High-quality magnetic field homogeneity is an essential prerequisite for most MR imaging and spectroscopy applications to guarantee, among others, the correct spatial registration, the best signal strength and optimal spectral resolution (1). Unfortunately, the homogenization of magnetic field distributions (i.e. shimming) of the mouse brain is a difficult task. Air-tissue interfaces e.g. between the auditory cavities and the brain create complex, strong and localized magnetic field distortions.

‘Passive shimming’, i.e. the strategic placement of magnetized pieces, has been shown to be capable of correcting a large part of the magnetic field distortions observed in the mouse brain (2). The labor-intensiveness of the method and the lack of flexibility to account for specific conditions in single experiments such as inter-subject variation in brain anatomy or head positioning might be reasons why passive shimming to date has not prevailed in experimental setups.

The most widely used shim method called ‘active shimming’ generates shim fields by driving coils with electric current. Fast and accurate field alterations are accomplished by changing the currents through the field producing coils and enable the method to account for experiment-specific conditions. To date, dedicated coils are used for active shimming, each of which provides a magnetic field shape to resemble one of the low order (n≤4) spherical harmonic (SH) functions. In practice, however, the available sets for active SH shimming are often times limited to first and second order SH terms only. The exclusive use of SH-shaped magnetic fields for shimming has been accepted by the MR community as a historical given for the last 60 years. If the SH functions are motivated in the literature at all, this is done by the notion that they form an orthonormal basis set satisfying Laplace's equation (3,4). An infinite series of SH terms with unconstrained amplitudes is indeed capable of modeling arbitrary field shapes. However, the realization of such field modeling system is not possible experimentally due to the infinite number of necessary coils and because efficiency (i.e. the generated magnetic field amplitude per unit current) faces increasingly strong limitations for higher order SH terms (4). The acceptance of the limitation of SH basis sets to low order terms corresponds to the assumption that the apparent field distortions can be satisfactorily modeled with the available low order SH field terms. However, given the large number of studies dedicated to whole brain shimming (2,5-11) this is clearly not the case in the human nor the rodent brain. Active shimming with low order SH basis sets is capable of compensating for shallow, large-scale magnetic field distortions, but it is not able to fully address the complex field distortions with localized hot spots as they are observed in the human (11) or the rodent brain (1).

Due to the lack of satisfactory whole brain SH shimming, in reality, the shim procedures as well as the subsequent MR experiments are often times limited to single brain slices (12,13) or cubic volumes (14-17) to better explore the capabilities of the SH functions. Efficient methods have been presented to provide fast and reliable magnetic field optimization in these selected geometries (12,13,16).

Dynamic shim updating (DSU) employs this principle for the homogenization of larger volumes by breaking them down into subvolumes such as single slices or voxels that are compatible with the MR experiment (5,7,18). The fast adjustment of volume-specific shim settings then allows the optimization of the magnetic field homogeneity in larger volumes based on the improvement in the constituent subvolumes within the same experiment. Switched SH shim coils typically generate a multitude of eddy current-related artifactual magnetic fields due to their proximity to the scanners' cold, conducting structures. Even if eddy current-related fields are compensated diligently (10) and although dynamic SH shimming is more powerful than static global SH shimming, it is still limited to relatively simple magnetic field distortions that do not adequately approximate the complex magnetic field distortions encountered in the mouse brain (19).

Due to limitations in whole brain shimming of the mouse, regions-of-interest for MR experiments are typically chosen small compared to the size of the brain and placed well inside the brain. The corresponding (static) shim areas, therefore, can avoid the vicinity close to the brain surface and more importantly the major distortion areas (14,20-24). Practical whole brain shimming of the mouse brain has not been achieved yet with any method, nor has whole brain slice shimming that is not limited to arbitrary, e.g. cubical or elliptical subsections.

The SH functions are only one of many possible basis sets for the description and synthesis of magnetic fields. The orthogonality of the SH terms has been accepted by the MR community as an essential prerequisite based on early papers (3). However, when magnetic field homogenization is achieved through the use of least-squares methods, orthogonality of the basis set is not a requirement. In this study, a paradigm shift away from the SH functions towards an alternate set of non orthogonal basis functions is proposed. SH shapes for shimming were replaced by a set of generic basis fields generated by circular, electrical coils placed on a cylindrical grid that provides highly localized gradient patterns. It is demonstrated that magnetic field modeling is possible with such an approach and that greatly improved shimming of the mouse brain can be obtained. After the introduction of the multi-coil (MC) concept for the synthesis of low order SH fields (25) here we show that shim fields generated by a set of independent non-SH coils allow improved slice and whole brain shimming of the mouse brain at 9.4 Tesla.

