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b0 Dependent Neuronal Activation in the Diffusion-Based Functional MRI
Progress in Medical Physics 2019;30(1):22-31
Published online March 31, 2019
© 2019 Korean Society of Medical Physics.

Hyug-Gi Kim1, Geon-Ho Jahng2

1Department of Biomedical Engineering, Graduate School, Kyung Hee University, Yongin, 2Department of Radiology, Kyung Hee University Hospital at Gangdong, College of Medicine, Kyung Hee University, Seoul, Korea
Correspondence to: Geon-Ho Jahng, (ghjahng@gmail.com)
Tel: 82-2-440-6187, Fax: 82-2-440-6932
Received January 14, 2019; Revised March 5, 2019; Accepted March 5, 2019.
This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

Purpose

To develop a new diffusion-based functional magnetic resonance imaging (fMRI) sequence to generate apparent diffusion coefficient (ADC) maps in single excitation and evaluate the contribution of b0 signal on neuronal changes.

Methods

A diffusion-based fMRI sequence was designed with single measurement that can acquire images of three directions at a time, obtaining b=0 s/mm2 during the first baseline condition (b0_b), followed by 107 diffusion-weighted imaging (DWI) with b=600 s/mm2 during the baseline and visual stimulation conditions, and another b=0 s/mm2 during the last activation condition (b0_a). ADC was mapped in three different ways: (1) using b0_b (ADC_b) for all time points, (2) using b0_a (ADC_a) for all time points, and (3) using b0_b and b0_a (ADC_ba) for baseline and stimulation scans, respectively. The fMRI studies were conducted on the brains of 16 young healthy volunteers using visual stimulations in a 3T MRI system. In addition, the blood oxygen level-dependent (BOLD) fMRI was also acquired to compare it with diffusion-based fMRI. A sample t-test was used to investigate the voxel-wise average between the subjects.

Results

The BOLD data consisted of only activated voxels. However, ADC_ba data was observed in both deactivated and activated voxels. There were no statistically significant activated or deactivated voxels for DWI, ADC_b, and ADC_a.

Conclusions

With the new sequence, neuronal activations can be mapped with visual stimulation as compared to the baseline condition in several areas in the brain. We showed that ADC should be mapped using both DWI and b0 images acquired with the same conditions.

Keywords : Brain function, Functional MRI, Diffusion, ADC, BOLD
Introduction

Functional magnetic resonance imaging (fMRI) has the ability to detect local signal increase secondary to behavioral tasks, has provided a powerful tool for cognitive neurosciences, and offers noninvasive access to the cortical and subcortical neuronal functions.1) fMRI relies on the common principle that regional blood flow and metabolism are modulated by neuronal activity.2) The blood oxygen level-dependent (BOLD) contrast is most common approach.3-5) During a brain stimulation, oxygen consumption increases and thus the blood flow increases largely, resulting in a net decrease of the blood deoxyhemoglobin concentration, which, in turn, leads to a measurable signal increase because deoxyhemoglobin is paramagnetic.1)

Diffusion MRI provides quantitative data on water molecular motion which is a very sensitive marker of tissue microstructures.6) The random displacements of water molecules provide valuable information on the cell size or cell orientation in tissues.7) MRI is only the means to observe in vivo diffusion, noninvasively. The diffusion-weighted imaging (DWI) signals are measured with and without applying diffusion gradients to map voxel-based apparent diffusion coefficient (ADC). DWI is routinely used in patients with brain strokes or tumors. Signal increase in DWI reflects a decrease of the ADC of water which has been ascribed to a phase transition of a fraction of water molecules from a fast to a slower diffusion pool in the pathological areas8,9) or cortex undergoing activation.10) In addition, diffusion-based fMRI was applied to measure neuronal responses after some tasks.6,10) The advantage of diffusion-based fMRI is that the signal contribution of blood itself can be reduced by the applied diffusion gradients, means that fMRI signal originates from neuronal tissues rather than blood vascular.

To calculate the ADC value from DWI, the signal, which is acquired without diffusion gradient and is called as b0, is used. The b0 signal may be affected on the signal alternations of diffusion-based fMRI experiments. Therefore, the purpose of this study was to develop a new diffusion-based fMRI sequence to generate ADC maps in a single excitation and to evaluate the b0 signal contribution on the neuronal changes.

