Shim Coil Design

Alexis Amadon¹ and Bruno Pinho Meneses²
¹CEA Neurospin, France, ²MR Magnet Technology, Siemens Healthineers, Eynsham, United Kingdom

Synopsis

Keywords: Physics & Engineering: Hardware, Physics & Engineering: High-Field MRI, Neuro: Brain

We review methods to design shimming coils, starting from basic Spherical Harmonic coils to anatomy-driven multi-coil arrays. Assuming cylindrical surface geometries for the sought shim currents, we focus on a recent simple numerical method to find the stream function (SF) producing whatever target field. Applying this to brain B0 maps, we perform a SF principal component analysis and translate its main components into current lines gathered around SF extrema. We then segment coil windings into multi-channels to add flexibility in the shimming process. The same strategy can be applied to design shim arrays dedicated to other parts of the anatomy.

Introduction

B0 shimming consists in homogenizing the static magnetic field in a region of interest (ROI), e.g. the human brain. As the B0 field strength increases, the disparities of magnetic susceptibilities between the diamagnetic biological tissue and the paramagnetic air contained in cavities like the lungs, inner ears or nasal sinus, cause a greater spread of B0 at the interfaces of exposed tissues (cf. Fig. 6). These translate into loss of spectrum resolution or image artefacts, such as distortions and signal dropout (due to intravoxel dephasing) in Echo-Planar images, or hypersignals in T1-weighted MP-RAGE images (e.g. Fig. 1). Thus we want to keep B0 as uniform as possible to minimize these artifacts. So shimming is required both at the main-magnet design stage, to correct for magnet manufacturing imperfections and reach B0 homogeneity in an empty sphere which will encompass the body under exam, and during examination to compensate for the above-mentioned unwanted gradients linked to the anatomy of the patient. Typically better-than-1ppm uniformity is required in the ROI, but magnet shimming and patient shimming are addressed separately: 1/ by passive shimming (with iron plates) or superconducting shim coils embedded in the helium vessel (sometimes referred as cold coils), 2/ by copper coils usually embedded in the gradient insert (warm coils). These sets of coils are usually designed to compensate for spherical harmonics (SH) of the B0 field spatial decomposition as they constitute a set of orthogonal solutions for the Laplace equation (ΔB0 = 0), which applies providing no magnetic source is present in the ROI. In this lecture, we will concentrate on the design of warm shim coils (however cold coil windings may be designed using some of the same methods). We will cover not only SH shim coil design, but also designs based on a particular anatomy (like the human brain), as these have recently proved to outperform the highest-order SH systems offered by manufacturers, at relatively low cost.

Historical developments

The Biot-Savart law yields the magnetic field spatial distribution generated by an arbitrary line of static current. We therefore need to solve an inverse problem: what current lines should we choose to compensate for some known initial non-uniform B0 distribution. Focusing first on low-order Spherical Harmonic compensation (cf. Fig. 2), we can start designing basic systems from circular loops or arcs of currents whose analytical solutions of the Biot-Savart law are known [Romeo 1984] (cf. Fig. 3). Then the Target Field Method (TFM) was introduced to solve the inverse problem analytically for infinite cylinder geometries using a Fourier-Bessel transform [Turner 1986]. It was later improved for finite length cylinders [Forbes 2001-03]. Then SH-based coils of general order and degrees could be found with a combined Target-Field and Stream-Function analytical approach [Brideson 2002] (cf. Fig. 4a).
The stream function (SF), noted ψ, is a fundamental quantity for our problem solving and is related to surface current densities through the equation: j = ∇ψ × n, where n is the unit vector normal to the surface. ψ is easier to calculate and manipulate than j as it is a scalar value (rather than a 3D-vector) defined on the surface which will bear the surface currents (cf. simple 1D example in top plots of Fig. 9). For arbitrary target fields (other than SH components), numerical methods like the Inverse Boundary Element Method (IBEM) are more appropriate, leaving aside shim coil design based on analytical solutions. The IBEM presents itself as a more versatile technique, used in the most diverse kinds of coil former geometries due to its discretization of the domain into a triangular mesh. It solves the inverse problem by means of a stream function with regularization based on power dissipation and/or magnetic field energy. Despite its initial formulation [Pissanetzky 1992], only in 2003 a first rigorous mathematical treatment of IBEM appeared [Peeren 2003], and a subsequent practical implementation of a gradient coil designed with IBEM was shown in [Lemdiasov 2005]. The method was then applied for the design of various novel coils in [Poole 2007] (cf. bottom of Fig. 4), and has not ceased to gain interest since. The numerical problem formulation can take several design aspects into account, including resulting forces, torque, shielding. Different norms were studied as cost functions in [Poole 2010, Poole 2014]. Nowadays, IBEM has been implemented in an open-source MR coil layout generator, CoilGen [Amrein 2022]. Adaptations of the method also led to the Equivalent Magnetization Current method [Lopez 2009a, Lopez 2009b], where a magnetization is attributed to nodes in a triangular mesh and the inverse problem is set to compute the optimal magnetization generating some arbitrary target field. The magnetization is actually equivalent to the stream function and can be used to calculate the current density over the coil former and perform discretization into winding. Based on this idea, a simple numerical method was recently proposed for cylindrical current surfaces, the Dipole Boundary Method (DBM) [Pinho-Meneses 2020b], which will be developed next for ease of comprehension and implementation.

