Mustafa Kaan Çan^{1}, Pei-Yan Li^{2}, Jiazheng Zhou^{2,3,4}, Pu-Yeh Wu^{2}, Yi-Tien Li^{2}, Risto Ilmoniemi^{1}, and Fa-Hsuan Lin^{1,2,5}

We propose single-channel head shim coils with either a helmet or a cylinder geometry. Shim current paths were designed by the stream function method with a realistic target field from a group of human subjects (n = 31). A smoothness constraint was included to decrease shim coil complexity. Helmet and cylinder shim coils with shim current amplitudes 4.2 and 6.9 A improved the standard deviation of residual magnetic field by approximately 17%, respectively.

The spatial homogeneity of the static magnetic
field, *B _{0}*, is one of the most fundamental and common
requirements for high-quality MRI. Living organism consists of various tissues
with different magnetic susceptibilities. These susceptibility differences
cause magnetic field distortions, hence, image artifacts and signal
degradation. In a human head, the most significant susceptibility variations
are due to the air–tissue interfaces around sinuses and auditory cavities

Here we propose shim coil designs to
reduce the image distortions in human head by using the target field method
with a stream function method^{2}. The target fields are obtained
from human subjects. We add a smoothness constraint in design in order to
decrease the complexity of shim current paths and to obtain a shim coil that is
more practical to build. We use simulations to calculate the performance of
shim coil designs with both helmet and cylinder geometries.

We used stream function method in order to
design shim current paths. In this method^{2}, a scalar field stream function
is defined on the coil surface such that curl of this field corresponds to the
surface current. The magnetic field was then derived from a matrix-vector formulation^{3,4}. Given a target magnetic field, the
stream function was solved from a system linear equation. In order to obtain a
spatially smooth stream function, we added a constraint that maximizes the
spatial smoothness of stream function values. We measured the smoothness of the
stream function values by a Laplacian operator. With the estimated stream
function, we took its iso-contours in order to obtain a wiring pattern of shim
current paths.

Off-resonance maps were measured from
31 participants using a dual-echo gradient-recall sequence (TE1
= 2 ms, TE2 = 4.46 ms) at 3 tesla (Skyra; Siemens, Erlangen,
Germany) with 2-mm isotropic resolution after a 2nd-order global
shimming. The phase accrued between two echoes was calculated at each image
voxel by first removing the phase related to
the coil sensitivity and then taking a weighted sum across receiver channels^{5}. The phase wrapping was reduced by the BEST-PATH algorithm^{6,7}. An off-resonance map was estimated
by the ratio between the unwrapped phase map and the difference between two
TEs. Off-resonance maps from all subjects were co-registered by FSL (FMRIB, Oxford, UK) to the standard brain MNI 305 atlas
(FreeSurfer, 2007) by only translation and rotation. We used the average
off-resonance map across all subjects for shim coil design.

Two different coil geometries were used: (i) a helmet geometry from a magnetoencephalography (MEG) system (MEGIN, Helsinki, Finland) and (ii) a cylinder geometry (15 cm radius; 25 cm height).

We calculated the sum of squared residual inhomogeneity and standard deviation of the residual inhomogeneity in order to measure the shimming performance. These metrics were separately calculated for global and slice-selective shimming.

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2. Pissanetzky S.Measurement Science and Technology.1992; 3:667-673.

3. Hoult D. I. & Deslauriers R.Journal of Magnetic Resonance, Series A.1994; 108:9-20.

4. Ungersma S. E., Xu H., Chronik B. A. et al.Magnetic Resonance in Medicine.2004; 52:619-627.

5. Luo J., Jagedeesan B. D., Cross A. H. et al.Neuroimage.2012; 60:1073-1082.

6. Schofield M. & Zhu Y.Opt Lett.2003; 28:1194-1196.

7. Bouwman J. G. & Bakker C. J.Magn Reson Med.2012; 68:621-630.

Figure 1: Shim
coil designs. (A) and (C) are shim coils
designed on a helmet geometry. (B) and (D) are cylinder geometry. (A) and (B) show the normalized stream
function distributions. (C) and (D) show the discretized current paths. In (C)
and (D), blue and red current paths denote opposite current directions. Blue
paths indicate counter-clockwise current flow and red paths indicate clockwise
current flow.

Figure 2: Residual
magnetic field distributions at three orthogonal central slices. (A): locations
and orientations of the slices. (B): the magnetic field distributions before
shimming. (C) and (D): the magnetic field distributions after helmet coil
shimming. (E) and (F) the magnetic field distributions after cylinder coil
shimming. In (C) and (E), the weighting of the smoothness constraint , while in (D)
and (F).

Figure 3: Standard
deviation of the magnetic field inhomogeneity before and after different
shimming with a helmet or a cylinder shim coil. Shimming performance was
averaged across four axial slices. Global shimming denotes the shimming with
the shim current path optimized for the whole head. Slice-selective shimming
(G) used the shim current path optimized in global shimming to shim each slice
separately. Slice-selective shimming (S) used the shim current path separately
optimized for each slice.