Julianna D. Ianni^{1,2} and William A. Grissom^{1,2,3,4}

^{1}Department of Biomedical Engineering, Vanderbilt University, Nashville, TN, United States, ^{2}Vanderbilt University Institute of Imaging Science, Vanderbilt University, Nashville, TN, United States, ^{3}Department of Radiology and Radiological Sciences, Vanderbilt University, Nashville, TN, United States, ^{4}Department of Electrical Engineering, Vanderbilt University, Nashville, TN, United States

### Synopsis

**A method for automatic correction of EPI trajectory errors is presented. The method is an iterative parallel imaging reconstruction which uses k-space data consistency to correct image artifacts. An advantage of the method is that it requires no calibration data and allows correction of non-static errors such as those due to gradient coil heating.**### Purpose

The purpose of this work is to automatically correct image errors due to eddy currents and resultant phase errors in echo planar imaging (EPI) reconstructions.

### Introduction

EPI
trajectories are typically affected by ghosting artifacts due to
trajectory errors. Common correction methods necessitate
acquiring calibration data and apply the same corrections to all
slices and dynamics within a scan. This does not
account for dynamic errors such as those from gradient coil heating. We propose to alleviate these artifacts without
calibration data by extending a previous method for
non-Cartesian trajectory error correction (TrACR-SENSE)

^{1}
to estimate EPI phase encode line shifts. The method works
by exploiting correlations between adjacent sampled lines in
multicoil EPI data.

### Theory

Errors
in EPI trajectories cause ghosting artifacts which can be represented
by a single 1-dimensional shift of each readout line in k-space, along the readout (assumed here to be $$$k_x$$$). We recover the
distortion-free image and shifts by minimizing an EPI TrACR-SENSE cost function: $$ \Psi({\pmb{f}},\pmb{\Delta k_x}) =
\frac{1}{2} \sum_{c=1}^{N_c}\sum_{i=1}^{N_{k}} \left \vert
d_{ci} - \sum_{j=1}^{N_s} e^{-\imath 2 \pi \left[( {k}_{xi} +\Delta
{k}_{xi}) x_{j} + k_{yi} y_{j} \right]} s_{cj} f_{j}\right
\vert^2 $$ where $$${\pmb{f}}$$$ is a length-$$$N_s$$$ vector of image
samples to be reconstructed, $$$\pmb{\Delta k_x}$$$ is a length-$$$N_{k}$$$
vector of trajectory errors to be estimated, $$$N_c$$$ is the number
of receive coils, $$$d_{ci}$$$ is coil $$$c$$$'s $$$i$$$th k-space
data sample, $$$\imath = \sqrt{-1}$$$,
$$$\left(k_{xi},k_{yi} \right)_{i=1}^{N_k}$$$ are the nominal k-space
locations, and $$$s_{cj}$$$ is coil $$$c$$$'s receive sensitivity at
spatial location $$$j$$$.
In
general we model the k-space trajectory errors $$$\pmb{\Delta k_x}$$$
as a sum of weighted error basis functions: $$ \Delta {k}_{xi}=
\sum_{b=1}^{N_b}{e}_{bi}w_b$$ where $$$N_b$$$ is the number of
error basis functions $$${{e}}_{b}$$$. For the case in which
odd lines are assumed to experience the same shift relative to even lines, only one error
basis function is required, constructed as: $$ \begin{align}
{{\pmb{e}}} &= \left[ \begin{array}{ccc} {q} \otimes
{1}_{N_{read}\times 1}\end{array}\right] \end{align} $$ where
$$$\otimes$$$ is the Kronecker product, $$$N_{read}$$$ is the number of
points in the readout dimension, and $$${{q}}$$$ is a vector the
length of the phase-encode dimension with elements: $$ \begin{align}
q_n= \left[ \begin{array}{cccc} 1 & for & n & odd \\ 0 &
for & n & even \end{array} \right] \end{align}$$ A more
flexible error model was also implemented, comprising weighted
triangular interpolation functions over the phase encodes to estimate
10 shifts, 5 each for odd and even lines.

### Methods

^{}
EPI
data were collected in a human volunteer on a 7 Tesla scanner
(Philips, Best, Netherlands) using the following parameters: 199 mT/m/ms max slew rate, 35 mT/m max gradient amplitude, 21×21cm FOV,
3mm isotropic resolution, TR 1744ms, TE 25ms, FA 63°. This was
repeated at acceleration factors of 2x, 3x, and 4x. A
SENSE^{2} reference
scan was also acquired. The signal model in the cost function was implemented using
NUFFTs^{3}. EPI-TrACR was
performed with each outer loop iteration consisting of 5 CG k-space
iterations and a CG-SENSE image update, using MATLAB's (Natick, MA, USA) lsqr function with a fixed tolerance of 10^{-3}. The RMS signal in
an ROI outside the brain was calculated, along with RMS noise in
an unaliased region. Final images were reconstructed with
SENSE and a fixed tolerance of 10^{-1} after
windowing the data. The scanner's reconstructions, which used
calibration data, were interpolated to the same grid.

### Results

EPI-TrACR
single- and multiple-shift reconstructions for each acceleration
factor are shown in Figure 1 alongside uncorrected and scanner reconstructions. Figure 2 plots RMS
ghosted signal versus acceleration factor for EPI-TrACR
reconstructions and uncorrected images. The RMS noise was 0.029.
Figure 3 shows representative error estimates. Compute time was
approximately 5 minutes for the single-shift model at 1x without
optimizing for speed. Faster reconstructions can be achieved by using
NUFFTs that perform gridding in a single dimension; NUFFTs
performed this way resulted in a 4.4x speed-up, which will accelerate
the EPI-TrACR reconstruction.

### Discussion and Conclusion

Ghosting
artifacts are reduced in the EPI-TrACR-corrected images, with
slightly better corrections in the multiple shift model, as shown
in Figure 1. As Figure 2 showed, the method performs best at
low accelerations, but was
able to perform significant correction at least up to 4x acceleration, with quality similar to the scanner’s calibrated reconstructions. Some differences between the EPI-TrACR and scanner reconstructions are most likely due to lack of coil noise correlation
matrices. EPI-TrACR is calibrationless, requires no additional probes
or measurements, and can make post-hoc corrections and correct
dynamic errors. In addition, corrections with more degrees of freedom
and higher accelerations may be possible by jointly estimating
errors from multiple slices and dynamics to boost SNR for the
estimation.

### Acknowledgements

This work was supported by NIH grant R01 EB016695.### References

1.
Ianni JD and Grissom WA. Trajectory Auto-Corrected Image
Reconstruction. Magn Reson Med 2015. in press. DOI 10.1002/mrm.25916.

2.
Pruessmann KP et al. SENSE: sensitivity encoding for fast MRI. Magn
Reson Med 1999; 42:952-962.

3.
Fessler, JA. Image Reconstruction Toolbox.
http://web.eecs.umich.edu/~fessler/code/index.html