Stream function solving with the Dipole Boundary Method

Suppose we target an arbitrary B0_target distribution inside the ROI, which would typically be the opposite of a measured B0 map in the case of an anatomy-driven shim coil design (modulo a constant). In DBM, the cylindrical surface is split into Nc elementary square cells. Each cell can be considered a magnetic dipole with a circular current running along its edges (cf. Fig. 10). It was demonstrated this current is actually the stream function we are seeking [Pinho-Meneses 2020b]. From Biot-Savart, the Bz field generated by each cell can be calculated (geometry in Fig. 11) and expressed in vector/matrix form: b = C ψ, where b is the vector spanning all B_z-values in the N_v voxels composing the ROI, ψ is the N_c-vector of unknown stream-function values (or cell currents), and C is the N_v x N_c matrix easily determined from the geometry of each cell with respect to every voxel. As the number of voxels N_v largely dominates the number N_c of cells, this is an over-determined linear problem, which needs regularization to avoid excessive currents and power. Overall power can be expressed as the volume integral P = integral(|J|²/σ dv), where J is the current density in the cylindrical copper structure, and σ is its conductivity. This in turn can be written as a function of ψ as J = j/d = (∇ψ × n)/d where d is the copper surface thickness considered. Thus P can be formulated as the quadratic expression: P = ψ^T R ψ where the N_c x N_c resistance matrix R can simply be found from differential calculus. Eventually we seek ψ that minimizes the objective function var(B0_target – C ψ) + λ ψ^T R ψ , where var() stands for the variance, and λ is the power-limiting regularization factor. This can be solved numerically via standard optimization solvers. Higher λ-values will limit power and the performance to reach the target field i.e. shimming. In fact, the full formalism is slightly more complex as boundary conditions need to fulfilled whereby no current should flow outwards or inwards of the cylinder from both ends, i.e. ψ should be kept constant on the cylinder boundaries. This can be expressed by the computation of a reduced vector ψ’, with ψ = Γ ψ’ where Γ is a simple matrix made of unit vector columns...

From Stream function to winding

In order to transform the virtual stream function into concrete copper wire windings, the wire gauge should first be chosen based on the target maximum current to be sustained. In addition, a minimum distance δ_w between wires needs to be specified which will dictate the current I_c that needs to flow in all wires to generate the sought B_z distribution: I_c = δ_w max|∇ψ|. From a nominal current I_c, a family of isoheight curves of ψ representing the geometric centers of the coil wires can be obtained as detailed in [Peeren 2003] (cf. Fig. 9). Lower δw will yield many loops close to one-another in the winding with less current and more fidelity to the target B_z distribution. However this will also mean more wire length, resistance, inductance, and manufacturing complexity. Indeed since all loops segmented with this Ic-based discretization need to undergo the same current, they should presumably be put in series with junctions and bridges along z (which do not perturb B_z) between neighboring and distant coils (cf. bottom middle plot of Fig. 9).
The DBM + winding method may be used for any target field, be it a particular SH order/degree compensation, or a B0-map characteristic of the ROI to be shimmed following low-order SH shimming. When applied to the B0 map of a human brain (e.g. in Fig. 6), the stream function ψ on the flattened cylinder looks like what is shown in Fig. 12a. Once δ_w is chosen, iso-I_c lines along the envelope of the SF plot represent close-to-ideal lines of currents needed to counteract the inhomogeneous B0-map inside the brain.

