Synopsis

Keywords: System Imperfections: Measurement & Correction, Low-Field MRI, B0-field map

Heavy B₀-field inhomogeneities in low-field MRI can lead to geometric distortions in the reconstruction, if not compensated. We simulated low-field data to evaluate different approaches for estimating the field map from phase difference maps. By using spherical harmonic basis functions as physical constraints in our neural network approach we could improve the estimation of B₀-field maps compared to other network architectures and methods not involving neural networks.

Introduction

Low-field MRI devices can be affordably manufactured and are mobile, which increases the accessibility to MR imaging. The open-source low-field system OSI² ONE ^1-3 is built from permanent magnets in Halbach configuration. These systems usually suffer from relatively high B₀-field inhomogeneities. In addition, permanent magnet properties depend on temperature which requires dynamic readjustments. If not compensated, the B₀-field inhomogeneities can lead to geometric distortions in the acquired images. Temperature-independent effects can be canceled by time-consuming and system specific shimming ⁴. We propose methods for system-independent corrections by using neural networks (NN) to overcome these challenges.

Methods

Spherical Harmonics
A phase difference map can be calculated from two acquisitions with different echo times and provides a first estimate of the magnetic field distribution. This B₀-field map can be improved by a fit of spherical harmonic (SH) functions to ensure spatial smoothness. Due to the low SNR in low-field MRI, the problem of finding the SH coefficients and the B₀-map from measurements is ill-posed. The problem can be solved by singular value decomposition (SVD), but due to the low SNR, this solution is not always optimal. Alternatively, the field map can be estimated by a neural network directly ⁵. Our approach overcomes the problem by estimating the coefficients from the measured phase difference maps using a NN. Hereby, the NN is constrained by a physical model. Using the estimated coefficients, we can calculate a high-quality B₀-field map. The network follows a U-Net architecture with an encoder path and a fully connected layer providing the SH coefficients. We compared our approach to a network that directly estimates the B₀-maps rather than the SH parameters. To exclude a possible performance increase of an NN-based method over another ⁶, we compared the two NN approaches at similar network capacity.

Finding the B₀-field Map
It has been shown that SH coefficients can sufficiently describe the field distribution of Halbach systems up to the order of four ⁴. Based on the sphere radius $$$r$$$, the polar angle $$$\varphi$$$, and the azimuthal angle $$$\theta$$$, the B₀-field map can be described by a weighted sum of real SH functions $$$Y_l^m$$$ and coefficients $$$c_l^m$$$. Since only projections for $$$\varphi=\frac{\pi}{2}$$$ are considered, the equation can be simplified.

$$\Delta B_0 (r, \theta, \varphi) = \sum_{l=0}^{L}\sum_{m=-l}^{m=l} (\frac{r}{r_0})^l c_l^m Y_l^m (\theta, \varphi)$$

Data
Based on a characterization of the low-field Halbach system⁷ we retrospectively simulated pairs of distorted low-field samples and ground truth images. Simulations are based on the FastMRI⁸ dataset and were performed in a similar fashion as described in Schote et al.⁹. The training dataset is supposed to cover a wide bandwidth of B₀-fields with respect to the field strength and inhomogeneity distribution to ensure the applicability on different systems. To obtain random field maps in the range of a predefined scale, we apply exponential weighting, dependent on the coefficient order. Since also the SNR differs between systems, we modified the ground truth data by varying degrees of Gaussian noise. The overall process, including data simulation, field map estimation, and image reconstruction, which is carried out with the estimated field map, is visualized in figure 1.

Results

Figure 2 depicts the field map results by SVD, U-Net^{10, 11} and SH-Net compared to the ground truth. Calculating the coefficients by SVD results in a smooth field distribution but leads to errors in the regions with low signal intensities (e.g. outside air). Within a circular region of interest (RoI), structural errors become visible in the U-Net approach. Combining the strength of both approaches, the SH-Net delivers a smooth B₀-field distribution with the best performance. Figure 3a shows the mean absolute error (MAE) within the RoI over a test dataset consisting of 180 samples. The SVD is compared to the U-Net architecture and our proposed architecture in terms of MAE. The results are evaluated at four different SNR levels. Both NN approaches lead to lower error boundaries and mean values than the SVD method. The MAE is the lowest and most stable with our SH-Net architecture for all the different noise levels. In Figure 3b we increased the capacity of the U-Net architecture. Thereby, an improvement of this architecture over our approach is gained only at low noise levels. Figure 4 compares the B₀-field informed reconstruction by time segmentation for all field map estimations. As can be seen, by the error maps in the last row, errors from the field map estimates are propagated to the reconstructed images. This includes structural errors from the U-Net architecture. In contrast, our NN yields smooth field maps due to the explicit use of the generative model.

Discussion and Conclusion

We could show the advanced performance of our proposed NN utilizing SH-model for B₀ estimation in low-field MRI. Especially in the regime of low SNR, the NNs were able to achieve better results. By introducing physical constraints imposed by SH basis functions, we could achieve superior performance compared to only using a NN. Accurate results can be obtained for all the different considered noise levels. Structural errors from the U-Net approach which could propagate to the reconstructed image are inhibited by explicitly involving SH basis functions as physical constraints in our methodology.

Acknowledgements

This work is part of the Metrology for Artificial Intelligence for Medicine (M4AIM) project that is funded by the Federal Ministery for Economic Affairs and Energy (BMWi) in the frame of the QI-Digital initiative.