Karl Landheer1 and Christoph Juchem1,2
1Biomedical Engineering, Columbia University, New York City, NY, United States, 2Radiology, Columbia University, New York City, NY, United States
Synopsis
It has recently been recommended that typical
preprocessing tools, such as linebroadening, zero-filling and apodization
(cutting), generally be avoided prior to signal quantification via consensus.
To date, little explanation has been provided against these tools which have
become commonplace. Here we demonstrate via realistic Monte Carlo simulations
that such preprocessing tools may reduce the precision of the extracted
parameters and artificially reduce the Cramér-Rao Lower Bounds and provide a theoretical outline for why they
should be avoided.
Introduction
Cutting/zero-filling and exponential
linebroadening are routine preprocessing tools used prior to quantification of
MR spectra as an attempt to tease out more information from the signal. These
tools have been implemented in packages such as LCModel1, TARQUIN2, jMRUI3, FID-A4, and INSPECTOR5. These steps are useful for visualization
of spectra but are not recommended prior to
quantification6,7. Here we demonstrate via Monte Carlo simulations that such preprocessing tools might reduce the precision of the extracted parameters and artificially reduce the estimated Cramér-Rao Lower Bounds (CRLBs). We provide a
theoretical outline for why these tools should be avoided prior to quantification.Methods
Spectral shapes were simulated for a
TE = 20 ms sLASER8 sequence for 18 metabolites and 10
macromolecules which were modeled as broad Lorentzian singlets via MARSS9, similar to what has previously been
performed8,11. The macromolecular (MM) signal was modeled
as the sum of these 10 macromolecule resonances with measured concentrations
and T2 values10. The
simulated spectral shapes were exponentially linebroadened by $$$\frac{1}{\pi T_2^m}$$$, where $$$T_2^m $$$ is the transverse relaxation constant for the
particular metabolite, and all metabolites were broadened by a Gaussian
linewidth of 8 Hz2 to resemble imperfect B0 conditions typically encountered in vivo. The
broadened spectral shapes were scaled by their respective concentration, and
corrected for T1 and T2 effects through the solution to the Bloch equation. A synthetic spectrum was used so that ground truth parameters were
known, which is not true in any experimental spectrum.
The effect these
preprocessing tools had on quantification was assessed by running three
different Monte Carlo simulations: 1) no preprocessing, 2) spectra were cut by
a factor of two and zero-filled back to the original length (i.e., 2nd
half of FID which is almost purely noise was set to zero) and, 3) spectra were
linebroadened by a 3 Hz exponential function. A total of 5,000 trials were
performed for each case, and spectra differed only by additive white Gaussian
noise. For each trial the spectra were quantified using a maximum-likelihood
estimator (MLE) in INSPECTOR and CRLBs were calculated12.Results and Discussion
The synthesized spectrum used for the
Monte Carlo simulations closely resembles experimental measurements with the
same sequence8 (Figure 1). Cutting/zero-filling has
little effect on measured standard deviations, while the 3 Hz exponential
linebroadening noticeably increased the resulting standard deviations (Figure
2). The estimated CRLB values were substantially artificially reduced by both preprocessing
steps and no longer become an accurate proxy for standard deviations (Figure 3).
In the case where the spectra has been cut/zero-filled by a factor of two the
apparent CRLB has been reduced by a factor of $$$\sqrt{2}$$$, whereas the 3 Hz exponential linebroadening reduces the CRLBs by a factor of 1.2 to 2.3 depending on the
specific metabolite (Table 1). These results can be understood from the
assumptions of the calculation of the CRLBs, namely that the parameters are sampled
from a normal distribution due to white Gaussian noise. Cutting/zero-filling
the spectrum in the time domain is effectively multiplying it by a step
function, hence its effect on the spectrum can be expressed as
$$S'(f)=S(f)+S(f)*\left(\frac{1}{2 \pi if}\right), $$
where $$$S'(f)$$$ is the spectrum after preprocessing, $$$S(f)$$$ is the spectrum before preprocessing and $$$*$$$ is the convolution operator. $$$S'(f)$$$ will no longer contain white Gaussian noise,
as there is clearly a correlation between spectrally neighboring noise points
(Figure 4A). Similarly, the effect of Lorentzian broadening can be expressed as $$ S'(f)=S(f)*\left(\frac{2 \pi LB}{(\pi LB)^2+(2 \pi f)^2}\right)$$
where $$$LB$$$ is the Lorentzian linebroadening width (Hz).
Once again this introduces correlation between spectrally neighboring noise
points (Figure 4B).
