Synopsis

Quantification of T₂ 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 T₂ 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 T₂ values from highly undersampled data.

Introduction

Quantification of T₂ 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 echoes⁴. The echo-modulation-curve (EMC) algorithm, can efficiently overcome these limitations to produce accurate T₂ values, which are moreover reproducible across scanners and scan settings⁵.

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 sensing⁶. It was shown that enforcing a fixed rank constraint along with sparsity constraints may outperform L+S, fixed rank only, or sparse only methods⁸.

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 T₂ 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 mm² (brain), and 1.3x1.3 mm² (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 r⁵. 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, T₂ maps were generated on a pixel-by-pixel basis using the EMC algorithm.
Analysis: Mean ± standard deviation of T₂ values were calculated for regions of interest within each anatomy, and relative errors were calculated with respect to the fully sampled results.

Results

T₂ 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 T₂ 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 T₂ 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.

T₂ 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 T₂ 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.

Discussion

Our results demonstrate that the combination of EMC and SPARK reconstruction achieves high accuracy and precision in T₂ 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.

Acknowledgements

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

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