Two-dimensional phase cycled reconstruction for inherent correction of EPI Nyquist artifacts

Abstract

The inconsistency of k-space trajectories results in Nyquist artifacts in echo-planar imaging (EPI). Traditional techniques often only correct for phase errors along the frequency-encoding direction (1D correction), which may leave significant residual artifacts, particularly for oblique-plane EPI or in the presence of cross-term eddy currents. As compared with 1D correction, two-dimensional (2D) phase correction can be much more effective in suppressing Nyquist artifacts. However, most existing 2D correction methods require reference scans and may not be generally applicable to different imaging protocols. Furthermore, EPI reconstruction with 2D phase correction is susceptible to error amplification due to subject motion. To address these limitations, we report an inherent and general 2D phase correction technique for EPI Nyquist removal. First, a series of images are generated from the original dataset, by cycling through different possible values of phase errors using a 2D reconstruction framework. Second, the image with the lowest artifact level is identified from images generated in the first step using criteria based on background energy in sorted and sigmoid-weighted signals. In this report, we demonstrate the effectiveness of our new method in removing Nyquist ghosts in single-shot, segmented and parallel EPI without acquiring additional reference scans and the subsequent error amplifications.

Introduction

The k-space data of single-shot and segmented echo-planar imaging (EPI) are acquired with fast-switching frequency-encoding gradients of alternating polarities. Because of eddy currents, field inhomogeneities and other hardware imperfections, the k-space trajectories corresponding to different gradient polarities are often mis-aligned, resulting in Nyquist artifacts in the reconstructed images.

Most Nyquist artifact reduction procedures to date are designed to correct for k-space trajectory shift along the frequency-encoding direction, and are referred to as one-dimensional (1D) correction in this paper. Using 1D correction procedures, the k-space misalignment is typically measured from non-phase encoded or phase encoded reference scans and subsequently used to correct for image-domain phase difference (a spatial function along the frequency-encoding direction) between 1D image-profiles corresponding to different frequency-encoding gradient polarities (1,2). Since the 1D phase difference may be estimated from just one or a few extra k_y lines embedded in regular EPI sequences, the 1D reference scan does not reduce the imaging temporal resolution (3).

It has also been shown that even without extra reference scans, the 1D image-domain phase difference between opposite gradient polarities can actually be estimated directly from EPI data (4–6), or by comparing images reconstructed from different subsets of the k-space data with parallel reconstruction (7–9). Alternatively, Foxall et al. have shown that Nyquist artifacts in single-shot EPI can be minimized with a 1D iterative phase cycling procedure without acquiring any reference scan (10). With iterative phase cycling, a series of images is first generated using different possible values of phase difference from which a final image with the lowest artifact level can be identified based on background signal levels. Clare has successfully extended the iterative phase cycling to 1D phase correction for segmented EPI (11).

Even though 1D correction techniques can usually reduce most Nyquist artifacts, in many cases the residual aliasing after 1D correction remains significant. Particularly, when the image-domain phase difference (between sub-sampled images corresponding to opposite frequency-encoding gradient polarities) changes along both frequency- and phase-encoding directions (e.g., in oblique-plane imaging (12) or in the presence of cross-term eddy currents (13)), the residual artifacts after 1D correction can be significant enough to make the reconstructed images unsuitable for clinical use. In these cases, a two-dimensional phase correction is needed.

Procedures for measuring and correcting for 2D phase errors differ from 1D phase mapping and correction in several ways. First, in terms of phase mapping, 2D phase errors in EPI data cannot be measured from a single over-sampled k_y line, as used in conventional 1D correction procedures. Instead, most previous 2D correction approaches require additional full k-space reference scans at a significant cost of temporal resolution (14–18). Even though it is possible to incorporate the 2D reference scan in dynamic EPI without increasing the total scan time (16), this imaging scheme is not generally applicable to different imaging protocols such as high-resolution imaging based on segmented EPI. As shown by Buonocore and Zhu (6), in theory the 2D phase errors can be estimated directly from a manually chosen region-of-interest (ROI) of raw EPI images. However, it is usually difficult to choose an ROI for reliable phase error estimation for multi-shot EPI with a large number of segments, particularly when the SNR is low (5). Furthermore, the ROI based phase estimation is not compatible with parallel EPI, where additional aliasing artifacts exist due to k-space under-sampling. Second, in terms of image correction, once 2D phase errors are measured from reference scans, artifact-free images need to be calculated by solving a set of linear equations through matrix inversion. If the acquired 2D phase maps are inaccurate (e.g., due to subject motion between reference and actual EPI scans), the error may be amplified depending on the condition number of the matrix inversion (17).

