Frederick C Damen^{1} and Kejia Cai^{1}

Radio Frequency (RF) receiving coil arrays improve the signal-to-noise ratio (SNR), and enable partial parallel imaging. However, these benefits often come at the cost of image non-uniformity. $$$B^-_1$$$ non-uniformity correction techniques are confounded by signal that not only varies due to coil induced $$$B^-_1$$$ sensitivity, but also due to true signal variations in proton density, susceptibility, and relaxation rates. Herein, we propose an empirical method that produces a $$$B^-_1$$$ non-uniformity-corrected complex image from the phased-array coil images themselves using minimal assumptions and without measuring the coil sensitivities. This method is validated using MRI of the abdomen, brain, and a homogeneous phantom.

The proposed correction method involves two main tasks: (a) estimating the spatial $$$B^-_1$$$ sensitivity for each coil element, and (b) correcting the coil images and combining them into a complex image.

(a)

Image voxels used in the estimation process were selected by thresholding a Z-score map (i.e., convolution using 5x5 kernel of mean/standard deviation) of the complex MRI coil image.

In the sensitivity model used, the recorded signal $$$r_c(x)$$$, for coils c, is the sum of the true signal $$$t(x)$$$ modulated by a smoothly varying coil sensitivity $$$b_c(x)$$$ and additive noise $$$n_c(x)$$$, i.e., $$$r_c(x) = t(x)b_c(x)+n_c(x)$$$. This model was transformed into two jointly additive models of magnitude ($$$m_c(x)=\ln \left|r_c(x)\right|$$$) and phase ($$$p_c(x)=\arg (r_c(x))$$$).

The coil sensitivity factor $$$\hat{b}_c(x)$$$ was estimated by performing $$$2^{nd}$$$ order 2D polynomial fit for $$$\hat{m}_c(x)$$$ and a $$$1^{st}$$$ order 2D polynomial fit for $$$\hat{p}_c(x)$$$, using singular value decomposition (SVD)[8,9,10], where $$$\hat{b}_c(x) = \exp(\hat{m}_c(x) + i\hat{p}_c(x))$$$.

Correction factor for each individual coil image $$$\alpha_c(x) = 1/\hat{b}_c(x)$$$.

(b)

Due to $$$\alpha_c(x)$$$ being determined on a per coil basis, $$$\alpha_c(x)$$$ is adjusted to normalize the corrections to a common reference; only Z-score selected voxels are used to determine this normalization. The true signal was estimated $$$\hat{t}(x)=\sum_c w_c(x) \alpha_c(x) r_c(x)$$$, where $$$w_c(x)$$$ is a Fermi filter used to remove heavily amplified voxels with very low SNR.

Improved corrections can be made by reiterating the aforementioned steps a and b, by first removing $$$\hat{t}(x)$$$ and $$$\hat{b}_c(x)$$$ from the processed signal, i.e., $$$r_c(x) = r_c(x)/\hat{t}(x)/\hat{b}_c(x)$$$.

**Image
Acquisition:** Abdominal
and brain MRI datasets were acquired on an AllTech EchoStar Spica
1.5T whole-body MRI system (AllTech Medical System, Solon, Ohio,
USA). For abdominal MRI, a 15-channel body coil was used with a
gradient echo sequence and parameters: TR/TE = 190/4.5 ms, FOV =
380x440 mm^{2},
matrix size = 256x256, and GRAPPA factor = 2. For brain MRI, a
10-channel head
coil was used with a fast spin echo sequence and parameters: TR/TE =
480/12 ms, FOV = 380x440 mm^{2},
matrix size = 256x256, GRAPPA factor = 2.
In
addition, a spherical phantom (11.3 cm in diameter) with homogeneous
water solution was scanned on a 3 Tesla GE MR750 (GE Healthcare,
Waukesha, WI) MRI scanner using an 8-channel head coil and a spoiled
gradient echo sequence with parameters TR/TE = 100/7.3 ms, FOV =
180x180 mm^{2},
matrix size = 256x256.

For all three experiments, the conventional sum-of-squares (SoS) method for combining phase array coil images was used for comparison. In addition, Percent Image Uniformity (PIU) [11] was performed to quantitatively compare the results from phantom imaging.

The Z-score map and a range of selection masks are shown in Figure 1. A threshold of 3.5 provides a balance between spatial coverage and anomalous voxel rejection.

For abdominal imaging, the original magnitude images, and the input/output images from each iteration of the algorithm are presented in Figure 2. We observed progressively improved image uniformity. Particularly, for the phase images, the first iteration successfully abated the branch cut issue as noted from coils 1, 3, 7, and 13. Our experience suggested that 2 to 3 iterations were adequate for correcting $$$B^-_1$$$ non-uniformity in both magnitude and phase images.

The corrected and combined images are presented in Figure 3a (magnitude) and Figure 3b (phase). For magnitude comparison, SoS image (Figure 3c) shows clear spatial non-uniformity.

The human brain images demonstrated similar results as shown in Figure 4.

For phantom imaging, Percent Image Uniformity of the spherical phantom using the proposed method was 96.4%, which was higher than 84.4% from SoS as shown in Figure 5.

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