Materials and Methods

1. Pulse sequence design

A diffusion-weighted spin-echo pulse sequence has a pair of diffusion gradients around the refocusing pulse. As mentioned before, DWI should be acquired at least twice with (b-value) and without (b0) applying diffusion gradients. Furthermore, to obtain an isotropic ADC value, DWI signals should be acquired with minimum three directions, which are slice-selection, phase-encoding, and frequency-encoding directions. Therefore, we have to scan at least four times to map isotropic diffusion. For the fMRI study, this scan must be repeated to several times with and without stimulations to map neuronal changes. This takes a long scan times so it is not suitable for fMRI and a subject can be moved during scans. Therefore, we developed a new sequence to minimize the scan time during diffusion-based fMRI by two approaches.

First, the diffusion-based fMRI sequence was designed with a single measurement. Therefore, the sequence was designed to acquire a single DWI with applying diffusion gradients in three directions at a time (Fig. 1a).11) To develop the sequence, the pulsed-gradient spin-echo (PGSE) pulse sequence was based.12) For DWI, the diffusion gradients were applied around the refocusing pulse on the three axes, but shifted in each axis that the diffusion gradient was not applied simultaneously. The diffusion gradient on the phase-encoding axis (Gy) was started after applying the first diffusion gradient on the readout-encoding axis (Gx). The diffusion gradient on the slice-selective axis was also started after applying the first diffusion gradient on the phase-encoding axis (Gz). In this way, three diffusion gradients do not interact with the diffusion pairs. The duration of the applied diffusion gradient in each axis was shorter than those applied on the general PGSE diffusion gradient. The diffusion weighting factor b is given by

b=γ2g2[δ2(Δ-δ3)+ε330-8ε26]

in which γ is the gyromagnetic ratio, g is the gradient strength and δ is the duration of the diffusion gradient, Δ is length of slope (ramp time), and ¥Ä is the diffusion time. In our experiments (Fig. 1b), δ was set to 8.27 ms, Δ was 35 ms, ε was 0.98 ms and the maximum gradient strengths were was 35.6 mT/m for both Gx and Gy and –35.6 mT/m for Gz. Diffusion gradients were applied along three orthogonal axes simultaneously (i.e., b=bx+by+bz with |bx|=|by|=|bz|). The b-factor for each axis was 200 s/mm2.

Second, b0 images, which were acquired without applying diffusion gradient, were acquired to twice to quantify the ADC map for each DWI scan (Fig. 2a). The ADC map can be calculated by combining at least two DWIs that are differently sensitized to diffusion, but remain identical with respect to the other parameters, for instance, the image S0 without diffusion weighting (b=0 s/mm2) and one DWI (b>0 s/mm2). We calculated an ADC value for each pixel with the following equation.

ADCi=-1bIn(SiS0)

Where So and Si were the signal intensities in a given voxel without (b0) and with applying the diffusion sensitization gradient at the time i respectively. Therefore, to map ADC, DWI with certain b-value and b=0 should be always acquired. However, it is inefficient and it needs long scan time to scan S0 image every time point with DWI. Therefore, the diffusion-based fMRI sequence was designed to scan optimally (Fig. 2a). One DWI with b=0 s/mm2 was obtained during the first baseline scan (b0_b) and then 107 DWIs with b=600 s/mm2 were imaged with altering the baseline and stimulation conditions. Finally, another DWI with b=0 s/mm2 was again obtained during the last stimulation scan (b0_a). Therefore, two b0 images were scanned with the baseline and activation conditions, separately. ADCs were mapped with three different ways: (1) using b0_b (ADC_b) for all 107 time points, (2) using b0_a (ADC_a) for all 107 time points, and (3) using both b0_b for the baseline DWI scans and b0_a for the stimulation DWI scans (ADC_ba) . Therefore, we had diffusion-based fMRI data for DWI which was the original dynamic diffusion-weighted serial images, ADC_b serial images, ADC_a serial images, and ADC_ba serial images. The diffusion-based fMRI sequence was implemented in the Philips pulse programming environment (Rel ver. 3.6.0).

