Evidence of the Diffusion Time Dependence of Intravoxel Incoherent Motion in the Brain

Abstract

Purpose:

To investigate the diffusion time (t_d) dependence of intravoxel incoherent motion (IVIM) signals in the brain.

Methods:

A three-compartment IVIM model was proposed to characterize two types of microcirculatory flows in addition to tissue water in the brain: flows that cross multiple vascular segments (pseudo-diffusive) and flows that stay in one segment (ballistic) within t_d. The model was first evaluated using simulated flow signals. Experimentally, flow-compensated (FC) pulsed-gradient spin-echo (PGSE) and oscillating-gradient spin-echo (OGSE) sequences were tested using a flow phantom and then used to examine IVIM signals in the mouse brain with t_d ranging from approximately 2.5 ms to 40 ms on an 11.7 Tesla scanner.

Results:

By fitting the model to simulated flow signals, we demonstrated the t_d dependency of the estimated fraction of pseudo-diffusive flow and the pseudo-diffusion coefficient (D*), which were dictated by the characteristic timescale of microcirculatory flow (τ). Flow phantom experiments validated that the OGSE and FC-PGSE sequences were not susceptible to the change in flow velocity. In vivo mouse brain data showed that both the estimated fraction of pseudo-diffusive flow and D* increased significantly as t_d increased.

Conclusion:

We demonstrated that IVIM signals measured in the brain are t_d dependent, potentially because more microcirculatory flows approach the pseudo-diffusive limit as t_d increases with respect to τ. Measuring the t_d dependency of IVIM signals may provide additional information on microvascular flows in the brain.

1. Introduction

Intravoxel incoherent motion (IVIM) signals originate from microcirculation of blood in the capillaries and small vessels that undergo pseudo-random motions within a voxel. The concept was introduced almost 30 years ago by Le Bihan et al. (1), which laid out the ground work for modeling and measuring IVIM signals. While the IVIM effect is often considered a confounding factor in estimating tissue apparent diffusion coefficient (ADC) from diffusion MRI signals acquired with relatively low b-values (e.g., < 500 s/mm²), it provides a unique way to characterize microcirculatory flows under normal and diseased states. Applications of IVIM imaging have quickly expanded in recent years in the brain, including stroke (2–4) and glioma (5–8), as well as other organs, such as the liver (9), kidney (10), and cancer in the pancreas (11), breast (12), and prostate (13).

The most commonly used model of IVIM signals is the biexponential model (1), given by $\frac{S}{S_0}=f\cdot e^{-b\cdot D^{*}}+\left(1-f\right)\cdot e^{-b\cdot D}$, where f and D* are the fraction of microcirculatory compartment and the so-called pseudo-diffusion coefficient. Both f and D* have been associated with important perfusion functions, such as cerebral blood volume and velocity (14). However, it has been pointed out that f and D* depend on both the properties of microcirculation and the choice of diffusion encoding gradient waveforms. For example, the observed IVIM signals depend on the average length of capillary segments and the average distance that blood travels within the diffusion time (t_d) (1,15). It is now well known that the ADC of tissue water depends on t_d. Recent progress on t_d-dependent diffusion MRI (16,17) opened new avenues to probe specific microstructural properties, such as cell size, membrane permeability, surface-to-volume ratio, and axonal diameter (18–21). The t_d dependency of IVIM signals may benefit investigations of vascular properties in a similar fashion.

Previously, Wetscherek et al. (22) demonstrated t_d-dependent IVIM signals in the liver and pancreas for t_d values ranging from 40 to 100 ms. The t_d dependency of IVIM signals in the brain, especially in the short t_d regime (10 ms or less), remains to be investigated. In light of the complex vascular geometry in the brain (23,24), e.g., the heterogeneous flow velocities (25,26) and relatively short segment length (27,28), the short t_d regime could be highly relevant for characterizing the brain vasculature.

