Bruno Pinho Meneses^{1,2} and Alexis Amadon^{1}

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 inserts^{1}
and multi-coil arrays^{2,3} (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 fieldmaps^{4} by
applying SVD over more tractable 2D stream functions (SF) tailored for each
fieldmap over a cylindrical surface with our original computational method^{5}.
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 ΔB_{0} 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 2^{nd}
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 wires^{6},
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 5^{th}-order was assessed without power limitation.

All simulations were performed in MATLAB®.

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

Applying 5^{th}-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.

The three-channel SF-SVD-coil insert provides shimming
performance equivalent to that of a complete 5^{th}-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.

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 (n^{th}
layer -> n^{th} 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 2^{nd}-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.