Lena V. Gast^{1}, Anke Henning^{2}, Bernhard Hensel^{3}, Michael Uder^{1}, and Armin M. Nagel^{1,4}

B_{0} inhomogeneities are a major challenge at ultra-high
magnetic field strengths (B_{0} ≥ 7 T), as they can lead
to strong image artifacts. In this work, a B_{0 }shimming routine that
is based on ^{23}Na B_{0} maps was implemented for applications
in X-nuclei imaging where no ^{1}H MRI data can be acquired and
therefore, no vendor-provided shimming routines can be used. The proposed ^{23}Na
ConsTru shimming routine showed an improvement in B_{0} homogeneity
comparable to the vendor-provided GRE Brain shimming routine both in phantom and in vivo
measurements using
^{23}Na B_{0} map acquisition times less than 1 minute.

The implemented B_{0} shimming routine consists of four steps:

^{23}Na image acquisition using a double-echo 3D density-adapted radial readout (DA-3D-RAD),^{3}resulting in two phase images $$$\Phi_1$$$ and $$$\Phi_2$$$ corresponding to echo times TE_{1 }and TE_{2}.- Phase unwrapping
^{4}of the images ($$$\Phi_{1,unwrapped}$$$, $$$\Phi_{2,unwrapped}$$$). - Calculation of the B
_{0}deviation map according to $$\Delta B_0 = \frac{\Phi_{2,unwrapped}- \Phi_{1,unwrapped}}{\gamma_{Na}\left(TE_2-TE_1\right)}\qquad\text{(Eq. 1)}$$ with the gyromagnetic ratio of sodium ($$$\gamma_{Na} = $$$11.27 MHz/T). - Solution of the shim problem
^{5}$$\left(A\cdot C\right)\cdot b = B_0\qquad\text{(Eq. 2)}$$ Here, $$$A$$$ is the matrix of the ideal shim fields described by spherical Harmonics, $$$C$$$ is the decomposition coefficient matrix modeling the real shim fields,^{5}$$$b$$$ is the vector of the shim currents to be determined and $$$B_0$$$ is the map calculated in Step 3. To solve Eq. 2, the ConsTru algorithm as proposed by Nassirpour et al.^{6}was chosen due to its robustness with respect to low SNR data as expected for in vivo^{23}Na B_{0}maps.

Measurements
were performed at a 7 T Magnetom Terra system (Siemens Healthineers, Erlangen,
Germany) equipped with third order shim coils. For the validation of the
implemented shimming routine, a double-resonant ^{32}Na/^{1}H
head coil (Rapid Biomedical, Rimpar, Germany) was used. B_{0} maps
of a spherical phantom containing 137 mM NaCl in 5% Agarose were acquired both
with the^{ 23}Na DA-3D-RAD sequence and a ^{1}H GRE-B_{0}-mapping
sequence. Parameters: ^{23}Na:
TR = 50 ms, TE_{1/2} = 0.3/5.8 ms, FA = 49°, nominal spatial resolution
Δx= (5 mm)^{3}, Gaussian Filter; ^{1}H: TR = 304 ms, TE_{1/2} = 2.99/4.60 ms, FA = 17°, Δx = (4 mm)^{3},
50 slices, FOV = 240x240x200 mm^{3},
T_{Acq} = 38 s. The dependency of the ^{23}Na shim on the B_{0}
map acquisition duration was examined by varying the number of radial
projections between 500 and 12,000, corresponding to acquisition times of 25 s
to 10 min. To assess the ^{23}Na shimming results, the vendor-provided GRE Brain shimming routine was used as reference (TR = 4.3 ms, TE_{1/2} =
1.02/3.06 ms, FA = 10°, Δx = (4.4 mm)^{3}, 52 slices, FOV =
282x282x274 mm^{3}, T_{Acq} = 10 s). Additionally, shimming with
B_{0} maps acquired using ^{23}Na MRI (T_{Acq} = 50 s
and 5 min) and shimming using the vendor-provided shim were performed on a
healthy volunteer. Reconstruction and post-processing of the sodium data sets was performed using MATLAB (TheMathworks, Natick, USA).

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- Chang P, Nassirpour S, Henning A. Modeling Real Shim Fields for Very High Degree (and Order) B0 Shimming of the Human Brain at 9.4 T. Magn Reson Med 2018; 79:529-540.
- Nassirpour S, Chang P, Fillmer A, Henning A. A Comparison of Optimization Algorithms for Localized In Vivo B0 Shimming. Magn Reson Med 2018; 79:1145-1156.