Accelerated Dynamic MRI Reconstruction with Sequential Low Rank Matrix Completion and Parallel Imaging

Eric G Stinson^{1}, Stephen J. Riederer^{1}, and Joshua D. Trzasko^{1}

Dynamic MR imaging is useful for a number of applications, including cardiac imaging, contrast-enhanced MR angiography, and perfusion imaging with Dynamic Contrast Enhanced MRI (DCE-MRI) or Dynamic Susceptibility Enhanced MRI (DSC-MRI). To obtain the necessary high spatial and temporal resolution undersampling is typically used. When multi-level sampling of uniform (traditional SENSE(1)) and non-uniform undersampling are combined, an image update time of $$$T_{update}<5s$$$ can be achieved while retaining spatial resolution.

Methods to account for the non-uniform sampling include "sample-and-hold view sharing" (SHVS) (2), where samples from neighboring time frames are shared to enhance image quality and spatial resolution (by increasing the coverage of k-space) at the expense of temporal resolution, or using the raw data to retain temporal resolution at the expense of spatial resolution. Other techniques use a sparse penalty to reconstruct undersampled images before a model-based or data driven reconstruction (3-4). Alternatively, a full regression-type reconstruction (5) can be performed to account for multi-level sampling at the expense of reconstruction time.

Here, we propose to use low-rank matrix completion (LRMC) (6) as a pre-processing step to fill undersampled accelerated k-space and retain both spatial and temporal resolution while taking advantage of the ease of implementation of this modular approach.

The full signal model for a time-resolved, multi-coil acquisition with multi-level sampling is shown in Equation 1.

$$ G \delta_t = \left( I \otimes \Phi_t F \right) S M X \delta_t + Z \delta_t \hspace{1cm} \text{(Eq. 1)}$$

where $$$\delta_t$$$ is Kronecker's delta, $$$G \delta_t$$$ is the received multi-coil signal at time $$$t$$$, $$$I$$$ is an identity matrix, $$$\Phi_t$$$ is the non-uniform sampling operator at time $$$t$$$, $$$F$$$ is the uniform (SENSE-type) sampling operator, $$$S$$$ contains the coil sensitivities, $$$M$$$ is the spatial support mask, $$$X \delta_t$$$ is the true image at time $$$t$$$, and $$$Z$$$ is zero-mean, Gaussian noise. Equation 1 can be used in solving the full regression problem efficiently (5), but a more efficient method may be to pre-process the data in the uniformly undersampled space before performing standard SENSE unfolding, under the assumption that a priori knowledge of the final image also informs the folded image and/or the undersampled k-space. The signal model for the preprocessing step in k-space is then

$$ G \delta_t = \Phi_t Y \delta_t + Z \delta_t \hspace{1cm} \text{(Eq. 2)}$$

where $$$Y \delta_t$$$ is now the multi-coil (only) uniformly undersampled k-space signal. Solving for $$$Y$$$ is the pre-processing step, followed by standard SENSE reconstruction. A schematic of this reconstruction procedure (with the traditional routes) is shown in Figure 1. Here a low-rank penalty is used as uniform undersampling in k-space, which results in object folding in image space, does not change the low-rank nature of the dynamic time series. Thus, the regression is of the form:

$$ \hat{Y} = \arg\min_{Y} \left\{ \| G - \Phi Y \|^{2}_{F} + \lambda Rank(Y) \right\} \hspace{1cm} \text{(Eq. 3)}$$

Cartesian k-space data was acquired with a variant of the CAPR (5,7) method optimized for DCE-MRI prostate imaging (pCAPR) where the uniform sampling operator is a SENSE-type operator, and the non-uniform sampling consists of 3 pseudorandom sets that alternate in a periodic fashion through time. These studies were performed at 3T (GE, Waukesha, WI) with a 32 channel phased array and scan parameters in Table 1.

LRMC was performed on the Casorati form of G in an iterative thresholding manner where at each iteration: 1) singular value hard thresholding (8-9), and 2) data fidelity enforcement via data replacement. In the dataset shown here, the visually optimized threshold value was equivalent to Rank = 2. The algorithm was implemented in MATLAB 2015b and used 50 iterations. Standard SENSE reconstruction (1,7) was performed on the acquired (non-view shared), LRMC, and SHVS data to obtain unfolded images of the prostate.

Funded by: NIH R21EB017840, NIH EB000212, NIH RR018898, DOD CDMRP W81XWH

1. Pruessmann KP et al. MRM 1999;42:952–962.

2. Riederer SJ et al. MRM 1988;8:1–15.

3. Liang D et al. MRM 2009;62:1574–1584.

4. King K et al. ISMRM Stockholm, Sweden, 2010. #4881.

5. Trzasko JD et al. ISMRM Toronto, Canada, 2015. #0574.

6. Cai J et al. SIAM J. Optim. 2010;20:1956–1982.

7. Haider CR et al. MRM 2008;60:749–760.

8. Jain P et al. Advances in Neural Information Processing Systems 23. 2010. pp. 937–945.

9. Recht B et al. SIAM Rev. 2010;52:471–501.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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