Andres Saucedo^{1}, Zohaib Iqbal^{1}, Manoj K. Sarma^{1}, and M. Albert Thomas^{1}

Magnetic Resonance Spectroscopic Imaging (MRSI) is a valuable tool to characterize metabolic concentrations and changes in several spatial locations in a single recording. However, the long acquisition time of conventional three-dimensional (3D) MRSI limits its practical application. Non-uniformly sampled 3D echo planar spectroscopic imaging (EPSI) has been proposed to accelerate the scan time, combined with compressed sensing (CS) to retain reconstruction fidelity. We apply the novel approach of reconstructing 3D EPSI data by applying TV, Wavelet-$$$\ell_{1}$$$, and TV + Wavelet-$$$\ell_{1}$$$ CS-based regularization on both the combined spectral and two undersampled spatial dimensions. These three reconstruction methods were evaluated in both simulated and in retrospectively undersampled data of a brain phantom.

CS
reconstruction was performed using the Split-Bregman^{7} algorithm with three
different types of sparsifying transforms applied on the y-z-f space for each point along the readout direction x, namely TV-, Wavelet-$$$\ell_{1}$$$-, and TV+Wavelet-$$$\ell_{1}$$$-based regularization. In addition to spatial sparsity, this approach
exploits the increased sparsity available in the spectral domain with respect
to the TV and/or wavelet transforms. The minimization is cast as $$ \min_{u} \mu\frac{1}{2}∥Fu−y∥^{2}_{2}+λTV(u)+γ∥Wu∥_{1} \quad \text{subject to} \quad ∥Fu−y∥^{2}_{2}<σ^{2}$$ where $$$ F $$$ is the undersampled Fourier transform, $$$ u $$$ is the desired data in y-z-f space, $$$ y $$$ is the acquired data in ky-kz-t, W is the wavelet transform (Daubechies 4, level 5), and $$$ \sigma $$$ is an estimate of the noise. The parameters $$$ \lambda $$$ and $$$ \gamma $$$ are set to zero to remove the $$$ TV $$$ or $$$ \|W \cdot \|_{1} $$$ regularization terms, respectively. Each reconstruction was run for 300 iterations or until the stopping criterion was satisfied. The regularization parameters were scaled according to the undersampled data, with the ratios $$$\mu:\lambda:\gamma \approx 1:0.01:0.01. $$$

A 3D EPSI sequence was used to acquire a fully sampled brain phantom data set on a 3T scanner with the following parameters: FOV = 240×240×120 mm^{3} , matrix = 32×32×8, spectral bandwidth = 1190 Hz, number of spectral points = 256, TE = 41ms, TR = 1.5s, number of averages = 8. This data was retrospectively undersampled along the spatial ky-kz dimensions at acceleration factors of 2x, 3x, 4x, and 5x. A 3D virtual phantom was created using a 32×32×8 Shepp-Logan spatial profile, in which each voxel contained a 256-length FID generated using the GAMMA library^{8}. Simulated metabolites included NAA, Cr3.0, Ch, Glx, mI, and Cr3.9. This virtual data was undersampled at 2x, 3x, 4x, and 5x.

Reconstruction performance was quantified using the normalized root-mean-square error (nRMSE) of metabolite concentrations and ratios, and was qualitatively evaluated with comparisons of spatial maps and spectra.

This study evaluates the novel approach of reconstructing 3D EPSI data by applying the TV and wavelet transforms on the three-dimension spectral-spatial set y-z-f, in which y-z are the undersampled dimensions. Results from the reconstructed spectra, metabolite ratios, and spatial metabolite maps generally indicate that Wavelet-$$$\ell_{1}$$$ regularization performs less optimally than regularization involving the TV term. As seen in Figures 3 and 4, the combination of TV + Wavelet-$$$\ell_{1}$$$ leads to lower nRMSE values in comparison to TV alone, while Wavelet-$$$ \ell_{1} $$$ regularization retains higher nRMSE's. The spectral quality and accuracy of metabolite maps are consistent with these trends. Both TV and TV + Wavelet-$$$\ell_{1}$$$ methods produce similar results, though the latter is more robust to higher acceleration factors, as indicated in Figures 3 and 4.

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Figure 1: Spatial NAA metabolite maps from TV, Wavelet-$$$\ell_{1}$$$, and TV+Wavelet-$$$\ell_{1}$$$ retrospective reconstructions (2x, 3x and 4x acceleration) of a brain phantom consisting of several metabolites at physiological concentrations. The three central slices with the most signal are shown.

Figure 2: nRMSE maps corresponding to the NAA metabolite maps of Figure 1. Results are shown for TV, Wavelet-$$$\ell_{1}$$$, and TV+Wavelet-$$$\ell_{1}$$$ retrospective CS reconstructions at 2x, 3x, and 4x acceleration. The nRMSE maps are masked to the volume-of-interest.

Figure 3: (A) Spectra obtained from TV, Wavelet-$$$\ell_{1}$$$, and TV+Wavelet-$$$\ell_{1}$$$ CS reconstructions of virtual phantom at 2x, 3x, and 4x acceleration. (B) Table showing NAA/Cr3.0, Cr3.9/Cr3.0, and Ch/Cr3.0 metabolite ratios from reconstructed virtual phantom at 2x, 3x, 4x, and 5x acceleration.

Figure 4: (A) Spectra obtained from TV, Wavelet-$$$\ell_{1}$$$, and TV+Wavelet-$$$\ell_{1}$$$ retrospective CS reconstructions of a brain phantom at 2x, 3x, and 4x acceleration. (B) Table showing NAA/Cr3.0, Cr3.9/Cr3.0, and Ch/Cr3.0 metabolite ratios from reconstructions of a brain phantom at 2x, 3x, 4x, and 5x acceleration.