Synopsis

Keywords: Shims, New Devices, Simulations

In this work, we propose a B₀ shim coil design method based on high-density wire pattern discretized from stream function optimization. We demonstrate that the magnetic field generated by the designed shim coil can fully match the theoretical magnetic field generated by continuous stream functions. We compared the performance of different current-carry surfaces over shim ability for stream function optimization. We also evaluated the impact of current limitation for the discretized coil on B₀ shim result and power dissipation.

Introduction

B₀ inhomogeneity caused by magnetic susceptibility differences among tissues and air grows with static magnetic field increases resulting in image distortions and signal losses, which is a significant drawback in Ultra-High Field MRI despite its high SNR.
Although 2^nd- or 3^rd-order Spherical Harmonic (SH) shimming coils have been integrated into most 7T human MRI scanner, residual B₀ inhomogeneity is relatively high for MR Imaging. Subject-based Stream Function optimization method¹ has been introduced to design Multi-Coil B₀ shimming arrays. However, existing practical implementations approximate the complex current pattern by single loops and result in advanced Matrix coils^2-5, in which shim performance may deviate from numerically optimized continuous stream functions.
To preserve theoretically achievable shim performance, we propose a B₀ shim coil design method to generate an optimal gradient-coil-like high-density wire pattern with more loops selected from stream functions for practical implementation. We simulated and evaluated the performance of cylinder (2D) and helmet (3D) current-carrying surfaces with a 3D gradient coil design method^6,7 on human head B₀ maps.

Methods

Surface: Two types of current-carrying surfaces are used for shim coil design (Figure 1). One is cylindrical with lengths ranging from 250mm to 500mm; another is a helmet-like surface composed of a cylinder and a partial sphere. Both have a radius of 170mm.
Database: 12 B₀ brain maps acquired on a 7T Philips system are randomly separated into the design-set (8 maps) and test-set (4 maps) in coil optimization. The 2^nd order SH B₀ shim is applied to these maps and set as the baseline $$$B_{z,0}(\vec{r})$$$. The inhomogeneity is quantified by standard deviation (SD) $$$\sigma(B_0)$$$, and the database has an average SD of 39.29Hz ± 4.51Hz.
Optimal stream function: Magnetic field can be simulated with Biot-Savart law from the stream function (SF) $$$\varphi$$$ of current density $$$\vec{J}(\varphi)$$$ on a given surface $$$S$$$, the simulation was based on the ThinWire toolbox⁸ which we modified to extend to 3D surfaces. Power dissipation is regularized to satisfy amplifier constraints and maintain safe temperature. Instead of calculating SF for each brain map and analyzing the principal component with SVD to get a single stream function⁴, we equally combined all maps in one design-set inside the cost function thus searching for only one optimal SF for all maps. Multiplier $$$m_i$$$ is pre-optimized by particle-swarm algorithm to balance inhomogeneity differences among maps. The SF is optimized by minimizing the residual magnetic field of the whole brain, with cost function$$\min_{\varphi}\sum_{i=1}^{8}\lVert m_{i}B_{z}(\varphi,\vec{r})+B_{z,0}^{i}(\vec{r})\rVert_{2}^{2}\text{,}\;\text{subject}\;\text{to}\;\frac{1}{\tau\sigma}\int_{S}\lVert \vec{J}(\varphi)\rVert_{2}^{2}dS<P_{assigned}.$$
Discretization and optimal current input: Resulting stream functions are discretized into wire patterns considering wire diameter and minimum distance to provide a practically implementable head shim coil. The wire patterns are grouped into individual shim channels, each of which will be jointly connected to one shim amplifier module when manufactured. The magnetic field strength of the k^th channel with one ampere input $$$B_{k,z}(\vec{r})$$$ is calculated from the realistic wire paths. Thus, the residual field map for subject $$$i$$$ is,$$B_{z,after}(\vec{r})=B_{z,0}^{i}(\vec{r})+\sum_{k=1}^{N_c}I_{k}B_{k,z}(\vec{r})$$with channel k’s input current $$$I_{k}$$$, and we minimize the inhomogeneity with ConsTru Algorithm⁹ under a given current limitation:$$\min_{I_{k}}\lVert B_{z,0}^{i}(\vec{r})+\sum_{k=1}^{N_c}I_{k}B_{k,z} (\vec{r})\rVert _2^{2}\text{,}\;\text{subject}\;\text{to}\;\max{(I_k)}<\text{current}\;\text{limitation.}$$The self-resistance $$$R_k$$$ of the coil is calculated with the conductivity of the wire, and the wire power dissipation is $$$P=\sum_{k=1}^{N_c}I_{k}^{2}R_{k}$$$.

Result and Discussion

We optimized stream functions (Figure 2) on different current-carrying surfaces, and then discretized them into wire patterns with grouped channels (Figure 3). The coil channel numbers are 28 (250mm), 25 (300mm), 24 (350mm), 26 (400mm), 22 (500mm), and 21 (Helmet). The similarities between the theoretical field $$$B_z(\varphi,\vec{r})$$$ and the discrete coil-generated field $$$B_{z,coil}(\vec{r})=\sum_{k=1}^{N_c}I_{k}B_{k,z}(\vec{r})$$$ are 0.9910 (250mm), 0.9767 (300mm), 0.9878 (350mm), 0.9877 (400mm), 0.9920 (500mm), and 0.9837 (Helmet). Thus, our proposed discretization method can preserve the theoretical shim ability.
We then optimized the shim channel-wise current input for each field map with various current limitations. The shim effect is evaluated by the inhomogeneity reduction compared to the baseline,$$\eta=\frac{\sigma(B_{z,0})-\sigma(B_{z,after})}{\sigma(B_{z,0})}\times100\;(\%).$$A channel-wise current limitation equal to or higher than 10A would provide much-improved shim performance compared to those lower than 10A (Figure 4(a) and Figure 5).
For the cylinder surface, the length should exceed 350mm for best performance, considering the performance gap between 300mm and 350mm (Figure 4(a)); while the power dissipation grows linearly over the length increase (Figure 4(b)).
The helmet surface has better B₀ shim performance than cylinder surfaces considering a 10A limitation and stays close to 400mm cylinder at 15A and 20A (Figure 4(a)), with similar power dissipation and requires fewer channels.
The inhomogeneity reduction plateaus as the power dissipation reach 40W (Figure 4(c)), regardless of surface shape and current limitation. Thus, the system power dissipation is estimated to be slightly under 40W for sufficient inhomogeneity reduction.

Conclusion

Our proposed multi-channel high-density wire pattern head B₀ shim coil insert has proven to preserve the numerically optimal shim performance based on continuous stream functions. For future designs and practical implementations, the current limitation is selected as 10A and the current-carrying cylinder length of 350mm or a helmet design is chosen. With around 25 channels a lower channel count is needed than in matrix coil designs for similar shim performance. This translates into less calibration effort and a lower shim amplifier module number making it more applicable for 7T or higher systems.