To address these limitations, we propose an improved procedure for estimating 2D phase errors and suppressing Nyquist artifacts without the need of reference scans (14) or user input (6). With this new procedure, the error amplification in previously reported 2D phase correction procedures can be avoided. Furthermore, this technique can be generally applied to both single-shot and segmented EPI acquisitions, with or without parallel imaging.

Theory and Methods

In the first sub-section, we will briefly review an EPI reconstruction procedure that performs the correction for 2D phase differences corresponding to opposite frequency-encoding gradient polarities assuming the 2D phase difference is known a priori. In the second sub-section, we will describe an iterative phase cycling method from which a series of phase-corrected images is reconstructed using various possible 2D phase error values when no reference scan is available. The image with the lowest Nyquist artifact level can then be identified with criteria such as low background energy. We will also describe a procedure to significantly reduce the computation cost for iterative phase cycled reconstruction.

EPI reconstruction with correction for 2D phase errors

Figures 1a1 and 1a2 schematically compare ideal and distorted k-space trajectories due to 2D phase differences corresponding to opposite frequency-encoding gradient polarities (using phase terms corresponding to positive frequency-encoding gradient as the reference), in a two-shot EPI acquisition. The reconstructed magnitude and phase images of a mathematical phantom (with displacement by 2.2 k_x-steps and 0.1 k_y-step between trajectories of opposite frequency-encoding gradient polarities) are shown in the corresponding columns in Figures 1b and 1c, respectively.

Figure 1.

a: Schematic diagrams of (a1) ideal and (a2) distorted k-space trajectories of two-shot segmented EPI. The acquired k-space data can be decomposed into four parts, as shown in (a3–a6). b,c: The magnitude and phase images reconstructed from k-space data [corresponding to (a1–a6)] with 2D FFT, respectively. d: Illustrates the image-domain complex signals [corresponding to (a1–a6)] as a function of parent image signals and 2D phase errors, with bold font representing parent image signals from the same location and regular font representing aliased signals from other three location.

The two-shot EPI data (Figure 1a2) can be decomposed into four subsets (or 2N subsets for N-shot EPI in general) corresponding to 1) segment #1 / positive frequency-encoding gradient (i.e., S1+: shown in Figure 1a3), 2) segment # 2 / positive gradient (i.e., S2+: shown in Figure 1a4), 3) segment #1 / negative gradient (i.e., S1−: shown in Figure 1a5), and 4) segment #2 / negative gradient (i.e., S2−: shown in Figure 1a6). The summation of four sets of images reconstructed from 25% k-space subsets with 75% zero-filled (Figures 1b3 to 1b6; 1c3 to 1c6) can reproduce EPI images with Nyquist artifacts (Figures 1b2; 1c2). In two-shot EPI images (e.g., Figure 1b2), signals from four voxels, separated by a quarter of FOV in the parent image (e.g., Loc 1 to 4 shown in Figure 1b1), are mixed due to the aliasing effect. The goal of the 2D correction is thus to determine those signals, spatially separated in the parent image (Figure 1b1), from the corrupted EPI data (Figure 1b2).

In Figure 1d P_n represents the signal from location n of the artifact-free parent image and θ =θ (x,y) is the phase difference between the two images reconstructed from full k-space data corresponding to opposite frequency-encoding gradient polarities, using phase terms corresponding to positive frequency-encoding gradient as the reference. The unknown phase difference θ (x,y) may be measured from reference scans (14), or mathematically constructed before performing iterative phase cycled reconstruction described in the next sub-section.

The signals from four locations (Loc 1 to 4) of four sub-sampled data sets (Figure 1b3 to b6; 1c3 to c6) are mathematically described in 16 rounded rectangles (with gray background) of Figure 1d. As shown in Figure 1d3, in each of the four locations the signal reconstructed from S1+ data is a linear combination of parent-image signals from all four locations. The signals reconstructed from S2+ data, because of the k-space trajectory shift along the phase-encoding direction (as compared with S1+), are the superpositions of parent and aliased images weighted by a certain phase modulation term per Fourier transformation (19), as shown in Figure 1d4. The signals reconstructed from S1− and S2− (as shown in Figures 1d5 and 1d6, respectively) are modulated by two factors: 1) linear phase variation along the phase-encoding direction due to different k-space trajectories of the chosen subsets, and 2) nonlinear phase variations along both frequency- and phase-encoding directions due to eddy currents and other hardware imperfections.