In this study, we hypothesized that IVIM signals in the brain are t_d dependent and used a three-compartment IVIM model to measure this effect. The model replaced the single flow component in the biexponential model with two components representing two types of microcirculatory flows according to Le Bihan et al. (1). We first used numerical simulations to evaluate the proposed model and then performed phantom and in vivo experiments. Oscillating-gradient and flow-compensated (FC) pulsed-gradient waveforms were employed to assess IVIM signals in the mouse brain at short to intermediate values of t_d (approximately 2.5–40 ms) and their t_d dependency.

2. Methods

2.1. Theory

2.1.1. Three-compartment IVIM model

Diffusion signal attenuations due to microcirculatory flow and random diffusion interacting with tissue microstructural barriers can be described using a two-compartment model (22):

$$\frac{S}{S_0}=f\cdot H\left(b,t_d,l,v, D_{blood}\right)+\left(1-f\right)\cdot e^{-b\cdot D}$$ [1]

where f ∈ [0, 1] is the fraction of signal attenuation associated with microcirculatory flow, and D is the ADC of tissue water molecules. The signal attenuation due to microcirculatory flow, H(b, t_d, l, v, D_blood), is formulated as a function of l, the average length of capillary segments; v, the average velocity of blood flow in the capillaries; b, the b-value;D_blood, the ADC of water molecules in the blood (29), which is approximately 1.6 × 10⁻³ mm²/s based on previous studies (30,31); and t_d. Note that Eq [1] does not consider non-Gaussian diffusion effects that occur at high b-values (32) and therefore is an approximation that is only valid for low or moderate b-values.

According to (1), two types of microcirculatory flows occur in the capillary bed:

1. When t_d is long enough for the microcirculatory flow to pass through multiple vascular segments, the signal attenuation can be modeled as H₁ = e^−D*·b, where D* is the so-called pseudo-diffusion coefficient. This case is referred to as the “pseudo-diffusive limit” here, similar to (33). Note that the effect of D_blood is included in the D* term here.

2. When the t_d is sufficiently short such that microcirculatory flow does not pass through more than one vascular segment and assuming that capillary segments are randomly oriented in space and distributed over 4π, we have $H_2=\left|sinc\left(cv\right)\right|\cdot e^{-D_{blood}\cdot b}$ (1,34), where c is the first-order moment of the diffusion gradient waveform. This case is referred to as the “ballistic limit” as in (33).

Here we assume that the IVIM term, H(b, t_d, l, v, D_blood), can be modeled as a weighted sum of the microcirculatory flows near the pseudo-diffusive and ballistic limits:

$$H\left(b,t_d,l,v, D_{blood}\right)=\frac{f_1}{f_1+f_2}{H}_1+\frac{f_2}{f_1+f_2}{H}_2=f_{df}\cdot e^{-D^{*}\cdot b}+\left(1-f_{df}\right)\cdot \left|sinc\left(cv\right)\right|\cdot e^{-D_{blood}\cdot b}$$ [2]

where f₁ and f₂ are the fractions of spins that undergo pseudo-diffusive and ballistic flows, respectively, and $f_{df}=\frac{f_1}{f_1+f_2}$ is the fraction of pseudo-diffusive flow among the total flow. By inserting Eq. 2 into Eq. 1, and taking into consideration that f = f₁ + f₂, we have

$$\frac{S}{S_0}=f_1\cdot e^{-D^{*}\cdot b}+f_2\cdot \left|sinc\left(cv\right)\right|e^{-D_{blood}\cdot b}+\left(1-f_1-f_2\right)e^{-D\cdot b}$$ [3]

For flow-compensated (FC) diffusion gradient waveforms, with a zero first-order moment (i.e., c = 0 and sinc(cv) =1), Eq. 3 can be simplified as

$$\frac{S}{S_0}=f_1\cdot e^{-D^{*}\cdot b}+f_2\cdot e^{-D_{blood}\cdot b}+\left(1-f_1-f_2\right)e^{-D\cdot b}$$ [4]

Thus, by using FC diffusion gradient waveforms, we can ignore the effects of v in the model fitting.