Since the concept of low SH order shimming does not allow satisfactory results for brain shimming in rodents (and humans), specific passive (2,6,8) and active (9,11) shim methods have been presented in which non-orthogonal, non-SH terms are added to the SH basis that are tailored to field distortions in the most problematic shim areas. The MC shim approach differed from these previous shimming techniques by the use of a generic coil matrix and by a complete refusal of SH shaped terms. The MC field shapes were not specifically tailored to the targeted field distortions like in (11) nor were the basis shapes required to be orthogonal. However, together they provided a repertoire of field shapes that included strong local gradient patterns close to the individual coils and shallow field shapes further away from the coils. The combination of individual coils with the MC approach, therefore, allowed the generation of magnetic fields with a degree of complexity that exceeded the capabilities of traditional SH shim coils. In fact, the low order SH terms just form a subsection of the repertoire of fields that can be generated with the MC approach. To this end, the SH fields could be discarded completely with the presented MC shimming, as they did not add to the available degrees of freedom.

The seven mice in this study had similar weight and predictable head positioning with inter-subject variation of ≤1 mm was achieved through the use of a bite bar. The essential patterns of the anatomy-based and susceptibility-induced field distortions before shimming were similar in all mice which was represented by the standard deviations of the different frequency measures being less than 10% of their average values. The requirements for SH shimming, however, showed major mouse-specific variations (although the limits of the dynamic range of the used SH shim system were never exceeded). In other words, SH shimming was not limited by the available shim field amplitudes, but by the inadequacy of the SH shapes. A down-scaling of the SH coil system from the size of the scanner bore to the size of the applied MC setup would lead to efficiency gains of the SH field generation, but the shapes of the generated fields as well as the shim performance would remain the same as SH fields are self-similar, i.e. they are scalable.

It has been demonstrated previously that the experimental generation of SH (10) and MC (25) fields can be predicted very accurately. The thorough calibration of the MC system is considered key for the determination of all 48 coil currents from a single reference field map. Similarly, the thorough calibration of the scanners' SH shim system allowed the reliable adjustment of the SH shim fields in a single step process. Multiple iterations as reported for SH shimming of the mouse brain in (20) were not necessary. The close congruence of theoretical predictions and experimental results in all mice and for all methods is represented in the low average deviations between them (Tab. 1, columns 3, 6 and 9). The variations of the reported frequency measures themselves, i.e. the S.D. of theoretical predictions and the experimental results in Table 1, therefore, can be attributed to inter-subject differences of the 7 mice with respect to the performance of the considered shim methods.

Spurious signals from outside the brain can pose problems for MR imaging and spectroscopy. Examples include the artificial excitation of the non-brain magnetization due to imperfect RF localization schemes or the spatial reconstruction of lipid signals from outside the brain to brain locations based on the different chemical shift. A commonly applied remedy is the active destruction of the non-brain magnetization before or during the experiment. The DMC shim fields in this study were tailored to the magnetic field distortions in the brain. Shim fields to address strong magnetic field gradients in the periphery of the brain also typically generated strong gradient terms in other parts of the considered slices outside the brain. Depending on the required shim fields, the outer brain components led to very effective phase spoiling that removed the largest part of the non-brain imaging signals (Fig. 3, column 4, slices 4-6) and spectroscopic acquisitions of these slices were essentially free of lipid contaminations (data not shown). Although phase spoiling of non-brain areas has not been the primary focus in this study, the DMC approach has the potential to include the generation of dedicated phase spoiling gradients outside the brain into the brain shimming routine.

Simulations have shown that the inclusion of the third order SH terms for static, global SH shimming of the mouse brain improves the frequency measures by only 7%. The reduction e.g. of the S.D. of the frequency distribution from 69 Hz to 62 Hz can hardly justify the additional effort and expenses that are necessary for the installation of the third order SH terms. The limited improvements of mouse shimming with the inclusion of further low order SH terms other than perhaps for very specialized applications is due to the complexity of the magnetic field artifacts in the mouse brain (Fig. 2C) which is beyond the modeling capability of low order SHs and much higher order SH terms would be required to describe them. The magnetic field shapes of the MC basis set scale from large to shallow local gradients based on the distance and relative positioning of the considered volume-of-interest to the individual MC coil. The availability of strong localized field gradients is the basis for the largely improved field modeling capabilities and the reduction of the homogeneity measures even in the static case (Fig. 2D). Simulations have furthermore shown that second SH order DSU can compete with the static MC shimming as presented in this study. However, the achievable frequency measures are on average still 52% broader than with DMC shimming for the considered axial slicing.

