Estimation of metabolite T _{1} relaxation times using tissue specific analysis, signal averaging and bootstrapping from magnetic resonance spectroscopic imaging data
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Abstract
Object A novel method of estimating metabolite T_{1} relaxation times using MR spectroscopic imaging (MRSI) is proposed. As opposed to conventional single-voxel metabolite T_{1} estimation methods, this method investigates regional and gray matter (GM)/white matter (WM) differences in metabolite T_{1} by taking advantage of the spatial distribution information provided by MRSI.
Material and methods The method, validated by Monte Carlo studies, involves a voxel averaging to preserve the GM/WM distribution, a non-linear least squares fit of the metabolite T_{1} and an estimation of its standard error by bootstrapping. It was applied in vivo to estimate the T_{1} of N-acetyl compounds (NAA), choline, creatine and myo-inositol in eight normal volunteers, at 1.5 T, using a short echo time 2D-MRSI slice located above the ventricles.
Results WM-T_{1,NAA} was significantly (P < 0.05) longer in anterior regions compared to posterior regions of the brain. The anterior region showed a trend of a longer WM T_{1} compared to GM for NAA, creatine and myo-Inositol. Lastly, accounting for the bootstrapped standard error estimate in a group mean T_{1} calculation yielded a more accurate T_{1} estimation.
Conclusion The method successfully measured in vivo metabolite T_{1} using MRSI and can now be applied to diseased brain.
Keywords
MR spectroscopic imaging Relaxation time T_{1} estimation Bootstrap Monte Carlo simulationIntroduction
Magnetic resonance imaging and spectroscopy is a powerful tool for non-invasively assessing anatomic and metabolic changes that occur in brain diseases. In clinical spectroscopic studies and especially for magnetic resonance spectroscopic imaging (MRSI) data acquisition, short repetition times (TR) are often required to meet scan time constraints, but accurate metabolite longitudinal relaxation time values (T _{1}) are then needed to correct the metabolite concentrations for the T _{1}-weighted effect. The metabolite T _{1} values are likely to be important for quantifying results that make comparisons between patients and normal controls. Moreover, the knowledge of ^{1}H metabolite longitudinal relaxation times can by itself give insight into the properties of a given region of interest.
In many previous studies, the estimations of the metabolite T _{1}s were performed using single voxel acquisitions. Short echo time spectra coming from either progressive saturation [1, 2, 3] or inversion recovery experiments [4, 5, 6] were collected and T _{1} values were usually derived from amono-exponential fit. The inversion recovery experiments typically used long repetition times (TRs are usually equal to 6 s) with varying inversion times [4, 5, 6], which is prohibitively long for MRSI experiments. These single voxel approaches assume a single T _{1} over the whole voxel regardless of its tissue composition. To obtain white matter or gray matter T _{1} values and good SNR, large (usually greater than or equal to 8 cc), and relatively heterogeneous single voxels were typically acquired. In most cases, gray matter (GM)-T _{1} results were obtained from voxels containing 60–70% of GM, while white matter were obtained from voxels containing 70–90% of WM. At the same time, different MRSI studies [7, 8, 9, 10] using linear regression demonstrated how metabolite concentrations (and thus metabolite signal intensities) can be different according to their tissue origin. Thus a common concern about the single voxel studies is whether the content of GM and WM in the examined voxel has an influence on the metabolite T _{1} results. Moreover attempting to reduce the size of the single spectroscopic voxel to reduce the voxel tissue heterogeneity would result in increasing the number of averages and the scan time. In contrast, MRSI techniques offer the possibility to acquire simultaneously several spectra over a wide brain region and at a resolution allowing tissue analysis. The first goal of this paper was, therefore, to develop a MRSI post-processing method to estimatemetabolite T _{1}s while accounting for the voxels’ tissue content. To date, no published studies investigated the use of MRSI data to estimate metabolite T _{1} relaxation times.
Then, as for any quantitative measurement based on model fitting, an assessment of the precision of the T _{1} estimation is desirable. A benefit of MRSI is that it provides several spectra and thus several data points for the metabolite T _{1} fit which can be resampled in a bootstrap manner to estimate standard error. Therefore, a second goal of this study was to develop an approach to obtain metabolite T _{1} standard error by bootstrapping.
The last contribution of the paper was to apply the new techniques to measure metabolite T _{1}s in different regions and tissues of the brains of healthy subjects.
- 1.