2.1.2. Model validation with simulation

We first examined whether the model can accurately estimate f_df and D* with FC gradients by fitting it to simulated IVIM signals with added noises. The flow signal attenuation profiles, H(b, t_d, l, v, D_blood), were generated using the tool developed by Wetscherek et al. (github.com/awetscherek/ivim_tools), which uses Monte Carlo simulations to generate diffusion signal attenuations with complete phase information (22). The parabolic velocity distribution was used in the simulation, which integrates over a range of flow velocities with N directional changes in the case of laminar pipe flow. Simulations were performed with the following parameters: v = 2 mm/s; characteristic time scale (τ = l/v)at 5, 10, 20, and 30 ms; gradient duration (T) = 5, 10, 20, 40, and 80 ms using an FC sequence (22); and b-values ranging from 0–500 s/mm². Rician noises (35) with a standard deviation of 0.02 were added to the signals. We then fitted Eq. 2 to obtain f_df and D* under the FC condition (c = 0) using MATLAB (www.mathworks.com). The fitting was repeated ten times with different noise realizations each time to obtain the means and standard deviations of the estimated f_df and D*.

2.2. Pulse Sequence

Here, we used the apodized cosine-trapezoid OGSE (36,37), FC-PGSE (22), and noncompensated (NC)-PGSE waveforms to access an extended range of t_d (Fig. 1A–C). The OGSE waveform is inherently flow compensated with a zero first-order moment and an effective t_d of 1/4 of the oscillating cycle (38). For the FC-PGSE waveform (Fig. 1B), the positive and negative lobes of the gradient waveform had the same amplitude. The duration of the negative lobe was set to 34% of the total duration T to achieve the zero first-order moment, considering both the diffusion gradient ramp time (set at 0.01T) and the duration of the refocusing pulse and crusher gradients (3.8 ms). In this study, we used T as the effective t_d for the FC-PGSE waveforms following (22). For the NC-PGSE sequence, we used Δ-δ/3 as the effective t_d under the narrow pulse condition (39). The spectra (40) of OGSE, FC-PGSE, and NC conventional PGSE waveforms (Fig. 1D–E) were calculated according to

$$F\left(\omega \right)=\gamma {\int }_{-\infty }^{\infty }{e}^{-i\omega t}{\int }_0^{t}g\left(\tau \right)d\tau dt$$ [5]

where γ is the gyromagnetic ratio and g is the diffusion gradient waveform.

Fig. 1:

Diffusion encoding waveforms used in this study and their spectra. Schematics of the diffusion encoding waveforms, including the oscillating-gradient spin-echo (OGSE) (A), flow-compensated pulsed-gradient spin-echo (FC-PGSE) (B), and noncompensated (NC) PGSE (C). For the OGSE, f denotes the oscillating frequency, and the duration of one oscillating period is 1/f. In FC-PGSE, T denotes the duration of the gradient with the polarity of the first 0.34 of the gradient opposite to the last 0.66 of the gradient. In NC-PGSE, δ and Δ denote gradient duration and separation, respectively. (D) Spectra of two OGSE waveforms (f = 50/100 Hz), which were normalized to the area under the curve of the individual spectra. (E) Normalized spectra of FC-PGSE waveforms with T = 10/20 ms and NC-PGSE waveforms with Δ = 40 ms.

2.3. Data Acquisition

All MRI experiments were performed on an 11.7 Tesla horizontal Bruker scanner (Bruker Biospin, Billerica, MA, USA, maximum gradient strength = 760 mT/m), which is equipped with a 72mm-diameter quadrature volume transmitter coil and several receive-only coils.