If the 2D phase errors (e.g., θ_n in Figure 1d where n = 1,2,3,4) are known a priori, then the artifact-free signals (e.g., four unknowns: P_n; n = 1,2,3,4) can be determined from measured signals (e.g., 4 signals of location 1 from S1+, S2+, S1− and S2− data sets: 4 red dots in Figure 1b) with linear equations (e.g., 4 rounded rectangles with red dots in Figure 1d). The linear equations corresponding to signals from other locations (e.g., location 2: indicated by rounded rectangles with green circles in Figure 1d) are redundant with the equations corresponding to location 1. In general, for N-shot EPI, artifact-free parent-image signals can be obtained by solving Equation 1 or equivalently Equation 2 in its matrix form.

$$u=Ep$$ [1]

$$\left[\begin{array}{c}{u}_1\\ u_2\\ ⋮\\ u_N\\ v_1\\ v_2\\ ⋮\\ v_N\end{array}\right]=\frac{1}{2\text{N}}\left[\begin{array}{cccc}{e}^{i\frac{\pi }{N}·0·0}& e^{i\frac{\pi }{N}·0·1}& \cdots & e^{i\frac{\pi }{N}·0·(2\text{N}-1)}\\ e^{i\frac{\pi }{N}·1·0}& e^{i\frac{\pi }{N}·1·1}& \cdots & e^{i\frac{\pi }{N}·1·(2\text{N}-1)}\\ ⋮& ⋮& \ddots & ⋮\\ e^{i\frac{\pi }{N}·(N-1)·0}& e^{i\frac{\pi }{N}·(N-1)·1}& \cdots & e^{i\frac{\pi }{N}·(N-1)·(2\text{N}-1)}\\ e^{i\frac{\pi }{N}·N·0}{e}^{i{\theta }_1}& e^{i\frac{\pi }{N}·N·1}{e}^{i{\theta }_2}& \cdots & e^{i\frac{\pi }{N}·N·(2\text{N}-1)}{e}^{i{\theta }_{2N}}\\ e^{i\frac{\pi }{N}·(N+1)·0}{e}^{i{\theta }_1}& e^{i\frac{\pi }{N}·(N+1)·1}{e}^{i{\theta }_2}& \cdots & e^{i\frac{\pi }{N}·(N+1)·(2\text{N}-1)}{e}^{i{\theta }_{2N}}\\ ⋮& ⋮& \ddots & ⋮\\ e^{i\frac{\pi }{N}·(2\text{N}-1)·0}{e}^{i{\theta }_1}& e^{i\frac{\pi }{N}·(2\text{N}-1)·1}{e}^{i{\theta }_2}& \cdots & e^{i\frac{\pi }{N}·(2\text{N}-1)·(2\text{N}-1)}{e}^{i{\theta }_{2N}}\end{array}\right]\left[\begin{array}{c}{P}_1\\ P_2\\ ⋮\\ P_{N-1}\\ P_N\\ P_{N+1}\\ ⋮\\ P_{2\text{N}}\end{array}\right]$$ [2]

where p in Equation 1 is a 2N × 1 column vector with its elements P_n representing unaliased complex parent image signals separated by $\frac{{\mathit{FOV}}_y}{2\text{N}}$ along the phase-encoding 2N direction; u in Equation 1 is a 2N × 1 column vector with its elements u_k and v_k representing aliased image signals corresponding to the positive and negative frequency-encoding gradient polarities of the k^th segment respectively (e.g., Figures 1b3 to b6); E in Equation 1 is a 2N × 2N matrix, with θ_n in Equation 2 representing the 2D phase errors at location n. Note that it is sufficient to solve Equation 2 for only

$\frac{x_{\text{size}}\times y_{\text{size}}}{2\text{N}}$ voxels of the 2N reconstructed u_k and v_k images to reveal unknowns P_n in the full FOV. Equivalently, one may perform the matrix inversion of reduced FOV images (i.e., without zero-filling the segmented k-space data as shown in Figures 1a3 to 1a6).

