Gradient-Echo and Spin-Echo MRI Data from a Phantom Model of Tissue Microstructure

Phantom Design

Spherical polystyrene microbeads of 10 µm, 20 µm and 40 µm nominal diameter were purchased from Microbeads AS (Skedsmokorset, Norway). The beads had a relatively narrow size distribution, with median diameters of 11.1 µm, 23.0 µm, and 40.4 µm respectively. A representative example of the size distribution is shown in Figure 2 of the related manuscript.

For each bead size, a sample of the beads was suspended in a solution of 2% gelatin doped with 0.07% gadobutrol, such that the beads occupied a tenth of the total volume. Each suspension was loaded into a 25-mL serological pipette, which was sealed at both ends with a glue gun. The pipettes were placed in a sonicator to eliminate microbubbles, then rotated continuously for several minutes to ensure uniform mixing of the beads, before being plunged into a bath of ice water to trigger rapid gelling. A fourth phantom containing only the gel medium was constructed as a control.

Pipettes were chosen as containers for the phantoms because they provide a good approximation to an infinite cylinder, thereby minimizing macroscopic field inhomogeneity along most of their length.

Phantom Arrangement

The phantoms were arranged side-by-side in a wrist coil in the following order*:

1. 20-µm bead suspension
2. gel-only control
3. 10-µm bead suspension
4. 40-µm bead suspension

On GRE images, phantoms 1-4 appear left-to-right, with 1 at the left.

On SE images, phantoms 1-4 appear bottom-to-top, with 1 at the bottom.

At the longest echo time, the gel-only control phantom has the highest intensity.

We chose the order above because the sensitivity of the coil is highest at the edges, and this arrangement helped boost the signal-to-noise ratio of the 20-µm and 40-µm bead suspensions, whose signal decayed fastest with echo time.

*In the experiment conducted on June 20, 2021, we imaged two additional phantoms containing 40-µm beads to test the sensitivity of the results to bead volume fraction. In one phantom (far left on the images) the bead density was higher by 2%, and in the other (far right) the bead density was lower by 2%.

Five experiments were performed, each with a fresh set of phantoms.

• For each experiment, the four phantoms were arranged side by side in a 15-channel wrist coil, with their long axes oriented along the long axis of the bore.
• A multi-echo GRE sequence was run in all five experiments. In the last three experiments, a customized SE sequence was added to the protocol (see the related manuscript for more detail).
• In every experiment, noise-only data were also acquired by setting the transmit voltage to zero. These data provided an estimate of the noise covariance among coil elements, which was used to optimize SNR in the reconstructed images.
• Immediately before and after each GRE and SE sequence, we ran an EPI diffusion sequence to estimate the apparent diffusion coefficient of the gel medium.
• In addition, we made independent measurements of the susceptibility difference between the beads and the gel medium using the method illustrated in Figure 3 of the related manuscript.

Files Containing Gradient-Echo Image Data

The files containing the multi-echo GRE image data are labeled using the following convention: gre_image_data_YYYY_MM_DD.mat, where YYYY_MM_DD is the study date. These files contain the following variables:

Variable Name	Size	Class	Description
complex_image_data_4d	224 x 224 x 80 x 32	complex double	Complex-valued images from 80 slices and 32 echoes
te_ms	1 x 32	double	Echo times in milliseconds

Files Containing Spin-Echo Image Data

The files containing the SE image data are labeled using the following convention: se_image_data_YYYY_MM_DD_run_Z.mat. These files contain the following variables:

Variable Name	Size	Class	Description
complex_image_data_3d	96 x 36 x 25	complex double	Complex-valued images from 25 echoes in a single slice
te_ms	1 x 25	double	Echo times in milliseconds

Summary of the differences among runs of the SE sequence.

Date	Run	TE Order	Plane	Position	Slice Thickness	Receiver Gain	Coil Group
2021/08/08	1	ascending	axial	H0.0 mm	10.0 mm	High	HW1
	2	descending	axial	H0.0 mm	10.0 mm	High	HW1
	3	ascending	oblique	H0.0 mm*	10.0 mm	High	HW2*
	4	descending	oblique	H0.0 mm	10.0 mm	High	HW2
	5	ascending	axial	H0.0 mm	10.0 mm	High	HW2
	6	descending	axial	H0.0 mm	10.0 mm	High	HW2
2021/09/26	1	ascending	axial	H0.0 mm	10.0 mm	High	HW2
	2	descending	axial	H0.0 mm	10.0 mm	High	HW2
	3	ascending	axial	H0.0 mm	10.0 mm	Low	HW2
	4	descending	axial	H0.0 mm	10.0 mm	Low	HW2
	5	ascending	axial	H0.0 mm	20.0 mm	Low	HW2
	6	descending	axial	H0.0 mm	20.0 mm	Low	HW2
2022/02/06	1	ascending	axial	H0.0 mm	10.0 mm	Low	HW2
	2	descending	axial	H0.0 mm	10.0 mm	Low	HW2
	3	ascending	axial	H20.0 mm	10.0 mm	Low	HW2
	4	ascending	axial	F20.0 mm	10.0 mm	Low	HW2
* On 2021/08/08, the table was repositioned between run 2 and run 3, so the slice positions are not comparable between runs 1-2 and runs 3-6.

Methodology

K-space data were saved at the end of each experiment and processed offline. Noise-only data were used to estimate the noise covariance among coil elements. For each slice and echo of the image data, the noise covariance matrix was used to combine the separate images from individual coil elements into a single complex-valued image with optimized SNR, Rician noise statistics, and a noise standard deviation of one.

