High spatiotemporal resolution dynamic contrast-enhanced MRI (DCE-MRI) is crucial for the accurate diagnosis of liver lesions. Previously, the TRACER¹ method used a nonlinear parallel imaging reconstruction of golden ratio variable density spiral data to achieve high spatial resolution and large coverage with a sub-second frame rate. PROUD² extended this work by using 2D patch based local similarity constraints and temporal regularization, leading to high overall CNR. However, because of their reliance on the similarity between successive frames, these methods are sensitive to large respiratory motion. To reduce motion blurring artifacts and retain high temporal frame rate, we developed a respiratory Motion corrected 3D Patch based Reconstruction of Under-sampled Data (M-PROUD).

In M-PROUD, the reconstruction problem was formulated as: $${\bf v}_{t=1,\ldots,T}^*=\substack{\tt\large argmin\\{\bf v}_t,{\bf\alpha}_{x,y,z,t},{\bf M}_t^{lt},{\bf M}_t^{pst},{\bf M}_t^{nst}\\t=1,\ldots,T}\left\{\sum_{t=1}^T\|{\bf U}_t{\bf FSv}_t-{\bf y}_t\|_2^2+\lambda_1\sum_{t=1}^T\sum_{x,y,z}\|{\bf R}_{x,y,z}{\bf v}_t-{\bf\alpha}_{x,y,z,t}{\bf P}_{x,y,z}{\bf D}_t\left({\bf M}_t^{lt}{\bf v}_r,{\bf M}_t^{pst}{\bf v}_{t-1},{\bf v}_t,{\bf M}_t^{nst}{\bf v}_{t+1}\right)\|_2^2\\+\lambda_2\sum_{t=1}^T\|{\bf v}_t-\left({\bf M}_t^{pst}{\bf v}_{t-1}+{\bf M}_t^{nst}{\bf v}_{t+1}\right)/2\|_2^2\right\},\,\tt{s.t.}\|{\bf \alpha}_{x,y,z,t}\|_0=1,\,\forall x,y,z\;\;\tt for\,t=1,\ldots,T$$ where $$${\bf v}_t$$$ was the image volume at time $$$t$$$, $$${\bf S}$$$ the multiplication with the coil sensitivity maps, $$${\bf F}$$$ the Fourier transform, $$${\bf U}_t$$$ the undersampling operator, $$${\bf y}_t $$$ the k-space data, $$${\bf D}_t$$$ a set of local dictionaries at time $$$t$$$, $$${\bf R}_{x,y,z}$$$ an operator which extracted a 3D patch located at $$$\left(x,y,z\right)$$$, $$${\bf P}_{x,y,z}$$$ an operator which extracted the local dictionary for this patch, $$${\bf \alpha}_{x,y,z,t}$$$ its dictionary coefficient vector, and $$$\lambda_1$$$ and $$$\lambda_2$$$ regularization parameters automatically obtained, as in PROUD². $$${\bf v}_r$$$ was a composite volume reconstructed from all acquired data. Compared to PROUD, this method used 3D instead of 2D patches and introduces three motion operators: $$${\bf M}_t^{lt},{\bf M}_t^{pst},$$$ and $$${\bf M}_t^{nst}$$$, which described the motion of $$${\bf v}_r, {\bf v}_{t-1},$$$ and $$${\bf v}_{t+1}$$$ relative to $$${\bf v}_t$$$ respectively. The solver for M-PROUD first assumed $$$\lambda_2=0$$$. Alternating optimizations were performed on two sub-problems: 1) optimize $$${\bf v}_t$$$ with fixed $$${\bf M}_t^{pst}$$$, and 2) optimize $$${\bf M}_t^{pst}$$$ with fixed $$${\bf v}_t$$$. The motion transformations were estimated with a gradient descent optimization scheme³ using residual complexity⁴ as the similarity measure, known to be robust against the contrast changes in DCE-MRI. In a second round, the solver included temporal regularization and alternate optimizations were again performed on the two resulting sub-problems.

Multiphase spiral LAVA in vivo data were acquired at 1.5T using 48 golden angle variable density spiral leaves per fully sampled volume, TR=6.1 ms, voxel size 1.4x1.4x5 mm, matrix size 256x256x44, 8-channel cardiac coil, and gadoxetate disodium injection. A total of 6 volumes (288 spiral leaves) were continuously acquired with a multiple breath-holds protocol, in which the patient was instructed to hold the breath as long as possible. A new breath-hold instruction was delivered each time the patient was observed to resume breathing but scanning was never interrupted. We used PROUD and M-PROUD to reconstruct the image sequence at a temporal frame rate of one spiral leaf, approximately 268 ms, resulting in 288 frames. Two periods with significant breathing acquired in between the breath-holds were selected for reconstruction.

Reconstruction was successfully performed of the images throughout the acquisition. The reconstructed images of a single slice in the middle the first breathing phase between two breath-holds are shown in Fig. 1. The zoomed-in regions indicate that the proposed M-PROUD significantly reduced motion blurring artifacts and improved visualization of the liver lesion near the anterior wall. Fig. 2 shows the temporal profiles during the two phases of significant breathing in a single cross section indicated by the dashed line. These profiles show that the proposed method captured the temporal variations more clearly than PROUD.

The preliminary data presented here show the feasibility of acquiring high frame rate (<1s) 4D liver imaging in the presence of substantial breathing motion by incorporating 3D motion estimation in the reconstruction algorithm. The resulting M-PROUD method significantly improved image quality, allowing a clear visualization of the liver lesion data throughout breathing. The method was designed to deal with respiratory motion larger than typically seen in free-breathing acquisition. This is necessary for the flexible imaging protocol used in this work. Despite the large respiratory motion in the resulting in- and exhalation phases, image quality was largely recovered using the proposed method.

Recently, XD-GRASP⁵ demonstrated promising performance for free breathing liver 4D DCE-MRI with 11-12 seconds frame rate. Our M-PROUD method further improved liver 4D DCE-MRI to a sub-second frame rate with high image quality.Respiratory Motion corrected Patch based Reconstruction of Under-sampled Data (M-PROUD) can preserve both spatial structures and temporal variations at a sub-second temporal frame rate in liver 4D DCE-MRI.