Iterative phase cycled reconstruction and minimization of Nyquist ghost artifact

Equation 2 can be used to reconstruct artifact-free images if the 2D phase errors are known a priori such as in the case of reference scans (14). However, 2D references scans are usually not preferred for reasons described in the Introduction section. Here we integrate the 2D EPI phase correction procedure with an iterative phase cycled reconstruction without the need of a priori 2D phase information. A series of images are first reconstructed with Equation 2, cycling through different possible values of 2D phase errors. An image with the minimal Nyquist artifact is then identified based on effective criteria such as the background energy level. For example, images from different columns of Figure 2a are reconstructed from the mathematical phantom (used to produce Figures 1b2 and 1c2) with five different possible values of phase gradient (centered at 2.2 kx-step displacement) along the frequency-encoding direction, and images from different rows are reconstructed with five different possible values of phase gradient (centered at 0.1 ky-point displacement) along the phase-encoding direction. The energy level measured from the background is plotted in Figure 2b, for 2500 images reconstructed with 50 different phase gradient values along each of the directions. It can be seen that the Nyquist artifact is effectively eliminated only when the chosen phase gradient values match the simulation input.

Figure 2.

a: A series of images reconstructed with Eq. 2, cycling through different possible values of 2D phase errors. b: The energy level measured from the background, for 2500 images reconstructed with 50 different phase gradient values along each of the directions

In reality, 2D phase errors often include constant and nonlinear terms (particularly along the frequency-encoding direction) and thus cannot be described with linear phase gradients as in our mathematical phantom. As a result, it will be very time consuming when all possible nonlinear patterns of 2D phase errors are included in the iterative phase cycled reconstruction. Furthermore, it usually requires human input to identify background regions from which the energy level needs to be calculated and compared.

To address these issues, we have designed a column-based procedure to reduce the computation cost of iterative phase cycled reconstruction for EPI Nyquist artifact removal without manually selecting background regions, as described below.

Here we implemented an iterative phase cycled reconstruction procedure assuming that all 2D phase errors are consisted of 1) a spatially-independent phase offset, 2) nonlinear terms along the frequency-encoding direction, and 3) linear terms along the phase encoding direction. We found that this model is highly sufficient to remove the Nyquist artifacts in our phantom and human EPI data. If needed, our procedure can also be extended to correct for nonlinear phase error terms along the phase-encoding direction, at a higher computation cost. The other assumption in our column-based procedure is that the scanned object is smaller than the FOV, which is valid for most MRI studies.

Specifically, we first used an iterative phase cycling scheme to process MRI signals from a single column along the phase-encoding direction located near the center of the FOV. The 2D phase errors in this chosen column (at location x₀ along the frequency-encoding direction) can be represented by Equation 3.

$$f(x_0,y)=C_1+C_2\times y$$ [3]

where C₁ includes the contribution from both 1) a phase offset that is uniform for the whole 2D image, and 2) nonlinear phase terms along the frequency-encoding direction; C₂ represents the linear phase gradient along the phase encoding direction.

We then generated multiple sets of 1D profiles with Equation 2, based on different possible values of C₁ (cycled between −π and +π in 50 steps) and C₂ (between −π per pixel and +π per pixel in 50 steps). In other words, 2500 sets of 1D profiles were generated from this chosen column. The question is how do we choose the profile (from 2500 profiles) that corresponds to the lowest level of Nyquist artifact? For most of MRI applications, we usually have a rough idea about the size of the scanned object in reference to the FOV. For example, for brain MRI the object likely occupies 70% to 95% of the FOV. This knowledge is actually sufficient to help identify the 1D profile (from 2500 profiles) that is among the lowest artifact level in four steps: 1) The 1D magnitude profiles generated from each of the phase patterns are sorted in an ascending order; 2) The sorted 1D profiles are multiplied by a sigmoid function, suppressing signals in ~80% of FOV_y; 3) The sorted and sigmoid-weighted signals are summed; and 4) The 1D profile with the lowest summed signal is identified from profiles corresponding to different combinations of coefficients C₁ and C₂.

For example, the left and right panels of Figures 3a show the 1D magnitude profiles of phantom EPI data (from one of the eight coils) generated from two different sets of C₁ and C₂ values in Equation 3, respectively. The sorted signals are presented by solid lines in Figure 3b, showing that more energy exists under the unsuppressed region of a sigmoid window (red dashed curve) in the left panel as compared with the right panel. The sigmoid-weighted signals are shown in Figure 3c. The value obtained from summing all sorted and weighed signals is higher in the left panel than in the right panel. Based on the intensities of the summed signals, we conclude that data in the left panels have a higher level of Nyquist artifact than data in the right panels. After identifying the 1D profile with the lowest artifact level from the phase cycled reconstructed data, the phase error patterns (represented by two coefficients C₁ and C₂ in Equation 3) in the chosen column can be determined.

Figure 3.