Coils optimized for the anatomy

SH-coils generally need one layer of conductor per SH order and degree (sometimes more as active shield coils may be added to prevent eddy currents in the main magnet vessel). Since a system which integrates a full n^th order needs (n+1)²-1 SH superimposed coils, the main MRI manufacturers may offer up to partial 3^rd-order shimming for Ultra-High-Field magnets. A supplier has even gone up to 4^th-order with two additional 5^th-order shims [Hetherington 2021] (cf. Fig. 5 - this cylinder encompasses 18 coils presumably on 18 different layers).
Nevertheless, such systems are heavy and not particularly optimized for the anatomy under scrutiny. This is why, for the past decade, efforts have been carried out to target the features of the human head anatomy more directly. First [Juchem 2010] proposed a set of 6 loops placed in front of the subject’s face to address the B0 inhomogeneities in the prefrontal cortex (Fig. 7). A bit later, the same authors proposed the first multi-coil array as a generalization of their primary 6-loop setup (cf. introduction of next section).
More recently, a couple of teams proposed more optimized anatomy-driven shim systems dedicated to the human brain [Pinho-Meneses 2020b, Jia 2020]. Such systems are based on collections of Ns brain B0 maps, typically a hundred. First a cylinder diameter is chosen between the RF coil and magnet bore as support for the shim wires. Proximity to the RF coil is sought to maximize shim performance for a given nominal current, or to minimize shim current for equivalent performance. For each of the individual B0 maps, an ideal stream function is found, e.g. with the method presented above and a given nominal power.
Then a Principal Component Analysis of the N_s vectorized stream functions ψ_s is performed to yield N_s Singular-Value-Decomposition stream functions ψ_SVD (Fig. 12c). The first component then captures the most variability among the subjects in the database (Fig. 14), while the last one may be specific to only one subject. Thus only the first few components are kept as potential cylindrical coils to be superimposed. However as they cannot occupy the same radius, the first SVD component is assigned to the chosen radius while higher SVD components are recalculated for outer radii, by using the B_z profile each one of them produces as a new target field. From these few ψ_SVD, windings are calculated from the method outlined above. In this way, we end up with a few-layer system, typically 3, corresponding to the number of novel shimming channels, shown in simulation to perform better than 16 unlimited-power SH coils [Pinho-Meneses 2020b]. Simulations performed by other groups at larger radii also point to improved performances compared to SH systems [Jia 2020]. Nevertheless, to the authors’ knowledge, no such design has yet been implemented probably due to the complexity of assembling wire loops in series with complex wire patterns (cf. Fig. 13). However there is a way out by accepting to increase the number of channels without increasing the number of layers.

SCOTCH : a Multi-Coil Array optimized for the human brain

Shim multi-coil arrays (MCA) have first been introduced in 2011 as a way to improve shimming of the human brain thanks to small circular loops regularly paving a cylindrical former at the top and bottom of an RF head coil [Juchem 2011] (cf. Fig. 8). One advantage of such systems is that the small size and proximity of the loops to the subject’s head typically require less current than standard larger SH coils. Therefore they can be driven by more affordable power supplies yet compatible with multiple channels that provide the degrees of freedom to improve shimming. The smaller inductance of the loops also makes them suitable for fast switching without having to worry about eddy currents in the main magnet vessel. With up to 48 multi-turn loops, this technology proves particularly useful for dynamic shimming (or slice-by-slice shimming), whereby shim currents are updated during acquisition of 2D Echo-Planar sequences, taking advantage of ROI reduction for each selected slice [Juchem 2015]. Other teams have developed matrix-like MCAs with the same concept, regularly covering the entire cylindrical surface outside the RF coil with identical loops [Aghaeifar 2018]. Even though these systems may significantly improve shimming beyond 2nd-order SH coils, their design is actually too simplistic to truly optimize shimming for the human brain.
This is why an MCA optimized for the brain was recently proposed, so-called SCOTCH, which combines the stream function SVD approach above with the multi-channel concept [Pinho-Meneses 2022]. Following the method described above, three SVD coils have first been designed as three cylindrical layers of optimal current lines. Yet to ease implementation, these current lines were gathered into bundles according to their separation around stream function extrema. This allowed a segmentation of each SVD coil into typically more than 10 bundles. For further simplification, each bundle was actually implemented as a multi-turn loop whose shape matched the outer-most current line of the bundle (such a loop approximatively yields the same Bz profile as the original bundle – cf. red lines in Fig. 14). In the end, the three-layer prototype comprised 48 loops of various size, shape, and location optimized for shimming the human brain (cf. Figs. 15-16). At 7 Tesla, with up to 3 A per channel (supplied by an open-source current driver [Arango 2016]), this was shown to be equivalent to a 96-channel matrix MCA, and to at least a full 6th-order SH system with unlimited power. It has thus reached the best performance to shim the whole human brain so far, and can be driven in dynamic mode for slice-by-slice shimming in 2D acquisitions [Pinho-Meneses 2020a]. A limitation of the cylindrical system in its current form is that it relies on a narrow 27-cm inner diameter, which implies the use of a dedicated compact RF coil isolated from the SCOTCH array with an RF shield [Luong 2022].
In the meantime, another MCA optimized for the human brain was proposed, but the shape of its 32 loops was set to squares [Aghaeifar 2020] (Fig. 17a), thereby limiting its performance compared to SCOTCH.

