Yan Zhang^{1}, Li An^{1}, and Jun Shen^{1}

^{1}National Institute of Mental Health, Bethesda, MD, United States

### Synopsis

Numerical simulations
of three-dimensionally localized MRS spectra have been very time-consuming for
multi-spin systems because the current state-of-the-art requires computation of
a large ensemble of spins pixel-by-pixel in three dimensional space. In this
abstract it was found that spatial coordinates of the full set of spin density
operators labeled by slice selection gradients can be projected onto one
dimension after slice selection as long as the crusher gradients are refocused.
Therefore, the conventional three-dimensional simulation can be converted to a
one-dimensional problem in cases such as the commonly used PRESS or STEAM. The
proposed method was implemented using a computer program written in Java
language.

### Introduction

Spatially-selected RF pulses lead to spatial dependence of spin
magnetization evolution. Consequently, numerical simulations of
three-dimensionally localized magnetic resonance spectroscopy (MRS) are time-consuming
for multiple spin systems because of the need to compute an ensemble of spins
in three dimensional space. In many cases
either ideal RF pulses or reduced spin density matrices are used for simulations.
As numerical simulation gains increasing popularity for designing and validating
in vivo MRS detection methods as well as for providing quantification basis
sets [1-3], improving the efficiency and accuracy of spin density matrix simulation
is much needed. This abstract reports a highly accelerated and accurate method
for simulating spatially localized MRS spectra.### Methods

The time evolution of spatially localized
multispin density matrix as the full solution to the Liouville–von Neumann equation
was analyzed. It was found that spatial coordinates of the full set of spin
density operators labeled by slice selection gradients can be projected onto
one dimension after slice selection as long as the crusher gradients are
refocused. Therefore, the conventional three-dimensional simulation can be
converted to a one-dimensional problem in cases such as the commonly used PRESS
or STEAM. The proposed method was implemented using a computer program written
in Java language. It was validated by comparing
with the conventional GAMMA [4] simulation and with phantom experiments. The time evolution of spatially localized
multispin density matrix as the full solution to the Liouville–von Neumann
equation was analyzed. The spatial coordinates of the full set of spin density
operators labeled by slice selection gradients were projected onto one
dimension after slice selection. The proposed method was implemented using a
computer program written in Java language. It was validated by comparison with the conventional GAMMA [4] simulations
and with phantom experiments. Furthermore, the simulated spectra were used as basis
set (18 metabolites) to fit in vivo short echo time spectra acquired from human
brain at 3 T.### Results

Figure 1 shows the comparisons
between three different simulations of PRESS-localized NAA and Lac spectra at 3
T; the simulation (red spectra) using GAMMA C++ library and three-dimension
computation, the simulation (green) using GAMMA C++ library and the proposed
one-dimensional projection method, and the simulation (black) using the
Java-developed program in this study and the proposed one-dimensional
projection. The voxel was represented by 100 × 100 × 100 equally spaced points.
As seen in Figure 1, they are completely identical. All computations were
performed on PC. The two simulations with the one-dimensional projection method
used 10 seconds or less for NAA (acetyl moiety and aspartate moiety were
combined) and Lac; in contrast, the conventional three-dimensional computation took
over 16 hours. Figure 2 shows the fit of
short TE phantom spectrum (black color) using the simulated basis sets (red
color). The estimated metabolite concentrations (relative to creatine) was in
good agreement with those of the listed ingredients of the phantom ( ±4%) which
consisted of NAA (12.5mM), creatine (10
mM), glutamate (12.5 mM), myo inositol (7.5mM), and lactate (5 mM). The
baseline was originated from the residual water after water suppression. Figure
3 shows an example of using the one dimensional projection method to simulate basis
sets (red) for fitting an in vivo short
TE spectrum (black) . The estimated metabolite concentrations (mM ± standard
deviation (SD)) averaged over twelve healthy subjects were: [tNAA] = 13.6 ±
4.9% , [tCr]= 9.5 ± 5.8%, [tCho] = 2.4 ±
10.1%, [Glu] = 11. 8 ± 6.7%, [Glx] =
14.3 ± 9.4%, [mI] + [Glycine] = 8.99 ±
7.7%, [mI] = 6.78 ± 12.5%, and [Glutathione] = 1.48 ± 23%.### Discusssion

The new method represents a dramatic improvement in efficiency and
accuracy for the simulation of spatially localized MRS. Thanks to the massive
increase in computational efficiency, more spatial points can now be used to define
the three dimensional voxel, leading to more realistic simulation. To the best
of our knowledge, the current work was
the first to show full spin density simulation with spatial points exceeding
100 in each of the three dimensions. Using the one-dimensional projection
method the computation time was as short as ~10 minutes with a typical desktop
computer for a six spin molecule.The resultant simulation agreed well with the phantom experiment
as demonstrated by Fig. 2. As simulation time of the one-dimensional projection method is
mainly consumed by computing the propagators which are prepared prior to
computing the spin density evolutions, simulating a 2D or higher dimensional
spectrum such as J-resolved spectrum can be made very efficientâ€•the preprepared
propagators can be repeatedly called back throughout the 2D or higher
dimensional loopâ€•and can be accomplished in a time frame similar to the case of
a 1D spectrum.### Conclusion

The
dramatically enhanced computational efficiency makes spin density matrix
simulation of multi-spin systems a convenient tool for designing and optimizing
MRS pulse sequences and for computing basis sets for spectral fitting. As many
experiments such as two-dimensional spectroscopy and T2 measurements demand highly extensive simulation the
proposed method is expected to greatly facilitate progress in those areas.### Acknowledgements

No acknowledgement found.### References

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