Increasing the SNR by averaging voxels according to their WM/GM content and their location (for example anterior vs. posterior) since the SNR of the metabolite signals is very low using the proposed acquisition parameters (number of excitations (NEX) and number of voxels in the slice) at 1.5 T.
- 2.
Estimating metabolite T _{1} for gray and white matter using a non-linear least squares algorithm. The underlying model function used in the fitting procedure associates WM/GM content of a voxel to the metabolite signal intensity.
- 3.
Using a bootstrap technique to assess uncertainty on the metabolite T _{1}s and taking into account this confidence when calculating a group mean value.
This paper presents the techniques developed to utilize MRSI data for metabolite T _{1} measurement. A validation of these techniques is then proposed through Monte Carlo data simulations, demonstrating the statistical performance of the proposed method. Finally the fitting procedure is applied to 2D conventional MRSI data acquired at 1.5 T from eight healthy subjects.
MR data acquisitions
Study subjects
A total of eight healthy volunteers (five females and three males, mean age 31.5±9.5 years) were examined to validate our method. Written and informed consentwas obtained from all participating subjects. The study was approved by the UCSF Committee on Human Research.
MR parameters
The healthy volunteers were scanned on a Signa 1.5 T clinical imager from GE Medical Systems (GE Healthcare Technologies, Waukesha, WI, USA) using a quadrature head coil. Short echo time (TE = 35 ms) 2D MRSI data sets (12×12, 1 cc resolution) were acquired using a PRESS volume selection at five different TRs (TE = 0.850, 1, 2, 4, 8s).The number of excitations (NEX) acquired were as follows: NEX=3 for TE = 0.850s, NEX=2 for TR = 1s, NEX=1 for TR = 2, 4 and 8s. Oblique Fast Spin Echo images were used to guide the positioning of the spectroscopic acquisition. Care was taken that the PRESS box avoided the ventricles and was centered in the anterior-posterior middle of the corpus callosum body. T _{1}-weighted 3D spoiled gradient recalled (SPGR) images were also acquired for segmentation of the anatomic images. The setup and data acquisition time for the anatomic and spectroscopic imaging was approximately 55 min.
Methods
The first goal of the analysis was the formulation of an approach to estimate regional metabolite T _{1}s in cortical gray matter and white matter.
The model function
The T_{1} estimation procedure
Metabolite T _{1}s are obtained after the following steps:
Step 1. Calculation of p_{n}^{WM} and p_{n}^{GM}
Step 2. Voxel averaging
Note that to perform regional analysis, this averaging procedure is applied on a specific part of the spectroscopic grid (for example in the anterior part of the slice) by working only with the voxels belonging to the region of interest.
Step 3. Quantification of S
- 1.
estimated relative Cramér-Rao lower bound [18] of the metabolite amplitude, (rCRB) < 15%.
- 2.
metabolite peaks that had smaller than 9Hz linewidth at half peak height.
After this selection, we have N _{final,TR} points at each TR for the metabolite T _{1} fit.
Step 4. Four-parameter fit
Estimating metabolite T_{1} uncertainties using bootstrap
To take into account the confidence in the T _{1} value estimation from a subject when calculating the mean T _{1} value over a group of subjects, we proposed using the estimated standard errors in a weighted average calculated as follows:
Monte carlo simulations
Parameters used in the Monte Carlo simulation studies
Metabolite | WM | GM | ||||
---|---|---|---|---|---|---|
Concentration (a.u.) | T_{1} (s) | SNR at TR = 8s | Concentration (a.u.) | T_{1} (s) | SNR at TR = 8s | |
NAA singlet (3 equivalent protons) | 7.5 | 1.55 | 2 | 9 | 1.45 | 2 |
Cho (9 equivalent protons) | 1.8 | 1.2 | 1.44 | 1.5 | 1.2 | 1 |
Cr-CH_{3} (3 equivalent protons) | 5.2 | 1.3 | 1.39 | 7.7 | 1.4 | 1.71 |
mI | 3.8 | 1.1 | 0.91 (at 3.56ppm) | 5 | 1.1 | 0.99 |
Glu (T _{1} not estimated) | 7 | 1.3 | / | 9 | 1.3 | / |
Background signal | WM/GM | ||
---|---|---|---|
α | ω (ppm) | SNR at TR = 8s | |
Nine gaussian components, time domain model: Σ _{i=1} ^{9} a _{i} exp(jωt)exp(−β ^{2} t ^{2})T _{1} = 0.2s and β = 50 Hz for all the components | 2.5 | 1.88 | 0.21 |
15 | 2.08 | 1.29 | |
2.5 | 2.39 | ||
2.5 | 2.55 | ||
2.5 | 2.71 | ||
10 | 3.09 | 0.86 | |
5 | 3.5 | 0.43 | |
5 | 3.66 | ||
10 | 3.91 |
Study 1: Effects of averaging
In the first Monte Carlo study, we evaluated the statistical performance of the method in terms of bias and standard deviation for SNR = 2 along with an increasing number of voxels in an average, N _{avg} = 2, 4 and 6, and the macromolecular background signal (see Table 1) added to the simulated short echo time signals. For each TR, N _{gen} = 120 voxels are created by the proposed averaging method from the original MRSI grid. We also tested our method in the case of no averaging. In this case, designated by N _{avg} = 0, only the 48 voxels (N _{gen} = N _{total}) in the MRSI grid contribute to the fitting method.
