Eduardo Coello^{1,2}, Ralph Noeske^{3}, Bjoern Menze^{1}, Axel Haase^{1}, and Rolf Schulte^{2}

This work
proposes an overdiscrete reconstruction for Single Voxel Spectroscopy (SVS). It
is demonstrated that in single voxel acquisitions benefit from the SNR and
linewidth improvement obtained by correcting for of B_{0}
inhomogeneities and the optimization of the Spatial Response Function (SRF), as
compared to regular signal averaging. This method, enables SV acquisitions in
challenging brain areas, i.e. where B_{0} shimming is sub-optimal, and
corrects for spectral artifacts such as peak aliasing.

The
overdiscrete reconstruction problem consist in finding the reconstruction
matrix F and apply it to each temporal point of the acquired signal, where $$F^{'} = TE^{H} (EE^{H} + \alpha \Psi)^{+},$$ where $$$T$$$ represents the target SRF matrix and $$$E$$$ the
encoding matrix, both expressed at $$$\zeta^{2}$$$-fold of the nominal resolution, $$$\Psi$$$ represents the noise covariance matrix, $$$\alpha$$$ is the regularization parameter to control the noise optimization and $$$^{+}$$$ denotes the pseudo-inverse. Correction for signal dephasing due to B_{0}^{ }inhomogeneities
is applied in an intermediate step by applying the phase correction factor
$$$e^{-i2 \pi\cdot t \cdot \triangle f_{0}(\mathbf r)}$$$, where $$$\triangle f_{0}(\mathbf r)$$$ corresponds to the local frequency shift in Hz at position $$$\mathbf r$$$. Finally,
coherent spatial averaging is performed while optimizing the Spatial Response
Function (SRF). This method has been evaluated for SVS through numerical
simulations and in-vivo experiments.

**Data acquisition:** SVS measurements of the brain of a healthy volunteer where
acquired using a 3T-MR750w (GE Healthcare, Milwaukee, WI) with a 12-channel
head receive coil. Three voxels were acquired: (SV1) Occipital lobe with
size=30x30x30mm^{3}, (SV2) Occipital lobe with size=20x20x20mm^{3}
and (SV3) Right-Frontal lobe with size=30x30x30mm^{3}. The acquisition
parameters were: TE/TR=35/1000ms, spectral bandwidth=2kHz. PRESS volume localization
was used to acquire a standard 64-average SVS and an MRSI grid with 8 phase
encodings in the SI and AP directions without changing the PRESS voxel nominal
resolution. Additionally, high-resolution B_{0}-field map and coil
sensitivity maps were also obtained.

**Data Processing:** The MRSI data was reconstructed with an overdiscrete
factor of $$$\zeta$$$=5 and Gaussian target function with $$$\sigma$$$=3 sub-voxels for
SRF optimization. Finally, standard SVS spectra were coherently averaged and reconstructed
using standard Fourier transform. Local B_{0}-field maps and coil sensitivity profiles
were extracted using the voxel location and size.

[1] Hurd R. Volume Spectroscopy Having Image Artifact Reduction. US Patent No. 5,804,966.

[2] Pruessmann K and Tsao J. Magnetic Resonance Imaging Method. US Patent No. 7,342,39.

[3] Kirchner et al. Reduction of voxel bleeding in highly accelerated
parallel ^{1}H MRSI by direct control of the spatial response function. Magn Reson
Med 2014.

[4]
Kirchner et al. Mechanisms of SNR and Line Shape Improvement by B_{0}
Correction in Overdiscrete MRSI Reconstruction. Magn Reson Med 2015.

Fig.
1. (A) Simulated metabolite peaks (green) and the result of a standard (blue)
and overdiscrete (red) reconstructions. (B) SNR value of the central metabolite
peak showing the potential 2-3-fold increase in SNR with the proposed
reconstruction. (C) The Full Width at Half Maximum (FWHM) representing the line
shape of the metabolite peak is kept unaffected after reconstruction.

Fig. 2. Spatial
response functions obtained for (A) PRESS voxel localization with the additional Chemical Shift Displacement to be considered, (B) phase encoded
MRSI with a matrix size of 8x8 keeping the voxel nominal size as in A and (C) optimized
Gaussian SRF achieved through the overdiscrete reconstruction.

Fig.
3. B_{0}-field map showing the
voxel location of (A) SV1, (B) SV2 and (C) SV3.

Fig.
4. Acquired SVS datasets of the brain in the occipital lobe with a voxel size
of (A) 30x30x30mm^{3}, (B) 15x15x15mm^{3} and (C) right-frontal lobe with a voxel of
30x30x30mm^{3}. Each dataset was reconstructed with standard SV (blue) and overdiscrete
MRSI (red). Improved in signal levels are observed specially for large voxel
sizes where B_{0}-field variations are larger.