The MC method is by no means limited to the details of the selected coil geometry. Simulations have shown that other matrix coil setups, e.g. with 8 rings of 6 coils, or even completely different coil configurations such as a series of coils spiraling around the used cylindrical former are also possible. Each geometry will have fundamental limitations on the magnetic field complexity that can be realized, but a wide range of configurations are possible to achieve satisfactory magnetic field homogeneity in the mouse brain. However, increased modeling capabilities are expected from further improvements of the MC matrix. A further miniaturization of the coils will be beneficial for the generation of even more localized field patterns that will be even better suited to correct highly confined field artifacts in the mouse brain. Similarly, a cone-shaped former for coil mounting will minimize the distances between the coils and the mouse brain and will facilitate the use of smaller coils. The choice of 48 coils in this study was determined by the available space surrounding the mouse head, the number of available power supplies and theoretical simulations predicting the benefits of (D)MC shimming. Simulations have shown, however, that (D)MC shimming of the mouse brain is also possible with the center 32 coils of the presented setup, i.e. if the outer two rows of coils are removed or not used. Global static MC shimming after removal of a third of the coil matrix is expected to still achieve a 22% average narrowing of the frequency measures compared to SH shimming, but a 23% broadening compared to static MC shimming with 48 coils has to be accepted (data not shown). The reduction of the peripheral 16 coils is expected to make essentially no difference for the outcome of DMC shimming, i.e. a significant reduction of the hardware requirements is possible at no cost if the amplifiers are switchable. Even if good shimming can be achieved over small volumes in uncritical brain areas, limited magnetic field homogeneity still poses the main bottleneck for MR spectroscopy of specific brain areas such as the olfactory bulb (21) or the brain stem of the mouse (28). The MC field modeling approach is capable of synthesizing advanced magnetic shim fields over the whole brain and in selected regions. It has been shown in this study that optimized shim fields generated with the MC approach in axial slices from the olfactory bulb, throughout the entire brain and up to the brain stem largely improved the quality of gradient echo imaging (Fig. 3). However, DMC shimming is by no means limited to axial slices and advanced shim fields can be synthesized in any other slice orientation for MR imaging and spectroscopic imaging or in other volumes such as cubic voxels for MR spectroscopy (data not shown). The dynamic application of MC fields as being presented in this study is possible at the same short time scale of the current switches. In fact, 10 μs is considerably shorter than minimum rise times of state-of-the-art gradient systems in the order of 100 μs and allows the incorporation of dynamic MC shimming into any MR sequence without duration penalty. The MC approach is expected to improve the outcome of MR experiments, especially if the entire brain is considered, and enable studies in brain areas that do not allow meaningful MR investigations so far because of limited magnetic field homogeneity.

Functional MR imaging based on gradient echo methods is particularly sensitive to magnetic field inaccuracies including those that are based on suboptimal shim (29). Global MC shimming has been shown to outperform global SH shimming and led to reduced signal dropout due to intra-voxel phase cancellation (Fig. 3). The dynamic updating of slice-specific MC shim fields allowed a further limitation of in-slice and through-slice field gradients and resulted in largely improved image quality. The major homogeneity improvements that can be realized with MC shimming are expected to directly translate in to improvements of the data quality for functional MR imaging in the mouse brain.

Dynamic MC shimming has been applied to the mouse brain at 9.4 Tesla in this study. The presented setup can be adopted to even higher scanner B₀ field strengths through an increase of the number of turns per coil or an increase of the available current range. Notably, the quality of the realizable shim fields was mostly limited by the available basis shapes and less so by the imposed current limitation. Simulations have furthermore shown that similar results can also be achieved in the rat brain. In the same vein, MC shimming has been shown theoretically to allow largely improved shimming of the human brain (30). The experimental realization of (D)MC shimming in the human brain is part of the ongoing research at the Yale MRRC.