Jessica A McKay^{1}, Steen Moeller^{2}, Sudhir Ramanna^{2}, Edward J Auerbach^{2}, Gregory J Metzger^{2}, Justin R Ryder^{3}, Kamil Ugurbil^{2}, Essa Yacoub^{2}, and Patrick J Bolan^{2}

Diffusion weighted imaging (DWI) acquired with SE-EPI is subject to Nyquist ghosts, which are commonly corrected using a three-line navigator. However, several alternative ghost correction strategies do not require any reference acquisition. These methods define a cost function that is minimized when the image is ghost-free. We propose a novel referenceless method called ghost/object minimization by defining the cost function as the summation over the image divided by its ½-FOV-shifted counterpart. In this work, we test noise sensitivity and demonstrate the feasibility of the ghost/object minimization using simulated ghosts and in vivo acquisitions including brain, prostate, breast, and liver DWI.

Diffusion
weighted imaging (DWI) acquired with single shot spin-echo echo-planar imaging (SE-EPI) is
subject to Nyquist (or N/2) ghosts caused by eddy currents, imperfect
gradients, and/or timing errors. An image with a typical first order ghost is
described by: $$M(x,y)=M_0 \left(x,y\right) \cos\left(2πκx+ϕ\right)+iM_0 \left(x,y-\frac{FOV}{2}\right) \sin\left(2πκx+ϕ\right)$$ where M_{0} is the ideal
image and κ and φ correspond to
the 1^{st} and 0^{th} order phase differences between alternating
lines in k_{y}-x-space. Nyquist ghosts are commonly corrected using a 3-line
navigator to measure the phase difference^{1}. However, several alternative
ghost correction strategies do not require any navigator acquisition,
which may allow for shorter TE. These methods work by defining a cost function f_{cost}(κ,ϕ) which is minimized when the image is ghost-free. Examples
include the entropy method^{2,3}, which minimizes the entropy in the image
domain, and the SVD method^{4}, which operates on k-space. The performance of
these methods is generally good but it can vary with geometry, signal to noise ratio
(SNR), and amount of ghost-object overlap.

In this work we introduce a new referenceless method called Ghost/Object minimization (G/O). We define the cost function as the summation over the image divided by a ½-FOV-shifted version of the image: $$$\sum_{x,y}^{}\frac{M\left(x,y-\frac{FOV}{2}\right)}{M(x,y)}$$$. In this work, we test the sensitivity to noise and demonstrate the feasibility of this technique in several applications using DW SE-EPI.

*Simulation*

A Shepp-Logan phantom with complex noise was simulated
with a realistic ghost by adding a phase ramp to alternating lines in the k_{y}-x
domain (κ=-0.42, ϕ=-0.15). The three methods were
tested by calculating each f_{cost} over a discrete parameter space (κ, ϕ), locating the minimum, and applying the correction (κ_{min}, ϕ_{min}) to the image. The
experiment was repeated with randomly varying ghost parameters and increasing noise (10
noise levels, 100 trials per level) to assess SNR sensitivity.

*In vivo*

To demonstrate in vivo feasibility, each referenceless method was retrospectively applied to DWI acquired in several body regions. Corrections were made independently for each channel, slice, and repetition for 1) brain DWI with simultaneous multi-slice (SMS, MB=4), 2) breast DWI with GRAPPA (R=2), 3) fully-sampled prostate DWI, and 4) liver DWI with R=3 and segmented ACS lines. For parallel imaging cases, the initial ghost correction was estimated using the auto calibration scans (ACS) and applied to the undersampled data. Ghost correction was then refined on a per-channel and per-repetition basis after GRAPPA reconstruction. Apparent diffusion coefficient (ADC) maps were generated using a pixel-by-pixel log-linear fit.

Figure 1 shows the cost functions measured on the simulated dataset. The bottom row demonstrates the image before correction, after correction by an incorrect solution that corresponds to a local minimum of SVD, and with the correct solution, i.e. the global minima of all metrics. The error as a function of noise level is plotted in Figure 2; the G/O method appears to be less sensitive to noise than Entropy or SVD.

Brain
data acquired with SMS (MB=4) b=0 s/mm^{2} images are shown in Figure 3
as a montage across all slices and a close-up of a subset, scaled to highlight
residual ghosts. The online reconstruction fails to fully correct the ghosts in
this case, while all three referenceless methods achieve almost complete ghost
suppression.

