Korbinian Eckstein^{1}, Siegfried Trattnig^{1}, and Simon Daniel Robinson^{1}

A wide range of MR sequences produce inhomogeneous magnitude images due to the coil sensitivity variation over the head, which is especially severe for ultra-high field strengths. The optimum solution would be a homogeneous reference coil, which however is not possible at 7T due to the shorter wavelength. To date, correction methods require a very long computation time rendering them impractical for on-console imaging. We propose a new magnitude inhomogeneity correction approach, which is based on simplified segmentation and fast interpolation to estimate the bias field. The resulting images show high homogeneity across all three dimensions without any visible artifacts.

**Purpose**

Inhomogeneity in magnitude images presents as a smooth intensity variation across the head which complicates visual inspection and leads to errors in automated processing such as segmentation. The origin of inhomogeneities is the spatially varying coil sensitivities $$$S_i$$$, where the measured magnitude $$$M_i$$$ can be written as (not considering noise);

$$M_i=S_i*K,$$

in which $$$K$$$ is the true imaged object and $$$i$$$ the coil number.

A common method for the combination of separate-channel
magnitude images over a number of coils is root sum-of-squares (RSOS). The
magnitude image from each coil serves as an estimate of the coil sensitivity.
Squaring the magnitude introduces weighting which yields close-to-optimum SNR but exacerbates inhomogeneity.
Taking the square root of the sum-of-squares images partially reduces
this^{1} as can be seen in
Fig.1.

$$\text{RSOS}=\sqrt{\sum{M_i^2}}=\sqrt{\sum{(S_i^2*K^2)}}=K\sqrt{\sum{S_i^2}}$$

If the sensitivities of the coils, which change in the presence of the object, are known, sensitivity-weighted combination can be performed:

$$\frac{\sum{(M_i*S_i)}}{\sum{S_i^2}}=\frac{\sum{(K*S_i^2)}}{\sum{S_i^2}}=\frac{K\sum{S_i^2}}{\sum{S_i^2}}=K$$

Several methods (e.g. Damen 2018^{2}) estimate the coil-sensitivities
to perform sensitivity weighted combination. This is time
consuming, however, as it requires the calculation of a smoothed sensitivity profile for
each channel, which requires segmentation and interpolation, followed by fitting or
smoothing. At high field strengths, the shape of the sensitivities is complicated due to the reduced wavelength and imhomogeneous profiles (Fig.1, sensitivities).

We propose a new homogeneity correction that operates on combined RSOS magnitude images and removes the bias field by estimating and dividing $$$\sqrt{\sum{S_i^2}}$$$ using similar, but simplified steps compared to the separate-channel approach described above.

The proposed method starts with a simplified segmentation that removes the highest and lowest values from the image in a box-wise approach (Fig.2). A modified moving average filter is then applied iteratively to smooth and interpolate the segmented image to obtain a sensitivity map for the combined image (Fig.3). Finally, the magnitude of each echo is divided by this combined sensitivity map to produce homogeneous magnitude images.

For each dimension, the modified moving average filter was
applied 8 times to approximate 3D Gaussian filtering. The size of the
individual moving average windows was chosen to obtain an effective sigma of 7mm. The
calculation steps are presented in a github repository^{3}.

Data was acquired from a healthy volunteer with a 7T MR whole body Siemens MAGNETOM scanner with a 32-channel Nova Medical head coil. The head was placed slightly off-center to achieve similar inhomogeneities as would be encountered with a larger head.

Scan 1 was 3D monopolar multi-echo gradient echo with a matrix size 416x375x224, TE=[4.5;9]ms, voxel size of 0.5mm, receiver bandwidth of 388 Hertz/pixel and GRAPPA of 2, slice and phase partial Fourier factors of 6/8 and TA=328s.

Scan 2 was a sagittal MPRAGE with matrix size 320x310x208, voxel size 0.75mm, receiver bandwidth 220 Hertz/pixel and TA=280s.

Figs. 4 and 5 show the correction method applied to the gradient echo and MPRAGE scans. The RSOS images show the inhomogeneous intensity which is characteristic at high field strengths. In the corrected images, no visible inhomogeneity is left and no artifacts due to the correction are observed.

The single-threaded prototyped version in MATLAB required 2min 20s to perform the correction of the dual-echo GRE scan of the size 416x375x224.

We have presented a simple method for correcting the homogeneity of magnitude images at ultra-high field which produces very homogeneous images without any visible artifacts. The method is heuristic, depending upon the masked tissue being similar throughout the brain, but it nonetheless works for images with very different tissue contrasts such as T1 and T2*-weighted GRE and MPRAGE acquisitions.

In the presence of large tissue regions, the inclusion of
SNR-based voxel measures^{2}, but applied only on the combined magnitude could improve
the segmentation without sacrificing the fast computation. Another possible improvement would be to perform a phase-corrected and
magnitude-weighted complex combination instead of the RSOS method as a first
step (as in ASPIRE^{4} and VRC^{5}) to improve the noise characteristic of the image.

Further work involves the implementation of the correction algorithm on the scanner image reconstruction computer and a more detailed assessment of the robustness in other body parts and in the presence of pathologies.

1. Belaroussi B, Milles J et al. Intensity non-uniformity correction in MRI: Existing methods and their validation. Medical Image Analysis 2006; 10:234-246

2. Damen FC and Cai K. $$$B_1^-$$$ non-uniformity correction of phased-array coils without measuring coil sensitivity. Magn Reson Imag 2018; 51:20-283.

3. Eckstein K, Magnitude Intensity Correction, (2018), GitHub repository; https://github.com/korbinian90/Magnitude-Intensity-Correction

4. Eckstein K, Dymerska B et al. Computationally Efficient Combination of Multi-channel Phase Data From Multi-echo Acquisitions (ASPIRE). Magn Reson Med 2018; 79(6):2996-30064.

5. Parker DL, Payne A et al. Phase reconstruction from multiple coil data using a virtual reference coil. Magn Reson Med 2014; 72(2):563-569

Figure 1: Two
approaches to creating a homogeneous magnitude: a) starting from the uncombined images and calculating
the sensitivity map for each channel to perform sensitivity-weighted
combination b) starting from the RSOS image and performing a correction
directly on the combined image.

Figure 2: Sample slices from the results of the simplified segmentation process. The images on the left are from the gradient echo acquisition and on the right from MPRAGE. The simplified segmentation selects some wrong voxels with similar values (arrows), but removes all the tissue with different intensity which would adversely affect the creation of a homogeneous magnitude.

Figure 3: Bias field estimation. The simple segmentation is performed on the RSOS image from the scanner and works well on the inhomogeneous magnitude using an “overlapping box” approach. The segmented image is interpolated and smoothed to obtain an estimation of the bias field √∑S_{i}^{2}. Note that the regular pattern in the estimated bias field corresponds to the position of the coils.

Figure 4: Correction of GRE image. Top row: images combined using RSOS show pronounced inhomogeneity in right posterior and occipital regions. Bottom row: inhomogeneity is effectively removed using the proposed method, which has RSOS images as input. The images have been scaled to the same contrast between grey and white matter.

Figure 5: Correction
of an MPRAGE image. The images in the top row are combined using RSOS. The
images in the bottom row were corrected images with the proposed method. Note that very low intensity values are removed in the process (area below the
eye-socket in the sagittal view).