2.3.1. Phantom experiment

A flow phantom was made with flowing water in a thin plastic catheter (inner diameter = 8 mm), which was wrapped around a glass tube (inner diameter = 20 mm) that contained static water. The flow in the catheter was controlled via an infusion pump (PHD 22/2000, Harvard Apparatus, Holliston, MA, USA), with flow rates set at 0, 2, and 4 mm/s. Images were acquired from the water phantom using a four-channel mouse brain phased-array receive-only coil (Bruker Biospin, Billerica, MA, USA). The diffusion gradient was applied along the horizontal direction (aligned with the direction of the flow) with a b-value of 200 s/mm² for OGSE, FC-PGSE, and NC-PGSE sequences. Two minimally diffusion-weighted (b0) images were acquired with an effective b-value of 1.5 s/mm² and a c-value of 0.088 s/mm in all three sequences due to crushers. OGSE data were obtained with an oscillating frequency of 100 Hz and one oscillating cycle; FC-PGSE data were obtained with a t_d of 10 ms, and NC-PGSE data were obtained with a t_d of 12 ms and a diffusion gradient duration (δ) of 4 ms. The other imaging parameters were the same for all sequences: four-segment EPI readout, echo time (TE)/repetition time (TR) = 38/3000 ms, one signal average, field of view (FOV) = 16 mm × 16 mm, in-plane resolution = 0.125 mm × 0.125 mm, and a single slice with a thickness of 0.5 mm. Scans were repeated eight times for each sequence.

2.3.2. In vivo mouse brain experiment

All experimental procedures were approved by the Animal Use and Care Committee at the Johns Hopkins University School of Medicine. During imaging, the mice were anesthetized with isoflurane (1–1.5%) together with air and oxygen mixed at a ratio of 3:1 via a vaporizer. The mice were restrained in an animal holder with ear pins and a bite bar. Respiration was monitored via a pressure sensor (SAII, Stony Brook, NY, USA) and maintained at 40–60 breaths per minute. After imaging, animals recovered within five minutes. Diffusion MRI data were acquired from adult C57BL/6J mice (male, three months old, n = 7) using a 15 mm planar receive-only surface coil, six directions, 16 b-values (25, 50, 75, 100, 125, 150, 200, 250, 300, 400, 500, 600, 700, 800, 900, 1000 s/mm²), two b0 images with an effective b-value of 1.5 s/mm², and a c-value of 0.088 s/mm (sinc(c·v)≈0.95) in all sequences. OGSE data were acquired with frequencies of 50 and 100 Hz (effective t_d ≈ 5 and 2.5 ms, respectively). FC-PGSE data were acquired with t_d = 10 and 20 ms. NC-PGSE data were acquired at δ/t_d = 4/40 ms. The other imaging parameters were kept the same for all scans: single-shot EPI, TE/TR = 58/3000 ms, four signal averages, in-plane resolution = 0.2 mm × 0.2 mm, five slices with a thickness of 1 mm, and a scan time of 20 minutes for each t_d. We used the same TE for all experiments.

2.4. Data Analysis

We used a two-step fitting procedure to estimate [f₁, f₂, D*, D] in the three-compartment model in Eq. 4. First, the fraction of tissue water and D were approximated using diffusion signals acquired with b ≥ 300 s/mm², using a log-linear fitting approach according to (41): $\text{log}\left(\frac{S}{S0}\right)=-bD+\text{log}\left(1-f_{tissue}\right)$. Then, f₁ and D* were fitted in a mono-exponential model using a constrained nonlinear linear-square fitting method (42) in MATLAB. Diffusion signals acquired with the NC-PGSE sequence at a t_d of 40 ms were fitted with the regular two-compartment model $\frac{S}{S_0}=f_1\cdot e^{-D^{*}\cdot b}+\left(1-f_1\right)e^{-D\cdot b}$, assuming that signals from ballistic flows in Eq. 4 are negligible at long t_d values in the mouse brain (i.e., f₂=0). This assumption was made based on the distance that blood flows within the t_d with respect to the capillary geometry in the mouse brain. Given the estimated capillary flow velocity v (1–2 mm/s) and l (0.01–0.03 mm) in the mouse brain (26,28), most flows should travel more than one vessel segment within a 40 ms period and no longer fit the description of ballistic flows. Any remaining signals from ballistic flows will be further attenuated due to the large first-order moment of the NC-PGSE sequence at long t_d.