2. Subjects

Sixteen healthy volunteers (7 males, 9 females) were recruited from the local community. None of the volunteers had neurological or psychiatry disease history. All young controls underwent Korean version of Mini-Mental State Examination (K-MMSE) test based on global cognitive ability and brain fMRI scans. One of the subjects had too many motion-related artifacts and therefore fifteen subjects were finally included in this study. Mean ages were 26.4±2.6 years (range: 20 to 30 years). Mean K-MMSE scores were 29.7±0.7 (the maximum score is 30). This study was performed with approval of the local institution review board and all subjects were provided prior written consent.

3. Functional magnetic resonance imaging paradigm

In this study, two functional sessions, which were BOLD fMRI and DWI-based fMRI, were performed from each subject. The paradigm for visual stimulation was used for two conditions which had a fixation for the baseline condition and a black-and-white checkerboard for the stimulation condition (Fig. 2b). A cross-sign was used for the fixation condition. This alternation was repeated three times for each session. The duration of each block was 63 seconds for the diffusion-based fMRI and 54 seconds for BOLD fMRI. Number of scans per each block was 18 for both diffusion-based fMRI and BOLD fMRI. There were 109 and 108 dynamic volumes for the diffusion-based fMRI and BOLD fMRI, respectively. The stimulation paradigm was presented using the Esys fMRI Manager Client system (Invivo, Orlando, FL, USA).

4. Magnetic resonance data acquisitions

MR imaging was performed on a 3 Tesla MRI System (Achieva; Philips Medical System, Best, the Netherlands) with an 8 channel phase-array sensitivity-encoding (SENSE) coil. The diffusion-based fMRI data were acquired with our sequence with the following parameters: repetition time (TR)=3,500 ms, echo time (TE)=70.41 ms, field of view (FOV)=230×230×120 mm, matrix size=64×49 mm, voxel size=2.4×2.4×6 mm3, slices=42, SENSE factor=2, and flip angle=90°. The max b-factor was 600 s/mm2. The scan time was 6 minutes 28 seconds.

To compare brain activations with the diffusion-based fMRI, BOLD fMRI data were also acquired using a gradient-echo echo-planar imaging sequence with the following parameters: TR=3,000 ms, TE=30 ms, FOV=230×120×230 mm, matrix size=64×64 mm, voxel size=2.4×2.4×6 mm3, direction=axial, slice thickness=6 mm without gap, slices=20, SENSE factor=2, and flip angle=90°. The scan time was 5 minutes 45 seconds.

In addition, volumetric three-dimensional T1-weighted images (3D T1WI) were acquired with the magnetization prepared rapid acquisition gradient-echo sequence for the fMRI processing step with the following parameters: TR=8.2 ms, TE=3.8 ms, inversion time=1,022 ms, flip angle=8°, SENSE factor=2.5, FOV=236×236×170 mm, matrix size=236×236 mm, slice thickness=1 mm, voxel size=1×1×1 mm3, and the number of slice=340.

5. Data processing

Statistical parametric mapping version 5 (Wellcome Department of Imaging Neuroscience, University College London, London, UK) software was used for data preprocessing and statistical analysis. First of all, the functional images of each subject were realigned to the first volume to minimize the effect of head motion during the brain scan and were created the mean image. And then, the functional mean image was co-registered to the corresponding structural 3D T1WI. After that, 3D T1WI and the corresponding fMRI data were spatially normalized onto the standard Montreal Neurological Institute template by using the non-linear transformations.13) The functional images were then smoothed with full width at half maximum of Gaussian kernel of 8×8×10 mm.

6. Statistical tests

For the individual level (1st level) analysis, the activation or deactivation with the visual stimulation compared with the baseline condition for each subject was calculated for each session by a general linear model approach14) based on a voxel-by-voxel calculation. Therefore, there were two contrast maps, activation or deactivation for each session of the diffusion-based fMRI or BOLD fMRI data.

For the group level (2nd level) analysis, the two contrast maps obtained from the individual level analysis were used to investigate average of all subjects between stimulation and baseline conditions with using one-sample t-test. For all fMRI data, the activated or deactivated map during the visual stimulation against the baseline condition was identified with P<0.001 with corrected for multiple comparisons using a false discovery rate and the extent threshold was 10 voxels. To find anatomical areas of the brain, the t-test results were standardized into Talairach coordinate15)by using the GingerALE and the Talairach Client (University of Texas Health Science Center San Antonio) software.