The coil sensitivities were estimated for each voxel by calculating the relative signals among coil elements in that voxel and averaging over all echoes. Due to the large number of echoes and the high signal within the phantoms across all echoes, this averaging process minimized the introduction of correlations between the coil sensitivity vector and the noise itself, thereby preserving the accuracy of the signal time course and the Rician character of the noise statistics.

Note that the signals from different coil elements in a given voxel differ both in magnitude and phase, so the coil sensitivities are complex-valued, as is the resulting combined signal. While the relative phase of the combined signal among echoes contains information about the physics of the system, most notably the local Larmor frequency in the case of the gradient-echo sequence, there remains an overall phase factor that can be set arbitrarily. This factor was chosen such that the phase of the combined signal on the first echo equaled zero.

The above method was applied to all voxels in the imaging volume. However, since voxels in the background contain no true signal, the algorithm used to calculate the coil sensitivities fails in those regions. As a consequence, the calculated values do not accurately reflect the coil sensitivities in the background, and they are not completely free of correlations with the noise itself (although averaging over echoes reduces the correlations substantially). As a consequence, the standard deviation of the noise in the background of the combined images is slightly greater than one. This is true for all echoes except the first, where the phase was set to zero as described above.

Files containing the complex-valued image data after coil combination are available in this repository for both gradient echo and spin echo sequences.

Files Containing Gradient-Echo ROI Data

The files containing the signal intensity time courses as a function of echo time from ROIs in the GRE images are labeled using the following convention:gre_ROI_data_YYYY_MM_DD.mat. They contain the following variables:

Variable Name	Size	Class	Description
uncorrected_intensities_X_sliceY	1 x 32	double	intensity time course in phantom X from slice Y, before correction for noise rectification
corrected_intensities_X_sliceY	1 x 32	double	intensity time course in phantom X from slice Y, after correction for noise rectification
roi_X_sliceY	224 x 224	logical	ROI in phantom X from slice Y
te_ms	1 x 32	double	Echo times in milliseconds
center_frequency_MHz	1 x 1	double	Larmor frequency at isocenter of scanner in megahertz
adc_um2perms	1 x 1	double	Apparent diffusion coefficient of gel medium in µm2/ms

Files Containing Spin-Echo ROI Data

The files containing the signal time courses as a function of echo time from ROIs in the SE images are labeled using the following convention: se_ROI_data_YYYY_MM_DD_run_Z.mat. They contain the following variables:

Variable Name	Size	Class	Description
uncorrected_intensities_X	1 x 25	double	intensity time course in phantom X, before correction for noise rectification
corrected_intensities_X	1 x 25	double	intensity time course in phantom X, after correction for noise rectification
roi_X	96 x 36	logical	ROI in phantom X
te_ms	1 x 25	double	Echo times in milliseconds
center_frequency_MHz	1 x 1	double	Larmor frequency at isocenter of scanner in megahertz
adc_um2perms	1 x 1	double	Apparent diffusion coefficient of gel medium in µm2/ms

Methodology

Magnitude images were obtained by calculating the absolute value of the complex-valued image data. For each slice, an ROI was selected in each phantom on the image with longest echo time. The ROIs were then copied to the corresponding images for all other echo times. The final echo was used for ROI selection because image artifacts (most notably point-like susceptibility artifacts on the gradient-echo images) were most prominent at long echo times, and clear visualization of the artifacts was essential to exclude them from the ROI.

The mean signal intensity within each ROI was calculated for each echo time and corrected for noise rectification (i.e. the bias introduced into the mean by the use of magnitude images rather than complex-valued images). Since the SNR was high across all echoes, the correction was very small, even at the longest echo time.

The gradient-echo sequence was run in a 3D coronal slab that encompassed the phantoms, and therefore yielded several slices for ROI selection. Large ROIs that extended along most of the length of each phantom were selected in each slice.

The spin-echo sequence was applied in a single 2D axial plane, which allowed for the selection of only one small ROI per acquisition. However, since the sequence was repeated several times (in different locations or with different chronological ordering of the echoes) several ROIs could be chosen for each phantom per experiment.

Files containing the signal intensity time courses for each ROI, with and without correction for noise rectification, are available in this repository for both gradient-echo and spin-echo sequences.

Files Containing Magnetic Susceptibility Data

The file is labeled: gre_ROI_data_YYYY_MM_DD.mat and contains the following variables:

Variable Name	Size	Class	Description
delta_chi_X_beads_wrt_gel	5 x 1	double	magnetic susceptibility difference in SI units between the polystyrene material of the beads and the surrounding gel medium, as measured using bead size X
study_date	5 x 1	cell	date of the experiment in which the measurements were made

Notes

The magnetic susceptibility difference between the polystyrene material of the beads and the surrounding gel medium was estimated using the method described in the related manuscript and illustrated in Figure 3 therein. The measurements were repeated for each experiment, since any inaccuracy in the concentration of gadobutrol in the gel medium would cause corresponding changes in the susceptibility of the medium. Slight variations in the magnetic susceptibility value were therefore expected among experiments, and were indeed observed.

However, since all beads were manufactured from the same material, the magnetic susceptibility value was not expected to differ significantly among bead sizes. Nevertheless, the smaller beads consistently yielded higher values than the larger beads. This may be due in part to less uniform mixing of the larger beads (which tended to sink more rapidly during phantom preparation than the smaller beads). Another possible factor may be contributions from higher-order cumulants to the phase evolution of the signal as a function of echo time. In calculating the frequency maps for the susceptibility measurements, the phase was fitted with a linear function of TE, thereby ignoring higher-order cumulants. Those cumulants would have been more important for the larger beads than the smaller beads.

In the interests of full transparency, all measurements are included in the file. However, we have more confidence in the results for the small (10-µm) beads than the larger beads for the reasons outlined above.