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.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.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.

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 SENSE² reference scan was also acquired. The signal model in the cost function was implemented using NUFFTs³. 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.

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.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.