Results

Fig. 3 shows the results of the voxel-based group analysis for 15 subjects using the one sample t-test for BOLD (Fig. 3a) and ADC_ba (Fig. 3b) data. Deactivated or activated areas during visual stimulation are overlaid on the standard axial T1 template and are shown with red or blue colors, respectively. The BOLD data had only activated voxels without any deactivated voxels. A strong activation was shown in the occipital lobe. However, ADC_ba data were observed both deactivated and activated voxels and activation areas were spread in several brain areas. There were no statistically significantly activated or deactivated voxels for DWI, ADC_b, and ADC_a.

In the BOLD data, BOLD signals were increased with the visual stimulation condition compared to the baseline condition, mainly in the right lingual gyrus and the left inferior temporal gyrus in the occipital lobe. The detailed results are summarized in Table 1. In the ADC_ba data, neuronal activations were increased or decreased with the visual stimulation condition compared to the baseline condition. The increased activation areas were in the right lingual gyrus of the occipital lobe which was the same area with the BOLD fMRI data. The Z-score was similar to that with the BOLD fMRI result. Additional increased areas were observed in the frontal lobe, the parietal lobe, the temporal lobe, the limbic lobe. With the ADC_ba data, we also found in deactivated areas in the occipital lobe, the frontal lobe, the parietal lobe, and the temporal lobe. The detailed results are listed in Table 2.

Discussion

1. Important to select the b0 scan to map apparent diffusion coefficient series for each condition

Result of the diffusion-based fMRI showed that neuronal activation depended on the b0 signal. In this experiment, we acquired two different b0 images: one during the baseline condition and another during the stimulation condition. ADC maps were obtained with DWI with b=600 s/mm2 using S0 acquired during the baseline condition, using S0 acquired during the stimulation condition, or using S0 acquired during both conditions. Our result showed that ADC mapped using both DWI and b0 images acquired with the same condition had high sensitive to map neuronal activations. Previous diffusion-based fMRI studies were not taken account into this issue.1,6,10) If the stimulation condition between the b0 and DWI scans does not match, then neuronal activation can be minimized.

There are several published papers to study BOLD, diffusion-weighted, and ADC-fMRI.16,17) Previous published studies were conducted as a stimulation of the visual system using a general diffusion gradient based sequence paradigm. The sequence, we developed, can be used to DWI scans with a relatively low b-value because of using the bipolar gradient for each axis. It may be problem to use it in the clinical applications such as strokes and tumors because it may be not sensitive to detect small lesions. However, in the fMRI study, we do not need to use a high b-value because only the vascular signal elimination is enough to obtain localized neuronal signals. In general, less than b=600 s/mm2 is enough to using in fMRI study.1,18) Therefore, our developed sequence is good enough to use in the fMRI study. However, the b=600 s/mm2 may be somewhat large.

2. Diffusion-based functional magnetic resonance imaging compared to blood oxygen level-dependent functional magnetic resonance imaging

The main advantage of using diffusion-based fMRI sequence over BOLD fMRI is that the signal acquired diffusion-based fMRI is originally from the neuronal tissue rather than macrovascular structures. Most of BOLD fMRI signals are usually came from the macroscopic vascular structure in the brain. However, a diffusion-weighted spin-echo pulse sequence has a pair of diffusion gradients straddling the radio frequency refocusing pulse. Therefore, vascular signal cannot contribute if the spins in the blood have unexpected both 90 excitation and 180 refocusing pulse. Therefore, the sources of signals from the diffusion-based fMRI are much more localized to the neuron compared to those of BOLD fMRI.

However, diffusion-based fMRI has limitation to a long acquisition time because at least four scans are required to map the isotropic ADC value for each stimulation condition. In addition, the echo time of the diffusion-based fMRI is relatively long compared to that of BOLD fMRI. This causes signal reduction in the diffusion-based fMRI. In this study, we designed the sequence to obtain the isotropic ADC value with in a single scan.11) This is a large advantage to enhance the temporal resolution of diffusion-based fMRI study. The proposed method can be also reduced the eddy current-related artifact because of applying the bipolar gradient for each axis. The eddy current-related artifact caused by the applied diffusion gradient is usually shown in the imaging distortion.19)