Quantitative analysis of the mouse brain data was performed in manually defined regions of interest (ROIs) that covered all brain tissues in the middle three slices, excluding the ventricles. f₁ and D* were extracted from the ROIs, and the voxels with fitting errors (those with f₁ values larger than 0.1 or smaller than 0.01) were excluded. The signal-to-noise ratio (SNR) was calculated as the ratio between the mean signal from the brain ROI and the standard deviation of the background signal in the b0 image.

Statistical tests were performed using GraphPad Prism (http://www.graphpad.com). Differences in f_df and D* due to changes in t_d and τ in the simulation test were evaluated using two-way Analysis of variance (ANOVA). Differences in the diffusivities measured at different flow rates in the phantom experiment were assessed with one-way ANOVA. The differences in IVIM parameters fitted at different t_d values in the mouse brains were also evaluated with one-way ANOVA.

3. Results

3.1. Simulation Results

For the FC-PGSE waveform, microcirculatory flow-induced signal attenuations were simulated with t_d values ranging from 5 to 80 ms and b-values between 0–500 s/mm² (22). Plots of simulated flow signal attenuations (small circles in Fig. 2A) with the characteristic timescale of the vasculature τ set to 20 ms, flow velocity v set at 2 mm/s, and added Rician noises with σ=0.02 (i.e., SNR = 50) showed that the IVIM effect increased with increasing t_d. Fitting results based on the model of ballistic and pseudo-diffusive flows (Eq. 2) (solid curves in Fig. 2A) agreed well with the simulation data.

Fig. 2:

Fitting simulated IVIM signals using the ballistic and pseudo-diffusive flow model (Eq. 2). (A) Simulated flow signal attenuations using a parabolic velocity distribution at a characteristic time scale (τ) of 20 ms, sampled using FC-PGSE waveforms with t_d values of 5, 10, 20, 40, and 80 ms (dots), b-values between 0 and 500 s/mm²), and added Rician noises (σ=0.02). The simulated signals were fitted with the proposed IVIM model in Eq. 4 (solid curves). (B) The pseudo-diffusion coefficients of blood (D*) and the fraction of pseudo-diffusive flow $(f_{df}=\frac{f_1}{f_1+f_2})$ were derived from the IVIM model fitted at different durations (t_d of 5–80 ms) and different τs (from 5 to 30 ms). The simulation was repeated 10 times, and the means and standard deviations of f_df and D* are presented.

Both the estimated D* and the fraction of pseudo-diffusive flow (f_df) increased with t_d in simulations with τ of 5, 10, 20, and 30 ms (Fig. 2B). Two-way ANOVA showed that the effects of both t_d and τ on the simulated flow signals were significant (p<10⁻⁴). Here, we used f_df instead of f₁, as the simulations only generated flow-related signals. The estimated D* was close to zero at a t_d of 5 ms and increased with t_d until reaching a plateau. The values of D* at plateau were proportional to τ. Note that $D^{*}=\tau \cdot \frac{v^2}{6}$ when t_d ≫ τ (1). When τ = 10 ms and v = 2 mm/s were used in the simulations, the estimated D* approximated the predicted value of 0.007 mm²/s for t_d greater than 20 ms. The relationship between the estimated f_df and t_d also depended on τ (Fig. 2B). For vasculatures with short τ (5 and 10 ms), the estimated f_df reached a plateau for t_d longer than 40 ms. For vasculatures with longer τ (20 and 30 ms), a plateau of f_df was not observed in the range of t_d = 5–80 ms. These results suggest that the t_d-dependent behaviors of IVIM signals could be used to distinguish brain vasculatures with altered τ, similar to what Wetscherek et al. showed previously for the liver and pancreas (22).

