Daniel Grzeda^{1}, Meirav Galun^{2}, Noam Omer^{1}, Tamar Blumenfeld-Katzir^{1}, Dvir Radunsky^{1}, Ricardo Otazo^{3}, and Noam Ben-Eliezer^{1,4,5}

Quantification of T_{2} values
is valuable for a wide range of research applications and clinical
pathologies. Multi-echo spin
echo (**MESE**) protocols offer significantly shorter scan-times, at the cost of
strong contamination from stimulated and indirect echoes. The
echo-modulation-curve (**EMC**) algorithm, can efficiently overcome these
limitations to produce accurate T_{2} values. In this work we
propose a new reconstruction algorithm based on Sparsity and Fixed Rank
constraints, denoted as SPARK.
We compare our
method against GRAPPA and show its superiority in the quantitative evaluation
of T_{2} values from highly undersampled data.

Quantification of T_{2} values
is valuable for a wide range of research applications and clinical pathologies^{1–3}. The use of single spin-echo (**SSE**)
protocol is unpractical due to its extensive scan time. Multi-echo spin echo (**MESE**)
protocols offer significantly shorter scan-times and lower diffusion effects, at the cost of strong contamination from stimulated
and indirect echoes^{4}. The echo-modulation-curve (**EMC**)
algorithm, can efficiently overcome these limitations to produce accurate T_{2}
values, which are moreover reproducible across scanners and scan settings^{5}.

Further acceleration of
quantitative mapping can be achieved by undersampling the spatial or temporal domains,
requiring designated reconstruction algorithms, and involving prior information
on the signal model^{6–8}.

Low-rank plus Sparse (L+S) signal-decomposition
was recently introduced as a powerful tool for reconstructing undersampled
dynamic MR images and has been demonstrated to enhance standard compressed
sensing^{6}. It was shown that enforcing a
fixed rank constraint along with sparsity constraints may outperform L+S, fixed
rank only, or sparse only methods^{8}.

In this work, the EMC model provides
us the signal’s rank, which is necessary to perform an iterative reconstruction
of highly undersampled MESE data using sparse and fixed low-rank constraints to
achieve highly accelerated mapping of quantitative T_{2} values. We
denote the suggested approach as SPARK (SPArsity and fixed RanK).

Methods

**MRI scans:** Two datasets
were scanned using a standard MESE sequence with fully sampled k-space.
Brain imaging was done using a 16-channel receiver coil and calf scans were
obtained using a flexible 4-channel receiver coil and 4 additional coils embedded
in the scanner bed. Scan parameters for both datasets were: NEchoes=30;
TE/TR=10/3000 ms, slice thickness=3 mm; in-plane resolution=1.1x1.1 mm^{2}
(brain), and 1.3x1.3 mm^{2} (calf). Only the first 20 echoes contained
significant information and used during reconstruction.

**Postprocessing:** Images
were reconstructed from retrospectively under sampled k-spaces using our
proposed method and standard GRAPPA. Reconstruction was performed by solving
the following optimization problem: $$$[L,S]=argmin_{L\in C, S}\frac{1}{2}||d-E(L+S) ||^2_2 + \lambda||S||_1$$$, where E is the acquisition operator, d is the under-sampled k-t
data and
is
the set of matrices of fixed rank r^{5}.
The Identity transform was used to enforce sparsity in the image domain of S. Fixed rank
value was estimated from the simulated EMC data prior to reconstruction, and variable
density schemes were used following the Compressed Sensing (CS) framework^{3,4,6}.

Following the iterative reconstruction, T_{2} maps were generated on a pixel-by-pixel basis using the EMC
algorithm.**Analysis:** Mean ±
standard deviation of T_{2} values were calculated for regions of interest
within each anatomy, and relative errors were calculated with respect to the fully
sampled results.

T_{2} values in the brain were
estimated for acceleration factors 2...5 in the putamen, head of caudate
nucleus, thalamus and white matter fascicles, with reference T_{2} values
(mean ±
SD) of 62.48±2.7,
66.5±3.0,
66.17±3.4
and 56.28±2.0
ms, respectively. The SD of SPARK’s evaluated T_{2}
values are similar to those of the fully sampled data (3.4, 2.4, 3.3 and 2.1%
at acceleration x5), ensuring high precision at
all acceleration factors and outperforming the conventional acceleration method,
GRAPPA, whose precision decreases from acceleration factor 3 and above, and is an
order of magnitude larger at factor x5 (24.4, 23.4, 23.8 and 17.8% at
acceleration x5). Mean relative errors at
acceleration factor x5 at the mentioned regions in the brain GRAPPA and
SPARK reconstruction are 32.1, 28.7, 26.4, 25.3% and 3.9, 3.4, 4.3, 3.1%, respectively.

T_{2} values in the calf were
estimated for the same acceleration factors in the tibialis anterior, gastrocnemius,
soleus, peronari longus and flexor longus muscles, with reference T_{2}
values (mean ± SD) of 30.6±2.9, 32.6±7.9, 35.0±6.3,
41.6±6.6
and 32.3±3.3
ms, respectively. The mean relative errors at
acceleration factor x5 at the mentioned regions in the calf muscle of GRAPPA
and SPARK are 18.4, 62.8, 50.6, 39.5, 49.1% and 3.0, 3.8, 5.3, 3.1, 4.4%,
respectively.

Our results
demonstrate that the combination of EMC and SPARK reconstruction achieves high
accuracy and precision in T_{2} quantification across different anatomies and
for a range of acceleration factors. The SPARK
algorithm’s performance on undersampled data is in good agreement with the
standard reconstruction from fully sampled data, provided we perform tuning of
the following parameters: λ for sparsity, c for singular value thresholding and
r for rank truncation. We have found that
accurate results can be achieved for fixed ranks of 5-10, hence being less
sensitive to the exact rank of the data. Future work on
SPARK aims to improve its robustness to the selection of tuning parameters.

ISF 2009/17; NIH NIBIB Biomedical Technology Resource Center (P41EB017183)

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