Study 2: Macromolecular background effect
We tested the macromolecular background effect on the T _{1} estimation in the second Monte Carlo study. We compared the statistical results obtained with N _{avg} = 6 with and without a macromolecule signal in the simulation. The weighted average using the bootstrapped standard error estimation is tested in these two cases.
Results
Monte Carlo studies
Metabolite | SE (%) | SE_{w}(%) | b(%) | b_{w}(%) | ||||
---|---|---|---|---|---|---|---|---|
w/o | w/ | w/o | w/ | w/o | w/ | w/o | w/ | |
NAA-WM | 7.51 | 12.84 | 6.94 | 11.58 | 1.96 | 5.33 | 0.54 | 0.41 |
NAA-GM | 9.28 | 17.65 | 9.12 | 15.96 | 1.95 | 0.82 | 2.02 | 0.73 |
tCho-WM | 12.14 | 14.99 | 10.18 | 14.97 | 1.83 | −5.260 | 0.14 | −8.20 |
tCho-GM | 25.44 | 42.47 | 24.60 | 37.31 | 6.21 | 13.82 | 3.37 | 9.74 |
tCr-WM | 12.84 | 14.08 | 12.06 | 13.32 | 2.31 | −8.041 | 1.93 | −7.63 |
tCr-GM | 14.83 | 18.38 | 13.46 | 17.28 | 2.74 | −3.01 | −0.70 | −6.83 |
mI-WM | 14.57 | 34.25 | 13.51 | 30.14 | 2.63 | 9.95 | 0.93 | 3.26 |
mI-GM | 25.18 | 59.20 | 20.95 | 42.59 | 3.97 | 15.30 | −1.50 | −1.75 |
Study 1: Effects of averaging
Figure 6 shows the mean and the standard deviation of the bootstrap estimates, as well as the gold standard of the SE for each metabolite, for a range of N _{avg}, SNR_{NAA} equals 2, and the set of 5 TRs. (*) indicates cases where the bootstrap estimate is biased by more than 100 %from the actual value.
For the four metabolites of interest (NAA, Cr-CH_{3}, Cho, mI), higher SE were found in the GM than in WM consistent with the discrepancy between the number of GM voxels versus the number of WM voxels available in the masks we used in the simulation. The SE globally decreases with the number of voxels used in an average N _{avg}. For N _{avg} = 6, the gold standard SE is below 20% of the true T _{1} for NAA, Cho and Cre in the WM.
For all of the considered metabolites in the WM and the GM and for N _{avg} ≥ 4, the SE estimated by the bootstrap technique was within 50% of the actual SE value. Less biased bootstrap estimates were generally obtained for N _{avg} = 2 and the bias between the bootstrap estimates and the actual SE value increased with N _{avg}. For mI, the bootstrap approach successfully estimated the SE with a large standard deviation in the WM and only for N _{avg} = 6 in the GM. For N _{avg} = 0, the bootstrap estimates have reasonable values in WM for NAA, Cho and Cre and failed to estimate the SE in the other cases.
