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.

Image results of a single time frame of the dynamic prostate exam are shown in Figure 2. Note that without view sharing, the spatial resolution suffers, but with SHVS or LRMC, the spatial resolution is recovered. Profiles across features in the images are shown in Figure 3, and show that LRMC has comparable spatial resolution to the SHVS. Figure 4 shows time courses for 3 different voxels depicted in the image at the bottom. Note that the temporal resolution of the SHVS is reduced, but the LRMC time course has high temporal resolution like the non-view-shared case.LRMC can be performed as a pre-processing step in reconstructing multi-level undersampled dynamic MRI data to retain the temporal resolution of the acquired data and recover the spatial resolution similar to the traditional SHVS technique.