Other types of shim arrays

Shim arrays based on the RF coil arrays have also been proposed in recent years: they mostly share the same loops as the ones used for RF reception; the RF loops can also become DC loops via chokes that bring the shim currents while preventing RF signal leakage towards the shim amplifiers [Han 2013, Stockmann 2016, Darnell 2017]. Although nice results have been obtained with such devices, especially for dynamic shimming, their suboptimal design for B0 shimming tends to pull them out of the scope of this shim coil design session.
For the sake of completeness, MCA with shimming loops orthogonal to the RF loops have also been proposed to limit the interaction with the RF loops with no need for an RF shield, while addressing the B_z component of interest more directly than when located along cylindrical formers [Zhou 2020]. A 7-channel orthogonal shim array was thus implemented above the forehead, somewhat improving shimming in the frontal lobe.

Conclusion

We have reviewed various methods to design shimming coils, starting from simple 0- and 1st-order SH coils to anatomy-driven shim multi-coil arrays. Assuming cylindrical surface geometries for the sought shimming currents, we focused our attention on a recent simple numerical method, the Dipole Boundary Method, to find out the stream function that produces whatever target B0 field pattern. Applying this to the brain anatomy, we used a principal component analysis of the stream function and translated the main components into current lines gathered around stream function extrema. From the identified bundles, we were able to segment close-to-optimal coil windings into multi-coil channels to ease implementation and add flexibility in the shimming process. As a result, we have learnt to build the best shimming coil ever designed for the brain. Of course, the same strategy can be used to design shim coil arrays dedicated to other parts of the human anatomy.

Acknowledgements

Thanks to Jason Stockmann, who provided his slides from last year as well as advice to get started.

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Figures

Figure 1: Example of T1-weighted image artifact due to B0-inhomogeneities above the nasal sinus: in this MP-RAGE sequence case, the inversion pulse failed in that region of the cortex which appears white [Damme 2021].

Figure 2: Examples of spherical harmonic functions on a sphere of unit-radius. In Y(l,m), l and m designate the order and degree, with m varying between -l and +l. Note l=0 corresponds to Z, Z2, Z3 components. Positive/negative l refer to the x/y-axes respectively (from [Pinho Meneses 2021]).

Figure 3: Simple first- and second-order SH-correction systems, merely designed with circular loops (for zonal components - degree m=0) and arcs of currents (for tesseral components – m≠0). Note the z-segments have no impact on the B0 field. Adapted from Koch K., et al., Prog. Nucl. Magn. Reson. Spectrosc. 2009 Feb. 1; 54(2) 69-96, and Juchem C., et al., NMR Biomed., 2021;34:e4350.

Figure 4: Sophisticated windings can better reach target fields. They are determined with various methods using the Stream Function approach. Upper left: TFM was used to compute a 2^nd order, 2^nd degree shim coil [Brideson 2002]. Upper right: similar winding patterns can be found with the more simple Dipole Boundary Method developed in this lecture [Pinho Meneses 2020b, 2021]. Here a 1^st-order y-gradient design is shown. Bottom: a shoulder-slotted gradient insert designed with IBEM, whereby any current-supporting surface can be defined by triangular meshing [Poole 2007].

Figure 5: Top: 18-channel Very-High-Order Shimming coil set aimed at correcting 3^rd and 4^th-order SH (plus two additional 5^th-order components) – from Resonance Research, Inc, Billerica, MA, USA. Bottom: The effect of VHOS on Echo-Planar images can be appreciated at 9.4T – top row = GRE reference brain image; middle row = EPI with 2^nd-order shimming; bottom row: EPI with VHOSH currents (in this case, 28 channels were used). Distortions are notably reduced, and signal is recovered in the lower part of the brain (adapted from [Chang 2018]).