Figure 7 shows the bias (denoted by b) on T _{1} values for the four metabolites of interest and for a weighted average calculated by Eq. 3 and denoted by b _{w}. b and b _{w} are displayed as percentages of the true metabolite T _{1} value. For (WM/GM)-T _{1,NAA} (WM/GM)-T _{1,Cr-CH3} and WM-T _{1,Cho} the bias is below ±10% of the true T _{1} value. We note that, in the case of our simulation, increasing the N _{avg} did not reduce the bias for WM-T _{1,Cr-CH3}. As seen in the next study, we think that this bias is more due to the interaction with the macromolecules than to a lack of SNR. Increasing N _{avg} seems to reduce the bias for (WM/GM) T _{1,NAA}, GM-T _{1,Cho}, GM-T _{1,Cre}, and (WM/GM) T _{1,mI}. For WM and GM-T _{1,mI}, the weighted average makes the bias below 10% of the actual value for N _{avg} ≥ 2.
The best bias and standard deviation trade-off is obtained for N _{avg} = 6 at the expense of slightly biased bootstrap estimates.
Study 2: Macromolecular background effect
Table 2 shows the statistical results (regular SE, SE_{w} calculated with Eq. 4, b and b _{w}) of the T _{1} estimation procedure for N _{avg} = 6, SNR = 2, with and without a background contamination in the signals to process. All the metabolite T _{1} estimations show the same trend with a larger standard deviation and a bigger bias in the case of macromolecular contamination as compared to the absence of a background signal. In the absence of a macromolecular background, the weighted average that takes into account the bootstrapped standard error successfully reduces the bias (b _{w} as compared to b) on the metabolite T _{1}.
In the presence of the simulated background signal, the SE of the T _{1} estimate increased by 10%(for WM-T _{1,Cr-CH3}) and more than doubled for the T _{1,mI}. The bias was also affected (but usually reduced) by using a weighted average. The GMT _{1,Cr-CH3} and WM-T _{1,Cho} estimation were more downward biased in the presence of a background signal when using a weighted average than when using the standard mean calculation.
Metabolite T_{1} estimation on in vivo data
For the eight subjects, the anterior-posterior center of the PRESS Box (12 × 12, 1 cc) was centered in the anterior-posterior middle of the corpus callosum body visualized in a sagittal plane. Voxels anterior to the center of the PRESS Box were analyzed as part of the anterior brain region (Ant.) and the rest of the voxels were evaluated as posterior voxels (Post.).
The bootstrap estimate of standard error on in vivo data
Anterior versus posterior metabolite T_{1}
From eight healthy volunteers, estimated T _{1}-relaxation times (in seconds) of NAA, Cho, Cr-CH_{3}, mI at 1.5 T, in pure WM and pure GM, using MRSI data, (N _{avg} = 6)
Metabolite | T1, N _{avg} = 6 | |
---|---|---|
Ant. | Post. | |
NAA | ||
WM | 1.38 ± 0.15 | 1.28 ± 0.10^{**} |
GM | 1.23 ± 0.36 | 1.31 ± 0.20 |
Cho | ||
WM | 1.20 ± 0.13 | 1.20 ± 0.05 |
GM | 1.18 ± 0.20 | 1.20 ± 0.10 |
Cr-CH_{3} | ||
WM | 1.26 ± 0.09 | 1.23 ± 0.08 |
GM | 1.19 ± 0.12 | 1.22 ± 0.06 |
mI | ||
WM | 1.29 ± 0.09 | 1.23 ± 0.10 |
GM | 1.19 ± 0.12 | 1.22 ± 0.09 |
The standard deviation in the GM was larger than in the WM, for most of the time, as in the Monte Carlo simulations. From these results, no significant differences were found in metabolite T _{1}s between GM and WM in the posterior region. In anterior region WM-T _{1,NAA}, \(WM-T_{1,Cr-CH_{3}}\), and WM-T _{1,mI} tend to be higher than GM-T _{1}s but this trend did not reach statistical significance (0.1 < P < 0.2). The T _{1} of NAA in the anterior part of the WM was significantly longer than in the posterior part of the WM (P < 0.05).
Discussion
This work presents a novel method for estimating metabolite T _{1}s using MRSI data, enabling estimates within tissue types (GM and WM) and across different regions (anterior and posterior were demonstrated here). The smaller voxel size of the MRSI data versus previous single voxel studies partially addresses the issue of large partial volume artifacts between gray matter and white matter. Additionally, incorporating the information regarding tissue type composition obtained from higher resolution MR images and the multiple voxel data obtained with MRSI allows better correction of partial volume artifacts than possible with the previous single voxel data. This method has been validated and tested on simulations and applied in vivo. We also proposed assessing a confidence interval in the fitted T _{1} results by introducing a bootstrap approach. We showed that a weighting average that takes into account the confidence assessment can reduce the bias on the estimated T _{1} value for a metabolite with a small SNR. This method relies upon a group mean metabolite T _{1} approach. The proposed algorithm yielded results that are in agreement with the literature and support the hypothesis of regional differences in T _{1} in the brain.
