Aaron A Pruitt^{1}, Ning Jin^{2}, Orlando Simonetti^{3}, Yingmin Liu^{3}, and Rizwan Ahmad^{3}

The accuracy of flow quantification in phase contrast MRI (PC-MRI) is limited by the presence of eddy current-induced background phase. A widely reported method to correct background phase relies on polynomial fitting of the pixels within regions of static tissue. However, separating regions with steady flow from static tissue can be challenging because such regions lack temporal variations that are often used to identify and eliminate pixels that are not static. In this work, we present and validate a processing method that identifies and eliminates outliers such as pixels belonging to venous flow and thus improves flow quantification.

Background phase correction often involves performing polynomial regression using pixels located within static tissue regions. To make this approach insensitive to pulsatile atrial flow, the polynomial regression can be performed using weighted least squares, where the weights are determined by the inverse of the temporal variance of the phase [1]. Recently, we proposed Weighted Regularized Least Squares (WRLS) that merges the weighted least squares with $$$\ell_1$$$ regularization of polynomial coefficients to guard against overfitting when high order polynomials are used [2]. These developments, however, do not account for the steady flow, e.g., in veins, which may have negligible temporal variance. In this work, we extend WRLS with an outlier rejection scheme, called WRLS+OR, to eliminate outliers including pixels with steady flow. The main steps in the implementation of WRLS+OR are as follows. Step 1: Apply magnitude thresholding to discard pixels with no or little flow and perform WRLS with 4th order polynomial. Step 2: Compute weighted residual $$$r=W(\theta-\theta_{fit})$$$, with $$$\theta$$$ being the measured phase map, $$$\theta_{fit}$$$ being the polynomial fit from WRLS, and $$$W$$$ being the inverse of the temporal standard deviation. Step 3: Compute z-scores for $$$r$$$ and identify pixels with z-score of more than 1 or less than -1. Step 4: For the pixels identified in Step 3, perform morphological filtering to discard single pixels that are not clustered. Step 5: Create a binary rejection mask for the remaining pixels in Step 4. These steps are also summarized in Figure 1. We have observed that one iteration (Steps 1 through 5) is adequate to reject most of the outliers, with marginal impact of applying additional iterations.

Eight PC-MRI datasets were acquired from five healthy volunteers (3 males, mean age 54.4±14.4). Imaging was performed on a 1.5 T clinical MR scanner (MAGNETOM Aera, Siemens Healthineers, Erlangen, Germany) with a body matrix and spine coils for signal reception. The imaging parameters for PC-MRI were: TE/TR = 2.47/4.64 ms, temporal resolution = 37.36 ms, BW = 450 Hz/Pixel, FA = 15°, IPAT = 2, imaging matrix = 192x130, VENC = 150 cm/s, and slice thickness = 6mm with a 90mm slice distance. Two parallel slices transecting the aorta with the upper slice placed at the level of the main pulmonary artery were acquired in two separate breath-holds (Figure 2). The flow volume in the descending aorta was compared across the two slices by computing the ratio $$$V_a/V_b$$$. The background phase was corrected using both WRLS and WRLS+OR.

Figure 1: Main steps involved in the proposed WRLS+OR method.
Here, WRLS represents weighted regularized least squares based polynomial fitting.

Figure 2: The locations of two imaging planes are shown with
the dashed lines. The flow quantification was performed in the descending aorta in
the two imaging planes. The flow in the superior plane was termed as $$$V_a$$$ while the flow in the inferior plane was
termed as
$$$V_b$$$

Figure
3: An example showing the impact of outlier rejection. A: Uncorrected phase image. B: Temporal variance map (dB) used in
WRLS. C: Rejection map based on
magnitude thresholding. D: The
expanded rejection map that also includes outliers identified by WRLS+OR. The pixels
with value 0 (black) are rejected. E:
WRLS-based background phase fitting that uses the rejection map in (C) and the variance map
in (B). F: WRLS-based background phase fitting that
uses the rejection map in (D) and the variance map in (B). The difference
between (E) and (F) at the location of descending aorta (contours) is 1%.

Figure 4: An example showing how WRLS+OR is able to identify
and reject major vessels. A: Uncorrected
phase map. B: Uncorrected phase map
with the pixels discarded via magnitude thresholding set to 0 (black). C: Uncorrected phase map where additional pixels identified by
WRLS+OR are set to 0. Such pixels are highlighted using arrows.

Figure 5: The ratio of flow in the descending aorta in two different image planes identified in Figure 2. The ratio is shown for the uncorrected
case, WRLS, and the proposed WRLS+OR.