Influence of Diffusion Time and Temperature on Restricted Diffusion Signal: A Phantom Study

Abstract

Purpose

Diffusion MRI is a physical measurement method that quantitatively indicates the displacement of water molecules diffusing in voxels. However, there are insufficient data to characterize the diffusion process physically in a uniform structure such as a phantom. This study investigated the transitional relationship between structure scale, temperature, and diffusion time for simple restricted diffusion using a capillary phantom.

Methods

We performed diffusion-weighted pulsed-gradient stimulated-echo acquisition mode (STEAM) MRI with a 9.4 Tesla MRI system (Bruker BioSpin, Ettlingen, Germany) and a quadrature coil with an inner diameter of 86 mm (Bruker BioSpin). We measured the diffusion coefficients (radial diffusivity [RD]) of capillary plates (pore sizes 6, 12, 25, 50, and 100 μm) with uniformly restricted structures at various temperatures (10ºC, 20ºC, 30ºC, and 40ºC) and multiple diffusion times (12–800 ms). We evaluated the characteristics of scale, temperature, and diffusion time for restricted diffusion.

Results

The RD decayed and became constant depending on the structural scale. Diffusion coefficient fluctuations with temperature occurred mostly under conditions of a large structural scale and short diffusion time. We obtained data suggesting that temperature-dependent changes in the diffusion coefficients follow physical laws.

Conclusion

No water molecules were observed outside the glass tubes in the capillary plates, and the capillary plates only reflected a restricted diffusion process within the structure. We experimentally evaluated the characteristics of simple restricted diffusion to reveal the transitional relationship of the diffusion coefficient with diffusion time, structure scale, and temperature through composite measurement.

Introduction

Diffusion MRI (dMRI) is a physical measurement method that quantitatively indicates the displacement of water molecules diffused in voxels.¹ The diffusion coefficient given by the dMRI contains information about water molecules diffusing in the direction and speed within the measurement object. Recently, many histological visualization studies have been reported in the medical field as the diffusion coefficient estimates the structure at the micron level.^2,3

The diffusion coefficients are quantitative values that are affected by several factors. The b-value and diffusion time are instrumental factors, whereas the structural scale and temperature are object factors. There have been many reports on the relationship between the b-value^4,5 and the constituent material of the measurements⁶ with the diffusion coefficient. From the root-mean-square displacement, diffusion time duration determines suitable diffusion scales to observe. However, the relationship between diffusion time and the diffusion coefficient has not been investigated sufficiently. Previous studies have used multiple diffusion times to observe neural tissues, such as the brain cortex.^7–9 These reports state that, in biological organisms, restricted and anisotropic diffusion occurs in cell membranes, nerve fibers, as well as other tissues. Therefore, understanding diffusion in simple structures, such as phantoms, is important to understand the complex diffusion processes in vivo.

The phenomenon of water diffusion has been reported in dMRI with phantoms on fibers^10,11 and glass capillaries.^12–14 These phantom studies were based on the estimation of axonal geometries in the cerebral white matter and a diffusion model in which the water molecules were observed inside and outside the restricted structures verified. No water molecules were observed outside the glass tube in the capillary late in this experiment, and simple restricted diffusion was investigated. It is necessary to consider the measurement environment in detail for the precise measurement of diffusion coefficients in multivariate analysis.

Temperature is a factor causing fluctuations in the diffusion coefficients in the measurement environment. From the Stokes-Einstein equation, the diffusion coefficient depends on the temperature; this has been confirmed by observed measurements.¹⁵ The temperature dependence of the diffusion coefficient has been used for temperature mapping^16–18 and apparent diffusion coefficients (ADC) representations of brain temperature variation. On the other hand, the effect of temperature dependence causes differences between the ex vivo phantom measurements at room temperature and the in vivo measurements at biological temperature. Moreover, water molecules in vivo have restricted diffusion, and their diffusion behavior differs depending on the diffusion spatial scale. However, the relationship between the structure size and temperature in the diffusion coefficient has not been investigated. By measuring the phantoms at different temperatures, a better understanding of the temperature dependence of restricted diffusion can be gained.

