A novel few-channel coil design for human brain shimming based on stream function Singular Value Decomposition
Bruno Pinho Meneses1,2 and Alexis Amadon1

1Neurospin/CEA-Saclay, Gif-sur-Yvette, France, 2Université Paris-Saclay, Orsay, France


A method for whole-brain shim coil design for Ultra-High field is proposed using Singular Value Decomposition of tailored stream functions to create a small set of independently driven coils providing the same performance as shim inserts with numerous high-order spherical harmonics coils.


High-order Spherical Harmonics (SH) shim inserts1 and multi-coil arrays2,3 (MCA) are popular approaches to reduce magnetic field inhomogeneity in the human at Ultra-High Field (UHF). However, their conception has neglected to capture inter-subject field similarities, which could be used to reduce the amount of coils needed to reach a given level of homogeneity.

Here we revisit the Singular Value Decomposition of 3D human brain fieldmaps4 by applying SVD over more tractable 2D stream functions (SF) tailored for each fieldmap over a cylindrical surface with our original computational method5. A reduced set of SF-SVD coils is extracted to compose a whole brain shimming system at UHF. For the sake of demonstration, the hereby study is limited to three independently-driven cylindrical coils, but could easily be extended to more channels.


We first built a database of 55 three-dimensional ΔB0 brain maps from informed and consenting adults. These maps were acquired at a 3T Siemens Magnetom-Prisma imager with 1.7-mm isotropic resolution and rescaled to 7T, since a shim system for UHF is intended. FSL’s brain extraction tool was used to exclude non-brain voxels and, as a baseline, a re-shim at 2nd order SH was performed in the brain mask. A sample of 25 maps (design set) was randomly picked and their residual fields used as target for tailored stream function computation and shim coil generation.

Defining three different cylindrical coil formers of 300-mm length and of 138-mm, 144-mm, and 150-mm concentric radii at the gradients isocenter, a set of 25 tailored shim coils are generated for each former. Each coil is designed to reduce as much as possible the inhomogeneity of the map used as target field, while keeping acceptable levels of power consumption (< 40 W)5. Field inhomogeneity is defined as the field standard deviation in each brain.

The three resulting sets of 25 stream functions are independently post-processed using Singular Value Decomposition. A new set of virtual SF-SVD coils is thus obtained at each cylindrical radius (Figure 1).

As higher singular values address the most common brain features, the stream functions of a given high SF-SVD rank are assumed to be similar when going from one radius to the next. Thus, for the 138-mm coil, only the stream function of highest singular value is kept; for the 144-mm radius coil, the second highest; and for the 150-mm radius coil, the third highest (Figure 2).

To guarantee field fidelity when discretizing current surface densities into wires6, and help reduce electric current, the amount of wires is maximized for a 20-AWG wire gauge and 5 wire layers per coil.

The magnetic field generated by each of the three SF-SVD coils is evaluated in the volume encompassing all brains; then for each fieldmap, three shim currents are computed from a Least Squares fitting algorithm. This was performed on the 25 brains of the design set and on the 30 remaining maps (test set).

For comparison, the performance of a SH-shim insert going up to 5th-order was assessed without power limitation.

All simulations were performed in MATLAB®.


Average global inhomogeneity of the 2nd-order shimmed database was 60±10Hz, regardless of the sample set.

Applying 5th-order SH shim over each set, 48±8Hz average inhomogeneity was achieved for all sets, bringing its reduction to 20% (Figure 3a).

SF-SVD coil shimming over the entire database(Figure 3b), design and test sets resulted in average inhomogeneity of 48±8Hz, 47±8Hz and 49±8Hz respectively (reduction was 20%, 21% and 18%). Average currents per coil were 2.3A, 1.3A and 1.2A for first, second and third SF-SVD coils respectively.

Figure 4 shows axial B0 brain slices of a random subject after shimming with either one of the discussed coil inserts.

Discussion and conclusion

The three-channel SF-SVD-coil insert provides shimming performance equivalent to that of a complete 5th-order shim insert with 27 coils and unlimited power. Its performance over the test set was only 3% worse than over the design set, which shows it was still effective over new brains, but also that more maps might be necessary in the design set.

Provided space is available for a thicker shim insert, more SF-SVD coils could be added at larger radii to increase performance, but preliminary simulations show that beyond 4 coils, the overall improvement is small (Figure 5).

Given an anatomical target such as the human brain, the SF-SVD coil design seems to be a great alternative to traditional SH coil or current MCA systems as they considerably reduce the space and cost required to reach a certain level of field homogeneity.


We would like to thank Guy Aubert for an early fruitful discussion leading to this work


1. J.W. Pan, K. Lo, H.P. Hetherington. Role of very high order and degree B0 shimming for spectroscopy imaging of the human brain at 7 Tesla. Magn Reson Med, 2012 Oct; 68(4): 1007-1017.

2. C. Juchem, et al. Dynamic multi-coil shimming of the human brain at 7 T. Journal of Magnetic Resonance, 2011 Oct; 212(2):280-8.

3. J. Stockman, L.L. Wald. In vivo B0 field shimming methods for 7 T. Neuroimage, 2018 Mar; 168:71-87.

4. E. Adalsteinsson, S.M. Conolly, et al. Design of Dedicated Shim Fields. International Society of Magnetic Resonance in Medicine, 1999; 447.

5. B.P. Meneses, A. Amadon. The Dipole Boundary Method : a simple approach to compute stream functions for shim coil design. ISMRM 2019

6. G.N. Peeren. Stream function approach for determining optimal surface currents. Journal of Computational Physics, 2003 Oct; 191(1):305-321


Figure 1: Stream Function SVD Coil generation pipeline: a coil former is used as input together with 25 fieldmaps and a set of 25 stream functions is post-processed into a new SF-SVD set. This process is repeated for three increasing radii; for each radius, a single SF-SVD coil is kept with the highest singular value corresponding to the layer (nth layer -> nth SV).

Figure 2: Iso-current lines of the three highest singular valued coils with increasing radii. The lines represent the path an electric current must flow to generate the desired magnetic field for shimming. For fabrication, the iso-current lines for a coil must be interconnected by paths oriented in the z direction and such that using a single current drive the same current will flow through all paths. The origin of the y axis faces the subject’s mouth and nose. This is where denser windings appear.

Figure 3: Histograms for 5th order SH shim and 3-channel SF-SVD coil shim performance over the 55 fieldmaps.

Figure 4: Fieldmap comparison after global shimming with the two different shim approaches. Baseline corresponds to 2nd-order shimming. Considerable improvement is seen in the pre-frontal cortex (upper row). For this particular subject, 3-channel SF-SVD shim seems to perform slightly better than 5th-order SH shim for all slices in display.

Figure 5: Box plots of the inhomogeneity levels for different shimming approaches with different numbers of coils. SF-SVD Coil Shim achieves with only 3 coils the same performance as 5th-order SH shim, but performance tends to flatten rapidly, with less than 1 Hz reduction in average after 4 SF-SVD coils.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)