3. Limitation of this study

First, the b-value of 600 s/mm2 was used in the diffusion-based fMRI. This value may be too large and most of neuronal activation areas may not detect with the large b-value. This causes that neuronal activation did not detect with DWI series. Therefore, in the future study, a relatively small b-value must be used to evaluate functional changes during brain stimulations. Second, TE for the diffusion-based fMRI was 70.41 ms, which is relatively long compared to that of BOLD fMRI (TE=30 ms). Therefore, to optimize the diffusion-based fMRI, the echo time should be reduced up to at least 40 ms at 3T MRI. To achieve this short echo time, the b-value should be also reduced to <100 s/mm2 for each axis. Finally, activated and deactivated areas with ADC_ba series should be further investigated to understand the mechanism of the diffusion-based fMRI. There were several activation areas in addition to the occipital cortex. We do not know why these areas had increased signals during the stimulation condition. Deactivation was caused by the downgrade neuronal function during the stimulation condition. Therefore, our result, which is shown in the deactivated areas during the stimulation condition, should be not artifact.

Conclusions

To evaluate the b0 dependence of the diffusion-based fMRI study, we developed a new diffusion-based fMRI sequence to generate ADC maps in a single excitation. With the new sequence, neuronal activations can be mapped with the visual stimulation condition compared to the baseline condition in the several areas in the brain. It does not need to obtain a DWI with b=0 s/mm2 in each time point. ADC should be mapped with using both DWI and b0 images acquired with the same condition. Our proposed method is possible to be short scan time by obtaining only two DWI with b=0 s/mm2 that one is during a baseline scan and another is during a stimulation scan. Therefore, our optimized sequence can be used in neurosciences and clinical populations.

Acknowledgements

This study was supported by a grant of the Korean Health Technology R&D Project, Ministry for Health, Welfare & Family Affairs, Republic of Korea (A092125) and the Convergence of Conventional Medicine and Traditional Korean Medicine R&D program funded by the Ministry of Health & Welfare through the Korea Health Industry Development Institute (KHIDI) (HI16C2352).

Conflicts of Interest

The authors have nothing to disclose.

Availability of Data and Materials

All relevant data are within the paper and its Supporting Information files.

Ethics Approval and Consent to Participate

The study was approved by the institutional review board (IRB approval number: KHNMC IRB 2009-056).

Figures
Fig. 1. Diagrams of the diffusion-based functional magnetic resonance imaging sequence (a) and the gradient for diffusion-weighting (b). Diffusion gradients were applied on the three axes to map isotropic apparent diffusion coefficient in a single acquisition. RF, radio frequency; Gx, readout direction; Gy, phase-encoding direction; Gz, slice-selection direction; TE, echo time; TR, repetition time; Echo, echo signal; EPI, echo-planar imaging; g, gradient strength; δ, duration of the diffusion gradient; ε, length of slope (ramp time); Δ, diffusion time.
Fig. 2. The scan orders (a) and the stimulation paradigm (b) for the diffusion-based functional magnetic resonance imaging (fMRI) acquisitions. The same stimulation paradigm was applied on the blood oxygen level-dependent fMRI scans. b0_b, the first baseline scan; b0_a, the last activation scan; DWI, diffusion-weighted imaging.
Fig. 3. Results of blood oxygen level-dependent (BOLD) (a) and ADC_ba (b) for activated and deactivated brain regions for visual stimulation over 15 subjects. The results for activation and deactivation are observed with P<0.001 for correcting multiple comparisons using a false discovery rate, respectively. In the BOLD data, increased (red) neuronal activations were found during the visual stimulation, and no deactivation areas were observed. In the ADC_ba data, increased (red) and decreased (blue) neuronal activations were found during the visual stimulation. ADC_ba was apparent diffusion coefficient (ADC) mapped with using both b0_b and b0_a for the baseline scans and activation scans, respectively. The color-coded maps are overlaid on the standard axial T1 template.
TABLES
Table 1

Significantly activated brain regions during the visual stimulation using the BOLD method in all subjects

RegionBATalairach coordinateZ-scoreCluster size

XYZ
Activation (stimulation “>” baseline)
Occipital lobe
Right
Lingual gyrus188.04–87.35–5.666.468,711
Left
Inferior temporal gyrus17–15.99–90.60–9.986.438,711
Sub-lobar
Right
Lateral geniculum-23.05–25.59–3.165.57136
Left
Lateral geniculum-–19.55–27.41–2.255.2286

Results of one-sample t-test with P<0.005, correcting for multiple comparisons using a false discovery rate and the extent threshold with 10 voxels.