Because CRLBs are
a fundamental bound on the standard deviation of parameters irrespective of
the method used for estimation13, coupled with the results obtained
here that the employed maximum-likelihood estimator algorithm effectively attains
the CRLBs (Table 1), this demonstrates that the signal is being used nearly as
efficiently as possible14. Thus, signal preprocessing methods
are fundamentally unable to yield any substantial information as the
information limit has nearly been attained. These preprocessing tools do,
however, invalidate the assumptions of the CRLB, specifically that the noise is white and Gaussian, which results in the assumption that the parameters are
sampled from a normal probability density function15, and hence only artificially reduces the CRLBs. Similarly, it has previously been
shown that increasing the number of points in the FID which are purely noise
has no effect on the CRLB7 (and hence quantification precision).
This is as expected as the Fisher information matrix (which is proportional to
the inverse of the CRLBs) is additive
and thus new data points cannot subtract
information. Although these points obscure visualization of the spectrum they
do not impede the MLE quantification. Note that cutting alone does not violate the
assumptions in calculating CRLBs, however cutting data points which contain
substantial signal would reduce the precision of the measured parameters.Conclusions
Cutting/zero-filling and and linebroadening,
although useful for data visualization, should not be used prior to
quantification as they yield no information while invalidating the assumptions
used in the calculation of the CRLBs, causing them to become artificially low.
These artificially low CRLBs could potentially result in false positives or
statistically under-powered studies.Acknowledgements
Special thanks to Martin Gajdošík, PhD, and Kelley Swanberg, MSc, for
fruitful discussions and input.References
1. Provencher, S. Estimation of
Metabolite Concentrations from Localized in Vivo Proton NMR spectra. Magn
Reson Med 30, 672–679 (1993).
2. Wilson, M., Reynolds, G., Kauppinen,
R. A., Arvanitis, T. N. & Peet, A. C. A constrained least-squares approach
to the automated quantitation of in vivo 1H magnetic resonance spectroscopy
data. Magn. Reson. Med. 65, 1–12 (2011).
3. Stefan, D. et al. Quantitation
of magnetic resonance spectroscopy signals: The jMRUI software package. Meas.
Sci. Technol. 20, 104035 (2009).
4. Simpson, R., Devenyi, G. A., Jezzard,
P., Hennessy, T. J. & Near, J. Advanced processing and simulation of MRS
data using the FID appliance (FID-A)—An open source, MATLAB-based toolkit. Magn
Reson Med 77, 23–33 (2017).
5. Prinsen, H., de Graaf, R. A., Mason,
G. F., Pelletier, D. & Juchem, C. Reproducibility measurement of
glutathione, GABA, and glutamate: Towards in vivo neurochemical profiling of
multiple sclerosis with MR spectroscopy at 7T. J Magn Reson Imag 45,
187–198 (2017).
6. Near, J. et al. Preprocessing,
analysis and quantification in single‐voxel magnetic resonance spectroscopy:
experts’ consensus recommendations. NMR Biomed. 1–23 (2020).
doi:10.1002/nbm.4257
7. Kreis, R. et al. Terminology
and concepts for the characterization of in vivo MR spectroscopy methods and MR
spectra: Background and experts’ consensus recommendations. NMR Biomed.
e4347 (2020). doi:10.1002/nbm.4347
8. Landheer, K., Gajdosik, M. &
Juchem, C. Semi-LASER Single-Voxel Spectroscopic Sequence with Minimal Echo
Time of 20 ms in the Human Brain at 3 T. NMR Biomed e4324 (2020).
9. Landheer, K., Swanberg, K. M. &
Juchem, C. Magnetic resonance Spectrum simulator (MARSS), a novel software
package for fast and computationally efficient basis set simulation. NMR
Biomed e4129 (2019). doi:10.1002/nbm.4129
10. Landheer, K., Gajdosik, M., Treacy, M.
& Juchem, C. Concentration and T2 Relaxation Times of Macromolecules at 3
Tesla. Magn Reson Med 84, 2327–2337 (2020).
11. Bolliger, C. S., Boesch, C. & Kreis,
R. On the use of Cramér-Rao minimum variance bounds for the design of magnetic
resonance spectroscopy experiments. Neuroimage 83, 1031–40
(2013).
12. Cavassila, S., Deval, S., Huegen, C.,
Ormondt, D. v & Graveron-Demilly, D. Cramér–Rao bounds: an evaluation tool
for quantitation. NMR Biomed 14, 278–283 (2001).
13. Beer, R. & Ormondt, D. Analysis of
NMR Data Using Time Domain Fitting Procedures. In-Vivo Magn. Reson.
Spectrosc. I Probeheads Radiofreq. Pulses Spectr. Anal. 201–248 (1992).
doi:10.1007/978-3-642-45697-8_7
14. Hstadsen, A. On the existence of
efficient estimators. IEEE Trans. Signal Process. 48, 3028–3031
(2000).
15. Cavassila, S., Deval, S., Huegen, C.,
van Ormondt, D. & Graveron-Demilly, D. Cramer-Rao Bound Expressions for
Parametric Estimation of Overlapping Peaks: Influence of Prior Knowledge. J
Magn Reson 143, 311–320 (2000).