The left and right panels of (a) show the 1D magnitude profiles of phantom EPI data (from one of the eight coils) generated from two different sets of C1 and C2 values in Eq. 3, respectively. The sorted signals and sigmoid window are presented by solid and dotted lines respectively in (b), and the sigmoid-weighted signals are shown in (c). d: Compares the uncorrected and phase-corrected two-shot EPI data, after combining data from all eight coils.

For two columns that are located immediately next to the previously processed column, we can safely reduce the range of phase cycling when searching for optimal C₁ and C₂ values that best suppress the Nyquist artifact, since the 2D phase errors are slowly varying in space. In our implementation, we reduced the range of C₁ and C₂ cycling by 5 times for each, so that only 100 profiles will be generated through matrix inversion based on Equation 2. This procedure can then be extended to all the columns in the slice, to the neighboring slices, and to the whole brain. Note that the 2D phase error patterns measured from one of the coils can actually be used to process the data obtained from other coils. Figure 3d compares the uncorrected and phase-corrected two-shot EPI data using this procedure, after combining data from all eight coils.

In comparison to the scheme illustrated in Figure 2, the developed column based data processing procedure has a lower computation cost. The computation time to reconstruct a segmented EPI data set, of 64 x 64 matrix size, is about 3 sec per slice using our Matlab implementation in a Linux PC (CentOS; 2.6 GHz CPU; 8GB memory).

Nyquist artifact removal for parallel EPI

Equations 1 and 2 can be used to reconstruct Nyquist-free EPI images. However, they cannot be directly applied to parallel EPI, where only subsets of the k-space data are acquired. For example, for single-shot parallel EPI with an acceleration factor of 2, the k-space data corresponding to Figures 1a4 and 1a6 are not available, and thus the system shown in Figure 1d becomes under-determined (with 4 unknowns and only 2 equations). In this case, we may integrate the phase-cycled reconstruction with the SENSE algorithm (21) to remove EPI Nyquist artifacts through iteratively solving Equation 4 with different possible values of 2D phase errors (for N-shot segmented EPI with an acceleration factor of M).

$$\left[\begin{array}{c}{u}_{M\times 0+1}^w\\ u_{M\times 1+1}^w\\ ⋮\\ u_N^w\\ v_{M\times 0+1}^w\\ v_{M\times 1+1}^w\\ ⋮\\ v_N^w\end{array}\right]=\frac{1}{2\text{N}}\left[\begin{array}{cccc}{S}_1^w{e}^{i\frac{\pi }{N}·(M\times 0)·0}& S_2^w{e}^{i\frac{\pi }{N}·(M\times 0)·1}& \cdots & S_{2\text{N}}^w{e}^{i\frac{\pi }{N}·(M\times 0)·(2\text{N}-1)}\\ S_1^w{e}^{i\frac{\pi }{N}·(M\times 1)·0}& S_2^w{e}^{i\frac{\pi }{N}·(M\times 1)·1}& \cdots & S_{2\text{N}}^w{e}^{i\frac{\pi }{N}·(M\times 1)·(2\text{N}-1)}\\ ⋮& ⋮& \ddots & ⋮\\ S_1^w{e}^{i\frac{\pi }{N}·(N-M)·0}& S_2^w{e}^{i\frac{\pi }{N}·(N-M)·1}& \cdots & S_{2\text{N}}^w{e}^{i\frac{\pi }{N}·(N-M)·(2\text{N}-1)}\\ S_1^w{e}^{i\frac{\pi }{N}·N·0}{e}^{i{\theta }_1}& S_2^w{e}^{i\frac{\pi }{N}·N·1}{e}^{i{\theta }_2}& \cdots & S_{2\text{N}}^w{e}^{i\frac{\pi }{N}·N·(2\text{N}-1)}{e}^{i{\theta }_{2N}}\\ S_1^w{e}^{i\frac{\pi }{N}·(N+M)·0}{e}^{i{\theta }_1}& S_2^w{e}^{i\frac{\pi }{N}·(N+M)·1}{e}^{i{\theta }_2}& \cdots & S_{2\text{N}}^w{e}^{i\frac{\pi }{N}·(N+M)·(2\text{N}-1)}{e}^{i{\theta }_{2N}}\\ ⋮& ⋮& \ddots & ⋮\\ S_1^w{e}^{i\frac{\pi }{N}·(2\text{N}-M)·0}{e}^{i{\theta }_i}& S_2^w{e}^{i\frac{\pi }{N}·(2\text{N}-M)·1}{e}^{i{\theta }_2}& \cdots & S_{2\text{N}}^w{e}^{i\frac{\pi }{N}·(2\text{N}-M)·(2\text{N}-1)}{e}^{i{\theta }_{2N}}\end{array}\right]\left[\begin{array}{c}{P}_1\\ P_2\\ ⋮\\ P_{N-1}\\ P_N\\ P_{N+1}\\ ⋮\\ P_{2\text{N}}\end{array}\right]$$ [4]