In this study, we characterized restricted diffusion using capillary plates of several sizes with uniform structures at various temperatures and multiple diffusion times. By measuring water diffusivity, we investigated the relationship between diffusion time, structural scale, and temperature dependence of restricted diffusion.

Materials and Methods

Capillary phantom

The capillary plates (Hamamatsu Photonics, Shizuoka, Japan) had circular holes with porosity of over 55% in 2 mm thick lead glass and were composed of water-impermeable capillaries arrayed in two dimensions (Fig. 1a). In this study, we used two plates, each with pore sizes of 6, 12, 25, 50, and 100 μm. We filled them with pure water in a centrifuge tube (Iwaki, Tokyo, Japan) with an internal diameter of 29 mm and vacuum degassed the phantom. We measured the free water inside the phantom without capillary plates.

Fig. 1.

(a) Capillary plates. The capillary plate observed without magnification, MRI, and microscopy. We used pore sizes of 6, 12, 25, 50, and 100 μm. (b) Phantom case in cross section. Isothermal water can be circulated around the capillary phantom (green). Direct monitoring of the temperature in the case by inserting a temperature probe (yellow) into the case inlet or outlet. (c) Constant temperature measurement system layout. The water temperature was adjusted outside the MRI room and the isothermal water was circulated around the phantom case through insulated tubes.

Constant temperature measurement system

We created a phantom case using a 3D printer to circulate isothermal water around the phantom (Fig. 1b). The temperature dependence was evaluated by circulating isothermal water (10ºC, 20ºC, 30ºC, and 40ºC) through the phantom case and the insulated tube. During the experiments, the temperature was directly monitored with a temperature probe (BIOPAC Systems, Goleta, CA, USA) placed at the outlet of the case. We adjusted the water temperature by checking the displayed values in the circulation system (Bruker Biospin). After adjusting the temperature in the circulatory system, isothermal water flowed into the MRI room through a tube, passed the phantom case, and returned to the system through the outflow tube (Fig. 1c).

MRI protocol

MRI was performed using a 9.4 Tesla system (660 mT/m, Bruker Biospin) and a quadrature coil with an inner diameter of 86 mm (Bruker Biospin). The imaging parameters for diffusion-weighted pulsed-gradient stimulated-echo acquisition mode (STEAM) were as follows: diffusion time (Δ-δ/3), 12, 25, 50, 100, 200, 300, 400, 500, 600, 700, and 800 ms; diffusion direction, 12; b-value, 900 s/mm²; gradient duration (δ), 2.4 ms; TR, 6000 ms; TE, 15 ms; FOV, 60 × 60 mm²; matrix size, 32 × 32; resolution, 1.875 × 1.875 mm²; slice thickness, 2 mm; number of averages, 1; and acquisition time, 7 h 40 min. All measurements were started at least three hours after the water temperature stabilized for the phantom to reach the set temperature.

Data analysis

The data were obtained from diffusion tensors analyzed using the diffusion toolkit¹⁹ to obtain the eigenvalues (λ₁, λ₂, and λ₃). The calculated radial diffusivity (RD) indicates the diffusion direction orthogonal to the running of the glass capillary in the phantom. RD was obtained using the following equation: RD = (λ₂ + λ₃)/2. Averaging reduces the distortion of the eigenvalue repulsion owing to noise.²⁰ The data were measured within a circular ROI placed on the phantom region of the RD image, and the mean values were plotted for each structural size and temperature. In addition, an exponential approximate curve was added as an eye guide using linear regression after logarithmic conversion. From the graphs, we evaluated the relationship between the diffusion time, structural scale, and temperature dependence. We calculated SNR based on the guidelines from the National Manufacturers Electrical Association (NEMA)²¹ and measured the mean signal value S at the same location of ROI and the average signal N at the four corners outside of the phantom of the diffusion-weighted imaging (DWI).

$$\mathrm{S}\mathrm{N}\mathrm{R}=\left(\mathrm{S}\sqrt{\pi /2}\right)/\mathrm{N}$$

Results

DWI and RD images at 10°C and 40°C in diffusion time 50 and 800 ms are shown in Fig. 2. The signal intensity varies in each axis in the DWI images. Table 1 presents the diffusion gradients and SNR for the 12 axes. At the lowest SNR (diffusion time, 800 ms; 40ºC), the average SNR of the axes was sufficient to confirm the quality as robust in the diffusion tensor analysis of raw data in this experiment.²²

Fig. 2.

