Amir Seginer^{1} and Assaf Tal^{1}

We compare, using numeric simulations, the point spread functions (PSF) and the SNR of different trajectories in k-t space for magnetic resonance spectral imaging (MRSI). This is a first step towards evaluating the trajectory of choice while balancing SNR efficiency, scan time, and localization of signal (resolution vs. bleed).

Different trajectories in k-t space have been proposed over the years for magnetic resonance spectral imaging (MRSI). In choosing a trajectory, three points are of importance: the trajectory efficiency, i.e., signal-to-noise ratio (SNR) per square root of time ($$$\mathrm{SNR}/\sqrt{t}$$$); the shape of the point-spread-function (PSF), i.e., the spectrum localization and signal bleed; and the minimum needed time of acquiring a complete data set.

Three families of trajectories were simulated: Echo Planar Spectroscopic Imaging (EPSI), Rosettes,^{1} and a concentric rings trajectory (CRT).^{2} These are compared to the Chemical Shift Imaging (CSI) method.

Trajectories where designed in Matlab (Mathworks, Natick, Mass.) for typical MRSI scenarios, including low vs. high gradient systems demands, and including spectral widths (SW) appropriate for 3T and 7T systems:

- Field-of-view (FOV): 220mm.
- Desired spectral-width (SW): 1250Hz (“3T”) and 2500Hz ("7T").
- Spatial resolution: 24×24.
- Desired spectral resolution: 256.
- Hardware limitations (maximum gradient amplitude and slew rate). Set 1: G
_{max }= 10$$$\mathrm{\frac{mT}{m}}$$$, S_{max}= 60$$$\mathrm{\frac{mT}{m\cdot ms}}$$$; Set 2: G_{max}= 80$$$\mathrm{\frac{mT}{m}}$$$, S_{max }= 200$$$\mathrm{\frac{mT}{m\cdot ms}}$$$. - Raster times: 10$$$\mathrm{\mu s}$$$ for gradients, 2$$$\mathrm{\mu s}$$$ for acquisition dwell time, and 1$$$\mathrm{\mu s}$$$ for ADC start position.
- Trajectory types: CSI; “Standard” EPSI, acquiring on both positive and negative readout gradients (with and without ramp sampling); flyback EPSI, acquiring only on the positive readout gradients and a shortest return gradient (with and without ramp sampling); Rosettes;
^{1}and CRT^{2}.

The “image” $$$\hat{\rho}$$$
was reconstructed from the “measured” signal
$$$S$$$
by iteratively minimizing
$$\left\Vert F^{\dagger}\left(F\hat{\rho}-S\right)\right\Vert =\left\Vert \left(F^{\dagger}F\right)\hat{\rho}-\left(F^{\dagger}S\right)\right\Vert \text{, }$$
where
$$$F$$$
is the (generally) non-uniform Fourier-transform from spectral image (SI) to k-t space, with
$$$F^{\dagger}$$$
its adjoint. Prior to reconstruction, the signal was apodized along time. For trajectories with a disk-shape support in k-space, the reconstruction was regularized by zero-filling the corners of k-space (filling to a square). In these cases, zero k-space corners were enforced on the final results by an FFT to k-space, application of a disk shaped mask, and an inverse FFT.
Noise was estimated by generating 100 noisy realization for each “experiment”. This was done for a circular disk “phantom” of diameter
100mm, at resonance, and having T^{*}_{2}=80ms.
The same reconstruction (including apodization) was used to determine the PSFs, however for a Dirac-delta signal (at resonance) with practically infinite
T^{*}_{2}. Sub-voxel behavior of the PSF was found by repeatedly shifting the delta function within the space-frequency “voxel” (10×10×10
sub-samples).

As a measure of PSF localization a 'bleed fraction' was defined: The ratio between the area under the PSF outside the central voxel/lobe to the area of the whole PSF (estimated at the xy plane, on resonance).

- Schirda CV, Tanase C, Boada FE. Rosette spectroscopic imaging: optimal parametersfor alias-free, high sensitivity spectroscopic imaging. Journal of Magnetic ResonanceImaging 2009;29:1375–1385.
- Furuyama JK, Wilson NE, Thomas MA. Spectroscopic imaging using concentri-cally circular echo-planar trajectories in vivo. Magnetic Resonance in Medicine 2012;67:1515–1522.

Table 1: comparison of mean (normalized) SNR in the phantoms and of the bleed fraction of the PSFs, for all four simulation cases performed.

Figure 1: Comparison of *spatial* PSF shapes, in absolute value; all set to the same peak height. Two cases of SW and
hardware limitations are shown. Insets are a zoom-in of the main PSF. Top row is
the PSF along the x-axis (readout), while the bottom is the y-axis
(phase encoding). On the y-axis EPSI (+variants) are phase encoded, just
like CSI, so the PSFs match exactly, unlike on x.

Figure 2: Comparison of typical *spatial* PSFs for trajectories with
different support in k-sapce. a) Rectangular support (CSI), and b)
disk-shape support (CRT). The CSI is symmetric under 90° rotations,
while CRT can be rotated at any angle. PSFs match the SW and hardware
given in Fig. 1a.

Figure 3: Top, (normalized) SNR comparison, in absolute value; for the same simulation cases as in Fig. 1. Shown is the start of the PSFs along the x-axis, as well as a zoom-in on the (normalized) SNR along the same axis for the phantom simulations. Bottom row shows graphically the bleed fraction. (See also Table 1.)

Figure 4: Comparison of the *spectral* PSF shapes, in absolute value; all set with the same peak height. The two cases are the same as in Fig. 1. Insets are a zoom-in of the main PSF. It can be seen that fly-back trajectories have the same behavior as that of CSI, while other trajectories (including EPSI) behave differently, having in general a lower base line.