There were no deactivation (visual stimulation<baseline) areas.

BOLD, blood oxygen level-dependent; BA, Brodmann area; -, not available.


Table 2

Significantly activated or deactivated brain regions during the visual stimulation for ADC_ba data in all subjects

RegionBATalairach coordinateZ-scoreCluster size

XYZ
Activation (stimulation “>” baseline)
Occipital lobe
Right
Lingual gyrus-13.58–65.541.906.471,427
1917.39–66.55–7.146.401,427
1924.74–68.80–3.625.931,427
Frontal lobe
Right
Precentral gyrus4454.5212.688.206.542,405
Superior frontal gyrus919.1759.0435.425.05509
24.8561.7426.764.99509
Inferior frontal gyrus4723.4513.94–22.834.1517
Superior frontal gyrus84.0140.0856.793.7815
Left
Precentral gyrus6–51.101.3512.55Inf.3,084
Superior frontal gyrus100.7763.5628.335.52509
10–34.3554.9521.514.3857
Middle frontal gyrus10–25.1062.1824.164.2857
Superior frontal gyrus9–17.8259.4033.033.8657
8–38.5322.1948.974.2341
Precentral gyrus4–57.09–16.4941.394.0225
Superior frontal gyrus6–3.4238.0858.283.8515
Parietal lobe
Left
Postcentral gyrus40–53.11–23.5617.365.5840
3–49.80–13.6750.793.9725
Temporal lobe
Left
Superior temporal gyrus13–56.90–40.6619.285.1852
Middle temporal gyrus22–54.83–37.243.434.1052
Superior temporal gyrus13–43.421.54–8.927.123,084
Limbic lobe
Right
Posterior cingulate306.11–55.029.985.07482
Left
Parahippocampal gyrus, amygdala-–28.59–7.51–13.137.734,307
Posterior cingulate29–6.86–51.4011.905.37482
23–1.35–57.1913.255.06482
Anterior cingulate33–1.1415.6518.354.7226
Sub-lobar
Right
Insula1343.22–8.8116.797.472,405
46.80–18.8423.105.782,405
Left
Insula13–41.80–0.218.95Inf.3,084
Deactivation (baseline ">" stimulation)
Occipital lobe
Right
Cuneus195.70–85.4132.324.1411
Left
Cuneus19–16.57–82.0937.664.3712
Frontal lobe
Right
Precentral gyrus435.51–20.1737.207.0868
Middle frontal gyrus933.8526.9436.236.921,078
926.5238.5033.606.911,078
616.783.9260.796.851,078
67.44–11.2962.795.2550
65.651.9462.215.4857
Left
Middle frontal gyrus9–21.6336.7234.427.101,388
8–1.3926.6041.016.421,388
8–14.3137.8441.856.301,388
6–1.76–10.8959.075.0250
Parietal lobe
Right
Inferior parietal lobule4040.84–52.9245.00Inf.414
46.47–50.5739.917.31414
Inferior parietal lobule3942.73–63.5938.615.19414
Superior parietal lobule724.01–66.9254.204.3081
Left
Inferior parietal lobule40–46.11–55.8539.656.10147
Supramarginal gyrus40–51.57–53.2632.595.78147
Inferior parietal lobule39–46.14–65.1738.764.98147
Precuneus7–25.88–76.9743.394.2221
Temporal lobe
Right
Transverse temporal gyrus4143.22–27.0911.455.23117
Middle temporal gyrus3952.45–54.745.385.2628
Left
Superior temporal gyrus22–55.07–59.1315.765.1929
Sub-lobar
Right
Insula1337.63–21.8215.467.13117

Results of one-sample t-test with P<0.001, correcting for multiple comparisons using a false discovery rate and the extent threshold with 10 voxels.

ADC_ba, apparent diffusion coefficient (ADC) mapped with using both b0_b and b0_a for baseline scans and stimulation scans, respectively.

BA, Brodmann area; -, not available; Inf., infinite.


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