Li Feng^{1}, Qiuting Wen^{2}, Hersh Chandarana^{3}, and Ricardo Otazo^{1,4}

Subspace-constrained reconstruction is a powerful technique to accelerate dynamic MRI. However, its performance is relatively limited for applications where a robust temporal model is not available. This work proposes to estimate temporal basis from undersampled dynamic golden-angle radial data without the need of a model or additional navigators, and to apply the estimated temporal basis for subspace-constrained reconstruction of undersampled dynamic images. The reconstruction algorithm also enforces an additional low-rank constraint on the resulting low dimensional dynamic images in the subspace. The proposed self-estimated subspace-constrained reconstruction technique was demonstrated for DCE-MRI of the prostate.

**
Self-estimated subspace reconstruction of dynamic golden-angle
radial MRI:
**Golden-angle
radial sampling offers a unique sampling geometry where undersampling is much
lower around the central k-space region, as shown in Figure 1. Therefore, even
at high acceleration rates, a low-resolution dynamic image-series can still be successfully
reconstructed with conventional sparse reconstruction methods, such as the Golden-angle
RAdial Sparse Parallel (GRASP) technique [8]. This reconstructed image-series,
despite reduced spatial resolution, still provides full temporal information
from which a temporal basis set can be estimated to represent the original high spatial resolution
image-series in a low-dimensional subspace. Figure 2a shows an example of low
spatial resolution DCE-MRI of the prostate (2D+time), where each temporal frame is
reconstructed using 13 golden-angle radial spokes with the GRASP method [8]. After
converting the image-series to a Casorati matrix (reforming each image as a
column), singular value decomposition (SVD) can be performed to obtain a full
basis set. The singular values (Figure 2b) of such decomposition show that image
information is mainly restricted to the first 4 main components, which suggests
that the entire image-series can be represented in a low-dimensional space
given by the first 4 principal components (Figure 2c). This observation is
further validated in Figure 3, where an undersampled full-resolution
image-series is compressed to a low-dimensional subspace with only 4 principal
components. More importantly, images in the subspace show dramatically reduced
streaking artifacts and improved image quality compared to images in their
original space. With this pre-estimated subspace, sparse image reconstruction can be performed by solving the following cost function:

$$d=\underset{d}{\text{argmin}} \frac{1}{2}\parallel EΦd-y\parallel _2^2+R(d) $$

where *y* is the sorted dynamic radial k-space, *E* is an encoding operator incorporating
coil sensitivities, Φ is the temporal basis of the subspace and *d* is the coefficients to represent the
image-series under the basis set Φ. An additional regularization function *R* can be applied to the coefficients *d* to further promote sparsity in the
subspace.

**Evaluation of reconstruction:**
The proposed method was tested in one prostate DCE-MRI dataset from
a prior study [9]. Relevant imaging parameters included: TR/TE = 4.12/1.96 ms,
matrix size=224x224x26, FOV=240x240 mm^{2}, voxel size=1.07x1.07x3.0 mm^{3}.
1755 spokes were acquired for each partition, with a scan time of 221 seconds. Subspace-constrained image reconstruction was performed after grouping every 13 consecutive spokes into
one temporal frame, with a temporal resolution of ~1.6 seconds/volume. The
temporal basis set was estimated from a low-resolution image-series reconstructed
using GRASP with the same temporal resolution but with a reduced image matrix
size of 48x48. The first 4 basis components were used to impose subspace
constraint with an additional locally low-rank constraint with a block size of 16x16
in the subspace [4,10,11]. This method, referred to as Subspace-GRASP, was
compared with standard GRASP reconstruction with a temporal total variation
constraint.

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