Fully reconstructed axial ADC maps are shown in Figures 4 and 5 from the breast, prostate, and liver DWI. In this breast case, both SVD and G/O eliminate the ghost, however there is a small residual ghost visible after the entropy correction. SVD performed poorly in all slices of this prostate case, while both entropy and ghost ratio fully suppressed the ghost. In the selected slice of liver DWI, SVD performs best, which is consistent for most but not all of the other slices.

We have demonstrated the feasibility of ghost correction with ghost/object minimization in a variety
of DWI data. A careful comparison of G/O with the linear navigator and other referenceless
methods was performed in breast R=3 data^{5}, but further work is
needed to compare in prostate, liver, and brain imaging. Although background
regions were assessed visually as a surrogate for ghost correction performance,
even inconspicuous residual ghosts can bias diffusion parameters in the tissue.

In general, entropy and SVD methods work well, but in some cases one performs noticeably worse. The addition of G/O provides a third independent option, which can be combined with entropy and SVD for increased robustness^{5}.

NIH P41 EB015894

NIH R21 CA201834

UMN Medical School Foundation

NIH/NIDDK R01 DK105953

- Heid, O. (2000). Method for the phase correction of nuclear magnetic resonance signals.
- Clare, S. (2003). Iterative Nyquist ghost correction for single and multi-shot EPI using an entropy measure. In Proceedings of the 16th Annual Meeting of ISMRM, Toronto, Canada, p. 1041.
- Skare, S., Clayton, D.B., Newbould, R., Moseley, M., and Bammer, R. (2006). A fast and robust minimum entropy based non-interactive Nyquist ghost correction algorithm. In Proc. Intl. Soc. Mag. Reson. Med, (Seattle, Washington), p. 2349.
- Peterson, E., Aksoy, M., Maclaren, J., and Bammer, R. (2015). Acquisition-free Nyquist ghost correction for parallel imaging accelerated EPI. In Proc. Intl. Soc. Mag. Reson. Med., (Toronto, Ontario), p. 0075.
- McKay JA, Moeller S, Zhang L, Auerbach EJ, Nelson MT, Bolan, PJ. (2017) Comparison of Referenceless Methods for EPI Ghost Correction in Breast Diffusion Weighted Imaging. In Proceedings of the 26th Annual Meeting of ISMRM [Submitted]

Figure
1. Cost functions measured on a Shepp-Logan phantom with a simulated ghost (κ = -0.42, ϕ = -0.15).
In
the bottom row, the phantom is shown **a)** before correction (black x), **b)**
after applying an incorrect solution that corresponds to a local minimum of
the SVD cost function (red x), and **c)** after a correct solution based on the
global minima of all three metrics (yellow x).

Figure
2. The average
mean squared error over 100 trials for increasing noise amplitudes (0 to
40% of the maximum signal). Ghost/object seems to be least sensitive to low SNR; entropy shows high sensitivity.

Figure
3. Axial b = 0 s/mm^{2} images of brain DWI acquired on Siemens 3 T Prisma with
SMS (MB = 4) scaled to highlight residual Nyquist ghosts. **Left**: a montage of
linear navigator results for all slices. **Right**: Montage zoomed in to focus on a
near-center slice (red box) for all four methods. In this case, the online
reconstruction fails (white arrows) to fully suppress the ghost in many slices
(left). Referenceless methods achieve better ghost correction to varying
degrees across slices (right). PE direction is anterior-posterior.

Figure
4. Axial ADC maps
of breast
DWI with b = 0,
100, 600, and 800 s/mm^{2} and
GRAPPA
(R =
2)
acquired on a Siemens 3 T Prisma^{fit}. Residual ghosts are apparent in the
background region after correction by Linear Navigator and Entropy (white
arrows). PE direction is right-left.

Figure
5. Axial ADC maps.
**Top:** fully-sampled
prostate DWI acquired
on Siemens 7 T Magnetom at b
= 0 and 1000 s/mm^{2}.
**Bottom:** undersampled (R =
3) liver DWI with segmented ACS lines (3 segments, EPI)
acquired on Siemens 3 T Prisma^{fit} at b
= 0, 50, 400, and 800 s/mm^{2} .
The
standard
navigator correction clearly fails (white arrow), and
SVD leaves residual ghosts
(white arrow) in
nearly all of the slices of the prostate images. The ghost correction
performance of all the methods is quite variable in the liver data. PE
direction is anterior-posterior in both cases.