DWI and RD of the capillary phantom at 10ºC and 40ºC in diffusion time 50 ms and 800 ms. Each image data is sorted from top to bottom as 6, 12, 25, 50, 100 μm, and free water. DWI outputs the image averaged over 12 axes with the signal value at b0 of each structure size as 100%, and the color bar is displayed at the bottom of the table. RD outputs map after diffusion tensor calculation. DWI, diffusion-weighted imaging; RD, radial diffusivity.

Table 1.

Diffusion gradients in 12 axes and SNR for each structure size at 20°C and 40°C(a) Diffusion time 50 ms

	Diffusion gradients			SNR
				20ºC						40ºC
	Gx	Gy	Gz	6 μm	12 μm	25 μm	50 μm	100 μm	Free water	6 μm	12 μm	25 μm	50 μm	100 μm	Free water
1	0.263	0.067	0.963	193	178	197	161	156	238	103	94	93	94	82	141
2	−0.426	0.141	0.894	236	252	234	185	153	263	153	139	141	109	100	152
3	−0.121	−0.560	0.819	319	337	246	202	178	265	212	212	155	120	105	143
4	0.742	−0.275	0.612	480	389	273	200	178	277	530	486	328	170	171	188
5	0.402	0.658	0.636	401	373	296	255	185	274	452	405	318	164	147	170
6	−0.371	0.751	0.547	569	462	367	247	186	262	710	603	409	205	156	171
7	−0.889	−0.091	0.449	789	785	447	253	177	223	861	754	504	226	164	199
8	−0.611	−0.724	0.321	1048	736	399	255	151	197	1045	973	540	279	159	200
9	0.374	−0.848	0.374	995	733	365	225	184	200	986	868	593	263	176	188
10	0.910	0.325	0.257	918	688	436	249	159	188	1275	1233	744	262	171	181
11	0.125	0.989	0.077	903	844	416	253	217	238	1469	1204	686	288	178	206
12	−0.861	0.503	0.079	869	715	352	244	178	230	1375	1234	625	301	165	220

Diffusion tensor eigenvalues

Figure 3 shows the eigenvalues λ₁, λ₂, and λ₃ from diffusion tensor at 10ºC, 20ºC, 30ºC, and 40°C with diffusion time. The principal eigenvalue λ₁ diffuses along the glass capillary within the phantom and can be regarded as free diffusion. Similarly, for free water at λ₂ and λ₃, there was no decay with diffusion time, indicating that free diffusion occurs without limitation in the direction of water molecule diffusion. On the other hand, λ₂ and λ₃ in the plates are orthogonal to the direction of running of the capillary; thus, restricted diffusion occurs depending on the capillary pore size, decreasing with diffusion time. In cylindrically symmetric diffusion with phantoms, the theoretical value λ₂ = λ₃; however, the results consistently showed λ₂ > λ₃ due to the presence of statistical sampling bias caused by noise, resulting in a systematic underestimation of λ₃.²⁰

Fig. 3.

Diffusion tensor eigenvalues of capillary phantom at each temperature as a function of diffusion time. From the top are 10°C, 20°C, 30°C, and 40°C, and each row from left: λ₁, λ₂, and λ₃. At all temperatures, λ₁ and free water (blue) did not decay, while λ₂ and λ₃ decreased in the order of structural scale from 6 μm (red), 12 μm (orange), 25 μm (yellow), 50 μm (green), and 100 μm (light blue).

