Introduction

Time-dependent diffusion MRI is sensitive to microstructural geometry (e.g. cell or axon size, density, and packing configuration) and such information is not obtainable by other imaging methods. The relationship between the time-dependent diffusion coefficient $$$D(t_{diff})$$$ and effective diffusion time $$$t_{diff}$$$ is complex, however, and theoretical expressions relating them require a priori assumptions about the underlying microstructure. For $$$t_{diff}\rightarrow0$$$, Mitra et al.¹ have shown $$$D(t_{diff})\sim\sqrt{t_{diff}}$$$, while for $$$t_{diff}\rightarrow\infty$$$ Novikov et al.² have shown a power-law relation can describe the $$$t_{diff}$$$ dependence as $$D(t_{diff})=D_\infty+const\cdot t_{diff}^{-\theta}\quad(1)$$
where $$$D_\infty$$$ is the long diffusion time limit, $$$\theta=(p+d)/2$$$, $$$d$$$ is the dimension and $$$p$$$ is the structural exponent determined by the spatial correlation of restrictions.
Using STEAM acquisitions for long $$$t_{diff}$$$ up to 600 ms , Fieremans et al.³ reported values of $$$\theta$$$ as 1 and 1/2 for radial (RD) and axial (AD) diffusivities, respectively, in healthy human white matter, which indicates different dominating structures in orthogonal directions.
In addition to the time domain analysis, diffusivity in the frequency domain may be described as: $$D(f)=D_0+DDR\cdot f^\theta\quad(2)$$
where $$$f\sim1/t_{diff}$$$ and $$$DDR$$$ is the diffusion dispersion rate. This analysis is particularly suitable for OGSE, where mean diffusivity (MD) has shown both $$$\theta=1/2$$$ and $$$\theta=1$$$, seemingly dependent on whether a measured PGSE value approximating $$$f=0$$$ is included^4,5. However, it is questionable whether Eq. 2 is valid in the diffusion time regime used. Moreover, there are a few other questions that remain unanswered: (1) is the power-law fitting reliable with a narrow frequency range in practice? (2) Can PGSE diffusivity be considered as the limiting value of $$$D(f\rightarrow0)$$$ in the power-law fittings? If so, (3) can the $$$t_{diff}$$$ of measured values bias the fittings?
To answer these questions, DTI data with a broad practical range of $$$t_{diff}$$$ (OGSE frequencies 50-133 Hz and PGSE $$$t_{diff}$$$ 10-40 ms) were acquired from non-human primate brains. MD/RD/AD were fit using a power-law with different $$$t_{diff}$$$ ranges. The results provide better understanding of power-law based $$$t_{diff}$$$ dependence in neural tissues.

Methods

Four squirrel monkeys were imaged using an Agilent/Varian 9.4T small animal scanner. Diffusion-weighted echo-planar images were acquired using TR/TE=5000/56 ms, transverse FOV 6.4x6.4 cm² and voxel size 1 mm³ isotropic. Reverse phase-encoded images with no diffusion preparation were acquired for susceptibility distortion correction. Each diffusion time had identical gradient tables comprising 32 uniformly spaced directions in pairs of parallel and antiparallel vectors with b=700 s/mm² and five b=0 s/mm² interleaved throughout, for a total of 69 acquisitions per diffusion time. 8 diffusion preparations were used: 4 PGSE with δ=3 ms and ∆=11, 21, 31, and 41 ms; and 4 trapezoidal cosine OGSE with $$$f$$$=50, 67, 100, and 133 Hz acquired with δ=15 ms, ∆=19 ms, n_cycle=1, 2 and δ=20 ms, ∆=24 ms, ncycle=1, 2. One subject also included 150 and 200 Hz with b=325, 180 (respectively); these points were not used in any fitting due to a differing b-value but are shown for comparison. Representative OGSE and PGSE images are shown in Figure 1.
The diffusion-weighted images were corrected for noise and Gibbs artifacts using MRTrix3 and for susceptibility and eddy-current distortions using FSL. Corrected intra-subject images were co-aligned by linear registration with FSL. The VALiDATe29 squirrel monkey brain atlas⁶ was registered to the mean b=0 image of each volume using nonlinear diffeomorphic registration with ANTS. Two ROIs in gray matter (the prefrontal (PFC) and visual (V1) cortex) and four ROIs in white matter (the body (BCC), genu (GCC), and splenium (SCC) of the corpus callosum and the gyrus of the internal capsule (ICg)) were identified for analysis using the VALiDATe29 labels.

Results

ROI-averaged DTI fits in the diffusion time domain (Eq. 1) are shown in Figure 2. Figure 3 shows power law (Eq. 2) fits for two sets of frequency domains: OGSE data from 50-133 Hz and 0-133 Hz using the 40 ms PGSE to approximate 0 Hz. Figure 4 shows power law fits assessing the impact in which PGSE data approximate as 0 Hz. Finally, Figure 5 shows the impact of different fixed values of $$$\theta$$$ on the goodness of fit (RMSE).

Discussion and Conclusion

This study shows that power law fitting in an intermediate diffusion time regime is not trivial. Fits using both Eq. 1 and 2 yield high variance between regions and values of $$$\theta$$$ greater than would be physiologically expected in terms of dimensionality and order, indicating the fitting range may be outside the long $$$t_{diff}$$$ regime assumed in the model. Figure 5 indicates this may be due to the fitting landscape – in this range of frequencies, the data are insensitive to the choice in power and therefore inverting Eq. 2 has low precision. The slope of Figure 4 indicates that, though inclusion of PGSE may stabilize the variance of $$$\theta$$$, approximating PGSE as 0 Hz may bias the fit. In conclusion, our results do not support the translation of the power-law fits derived from long $$$t_{diff}$$$ regimen to an intermediate regime (1.9 – 40 ms). The high variance in power-law fits suggests that the power-law fitted $$$\theta$$$ may not be a reliable indicator to the underlying microstructure, and caution should be taken when interpreting the power-law fitted $$$t_{diff}$$$-dependent data with intermediate $$$t_{diff}$$$ regime.

Acknowledgements

No acknowledgement found.