Diffusion weighted image (DWI) is a powerful and non-invasive tool in the detection of diffusion processes of water molecules in biological tissues ¹. The departure from purely mono-exponential decay of DWI signal in biological tissues has been linked with anomalous diffusion. Recently, a fractional motion (FM) based diffusion MRI theory ² was proposed, which was claimed to be a proper model in the description of diffusion processes in biological systems ³. FM related parameter maps revealed great contrast between different brain tissues through fitting DWI data in healthy subjects. However, the tensorial properties of FM related parameters is still unknown. In this work, diffusion magnetic field gradients were applied in several non-linear directions to acquire DWI images. Then, the FM-based parameters were obtained in each diffusion direction and their anisotropic properties were further investigated.

According to the previous study, the diffusion weighted signal of Stejskal-Tanner sequence ⁴ can be computed as:

$$S/S_{0}=\exp(-\eta\cdot D_{\alpha,H}\cdot \gamma^{\alpha}\cdot G_{0}^{\alpha}\cdot \Delta^{\alpha(1+H)}) (1)$$

where $$$G_{0}$$$ is the amplitude of S-T diffusion gradients, $$$\delta$$$ is the pulse duration, Δ is the time between onsets of the first and second gradient pulses. $$$D_{\alpha,H}$$$ is generalized anomalous diffusion coefficient.

And $$$\eta$$$ is a dimensionless number which can be formulated as:

$$\eta=\frac{1}{(\mu+1)^{\alpha}}\left[\int_{0}^{\delta/\Delta} \mid (\frac{\delta}{\Delta}+1-u)^{\mu+1}-(1-u)^{\mu+1}-(\frac{\delta}{\Delta}-u)^{\mu+1}\mid^{\alpha}du+\int_{\delta/\Delta}^{1}\mid (\frac{\delta}{\Delta}+1-u)^{\mu+1}-(1-u)^{\mu+1}\mid^{\alpha}du+\int_{1}^{1+(\delta/\Delta)}(\frac{\delta}{\Delta}+1-u)^{\alpha\mu+\alpha}du \right] (2)$$

where $$$\mu=H-1/\alpha$$$.

The diffusion weighted images (DWI) of the whole brain were acquired using a Stejskal-Tanner spin-echo DWI sequence on a 3T GE Discovery MR750 MRI scanner at the Center for MRI Research of Peking University. Sequence parameters were as follows: TR/TE = 5400 ms/115 ms, in-plane matrix = 80 $$$\times$$$ 80, FOV = 24 cm,37 axial slices of thickness 3.0 mm.Nine diffusion gradient amplitudes $$$G_{0}$$$ were chosen from 15 mT/m to 49 mT/m and the four varied separations Δ were from 27.1 ms to 54.1 ms. In summary, there were thirty-six different corresponding b-values: from 150 sec/mm² to 3700 sec/mm² in a single direction. Nine non-collinear directions of diffusion gradients were applied to study the tensorial properties of FM related parameters.

DWI images were first corrected for eddy currents distortions and head motions using FSL (http://www.fmrib.ox.ac.uk/fsl). To obtain the FM-based anomalous diffusion indices $$$H$$$ and $$$\alpha$$$ maps along nine non-collinear directions, the signal attenuation of each voxel were fitted along each direction separately to Eq. (1) using a nonlinear least-squares fitting algorithm. Tensors of both $$$H$$$ and $$$\alpha$$$ were then obtained from those parametric maps of nine non-colinear diffusion directions. Furthermore, fiber tractography of $$$H$$$ was done.

Anomalous diffusion parametric $$$H$$$ and $$$\alpha$$$ maps are computed along each diffusion gradient’s direction and shown in Fig. 1. Both maps of $$$H$$$ and $$$\alpha$$$ reveal good contrast among gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF). It is shown in Fig. 1 that CSF has the largest value for both $$$H$$$ and $$$\alpha$$$ maps. The $$$H$$$ and $$$\alpha$$$ values in CSF regions close to 2.0 and 0.5， which means that the diffusion process here is approximate to Brownian motion. The FA maps and direction encoded colormaps of V₁ for $$$H$$$ and $$$\alpha$$$ are shown in Fig. 2. In general, the anomalous diffusion indices $$$H$$$ and $$$\alpha$$$ can demonstrate different tissue structures. It can be seen that the FA map of $$$H$$$, which is similar to the diffusivity tensor, reveals better image contrast than that of $$$\alpha$$$. The whole brain tractography based on $$$H$$$tensor is shown in Fig. 3. It is shown that the $$$H$$$tensor-based tractography is capable of revealing the basic fiber tract trajectories, which indicates that the $$$H$$$ tensor reflects the microscopic tissue structures.

Several anomalous diffusion models, both empirical and theoretical, were proposed to fit the DWI signal decay. Only few of them, which were mainly focused on the stretched exponential model, have investigated their anisotropic properties. Here, the tensorial properties were investigated for FM-based MRI. The Hurst exponent $$$H$$$, which demonstrates the similarity of particle trajectories, reveals great anisotropic property. It indicates that particle trajectories have varied similarity property in different directions. However, the FA map of $$$\alpha$$$ is not as well-contrasted as that of $$$H$$$. The reason of this phenomenon needs to be further investigated.

In sum, the tensorial properties of FM-related parameters were studied in this work and it was found that those parameters are direction dependent.