Structural scale dependence

Figure 4 shows RD plotted at each temperature with diffusion time for all capillary sizes. Larger structural scales shifted toward more gentle decay curves. The same RD data re-plotted for each capillary size at respective temperatures are shown in Supplementary Fig. 1. From Supplementary Fig. 1, RD decreased in the short diffusion time phase for smaller capillary size at all temperatures. Especially up to diffusion time of 100 ms, RD decreased by 60%–85% at 6, 12, and 25 μm, while at 50 and 100 μm, RD decreased by only 15%–40%. For free water, RD did not change with diffusion time due to free diffusion of the water molecules. RD became constant in order of smaller structural scale with diffusion time.

Fig. 4.

RD as a function of diffusion time at each temperature for the respective capillary size. Upper row from left: 6 μm, 12 μm, 25 μm; lower row from left: 50 μm, 100 μm, free water. Temperatures were plotted at 10°C (light blue), 20°C (green), 30°C (orange), and 40°C (red). RD, radial diffusivity.

Supplementary Fig. 1.

Re-plotted data from Fig. 4 as the radial diffusivity of capillary size at respective temperatures as a function of diffusion time. From left at 10°C, 20°C, 30°C, and 40°C, at all temperatures from bottom to top: 6 μm (red), 12 μm (orange), 25 μm (yellow), 50 μm (green), 100 μm (light blue), and free water (blue). The attenuation was greater for the smaller structure scales at each temperature.

Supplementary Fig. 2 shows RD re-plotted for each diffusion time with increasing temperature. For small capillary sizes, such as 6–12 μm, RD converged to the same value after diffusion time 50 ms. In addition, for 25 μm, the values converged after diffusion time of 400 ms.

Supplementary Fig. 2.

Re-plotted data from Fig. 4 as radial diffusivity (RD) of capillary size at respective diffusion times as a function of temperature. Upper from left to diffusion time 12 ms, 25 ms, 50 ms, 100 ms, middle from left to diffusion time 200 ms, 300 ms, 400 ms, 500 ms, and lower from left to diffusion time 600 ms, 700 ms, 800 ms. All diffusion times, from the top down, free water (blue), 100 µm (light blue), 50 µm (green), 25 µm (yellow), 12 µm (orange), and 6 µm (red). The slope of the lines indicates the rate of RD shifts due to temperature change.

Temperature dependence

From Fig. 4, the spread of the graphs indicates the effect of temperature change. Increasing the structural scale caused a larger difference in measurements, and fluctuations in RD varied with temperature. For all capillary sizes except for free water, the RD difference between the temperatures decreased with diffusion time. The RD distribution for different temperatures and structural scales for each diffusion time is shown in Fig. 5 as contour plots. At short diffusion time, RD ranged widely, following the structural scale. With a longer diffusion time, the RD distribution became flat, and there was only a slight difference in RD with temperature, unless it was close to free diffusion. In Supplementary Fig. 2, the slope of the graph indicates the strength of the temperature dependence of the RD, and RD fluctuation occurs with temperature as the slope increases. At 6–12 μm, there was almost no change in RD with temperature for all the diffusion time. With increasing diffusion time, the fluctuation of RD with temperature decreases for smaller structural sizes.

Fig. 5.

Contour plots of the relationship of RD with temperature and structure scale for each diffusion time. The distribution of RD follows the color bar in the lower right. Upper left to diffusion time 12 ms, 25, ms, 50 ms, 100 ms, middle left to diffusion time 200 ms, 300 ms, 400 ms, 500 ms, and lower left to diffusion time 600 ms, 700 ms, 800 ms. RD was widely distributed in shorter diffusion time and larger structure scale. RD, radial diffusivity.

Discussion

In this study, we measured the RD of glass capillary plates of several sizes at different temperatures using multiple diffusion times. From the observed data, we investigated the relationship between the structural scale, temperature, and diffusion time with the diffusion coefficients in RD.