3.2. Sequence validation using a flow phantom

To validate the effectiveness of flow compensation in OGSE and FC-PGSE sequences, diffusivities were measured in the phantom with both flowing and static water using the FC-PGSE, OGSE, and NC-PGSE sequences. With the flow direction mainly along the X-axis (Fig. 3A), diffusivities measured using NC-PGSE along the X-axis increased rapidly as the flow rate increased from 0 to 4 mm/s (Fig. 3B), whereas diffusivities measured using OGSE and FC-PGSE did not change with the flow rate (p=0.09 and p=0.12, respectively, one-way ANOVA). Diffusivities measured in the static water (Fig. 3C) that were beneath the catheter showed nearly identical values at all flow rates using the three diffusion encodings (p>0.05). These results demonstrate that the FC-PGSE and OGSE diffusion encodings were flow compensating.

Fig. 3:

Validation of flow compensation using a phantom. (A) A T2-weighted image of the flow phantom with water flowing in a catheter along the × direction and a glass tube of static water lying underneath. (B-C) Diffusivities (D) measured using the NC-PGSE waveform at δ/Δ of 4/12 ms, the OGSE waveform with an effective t_d of 2.5 ms, and the FC-PGSE waveform at t_d of 10 ms, in the catheter with flowing rates at 0 mm/s, 2 mm/s, and 4 mm/s (B) and in static water (C). The measurements were repeated 8 times for each setup. * denotes p<0.001 with one-way ANOVA.

3.3. IVIM measurements in the mouse brains

Diffusion MRI signals were acquired from adult mouse brains (n=7) at five different t_d values using the validated OGSE, FC-PGSE and NC-PGSE sequences. The SNR of the b0 images was 53.5±9.9 across all sequences. The signal attenuation profiles demonstrated t_d-dependent decay patterns (Fig. 4A). In the low b-value regime (b<200 s/mm² in Fig. 4B), the signal decay was faster at longer t_d (e.g., the red curve for NC-PGSE with a t_d of 40 ms dropped faster than the blue curve for OGSE with an effective t_d of 2.5 ms), whereas in the relatively high b-value regime (b>500 s/mm²), the signal decay was faster at shorter t_d. This trend was clear by examining the data at b of 50 s/mm² versus b of 1000 s/mm² (Fig. 4C–D). While the observed t_d dependence of signals in the high b-value regime mostly reflected the scale of microstructural barriers in the tissue compartment, the inverse t_d dependence of signals at low b-values suggested that the IVIM effects were more pronounced at long t_d, similar to the simulation results in Fig. 2A.

Fig. 4:

t_d-dependent IVIM signals in the mouse brain. (A) Diffusion signal attenuations (S/S₀) acquired from adult mouse brains (n=7) with approximate t_d of 2.5 ms and 5 ms using OGSE with f=100/50 Hz, t_d of 10 ms and 20 ms using FC-PGSE, and t_d of 40 ms using NC-PGSE, and b-values from 0 to 1000 s/mm². Solid lines denote the fitting results with the proposed three-compartment IVIM model. The dashed window shows an enlarged view of signals at b-values of 0–150 s/mm² (B), which demonstrated larger signal attenuation at longer t_d, indicating a larger proportion of pseudo-diffusive flows at longer t_d. (C-D) The relationships between signal attenuation and t_d at low and high b values are reversed, indicating different mechanisms at work. * denotes p<0.001 with one-way ANOVA.

The voxelwise f₁ maps in Fig. 5 show increased fractions of pseudo-diffusive flows as the t_d increased from 2.5 ms to 40 ms. Blue contours on the t_d=2.5 ms images denote the manually masked brain parenchyma used in the ROI analysis. Quantitative analysis revealed significant increases in f₁ and D* values with increasing t_d (p<0.001, Fig. 6). For the OGSE measurements, the estimated f₁ values were significantly higher at 50 Hz than at 100 Hz (p < 0.001, paired t-test). For the FC-PGSE measurements, both f₁ and D* values were significantly higher at t_d =20 ms than t_d =10 ms (p < 0.015, paired t-test). f₂ was only fitted for signals acquired with t_d from 2.5 ms to 20 ms from OGSE and FC-PGSE sequences (assuming f₂ is negligible for NC-PGSE with a t_d of 40 ms), which did not show significant changes (p=0.1). As a result, the fraction of the tissue component (f_tissue = 1 -f₁ -f₂) declined with t_d (p<0.001, Fig. 6). The tissue diffusivity D decreased with t_d as expected.

