Myocardial stiffness is an important determinant of cardiac function. Cardiac magnetic resonance elastography (MRE) is a noninvasive technique to estimate stiffness of the myocardium.² The acquisition time for cardiac MRE can be prohibitively long for routine clinical use. We propose a compressive recovery method, called composite regularization with constant magnitude (CRCM) to accelerate cardiac MRE by exploiting structure that is unique to MRE.

Theory: Compressive sensing (CS) inspired methods have been extensively applied to accelerate MRI. Most CS-based techniques used in MRI exploit sparsity in a single sparse representation. However, recent evidence suggests that multiple sparse representations are better suited to express and exploit rich structure in MR images. When using multiple sparse representations, the level of sparsity may vary across different representations. Therefore, adapting the thresholding rule (regularization strength) for each representation may yield better results.³ For MRE, in addition, the images collected at different offsets (phase difference between the externally applied mechanical vibration and motion encoding gradient (MEG)) should have the same magnitude. In CRCM, we utilize both composite regularization that adapts to the inherent level of sparsity in each representation and a constraint that enforces magnitude consistency across different offsets. CRCM can be expressed as $$\hat{x} = \underset{x} {\mathrm{argmin}}\enspace \|y-\Phi x\|_2^2 + \sum_{i=1}^{D} \lambda_i\|\Psi_ix\|_1^1\enspace \text{s.t.}\enspace |x_j|=\bar{|x|} \enspace \forall \enspace j,$$ where $$$x$$$ is the complex-valued image, $$$y$$$ is the measured noisy data, $$$\Phi$$$ is the measurement matrix, $$$\Psi_i x$$$ is the i^th sparse representation of $$$x$$$, $$$D$$$ is the total number of sparse representations, $$$x_j$$$ is the image corresponding to the j^th phase offset, and $$$\bar{|x|}$$$ is the magnitude image averaged across different offsets. The $$$\lambda_i$$$ values were iteratively adapted as described by Ahmad et al.³ For a given set of $$$\lambda_i$$$, $$$x$$$ was iteratively solved using alternating minimization.

Phantom imaging: A cylindrical phantom with uniform, known (5.1 kPa) stiffness was imaged on 3T scanner (Tim Trio, Siemens Healthcare, Erlangen) using GRE MRE sequence. A fully sampled dataset was collected from a 5 mm thick slice. Other imaging parameters included: matrix size 128x128, FOV 280x280 mm², excitation frequency 60 Hz, motion encoding gradient (MEG) frequency 60 Hz, number of offsets 8 (4 with positive MEG and 4 with negative MEG). The collected data were retrospectively downsampled at R = 2, 3, 4,…,10 using VISTA.⁴ CRCM was applied to recover 4D image (128x128x4x2) from undersampled k-space data. Non-decimated 4D wavelet transform was used to create 16 sub-bands, with each sub-band treated as a separate sparse representation. After image reconstruction using CRCM, the resulting first harmonic displacement images were converted to a stiffness map using local frequency estimation algorithm implemented in MRELab (Mayo Clinic, Rochester, MN). Central 64 k-space lines from one MEG value were used to compute kernels for GRAPPA and sensitivity maps for SENSE-based CRCM.

In vivo imaging: A healthy volunteer was imaged on a 1.5T scanner (Aera, Siemens Healthcare, Erlangen) using a prospective, cine, cardiac gated GRE MRE sequence. Three fully sampled datasets were collected in a short-axis view of the heart in three separate breath-holds, each with a spatially different MEG direction. Other imaging parameters included: matrix size 128x128, FOV 380x380 mm², excitation frequency 80 Hz, MEG Frequency 160 Hz, number of offsets 8 (4 with positive MEG and 4 with negative MEG), number of cardiac phases 4. The collected data were retrospectively downsampled at R = 2, 3, 4, 5, 7, and 10 using VISTA. To reduce computation time, each cardiac phase and encoding direction was reconstructed separately. CRCM was applied to recover 4D image (128x128x4x2) from undersampled k-space data. The rest of the image recovery details were similar to the ones used for phantom imaging.

Figures 1 and 2 show phantom imaging results. Figure 1 shows root mean square error (RMSE) and mean stiffness (averaged over pixels) as a function of acceleration rate for GRAPPA and CMCR. The error bars indicate ($$$\pm \frac{1}{2}$$$std.) stiffness variations within the image. Figure 2 shows the stiffness maps for different acceleration rates. Figures 3 and 4 show similar results for the in vivo data.Compared to GRAPPA, CRCM allows higher acceleration rates while maintaining the quantitative accuracy of MRE-based stiffness measurements. Since cardiac MRE is performed in a hybrid domain (space, time, offset, etc), it can specifically benefit from CRCM, which exploits both multiple sparse representations and disparity in the level of sparsity across the representations. Additionally, CRCM enforces magnitude consistency that is unique to MRE data. Future work will include joint processing of all cardiac frames and encoding directions and comparison to other CS methods.