Diffusion time

From the Einstein–Smoluchowski equation,²³ the root-mean-square displacement (MSD) is $\mathrm{M}\mathrm{S}\mathrm{D}=\sqrt{6\mathrm{D}\mathrm{t}}$ in three dimensions of water molecules diffusing after t (s), where the diffusion coefficient of water at 20°C is D = 2.0 × 10⁻⁹ (m²/s) based on previous studies.²⁴ We used this equation in the present experiment. The t (s) obtained is the time for water molecules to reach the structure wall depending on the structure scale and start to decrease RD, increasing exponentially to 3.0 ms at 6µm, 12 ms at 12 µm, and 833 ms at 100 µm. t (s) is the theoretical value of diffusion time for decreasing RD, indicating that the diffusion time required to estimate the structure depends on the structural scale. In smaller structures, RD becomes constant immediately after decreasing because the distance to the structure wall is shorter, indicating that almost all water molecules inside the structure hit the inner wall and reach a steady state in a short time. In contrast, at larger structural scales, the distance to reach the structural wall increases; therefore, the time needed to decrease RD increases with distance. In previous studies,^11,25 it has been theoretically predicted that the decay curve of the diffusion coefficient changes with the structural diameter. In this experiment, the decay curve of RD changed to a gentle curve with an increase in the structural scale; the process of diffusion time elongation for the RD to become constant was confirmed by actual measurements (Fig. 4).

From the Monte Carlo simulations of intra-axonal diffusion,^11,25 the behavior at long diffusion times differed between periodic and random structures, and suggested that long diffusion times have higher sensitivity to fluctuations in structure. We also found faster decay than the inverse power law at 6–50 μm, which satisfied the theoretical diffusion time value required for structure estimation under our experimental condition of 800 ms, reflecting the periodic configuration of the capillary plate. For the fiber phantom (diameter, 17.2 ± 2.6 μm),¹¹ the diffusion coefficient perpendicular to the fiber changed with the logarithmic singularity as a function of the inverse diffusion time. In this study, similar logarithmic changes were observed in the RD at 50 and 100 μm; however, only linear increases were obsered before reaching the singularity at 6, 12, and 25 μm. In the fiber phantom, the diffusion coefficient was affected by water molecules external to the fibers. In contrast, in the capillary plate, the data reflected only restricted diffusion because there are no water molecules outside of the glass tubes.

In the STEAM sequence used in this experiment, the diffusion time becomes longer by increasing the mixing time (TM).²⁶ However, as shown by the SNR calculation equation in STEAM,²⁷ the pulsed gradient stimulated echo (PGSTE) signal obtained after the third 90° pulse is attenuated because T1 relaxation was accelerated during TM. Using an excessively long diffusion time (TM) in the structural scale reduces the signal and may result in errors of diffusion coefficient measurement. In this experiment, the phantom was placed vertical to the static magnetic field of the MRI machine. Therefore, from the axis information shown in Table 1, the first axis of 12 axes indicates SNR in approximately the axial diffusivity (AD) direction, and the 12th axis in approximately the RD direction. Table 1 shows that the SNR of AD (first axis) and the free water in RD decreased as diffusion time was longer, regardless the structure scale of the capillary plate. The free diffusion in the AD direction suggests that the PGSTE signal decreases with diffusion time (i.e., TM) in accordance with the theory.²⁶ In contrast, in the RD direction (12th axis), SNR changes depending on the structure scale and temperature. At room temperature (20°C), the SNR improved with longer diffusion time. Diffusion time of 800 ms satisfies the theoretical condition required for structure estimation at 6–100 μm. At this time, water molecules reach the structure wall and the diffusion coefficient decreases based on Neumann’s boundary condition, and thus the PGSTE signal strengthens and the SNR improves. The characteristics of SNR transitions vary depending on the diffusion time and structure scales.

To detect the displacement of microscopic structures at 6–12 μm, a very short diffusion time of ≤ 10 ms is required from the theoretical value calculated from the MSD. However, the STEAM used in this experiment cannot achieve a very short diffusion time owing to the upper limits of the magnetic field strength of the motion probing gradient (MPG). Therefore, as in previous studies,^28,29 a few milliseconds of diffusion time was performed using oscillating gradient spin-echo to detect microstructural displacements at the several-micrometer level.

