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Fast $$$T_{2}^{*}$$$ and QSM mapping using temporal CS with a Complementary Stochastic Sampling Scheme
Charles Iglehart1, Ali Bilgin1, and Manojkumar Saranathan2
1Electrical and Computer Engineering, University of Arizona, Tucson, AZ, United States, 2Department of Medical Imaging, University of Arizona, Tucson, AZ, United States

Synopsis

We develop the motivation for and demonstrate the functionality of a temporal model-based complex multi-echo reconstruction algorithm coupled with a Complementary Stochastic Sampling procedure allowing for temporal sparsity to be leveraged . We demonstrate results for magnitude and phase images as well as parameter maps.

Introduction

$$$T_{2}^{*}*$$$ and quantitative susceptibility mapping (QSM) are increasingly used clinically for investigation of iron deposition in Parkinson's disease (1,2), hemorrhage detection (3,4), and as potential age-related biomarkers of neurodegeneration (5-7). Parametric mapping requires 3D multi-echo gradient-recalled echo (ME GRE) MRI acquisitions, which prolong long scan times at high spatial resolution (1 mm or higher). Thus, scan time reduction using k-space undersampling with high fidelity magnitude and phase reconstruction is critical. Some temporal subspace-based compressed sensing (CS) algorithms have been developed (8,9) using simulated signal evolutions to recover $$$T_{2}$$$ maps from undersampled k-space data. These are not easily translatable to complex-valued reconstruction due to scaling issues. Ong et al (10) provide a general framework for simultaneous magnitude and phase reconstruction from partial Fourier acquisitions, but only for single-echo reconstruction. Huang et al (11) developed a model-based method for simultaneous, complex-valued reconstruction from undersampled k-space data but do not show results for phase-echo images or QSM . Additionally, these CS schemes apply the same (ky,kz) sampling pattern at each echo, eliminating potential gains from temporal sparsity. In this work, we propose a Complementary Stochastic Sampling (‘COSTS’, 12) strategy with a novel interleaved, complex-valued reconstruction using separate subspace-based temporal signal models for magnitude and phase. We demonstrate results for complex multi-echo reconstructions as well as derived parameter maps from in vivo data.

Methods

Figure 1 shows the proposed complex, multi-echo reconstruction using an iterative optimization that interleaves Alternating Direction Method of Multipliers (‘ADMM’, 13)-based magnitude and phase updates. Both magnitude and phase objectives are formulated using temporal model coefficients to promote sparsity. To model the temporal behavior of the magnitude images, a subspace basis $$$\Phi$$$ is computed via singular value decomposition of an ensemble of 50,000 simulated monoexponential signal evolutions with $$$T_{2}^{*}$$$ randomly selected from [0.001s,1.0s] (Figure 1A). A two-parameter linear fit is used to model the temporal behavior of unwrapped phase for voxel across the echo dimension, (Figure 1B). K-space undersampling is performed via COSTS (Figure 1C). Sampling is restricted to an annular region in order to maintain spatial resolution while avoiding corners in (ky,kz). Within the annular region, each (ky,kz) point is sampled at exactly one echo, ensuring temporal as well as spatial incoherence. The central region of k-space is fully sampled. The magnitude objective function is constructed as shown in Figure 1(D): $$$\alpha$$$ is the stack of PCC maps; $$$\Phi_{K}^{H}$$$ is the truncated temporal subspace basis; $$$S$$$ is an ESPIRIT (14) sensitivity encoding operator; $$$F$$$ is the forward Fourier operator; $$$P$$$ COSTS sampling operator; y is the acquired, undersampled k-space data; $$$\lambda_{m}$$$ is a regularization parameter; $$$R_{r}(\alpha)$$$ is an operator enforcing local low-rank structure in the magnitude PCC maps. The magnitude objective is iteratively optimized in ADMM, with the resulting PCC maps projected into the time domain to obtain reconstructed multi-echo magnitude images. These magnitudes are subsequently multiplied by the initial multi-echo phase estimate obtained from the zero-filled, undersampled k-space data to seed the phase objective function. Figure 1(D) shows the phase objective: $$$\Psi$$$ projects the estimated model coefficients into the time domain; $$$F^{H}$$$ and $$$S^{H}$$$ are the adjoints of the Fourier and ESPIRIT operators; $$$\Theta$$$ is an operator that outputs the unwrapped phase of complex multi-echo images; $$$\lambda_{p}$$$ is a regularization parameter, with phase regularization performed via the $$$L_{1}$$$ norm of total variation. As with magnitude, the phase objective is iteratively optimized in ADMM. The resulting phase model coefficients are projected into the time domain, with multi-echo images used to seed subsequent iterations of magnitude reconstruction. Magnitude and phase updates are interleaved until convergence or stopping criteria have been encountered. Multi-echo magnitude, phase, $$$M_{0}$$$, and $$$R_{2}^{*}$$$ (the latter two having been computed from the reconstructed magnitude images via ordinary least squares) are evaluated against ground truth from fully sampled data. All reconstructions and analyses are performed in Matlab using a modified version of BART Toolbox (15) as a framework. Fully-sampled ME-GRE datasets from the Cornell QSM toolbox (16) were used for retrospective undersampling. Dataset 1: Siemens 3T scanner, TR/TE 5.5/5.9ms, ETL 8; Dataset 2: GE 3T scanner, TR/TE 4.9/5.2ms, ETL 9; both datasets with voxel size 0.94 x 0.94 x 2.0 mm and 15° flip angle.

Results

Figure 2-3 show reconstructed magnitude and phase for a single subject using acceleration ratios of 4 and 8, respectively. Figure 4 shows $$$M_{0}$$$ and $$$R_{2}^{*}$$$ maps for both accelerations. Figure 5 shows QSM for an ROI over a mid-axial slice.

Discussion

For R = 4 (Fig 2), both magnitude and phase reconstruction reproduce finely detailed structures in the brain. Phase errors are small, primarily concentrated in CSF. For R=8 (Fig 3), the magnitude reconstruction is still accurate, with corresponding accuracy in the parameter maps; the phase images are blurrier than in the R = 4 case. The reconstructed parameter maps show larger errors in $$$R_{2}^{*}$$$, concentrated largely in CSF, while $$$M_{0}$$$ is still high quality. The accelerated QSM (Fig 5) shows good correspondence with truth with some blurring at R=8.

Conclusion

We have demonstrated a multi-echo, complex reconstruction technique using temporal models coupled with COSTS. The method shows promising accuracy in multi-echo magnitude and phase images and $$$R_{2}^{*}$$$ maps, even at higher R values.

Acknowledgements

(None.)

References

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Figures

Figure 1. (A) Formation of magnitude temporal subspace basis from randomly generated T2* decay curves; (B) Two parameter phase temporal model; (C) Illustration of simultaneous spatial and temporal complementarity of COSTS sampling patterns; (D) Magnitude and phase objective functions used for reconstruction from undersampled k-space data

Figure 2. Magnitude and phase results for complex multi-echo reconstruction for echoes 1,4 and 8 using R = 4 COSTS undersampling.

Figure 2. Magnitude and phase results for complex multi-echo reconstruction for echoes 1,4 and 8 using R = 8 COSTS undersampling.

Figure 4. M0 and R2* results for R = 4 and 8.

QSM results for a mid-axial slice for fully sampled data vs. R = 4,8.

Proc. Intl. Soc. Mag. Reson. Med. 30 (2022)
2349
DOI: https://doi.org/10.58530/2022/2349