where $S_n^w$ represents the known coil sensitivity profile for coil number w at location n θ_n represents the 2D phase errors at location n; and $u_k^w$ and $v_k^w$ represent aliased image signals, measured from coil number w, corresponding to the positive and negative frequency-encoding gradient polarities of the k^th segment. Note that, when including data from all W coils, there are 2N unknowns and $\frac{2\text{N}}{M}\times W$ linear equations in Equation 4, which is solvable when W > M.

The first set of human brain EPI data presented here was acquired from a healthy volunteer at 3 Tesla with a quadrature head coil. Four whole-brain images were obtained with spin-echo EPI consisted of 1, 2, 4 and 8 segments. Other scan parameters included: FOV = 24 cm x 24 cm, matrix size = 64 x 64, slice thickness = 4 mm, TR = 2 sec, and TE was set to the minimal value available (34, 26, 20, and 16 msec, respectively). The subject’s head position was tilted from the ideal position (yaw: ~20°; pitch: ~10°), so that a double-oblique plane was chosen to generate images that correspond to a regular axial-plane. The acquired data were corrected with either conventional 1D correction (4), or our new 2D correction method. The sigmoid function was chosen to suppress signals in 85% of the FOV (i.e., assuming that the scan objects occupied 85% of the FOV). The reconstructed images were then compared in terms of the residual artifact level in the background.

The second and third sets of human brain EPI data were obtained from a healthy volunteer at 3 Tesla with an eight-channel coil. T2*-weighted images were acquired with gradient-echo full-Fourier EPI of 2 segments, and scan parameters included: FOV = 24 cm x 24 cm, matrix size = 160 x 160, slice thickness = 2.4 mm, TR = 3 sec, and TE = 35 msec. T1-weighted images were acquired with inversion-recovery prepared spin-echo partial-Fourier EPI of 2 segments, and scan parameters included: FOV = 24 cm x 24 cm, matrix size = 160 x 160, slice thickness = 2.4 mm, inversion time = 1 sec, TR = 5 sec, and TE = 67 msec. These two sets of images have identical voxel size (1.5 mm x 1.5 mm x 2.4 mm) and distortion patterns, and can be directly compared with each other for multi-contrast evaluation.

For T1-weighted partial-Fourier EPI, the 2D correction was performed on data from one of the coils, and phase errors were characterized using the iterative phase cycling scheme described in the Theory section. The derived information was then applied to remove Nyquist artifacts in data from each of the eight coils using Equation 2. The phase corrected partial-Fourier data were extended to full-Fourier data using Cuppen’s algorithm (20). Data from multiple coils were then combined, with sum-of-squares, to form a composite magnitude image. A very similar 2D phase correction procedure was applied to remove Nyquist artifacts in T2*-weighted full-Fourier EPI data, except that the Cuppen’s algorithm was not needed for full-Fourier images.

The fourth set of human brain images was obtained from a healthy volunteer at 3 Tesla with an eight-channel coil, using single-shot T2*-weighted parallel EPI sequence with acceleration factor of 2. Images in oblique-plane (perpendicular to the AC-PC line) were acquired with the following parameters: FOV = 24 cm x 24 cm, matrix size = 160 x 160, slice thickness = 3 mm, TR = 2 sec, and TE = 35 msec. Coil sensitivity profiles were estimated with a T-SENSE scheme (22). The subsequently acquired parallel EPI images were then processed iteratively with Equation 4, to estimate 2D phase errors and to remove the Nyquist artifacts.

Results

The human brain EPI data in double oblique plane, acquired with 1, 2, 4, and 8 segments are shown in Figures 4a to 4d, respectively. The images in the left column of Figure 4 were reconstructed directly from the k-space data with Fourier transform without any phase correction, and show strong Nyquist artifacts in all four data sets. The images in the middle column of Figure 4 were reconstructed with 1D phase correction and exhibit significantly reduced artifacts as compared with uncorrected images. However, residual artifacts remain visible and may interfere with the parent image signals, as indicated by arrows. The images in the right column of Figure 4 were reconstructed with the new 2D phase correction technique without the need of any reference scan. It can be seen that Nyquist artifacts are much better suppressed with 2D correction than with the conventional 1D correction method.