Methodology
- 1.
The SNR, as expected, appeared to clearly play a role both on the bias and the standard deviation of the T _{1} estimation. In Figs 6 and 7 the results for NAA and Cr-CH3 which have the greatest SNR in our simulation, present good biases and standard deviations on T _{1} estimation compared to the ones for Cho (especially in GM) and mI. In order to realize a robust four-parameter fit with low SNR, non-reliable voxel quantification results should be rejected from the analysis. The use of criteria, such as Cramér-Rao lower bounds [18] or linewidth thresholds, is necessary to determine the quality of the metabolite amplitude quantification and to perform voxel selection.
- 2.
Increasing the number of voxels used in an average N _{avg}, improves SNR and so typically reduces both bias and standard deviation. Note that, by averaging the voxels, the independence between the averaged voxels is reduced and the bootstrap technique can tend to underestimate the real standard error. Also note that this averaging is performed while taking into account the WM-GM distribution and the introduced dependence tracks the initial tissue content. It was also shown by the Monte Carlo simulation results that, for an original SNR of two, the use of six voxels in an average corresponds to a good trade-off between the SNR gain and the lost of voxel independence and leads to reliable metabolite T _{1} estimations. Furthermore, this voxel averaging introduces some partial voluming with CSF to the generated voxels, especially for mostly GM voxels originating from the thin cortical ribbon at the midline. Nevertheless, as the percentage of GM and WM (and thus, CSF) in the voxel are explicitly taken into account during the fitting procedure, the regression presented in Eq. 1 enables a correction of this partial voluming effect.
- 3.
The number of voxels available for a specific tissue type influences the standard deviation of the T _{1} estimation. Consequently, the dispersion of the results is larger for the GM than for the WM values from both our simulation and the in vivo data.
The first Monte Carlo simulations showed that for an original SNR_{NAA} of 2 in the MRSI data, an almost unbiased estimation of the T _{1,NAA}, T _{1,Cho}, T _{1,Cr-CH3} is possible. A weighted average, taking into account the bootstrap estimates of the uncertainty on T _{1} could also yield to an unbiased estimation of T _{1,mI}. This approach tends to reduce the dispersion due to bad data points and makes a group mean value more accurate.
We conclude from the second Monte Carlo study that the presence of a macromolecular background signal has an important effect on the dispersion and the bias of the results and should be considered as another effect, besides the set and number of TRs and the type and parameters of the sequence used [25], leading to the discrepancy between the different published T _{1} values. The bootstrapped standard error estimation is also hampered by the macromolecular contribution. In the proposed simulation, all the voxels were contaminated in the same way and the background signal was arbitrary and particularly elevated under the mI making its T _{1} estimation harder.
Finally, the proposed bootstrap procedure is a novel method to estimate metabolite T _{1} standard error and is enabled in this study by the use of MRSI data. The bootstrap technique is a non-parametric method that does not require a complex implementation. This approach may be beneficial in future uncertainty and/or bias estimation studies of spectroscopic quantification.
In vivo results
The range of our results for the four metabolites is in good agreement with existing literature [1, 2, 3, 4], when using the weighted average and N _{avg} = 6. The method gives reliable results, for an original SNR around 2, when processing equivalent 6 cc voxels, which corresponds to an SNR around 4.9 for NAA. Voxel averaging combined with the quantification procedure, which constrained the additional damping factor allowed for each metabolite, was required to achieve the accuracy of the results.
We found significantly greater WM-T _{1,NAA} values in the anterior (frontal) part (1.38 ± 0.15, mean ± SD) than in the posterior part of the slice (1.28 ± 0.10), but we were not able to see this result for the GM or for the other metabolites. Brief et al. [1] also reported similar results between WM frontal (1.59 ± 0.10), and WM parietal (1.35) regions. In the literature, WM-T _{1,NAA} values can differ to some extent. Our anterior WM T _{1,NAA} is lower than the one reported by Brief et al. or Kreis et al. [3] (1.88 ± 0.09), but still higher than others, as for example the value reported by Ethofer [2] (1.19±0.09 in the fronto-parietal region). This discrepancy may be partly due to the way the macromolecular background signal was fitted.
