Wave-encoding¹ has been demonstrated to improve the performance of accelerated MRI reconstruction by distributing the under-sampling in both phase-encode and readout directions. Wave-CS² combines wave-encoding with compressed sensing and has been shown to significantly reduce aliasing artifacts in brain imaging. However, a separate calibration scan is required. Subject motion between the calibration scan and the wave-encoded acquisition may introduce sensitivity variation and therefore degrade image quality. In this work, we propose a motion-robust auto-calibrating Wave-CS technique for abdominal and pelvic imaging.

The concept is to use the wave-encoded center k-space and the known point-spread-function (PSF) of wave-encoding to reconstruct a Cartesian central k-space for calibration. In detail, the wave-modulated signal $$$S_{wave}$$$ and the Cartesian signal $$$S_{Cartesian}$$$ have the following relationship in the kx-y-z domain¹:$$S_{wave}[k_x,y,z]=PSF[k_x,y,z]\cdot S_{Cartesian}[k_x,y,z]$$The PSF can be expressed as: $$\begin{aligned} PSF[k_x,y,z] &=\exp(-i\gamma\int g_y(\tau)y+g_z(\tau)z d\tau) \\ &=\exp(C_1\cdot y+C_2\cdot z) \\ &=\exp(C_1\cdot \Delta y \cdot i_y+C_2\cdot \Delta z \cdot i_z)\end{aligned} $$ where $$$C_1$$$ and $$$C_2$$$ are constants associated with gradients $$$g_y$$$ and $$$g_z$$$, $$$y$$$ and $$$z$$$ are positions in image space, $$$\Delta y$$$ and $$$\Delta z$$$ are the corresponding spatial resolutions, and $$$i_y$$$ and $$$i_z$$$ are spatial indices. The wave-encoded central k-space can be treated as an independent low-resolution k-space modulated by the same PSF with lower resolution. Thus, by replacing $$$\Delta y$$$ and $$$\Delta z$$$ with $$$\Delta y_{calib} = \frac{N_y}{N_{y, calib}}\cdot \Delta y$$$ and $$$\Delta z_{calib} = \frac{N_z}{N_{z, calib}}\cdot \Delta z$$$, where $$$N_y$$$ and $$$N_z$$$ are the acquisition matrix size, $$$N_{y, calib}$$$ and $$$N_{z, calib}$$$ are the calibration matrix size, and $$$\Delta y_{calib}$$$ and $$$\Delta z_{calib}$$$ are the spatial resolution of the calibration k-space, we can obtain a set of low-resolution PSF (Fig.1b), and subsequently reconstruct the Cartesian central k-space using inverse Fourier transform of the PSF. The resulted Cartesian center can then be used for calibration.

To test the accuracy of this approach, five phantom experiments with both fully sampled Cartesian acquisition and wave-encoded acquisition were conducted. The normalized root-mean-square-error (RMSE) between low-resolution images from the calibration data in Cartesian acquisition and those reconstructed by the proposed method were calculated as an indicator of calibration accuracy (Fig. 2). Two volunteer scans (Figs. 3 and 4) were acquired on a 3T GE MR750 scanner (GE Healthcare, Waukesha, WI) with wave encoding (3 cycles of sinusoids, 4mT/m amplitude) using a 32-channel cardiac coil (Invivo Corp., Gainesville, FL) and a 32-channel torso array coil (NeoCoil, Pewaukee, WI), respectively. In the second scan, spatial selective excitations in frequency-encoding directions were used to maintain a reasonable over-sampling factor of ~1.5. Down sampling using a VDRad trajectory³ was simulated (Fig. 3) and implemented (Fig. 4) in a 3D SPGR sequence with a 16×16 calibration region at a reduction factor of 5.1 and 6.2, respectively. Sensitivity maps were estimated using ESPIRiT⁴ from Cartesian acquisition directly and from wave-encoded acquisition with the proposed method. CS-SENSE reconstruction using the same sampling pattern was implemented for the auto-calibrating method and compared with the conventional Wave-CS method. Acquisition parameters for the 3D SPGR sequence were: TR/TE 12/2.2ms, FA 15°, and BW $$$\pm$$$125kHz. Acquisition matrices were 308(k_x)×128(k_y)×128(k_z) (with partial readout factor 0.6) for FOV 420mm (R/L)×210mm (A/P)×210mm (S/I), and 256(k_x)×256(k_y)×64(k_z) for FOV 400mm (S/I)×400mm (R/L)×256mm (A/P).

Normalized RMSE for low-resolution calibrating images decreases with increased size of calibration for the auto-calibrating method (Fig. 2). Artifacts due to motion and under-sampling occur in both reconstructions (Fig. 3). Auto-calibrating Wave-CS reduces the artifacts in both free breathing and breath-holding cases, in comparison with conventional Wave-CS approach (Figs. 3 and 4).To maintain a low RMSE of less than 2%, a calibration size of ~20×20 is necessary. Using larger calibration size would be helpful until the calibration size reaches 40×40. When the calibration size is larger than 40×40, the non-zero RMSE is mainly caused by inaccurate PSF estimation of wave encoding. In comparison with conventional Wave-CS, the proposed auto-calibration approach can reduce the motion artifacts and the aliasing artifacts due to sensitivity variations between the calibration and the wave-encoded acquisitions, as pointed by yellow arrows in Figs. 3 and 4. As the acquisition parameters were designed for post-contrast T1 weighted acquisitions, the image quality may improve in clinical contrast-enhanced scans.In this work, we evaluated the feasibility of auto-calibrating Wave-CS for potential applications with subject motion. Applying this method in dynamic contrast enhanced abdominal imaging will be our future work.