Even with highly accurate prospective motion correction (PMC)¹, for large motion, the corrected images can be degraded by residual artifacts. The change in B0 field (∆B₀) and gradient nonlinearity (GNL) are important sources of distortion and may impact PMC data. This work proposed an image reconstruction to mitigate the geometric distortion in PMC data by integrating corrections of GNL and ∆B₀ into the augmented SENSE reconstruction².

Data acquisition: Experiments were performed on a 7T MR (Siemens) equipped with a 32-channel head coil. The application of the proposed method to real motion with PMC enabled was performed using 3D MPRAGE with matrix; 256x256x176, voxel size; 1 mm3, TR/TE/TI; 1800/1.99/1050 ms, and BW; 200 Hz/pixel. After the first half of the acquisition, the volunteer was instructed to perform a one-time head rotation around the z-axis. The motion pattern from the tracking log file is shown in figure 1a. The field maps were also measured before and after motion using dual-echo 3D FLASH with the same resolution as 3D MPRAGE, TR/TE1/TE2; 10.00/3.06/5.84 ms, and BW; 250 Hz/pixel.

Image reconstruction: We aimed to obtain the distortion-free PMC MR images () by minimizing the residual sum of squares (Eq. 1) using the Conjugate Gradient (CG) method³,

$$min_v\left\{∑_{i,j}‖m^{i,j}−M^iFG^i_{reg2warp}C^{i,j}v‖_2^2+λ‖Lv‖_2^2\right\} ----(1)$$

where $$$m^{i,j}$$$ is MR measurement data at pose $$$i$$$ and coil $$$j$$$ which is aligned to a Cartesian grid by PMC. Coil sensitivity ($$$C^{i,j}$$$) was estimated from the central 64x64x64 k-space data of 3D FLASH (TE1) acquired using a cosine tapper window⁴, $$$M^i$$$ is a Cartesian undersampling mask (1=sampled, 0=otherwise), $$$F$$$ is fast Fourier transform. $$$G^i_{reg2warp}$$$ is a 3D regular Cartesian to warped coordinate operator transferring the image data from regular to warped coordinate systems. Following the nonuniform FFT (NUFFT) algorithm⁵, we implemented the operator using a Kaiser-Bessel interpolation with optional 2x oversampling. The warped coordinates corrupted by GNL and ∆B₀ at any pose $$$i$$$ were prepared prior to solving Eq. 1 using the vendor-provided Spherical Harmonics expansion of the gradient distortions and the dual-echo field map method⁶, respectively. The regularization parameter $$$λ$$$ is a positive real constant that was adjusted manually in this study, and $$$L$$$ is here a tri-diagonal (1 -2 1) sparse matrix⁷.

Figure 1a shows the motion pattern when the subject performed the head rotation during the data acquisition. A maximum rotation around the z-axis of approx. 23 degrees was detected. Figure 1b demonstrates the effectiveness of the proposed method for undersampled k-space data, the full PMC k-space data were artificially accelerated by factor 2x2 along both phase directions (phase and slice directions). Although, the conventional CG-SENSE [4] provided acceptable images (1st column), the remaining aliasing artifacts due to coil sensitivity misalignment and also blurring caused by ∆B0 and GNL are clearly visible as shown in the red circles. Note that sensitivity maps specific to the first half of the acquisition were used in conventional CG-SENSE. Superior image quality with very little remaining artifacts was achieved after applying the augmented CG-SENSE with integrated ∆B₀+GNL corrections (2nd column).In this study, PMC had a high accuracy in detecting the motion since the residuals of rigid motion errors were not visually identifiable after correcting non-rigid distortion by the proposed technique. We also found that using a tri-diagonal sparse matrix⁷ in the regularization provided smoother solution than using an identity matrix. It accounted for the high intensity variations in strong field inhomogeneity regions. Using this regularized matrix with an automatic λ selection technique⁸ may be practical. The requirement for a field map per motion pose is a limitation of the method. Using a magnetic field model⁹ which neither requires additional scan time nor suffers from low SNR at air/tissue boundaries may be feasible. In conclusion, GNL and ∆B₀ can induce residual artifacts even with perfect prospective motion correction. These artifacts can be alleviated by the proposed technique when the relevant field information is available.