Figure 4.

The human brain EPI data in double oblique plane, acquired with 1, 2, 4, and 8 segments are shown in (a–d), respectively. Left column: reconstructed without any phase correction; middle column: reconstructed with 1D phase correction; right column: reconstructed with the new 2D phase correction technique without the need of any reference scan.

The display scale of the four-shot segmented EPI obtained with 1D and 2D correction were then adjusted (with power of 0.2) so that both residual Nyquist artifacts and background noises are visible. As shown in Figure 5a, the 1D phase corrected images have noticeable residual artifacts in 12 chosen slices. On the other hand, using the developed 2D phase correction method, the majority of the Nyquist artifacts can be suppressed more effectively (Figure 5b). The ghost-to-noise ratios measured from the manually chosen ROIs in all 12 slices, e.g., the yellow area (ghost) and the red area (noise) for 1 of the slices, were 4.6 in 1D phase-corrected images, and 2.3 for 2D phase-corrected images.

Figure 5.

a: Four-shot segmented EPI, of 12 slices, obtained with 1D phase correction. The display scale was adjusted (with power of 0.2) so that both residual Nyquist artifacts and background noises are visible. b: The images reconstructed with our new 2D phase correction method.

We would like to point out that even though we chose a single sigmoid function to suppress signals in 85% of the FOV for all of the slices, the residual artifacts are low in almost all of these 12 slices in Figure 5b regardless of the object size in reference to the FOV. We have also evaluated the image quality obtained with different sigmoid profiles (e.g., to suppress signals in 70%, 80% or 90% of the FOV) and similar artifact suppression efficiency can still be reliably achieved.

The phase-corrected T2*-weighted and the inversion-recovery prepared EPI images, of three selected slices, are shown in the top and bottom rows of Figures 6 respectively. It can be seen that the achieved image quality appears similar to that obtained with conventional spin-warp imaging.

Figure 6.

The top and bottom panels show 2D phase-corrected T2*-weighted and inversion-recovery prepared high-resolution EPI images, respectively.

The top panel of Figure 7 shows undersampled EPI of three slices from one of the coils (without performing parallel reconstruction), where the Nyquist artifact and the aliasing artifact due to under-sampling overlap. The bottom panel shows images reconstructed with iterative phase-cycled and SENSE reconstruction based on Equation 4. It can be seen that the Nyquist artifact in parallel EPI can be successfully removed using the developed methods.

Figure 7.

The top panel shows undersampled EPI of three slices from one of the coils (without performing parallel reconstruction), and the bottom panel shows images reconstructed with iterative phase-cycled and SENSE reconstruction.

Discussion

It has been shown that when the phase error terms vary spatially along both frequency- and phase-encoding directions the resultant Nyquist artifacts in EPI cannot be effectively removed with 1D phase correction. This is particularly problematic for oblique plane EPI, in which two or more physical gradients are combined to generate fast-switching frequency-encoding gradient. Oblique-plane scans are regularly used for brain imaging (e.g., parallel or perpendicular to AC-PC line) and cardiac EPI (e.g., perpendicular to the long cardiac axis) among others. Therefore, we expect that our new 2D phase correction for effective EPI Nyquist artifact removal will prove valuable for many clinical applications.

In our implementation, the 2D phase errors are completely characterized with iterative phase cycling without the need for any extra reference scan. Note that our new technique is also compatible with reference-scan based phase correction. For example, the 1D reference scan (e.g., embedded in EPI scans without scan time penalty (3)) can be used to first correct for the global phase offset and 1D nonlinear phase variation along the frequency-encoding direction (i.e., C₁ in Equation 3). The 1D phase corrected EPI data can then undergo the same iterative phase cycled reconstruction and 2D phase correction. In this way, the range of the phase cycling, and thus the computation cost, can potentially be significantly reduced.

In Equation 3, we assumed that the 2D phase errors vary linearly along the phase-encoding direction, and we found that this model is sufficient to remove the Nyquist artifacts in our EPI data. If there exists nonlinear phase gradient along the phase-encoding direction, then we may extend the linear model in Equation 3 to include nonlinear terms (e.g., f(x₀, y)=C₁+C₂× y+C₃ × y²). Iterative phase cycled reconstruction based on this model obviously requires a higher computation cost. In this case, the strategies for reducing the computation time, as described in the previous two paragraphs, will be important.

