Michael J van Rijssel^{1}, Frank Zijlstra^{1}, Peter R Seevinck^{1}, Peter R Luijten^{1}, Dennis W J Klomp^{1}, and Josien P W Pluim^{1,2}

The majority of diffusion acquisitions is affected by geometrical distortions due to susceptibility induced off-resonance effects in the EPI readout. This hampers the use and effectiveness of these images in multiparametric cancer protocols, especially in lipid-rich environments such as the human breast where tissue interfaces cause large but local discontinuities. Preliminary results show that improvements upon existing correction techniques can be made by using high-resolution off-resonance information in distortion correction algorithms.

In order
to exclude effects caused by subject motion, MR experiments were performed on a
motionless pork chop. All experiments were performed using a unilateral breast
coil setup on a 7T whole-body MR system (Achieva; Philips, Cleveland, Ohio,
USA). The protocol involved two fat-suppressed spin-echo EPI acquisitions with
opposed phase encoding directions (2x2x3 mm^{3}, bandwidth/voxel 20 Hz),
a non-accelerated spin-echo acquisition with otherwise equal imaging parameters
as reference, and two dual-echo ΔB_{0} measurements at different
resolutions (2x2x3 mm^{3} and 0.7x0.7x3 mm^{3}). In order to
avoid lipid bias in the off-resonance measurements, the echo times at which the
ΔB_{0} map was measured were chosen such that a lipid model consisting
of the 10 largest resonances was in phase with the water signal.^{2}

Distortion
correction was performed using an inverse problem approach resembling
techniques described earlier.^{3, 4} The forward model was described using fast
steady state simulations (FORECAST method), restricted to distortions in the
phase encoding direction (EPI direction).^{5} This formulation
allows efficient evaluation of $$$A(\Delta B_{0})\cdot x$$$ (the forward model) and $$$A(\Delta B_{0})^{H}\cdot x$$$ (the conjugate transpose of the forward model),
where A is the transformation matrix that maps undistorted to distorted space
and with the ΔB_{0} map and corrected image at equal or higher
resolution than the measured EPI image. (Direct computation of A was avoided
since it is very large and sparse.) The inverse problem was then formulated as
a damped least squares problem: $$$Im=\begin{matrix} argmin\\x \end{matrix}\left \| \begin{bmatrix} A\\ \lambda I \end{bmatrix} x - \begin{bmatrix} b\\ 0 \end{bmatrix} \right \|_{2}$$$, where
Im is the corrected image, b is the measured EPI data, I is the identity
matrix, and the regularization parameter λ was empirically set to 10. This problem
was solved using the LSQR algorithm.^{6}

EPI distortion correction was applied to the
same pair of oppositely encoded images using the measured ΔB_{0} map at the EPI resolution (2x2x3
mm^{3}) and at a considerably higher in-plane resolution (0.7x0.7x3 mm^{3})
that better captures local ΔB_{0} discontinuities at tissue interfaces.
The resulting images were compared visually.

1. van
Rijssel M J, Zijlstra F, Seevinck P R, et al. Susceptibility-induced local ΔB_{0}
variations are essential for predicting EPI distortions in the breast. Proc.
Intl. Soc. Mag. Reson. Med. 2017; 1389.

2. Boer
V O, Luttje M P, Luijten P R, et al. Requirements for static and dynamic higher
order B_{0} shimming of the human breast at 7 T. NMR Biomed. 2014;27(6):625-631.

3. Andersson J L R, Skare S, and Ashburner J. How to correct susceptibility distortions in spin-echo echo-planar images: application to diffusion tensor imaging. Neuroimage. 2003;20(2):870-888.

4. Munger P, Crelier G R, Peters T M, et al. An inverse problem approach to the correction of distortion in EPI images. IEEE Trans Med Imaging. 2000;19(7):681-689.

5. Zijlstra F, Bouwman J G, Braskute I, et al. Fast Fourier-based simulation of off-resonance artifacts in steady-state gradient echo MRI applied to metal object localization. Magn Reson Med. 2017;78(5):2035-2041.

6. Paige C C, and Saunders M A. LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares. Acm Transactions on Mathematical Software. 1982;8(1):43-71.