Biological cells are composed of various structural scales, ranging from neural cells at several micrometers,³⁰ to hepatocytes at approximately 20 μm,³¹ to muscular cells at 50 μm or more.³² Therefore, when setting diffusion time far from the theoretical values calculated from the structural scales, the acquired diffusion coefficient may not accurately reflect the information of the target tissue. For microscopic structures below 50 μm, the diffusion coefficients converged to the same value (Supplementary Fig. 2) and micron-level structural variations could not be detected. Meanwhile, for relatively large scales, such as 100 μm, the true diffusion coefficient for restricted structure cannot be obtained because water molecules within the structure have not reached the inner wall. Moreover, biological cell membranes are selectively permeable³³ and some cells have water channels for active water exchange inside and outside the cell.³⁴ For dMRI in biological subjects with various structural scales, it is important to set the diffusion time adapted to the scale of the target organization.

Temperature

Previous studies using nuclear magnetic resonance (NMR) to investigate the water self-diffusion coefficient at various temperatures^15,24 showed that the diffusion coefficient decreases as a gentle curve to the Arrhenius equation with an increase in temperature, indicating that diffusion is a physical phenomenon of thermal activity with temperature dependence. Applying the results of this study (Fig. 4, free water) to the Arrhenius plot, we obtained a gentle curve that was slightly higher at 10°C and lower at 40°C than the recommended value,²⁴ suggesting that water molecule diffusion was measured physically, as in previous studies.

Fluctuations in the diffusion coefficient with increasing temperature tended to occur at larger structural scales (Supplementary Fig. 2). Increasing the temperature causes active thermal motion of water molecules, which enhances the Brownian motion by promoting collisions between the water molecules. Therefore, water molecule movement by diffusion within the same time increases at high temperatures compared with that at low temperatures.

Therefore, a larger diffusion space can accurately reflect fluctuations in water molecule migration with temperature. Conversely, for smaller-scale structures, most water molecules in the structure hit the inner wall in a short time because of the short distance to the wall. As a result, even if Brownian motion is not enhanced, most water molecules in the structure easily reach the structure walls, and the diffusion coefficient fluctuations were less dependent on temperature.

Fluctuations in the diffusion coefficient with temperature easily occur with a shorter diffusion time. When the diffusion time is short, the phase until all water molecules reach the structural wall reflects easily because it is before many water molecules collide with the wall. With a longer diffusion time, the effect of temperature dependence diminishes because the water molecules hitting the structural walls reach the upper limit.

Based on the above, experiments on biological subjects need to consider changes in diffusion coefficients in the same region caused by the temperature difference between in vivo (40°C) and ex vivo (25°C). The degree of change in the diffusion coefficient with temperature depended on the target structural scale and diffusion time setting. Especially for larger scales, such as myocytes and hepatocytes, the temperature dependence of diffusion coefficients and diffusion time settings must be taken into account. Previous studies³⁵ have confirmed that RD in the brain fluctuates with temperature between in vivo and ex vivo conditions, and the rate of change with temperature was greater in gray matter than that in white matter. Applying the contour plot of diffusion time 12 ms in Fig. 5, which is close to the conditions of the previous study,³⁵ it is considered that gray matter is larger in scale with a greater change rate. The difference in the rate of change in the diffusion coefficient with temperature may have resulted from a few micrometers in structural scale between white matter composed of nerve fibers and gray matter composed of cell bodies.³⁶