Single-shot EPI has been a powerful tool for functional MRI, dynamic contrast enhanced imaging, and diffusion tensor imaging. On the other hand, even though segmented EPI has great potential for producing multi-contrast data, it has not yet been widely used clinically, in part due to the challenges in suppressing the undesirable Nyquist artifacts. With the effective artifact correction technique shown in this paper, segmented EPI based structural imaging may potentially provide image quality comparable to spin-warp imaging but with shorter scan time.

Even though in our implementation we assume that the scanned object is smaller than the FOV, our data show that the implemented methods may still perform well when the scanned object is slightly larger than the FOV as long as the wrap-around is insignificant. For example, in Figure 5b, the Nyquist artifact can be removed even though some nose signals are outside the FOV. Note that there exist low intensity areas (e.g., the nasal cavity in Figure 5b) in these images, making it possible to correctly estimate the phase errors through minimizing the total sorted sigmoid-weighted signals for those columns with insignificant wrap-around. In general, for images with significant wraparound and without low intensity areas, the developed iterative phase-cycled reconstruction and the parallel reconstruction need to be integrated to generate images free from Nyquist artifact (17).

Foxall et al. (10) first showed that the iterative phase cycling is an effective way to estimate the phase errors, through minimizing signals for just a few voxels (or even a single voxel) in the background area (e.g., Figure 2 in Ref. 10). However, note that the reconstruction framework developed by Foxall et al. (e.g., Equation 2 in Ref. 10) can only correct for 1D phase errors. In order to reconstruct EPI images free from 2D phase error induced Nyquist artifact, a matrix inversion (as shown in our Equations 1 and 2) is needed. Another novel component in our 2D phase-cycled reconstruction is the sigmoid-weighting and sorting procedure, which makes it possible to estimate 2D phase errors without the need to manually select background ROIs.

Buonocore and Zhu (6) reported that 2D phase errors can be directly estimated from a manually-chosen ROI of raw EPI images, without an additional reference scan. However, their approach has a few limitations. First, it is not always easy to identify an ROI for reliable phase error estimation, particularly for multi-shot EPI with a large number of segments. Second, for parallel EPI, it is very difficult to perform ROI based phase error estimation, because of additional aliasing artifacts originating from k-space under-sampling. Third, the phase error estimated from a manually-chosen ROI may be noisy, resulting in error amplification in images reconstructed through a matrix inversion. In comparison to the method reported by Buonocore and Zhu, our phase-cycled reconstruction algorithm does not require user input, and can be more reliably applied to EPI with a large number of segments (e.g., 8-shot EPI shown in Figure 4d). Furthermore, the Nyquist-artifact removal based on our 2D phase-cycled reconstruction is applicable to both non-parallel and parallel EPI (e.g., Figure 7).

It should be noted that the optimization procedures based on the phase-cycled reconstruction may have multiple solutions (i.e., multiple local minimal points), and each of the solutions represents an artifact-free image with a certain displacement, along the phase-encoding direction, from the true parent image. For example, for single-shot EPI, the parent-only image and the ghost-only image have the same energy level, as measured by our sigmoid-weighting and sorting procedures. Therefore, if the ghost-image signals are stronger than the parent-image signals in raw EPI data, it is possible that the image generated from the phase-cycled reconstruction will be shifted by half of the FOV from the true parent image. In this case, we may need to introduce an additional criterion, preferring a solution that corresponds to an image with more energy located near the center of the FOV (i.e., true parent image).

Because it does not need additional reference scans, our new 2D phase correction technique can be retrospectively applied to suppress Nyquist artifacts in multiple existing phantom and human EPI data sets previously acquired with different scan parameters (e.g., spin-echo and gradient-echo EPI; single-shot and segmented EPI; full-Fourier and partial-Fourier EPI; quadrature coil and phased array coils; 1.5 T and 3 T). We found that the new 2D phase correction method can consistently better suppress Nyquist artifacts than the 1D correction.

The developed phase cycled reconstruction scheme may be further extended to identify and remove artifacts originating from other types of intra-scan phase inconsistencies, in addition to the Nyquist artifacts in EPI. For example, motion-induced phase variations in diffusion-weighted segmented EPI may potentially be estimated and corrected with our new technique, without requiring a navigator echo. Our team is currently working on generalizing the phase cycled reconstruction to address various types of phase related artifacts in EPI and other types of accelerated scan (e.g., spiral imaging).

In conclusion, in this paper we present a reference-less 2D phase correction technique, for reducing EPI Nyquist artifacts. This technique can generally be applied to single-shot and segmented EPI with or without parallel imaging. The developed method is significantly superior to 1D phase correction, particularly for oblique-plane imaging.