Figure 6: Top: Model of a human brain with air cavities adjacent to it (in blue). Bottom: ΔB0 map of that human brain model (computed from susceptibility Laplacien and scaled at 11.7T [Pinho-Meneses 2021]): after 3^rd order SH shimming, strong inhomogeneities remain around the inner ears and frontal/nasal sinus (results of simulation). A device dedicated to brain shimming should focus on such regions.

Figure 7: (b) When properly located, a simple current loop in the xz plane can generate a B_z-component of the magnetic field which roughly compensates the susceptibility-gradient ΔB0 inhomogeneity above the nasal sinus. (c) Prefrontal cortex shim coils used in [Juchem 2010] lead to signal recovery in FLASH acquisitions above the nasal sinus, but enhance signal dropout elsewhere (arrow pointing to ear canal region in the bottom).

Figure 8: (a) The Shim Multi-Coil Array (MCA) introduced by [Juchem 2011], so-called DYNAMITE, gives less distorted non-accelerated EPI images compared to 3^rd-order shimming at 7T (b). (c) Simulated fieldmaps from static and dynamic SH and DYNAMITE shimming are compared. Images adapted from [Juchem 2011] and [Juchem 2015].

Figure 9: Knowledge of the stream function ψ = I on the current-carrying surface (top curve, I(x) at y=0.5) leads to determination of the surface current density j (j_y curve here), which can then be approximated by segmenting conducting lines carrying current I_c at y=0.5 (left bottom) [Peeren 2003]. The smaller the I_c step defined at top, the more current lines, and the more faithful the current density along x. Bottom middle: implementing iso-current I_c lines requires joining them in series. Bottom right: generalization to 2D: lines represent iso-contours of the stream function ψ.

Figure 10: Consider the ROI to be shimmed is surrounded by a cylindrical surface. The DBM starts by discretizing the cylindrical current-carrying surface into small square cells. Each cell is characterized by a loop current I which is equivalent to the stream function when the cell becomes infinitesimal [Pinho-Meneses 2021].

Figure 11: The B_z-component of the magnetic field generated in point k by of a square current cell n can easily be computed from the Biot-Savart equation and geometric considerations [Pinho-Meneses 2021].

Figure 12: Top: Typical Stream Function (SF) obtained on a cylindrical surface following optimization to target a specific human brain B0 field map (Subject-Optimized SF or SO-SF). After projection onto the cylindrical surface, the iso-SF lines represent the current lines sought. The presence of mountains and troughs indicate opposing current lines (red vs blue). Bottom: A Principal Component Analysis is then carried out on a set of SO-SFs obtained from tens of brain B0-maps, which yields as many SVD components as the number of subjects in the database.

Figure 13: An SVD coil implies a single current runs in all surface windings, requiring bridges and bypasses joining all loops. This complexity can become a real challenge at the manufacturing stage [Pinho-Meneses 2020b].

Figure 14 : Representation of iso-SF lines on to the flattened cylindrical surface (the ordinate axis represents the main magnet z-axis) for the first SVD-component. In addition, this SVD-SF has been symmetrized with respect to the central sagittal plane (Φ = 0), assuming the human brain is symmetrical in average with respect to susceptibility artefacts. The red lines at the bottom (top) of every mountain (trough) serve as central pathways of coil windings in a segmented version of the SVD coil into multiple channels.

Figure 15: The segmentation of the first three SVD-component coils of Fig. 12 into simple loops yields three layers of an MCA where independent loops overlap for maximum shimming flexibility [Pinho-Meneses 2021].

Figure 16: Left: Some of the coils following the pathways presented in the previous figure, facing the mouth and nose of the MRI subject. The loops have become 20-turn windings to increase shimming efficiency. Right: First and second layer of coils of the SCOTCH array [Pinho-Meneses 2021].

Figure 17: (a) Rectangular-loop-based MCA developed in [Aghaeifar 2020]. (b) Similar set of DC rectangular coils placed in front of the lower part of the head to correct for prefrontal artefacts. The other coils with PCBs are AC/DC coils primarily placed for NMR signal detection, but also used for shimming (from [Arango 2020]).

Proc. Intl. Soc. Mag. Reson. Med. 32 (2024)