The SNR in the RD direction (12th axis) from Table 1 varied with structural scale and temperature. At 40°C, the SNR decreased at long diffusion times in small structure scales of 6 and 12 µm. These results suggest that the attenuation of the PGSTE signal was caused by the increase in water molecule movement in the capillary plate planes and along the long axis direction beyond the setting slice thickness as the water molecule diffusion velocity increases with increasing temperature. This finding is consistent with the SNR decrease in the AD direction (Axis 1) when the temperature changed from 20°C to 40°C. Based on our results, substituting the diffusion coefficient of water at 40°C as D_temp40 = 2.8 × 10⁻⁹ (m²/s) and $\mathrm{M}\mathrm{S}\mathrm{D}=\sqrt{6\mathrm{D}\mathrm{t}}$, we obtain a 3D MSD of 0.12 mm at t (diffusion time) = 800 ms, and predict that water molecules move about 9 − 19 times farther than the structural scale of 6 − 12 μm. Moreover, the decrease in signal intensity may cause errors in the calculation of diffusion coefficients. We observed a slight increase in RD with longer diffusion time at 6 − 50 μm in Fig. 4, and the growth rate was larger at high temperatures. This result suggests that the diffusion coefficient was probably calculated larger by signal attenuation, as SNR at long diffusion times decreased with higher temperature in small structure scales.

Diffusion coefficients vary complexly depending on the device settings (e.g., diffusion time, b-value, and voxel size)³⁷ and the subject (e.g., structure scale, temperature, viscosity, and molecular weight); therefore, it is difficult to determine the single factor that causes the clinical changes in the diffusion coefficients. On the other hand, water molecule diffusion is a physical phenomenon, and we obtained data based on the diffusion theory using uniformly restricted structures. For the accurate assessment of biological diffusion properties in dMRI, it is necessary to collect experimental data under multiple diffusion times and b-value conditions using a subject, such as phantoms, in which the parameters can be comprehensively combined.

Conclusion

In this study, we evaluated the characteristics of restricted diffusion by obtaining data from diffusion phenomena only in a structure using capillary plates. Our experiments showed the transitional relationship of the diffusion coefficient between diffusion time, structure scale, and temperature through a composite measurement environment. We obtained data to better understand the relationship between diffusion time, structure scale dependence, and temperature dependence in restricted diffusion.

(b) Diffusion time 800 ms

	Diffusion gradients			SNR
				20ºC						40ºC
	Gx	Gy	Gz	6 μm	12 μm	25 μm	50 μm	100 μm	Free water	6 μm	12 μm	25 μm	50 μm	100 μm	Free water
1	0.263	0.067	0.963	117	94	119	110	92	164	40	40	40	41	46	56
2	−0.426	0.141	0.894	142	114	147	139	100	112	50	50	63	63	48	63
3	−0.121	−0.560	0.819	222	183	200	201	126	154	93	76	77	96	69	64
4	0.742	−0.275	0.612	387	339	407	316	208	161	246	185	223	174	144	88
5	0.402	0.658	0.636	353	250	344	306	197	164	229	157	230	155	134	79
6	−0.371	0.751	0.547	469	369	487	382	218	179	295	229	340	230	177	87
7	−0.889	−0.091	0.449	593	459	585	422	220	153	428	346	399	377	243	105
8	−0.611	−0.724	0.321	756	664	697	503	277	181	687	615	619	555	315	121
9	0.374	−0.848	0.374	784	605	726	536	288	198	567	433	542	475	265	112
10	0.910	0.325	0.257	864	577	714	653	344	218	848	510	713	682	359	142
11	0.125	0.989	0.077	1034	682	862	712	394	255	1028	796	978	918	536	153
12	−0.861	0.503	0.079	1031	716	848	540	377	230	1147	994	1011	826	534	149

SNR was calculated based on NEMA²¹ using $\mathrm{S}\mathrm{N}\mathrm{R}=\left(\mathrm{S}\sqrt{\pi /2}\right)/\mathrm{N}$ from the average signal S in the phantom and the noise average signal N in air. NEMA, National Manufacturers Electrical Association.

Funding Statement

This work was supported by the program for Brain Mapping by Integrated Neurotechnologies for Disease Studies (Brain/MINDS) from the Japan Agency for Medical Research and Development (AMED) (Grant Number JP21dm0207001 and JP22dm0207001 to Hideyuki Okano), JSPS KAKENHI (Grant Number JP20H03630 to Junichi Hata), and “MRI platform” (Grant Number JPMXS0450400622 to Junichi Hata) as a program of Project for Promoting public Utilization of Advanced Research Infrastructure of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

Supplementary Information

Supplementary files are available online.