Fig. 5:

Maps of estimated f₁ using the three-compartment IVIM model at different t_d in one representative mouse brain. Regions of interest (ROIs) that covered the brain tissues in the middle three slices are delineated with blue contours on the f₁ maps at a t_d of 2.5 ms.

Fig. 6:

Estimated f₁, f₂, D*, f_tissue, and D measured from seven mouse brains at t_d from 2.5 ms to 40 ms. * denotes p<0.001 with one-way ANOVA.

4. Discussion

In this study, we investigated t_d -dependent changes in IVIM signals in the mouse brain using a three-compartment IVIM model that separated pseudo-diffusive microcirculatory flows from flows near the ballistic limit. Both simulations and in vivo results suggested that, for t_d smaller than or approximately equal to the characteristic timescale of the underlying vasculature (τ), the fraction of pseudo-diffusive microcirculatory flow (f_df) in the brain increased as t_d increased. Similarly, the estimated D* also increased with t_d and approached $(\tau \cdot \frac{v^2}{6})$ at sufficiently long t_d. The time dependence of IVIM signals may be used to estimate τ of the vasculature.

To simplify the three-compartment model in Eq. 3, we used FC-gradient waveforms to measure IVIM signals. As the FC-gradient waveforms have zero first-order moment, we can eliminate the effects of v in Eq. 3, which is convenient, as v is often unknown and may vary spatially. The oscillating-gradient waveform allowed us to measure the diffusion signal at very short t_d values that were not accessible with pulsed gradients. To reach long t_d (t_d ≫ τ) while keeping the echo time consistent, we used the conventional NC-PGSE sequence, assuming that ballistic flow is negligible at the longest t_d. With these diffusion encoding strategies, IVIM signals were measured across a relatively wide range of t_d values. However, the exact t_d values for the three diffusion gradient waveforms used here could not be clearly defined. For instance, a recent discussion in the diffusion MRI community suggested that the relationship between the frequency of oscillating gradients and effective t_d depends on specific microstructural environments (43). Therefore, a direct comparison of OGSE and FC-PGSE IVIM measurements at equivalent t_d remains a challenge. Nevertheless, comparisons between IVIM signals measured using the same gradient waveform (OGSE 100 Hz versus 50 Hz and FC-PGSE t_d=10 ms versus 20 ms) showed significant t_d-dependent changes. Together with the general trend, our results support the hypothesis that IVIM signals in the brain are t_d dependent.

Spectral domain presentation may better capture the intrinsic differences between the diffusion waveforms than t_d alone. Fig. 1D–E shows distinct diffusion spectra of the diffusion encoding schemes used in this study. The spectra of the OGSE and FC-PGSE sequences were zero at 0 Hz (Fig. 1D), whereas the spectrum of the NC-PGSE sequence had its peak at 0 Hz (Fig. 1E). The OGSE sequence had peaks at 100 Hz and 50 Hz in its spectrum (Fig. 1D), and the FC-PGSE sequence had peaks at lower frequencies of approximately 40 Hz and 20 Hz in its spectrum (for t_d = 10 and 20 ms, respectively). These spectra suggest that different waveforms have different sensitivities to water diffusion and microcirculatory flow. As explained by Does et al. (44), restricted diffusion and flow had different velocity autocorrelation functions and thereby showed entirely different (opposite) spectral profiles. For example, for flow that has coherent velocities over a period of time, its spectrum peaks at zero frequency and decays towards higher frequencies, whereas for restrictive diffusion, the spectrum increases towards higher frequency. The microcirculatory flow in the capillary bed is likely to have both attributes, which may result in a more complicated spectrum. By measuring t_d-dependent IVIM effects, we could potentially characterize the spectrum of blood microcirculation.

