Difference Time Domain (FDTD) technique, for each current mode, as well as detailed knowledge of the dielectric structure and spin distribution of the object.
In simulating the ultimate SNR we are effectively using an infinite number of coils surrounding the object, and we expect a very high signal close to the surface. As a consequence, the ultimate SNR rises rapidly at the edges, and relative coil performance is higher near the center of the object. A similar spatial variation was reported in a previous paper which showed coil performance maps in the case of non-parallel MRI 6
. The performance values presented in that study are slightly lower than those found for the unaccelerated case in this investigation, but the results are not directly comparable as the phantom, the image section, and the imaging system were different. Coil performance maps for parallel MRI were presented for the first time at the 2006 meeting of the International Society for Magnetic Resonance in Medicine 9
, for the case of a cylindrical phantom. In that preliminary work, we reported performance values of less than 50% in the center of the sample, for a coil array with fewer elements that was more poorly matched to the phantom than was the case in this work.
In the current work, the overall maximum performance, corresponding to the unaccelerated case with Phantom 1, was more than 80% in the central portion of the phantom, implying that there is little room for improvement in coil array design for the case of fully-sampled acquisitions. A slightly higher performance in the center of a uniform spherical sample was predicted by a prior simulation study which modeled receive arrays of circular coils distributed around the surface of the sphere 26
. In that work the electrical properties of the sphere were chosen to approximate values in the human brain and they were consistent with the properties measured for Phantom 1. In order to investigate whether the higher performance values in the center were due to the fully encircling configuration of the coil elements in Ref. 26
, as opposed to the open-faced configuration of our 32-element array, we also simulated (see ) the performance with respect to the ultimate intrinsic SNR of a 32-element array, with loop coils arranged similarly to the head array in . The simulated unaccelerated peak performance of 97% is similar to the values reported in Ref. 26
, which suggests that removing coils at the bottom and front of the array has little effect on the SNR in the central region of the phantom. The difference in values between the map in and the corresponding map in must be then linked to other factors.
- A minor contribution may traced to our expression for voxel volume in Eq. . In reality the signal contributions to a given reconstructed voxel value are weighted by the point-spread function (PSF), whose integrated volume may differ somewhat from simple cuboid voxels. However, the Cartesian SENSE reconstruction we used maintains a nearly sinc-shaped PSF, whose integrated volume is the same as that for a cuboid voxel. To the extent that the phantom is uniform, that the sensitivities vary slowly within a given voxel, and that partial volume effects are limited to positions near the edges of the object, the voxel volume is not expected to differ from that used in Eq. .
- Another minor effect may be related to our assumption of uniform noise factor for all receiver channels. In practice, variations in noise factor are expected to be modest (e.g. ~10%), resulting in still smaller variations in measured SNR for any given channel (~5%). Meanwhile, differing values in different receiver channels will be averaged to some extent by inclusion in the noise correlation matrix, resulting in even smaller effects on combined SNR.
- An additional complication is that, while we do account for body-derived noise correlation between array elements (see Eq. ), we do not model inductive coupling, which may also contribute to noise statistics. In fact, measured noise correlations between coil elements are on average higher than predicted by our DGF method (ratio of off-diagonal to diagonal elements of the noise correlation matrix 55% maximum and 9% average for simulations versus 53% maximum and 12% average for experiments). To the extent that this increased correlation reflects pure linear inductive coupling, it is not expected to have a significant effect on SNR when the noise correlation matrix is included in the reconstruction 27. However, some degradation due to extensive coupling cannot be ruled out.
- The most significant contribution to the discrepancy between simulated and measured performance is likely due to the presence of additional noise sources that are not taken into account by our DGF formulation. The weights used for our 32-element DGF calculations produce a relatively simple current distribution inside the conductors, neglecting current concentration effects along edges and around capacitors.
The particular contributions of each of these effects to the observed discrepancy between simulated and experimental coil performance may be investigated in detail. However, one of the explicit advantages of ultimate performance maps is that, when combined with appropriate simulations of particular array geometries, they can indicate now much of the room for improvement in coil performance relates to practical as opposed to theoretical limits.
Phantom 2 was included in this study to illustrate the effects of electrical properties upon coil performance, and also to provide data from a phantom which is widely available to the research community. Absolute SNR values were higher for Phantom 2 (see ) due to its smaller electrical conductivity, which results in lower sample noise (see Eq. 
). On the other hand, smaller electrical conductivity corresponds to larger standing wave effects, which are partly responsible for the lower performance of the 32-element head array with Phantom 2. In fact, as the array was tuned and matched for the human head, better loading with Phantom 1 might have positively affected the performance results because the array was designed to provide the correct noise match to the preamplifiers with a human head-type load, and because the higher conductivity of Phantom 1 increases loading and leads to a smaller relative contribution of coil noise. The fact that coil performance was lower with Phantom 2 does not imply that the head array design is suboptimal for its target clinical applications, as Phantom 2 is filled with a solution whose electrical properties are not meant to approximate human tissue, but rather to minimize susceptibility artifacts.
The average performance of the coil array (see and ) was almost constant for 2-fold acceleration, but rapidly decreased for larger acceleration factors. This suggests that 32 elements, arranged as in , are not enough for effective highly-accelerated parallel imaging and that larger arrays, or a different configuration of the conductors, are needed in order to approach the acceleration efficiency of the ultimate intrinsic case.
The examples shown in this work demonstrate that the method proposed here can serve as a tool for the evaluation of coil designs. Ultimate intrinsic SNR defines an absolute performance target for coil designers, and coil performance maps provide useful and immediate feedback on how far a particular array configuration is from this target. Ultimate SNR need be computed only once for each particular geometry and object composition, which facilitates its use as an effective evaluation instrument for coil engineers.
Given the comparatively large set of basis functions required for convergence of the ultimate intrinsic SNR at multiple positions, it is unlikely that all the modes used to simulate the optimum can be practically realized with an actual coil array. However, the asymptotic growth of the ultimate intrinsic SNR with increasing number of modes (see Ref. 6-8
), and our results showing that 32 elements nearly saturate the ultimate SNR in the unaccelerated case, demonstrate that a smaller subset of the larger basis can capture the dominant SNR behavior. It may even be possible to build an array targeted to that subset of modes in order to approach the simulated optimal sensitivity patterns, using the weighting factors generated by the ultimate SNR optimization to directly compute these ideal EM fields. For example, the distributed optimal current pattern shown in could in principle be approximated with a comparatively small number of concentric loops. However, the optimum will be different for every point in the sample and thus only a tradeoff solution might exist for the conductor patterns needed to approach optimal performance at a large number of positions. Genetic algorithms have been already applied to a similar optimization problem 28
and, with an accurate and robust parameterization, they might be employed also for the design of optimized coil arrays for different parallel MRI applications.
Since such arrays should in principle perform very close to the optimum, any substantial discordance could be linked to noise other than that coming from the sample. The development in recent years of arrays with very many elements 29
has raised to practical priority the question of what is the smallest size for array elements before the final SNR begins to be dominated by the noise coming from the electronic components. Compared to other methods previously proposed to calculate ultimate intrinsic SNR (see Ref. 6-8
), the DGF approach used in this work allows convenient modeling of arbitrarily-shaped finite arrays within the same framework.
In conclusion, the capability of parallel MRI to accelerate image acquisitions is fundamentally limited by electrodynamic constraints, but the knowledge of such limitations can be exploited to evaluate current coil design and eventually to guide the development of innovative receivers that may operate close to the optimum performance.