The effect of regional variation in metabolite T _{1} values may be necessary to take into account in the estimated metabolite concentration, depending on the ratio of TR/T _{1} used, on the ratio of the regional T _{1}s, and on the accuracy of the metabolite amplitude estimation. In our study, a regional T _{1} variation for NAA of 7.8% would result in a difference of only 5% in the NAA signal amplitude for a short TR of 1 s. Considering the biological variability and the accuracy achievable for the NAA signal amplitude estimation, this difference might be negligible for a concentration estimation point of view. In the case of healthy versus diseased brain, the metabolite T _{1} difference may be larger than 7%. The measured NAA signal amplitudes could have important differences (greater than 5%), due solely to the T _{1} variation and not to a tissue concentration difference. Conversely, the T _{1} differences could mask the concentration changes due to the disease. Of course, when the repetition time exceeds three expected T _{1}(TR > 3T _{1}), the difference in regional T _{1} can range from 0 to as much as 50%, and the difference (due to the T _{1} weight) in metabolite signal amplitude will remain below 5% of the actual concentration value. Then the regional T _{1} variation will have effectively no effect on the estimated concentrations. In practice, the use of a long TR increases scan time, especially for MRSI acquisition and is therefore avoided.
The T _{1} value found for NAA in the WM of the posterior part of the slice (1.28 ± 0.10) is close to the values reported by Rutgers et al. [5,26] (1.30 ± 0.14) in the centrum semiovale. The T _{1} relaxation times found for the other metabolites Cho, Cr and mI, are also essentially the same as other reported values [2,5,26].
Especially in gray matter, where the glutamate signal is higher, the macromolecular signal, the NAA and the glutamate signals and some contribution from metabolite present at low concentration such as GABA are unknowingly entangled at 2 ppm. Moreover, the amount of macromolecule signal compared to the metabolite signal differs at each TR, as macromolecules have a shorter T _{1} than metabolites. As a consequence, the variability of the quantification results increases. We think that the higher standard error found for the GM-T _{1,NAA} is partly due to this variability and partly due to the few number of available gray matter voxels.
We also observed without reaching statistical significance that, as opposed to water T _{1}, the T _{1} for NAA, Cr-CH_{3} and mI could be greater in the white matter than in the gray matter in the anterior part while no difference was seen in the posterior part. Although a difference of WM/GM voxel distribution in the anterior and posterior regions (see Fig. 1) might have influenced this result, this observation supports the assumption of different underlying mechanisms for water and metabolite relaxation times. While tissue composition and difference in anisotropy may be involved in water T _{1} relaxation process, the intra-cellular metabolite T _{1}, may be more dependent, as suggested by Ethofer et al., on micro-structural characteristics and viscosity properties.
Conclusion
Brain metabolite T _{1} measurements were calculated using a novel MRSI voxel averaging and bootstrapping approach. The proposed method takes advantage of the multi-voxel acquisition provided by MRSI and enables the investigation of regional variations in metabolite T _{1} values. It also introduces a bootstrap technique for estimating a standard error on metabolite T _{1}s. Significant differences were found between anterior WM-T _{1,NAA} and posterior WM-T,_{1,NAA}. This result emphasizes the need to take into account tissue and regional T _{1} differences in MRSI metabolite quantification. Finally, the method only requires a multi-WM/GM voxel acquisition and is not restricted to short echo time 2D MRSI acquisition. The presence of a macromolecular background made the metabolite T _{1} estimation less accurate and substantially increased the dispersion. The principle of the method can be applied and extended to other field strengths (to increase the SNR) or to other types of data acquisition that present less macromolecular contamination such as TE-averaging [27], longer TE acquisition or to using localization in a third spatial dimension (3D-CSI).
Notes
Acknowledgement
We thank Dr. Ying Lu, Associate Professor in Residence of Radiology and Biostatistics at UCSF, for his expert assistance in the statistical aspects of this project and are grateful to Sung Won Chung and Yan Li for helpful discussions. This study was supported in part by Research Grant RG-3517A2 from the National Multiple Sclerosis Society (PI: Daniel Pelletier). Dr. Daniel Pelletier is a recipient of the Harry Weaver Neuroscholar Award from the National Multiple Sclerosis Society (JF-2122-A).
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