With three different gradient encoding strategies, we acquired IVIM signals in the mouse brains from short to intermediate t_d. The estimated f₁ values between 4 and 5% agreed with the perfusion fraction measured in other brain studies (45,46). The results indicated that the fraction of pseudo-diffusive flow increased significantly with increasing t_d, likely because as t_d increases, a higher fraction of microcirculatory flow reaches the pseudo-diffusive limit. Our results agree with a recent report by Fournet et al. (34), in which they demonstrated that D* of the slow flow component increased with t_d (14–34 ms in a PGSE sequence), the fast flow component decreased with t_d, and the total fraction of IVIM effect increased with t_d. The fast flow component in (34) may correspond to the ballistic flow, and D* of the slow flow component may correspond to the D* of the pseudo-diffusive flow in our model. Note that in our model, D* included the D_blood term. While D_blood was originally thought to be a constant, a recent study using an FC-gradient waveform showed changes in D_blood when t_d was increased from 40 to 100 ms (31), which may also contribute to the t_d dependency of IVIM signals in the brain. The potential causes of the counterintuitive decrease in f_tissue in our results at longer t_d were not clear. Both the increase in IVIM effect at longer t_d as shown in (34) and the decrease in diffusivities in the tissue compartment may have contributed to this result.

The t_d dependency of IVIM signals in the brain may be useful to study changes in capillary flow and geometry under pathological conditions, e.g., the aberrant vasculature in brain glioma (47). The imaging sequences and IVIM model proposed here can be easily translated to human studies, as the FC-PGSE and OGSE sequences have been implemented on clinical MRI systems (38,48). Unlike studies on tissue microstructures that require strong diffusion weightings, most IVIM studies only use relatively low b-values. High-performance gradient systems currently available on clinical MRI systems will likely meet our needs to perform IVIM studies at relatively short t_d.

One limitation of the model used here is that it only considers two cases of microcirculatory flows, while the intermediate flow (flow that passes one or two vascular segments) may not fall into either category. The intermediate flow may be a confounding factor to the fitting of model parameters, e.g., f_df and D*. Our model only focused on the capillary flow, while flows in the small vessels that have intermediate τ and may not satisfy the assumption for the biexponential IVIM model (t_d/τ ≥7) were not considered in the current study. In addition, exchanges between vascular and tissue compartments were not considered in our model. Higher b-values and additional components may be needed if the effect of exchange becomes prominent (21,32).

In our mouse brain experiment, t_d longer than 40 ms was not investigated. The shortened blood and tissue T₂ values at 11.7 Tesla limited the range of TEs we could achieve, which prevented us from observing the plateau of f₁ and D* at longer t_d. To achieve long t_d with relatively short echo times, stimulated echo sequences can be used but will incur significant SNR penalties compared to the PGSE sequences. Another issue that may play a role in the measurements is the difference in the T2 relaxation times between the blood and tissue water, especially in the high field. It is known that the T2 relaxation time of blood rapidly decreases with increasing field strength (49). A potential solution is to acquire IVIM signals at different TE values and to incorporate the different T2 values into the model (50) or to use theoretical T2 values from existing literature to correct for the T2 effect (34). In addition, we only examined the overall IVIM profiles in brain tissues without looking into individual structures due to the limited resolution and contrast in the IVIM maps.

5. Conclusion

We investigated the t_d dependency of IVIM signals in the mouse brain at short to intermediate t_d values using flow-compensated oscillating-gradient and pulsed-gradient sequences. Both the simulation and in vivo mouse brain experiments suggested that the fraction of pseudo-diffusive flow and pseudo-diffusion coefficient increased as t_d increased from approximately 2.5 ms to 40 ms. The t_d-dependent IVIM measurements may be used to characterize microcirculation and vasculature in the brain.