Emilie T. McKinnon^{1,2,3}, Jens H. Jensen^{1,3}, and Joseph A. Helpern^{1,3}

A better understanding of the complex water diffusion dynamics in the extra-axonal environment may aid in the mathematical modeling of the diffusion MRI signal and have application to pathologies that specifically impact glial cells and the surrounding extracellular space. Here we employ a novel method of combining diffusion MRI data for weak and strong diffusion weightings to show that extra-axonal water diffusion anisotropy strongly correlates with intra-axonal diffusion anisotropy and takes on large values in voxels with highly aligned axons. This connection suggests that the geometrical alignment of axonal fibers is important for both intra-axonal and extra-axonal water diffusion.

dMRI data from one healthy adult (30 yrs) were acquired on a Siemens Prisma scanner for 4 b-value shells with b = 0, 1000, 2000 and 6000 s/mm^{2} and 30, 30 and 256 diffusion encoding directions, respectively. Additional imaging parameters were TE = 98 ms, TR = 5100 ms and voxel size = (3 mm)^{3}. Estimations of the total diffusion tensor, $$$\bf{D}$$$ , were obtained through conventional calculations by using data from the first two shells^{8}. The diffusion tensor for the intra-axonal compartment, $$$\bf{D_a}$$$, was calculated, up to an overall scaling factor, from the highest b-value shell with FBI. This yielded $$${\bf A}\equiv{\bf D_a}/D_a$$$ (1), with the unknown scaling factor, $$$D_a$$$, corresponding to the intrinsic intra-axonal diffusivity. Under the assumption that myelin water can be neglected due to its short T2, $$$\bf{D}$$$ is related to $$$\bf{D_a} $$$ and the extra-axonal diffusion tensor, $$$\bf{D_e}$$$ by

$${\bf D}=f \cdot{\bf D_a}+ (1-f)\cdot{\bf D_e}\space\space(2), $$

where $$$f$$$ is the axonal water fraction. This can be rearranged into $${\bf D_e}=\frac{{\bf D}-f\cdot{\bf D_a}}{1-f}\space\space(3) $$

Since FBI also yields estimates for the microstructural parameter $$$\zeta=f/\sqrt D_a$$$^{7}, it is useful to rewrite Equation (3) in the form

$${\bf D_e}=\frac{{\bf D}-\zeta\cdot\sqrt D_a^3\cdot{\bf A}}{1-\zeta\cdot\sqrt D_a}\space\space(4) $$

Thus given a value for $$$D_a$$$, one can calculate $$$\bf{D_e}$$$ along with its associated anisotropy FAE. Here we calculate FAE in healthy WM for a physically plausible range of (1, 1.5, 2, and 2.5 µm^{2}/ms)^{4-6,9,10}. Voxels with a mean diffusivity lower than 1.5 µm^{2}/ms and a mean kurtosis larger than 1 were considered to be WM.

The average FAA of the intra-axonal water in WM is found to be 0.54±0.13, and the distribution of FAA values is shown in Figure 1. Not surprisingly, the FAE is lower than the FAA at 0.40±0.14, 0.37±0.13, 0.33±0.13 and 0.26±0.13 for $$$D_a$$$ = 1, 1.5, 2, and 2.5 µm^{2}/ms (Figure 2). The large standard deviations are indicative of a heterogeneous microstructural environment, and the regional variability is illustrated in Figure 3 using voxelwise parametric maps of FAA and FAE. The relationship between FAA and FAE is shown in Figure 4, revealing strong correlations for all the $$$D_a$$$ values considered. The coefficients of determination decrease as is increased, with r^{2} values of 0.88, 0.85, 0.79 and 0.61 for $$$D_a$$$ = 1, 1.5, 2, and 2.5 µm^{2}/ms, respectively.

Figure
1: Distribution of FAA
in the WM for one healthy control. The average±standard deviation)
FAA is 0.54±0.13.

Figure
2: Distribution of FAE
in the WM for same subject as in Figure 1 for a range of Da values (1,
1.5, 2, 2.5 µm^{2}/ms).
The average±standard deviation)
FAE is 0.40±0.14, 0.37±0.13, 0.33±0.13 and 0.26±0.13, respectively, for the different Da. The widths of the distributions reflect the high degree
of regional variability for extra-axonal environment.

Figure
3: Parametric maps showing
the FAA (left) and the FAE (right) for an identical anatomical slice. FAE was
calculated using Da of 2 µm^{2}/ms. In
general, the anisotropy of the intra-axonal compartment exceeds that of the
extra-axonal compartment, with the largest FAE values occurring in voxels with highly
aligned axonal fiber bundles (e.g. corpus callosum, posterior limb internal
capsule).

Figure
4: Relationship between
FAA and FAE, calculated from **De**, for a
range of Da values, using data from one healthy control.
FAA and FAE are highly correlated with r^{2}
= 0.88, 0.85, 0.79, 0.61,
depending on Da . These results strongly suggest that the geometrical
arrangement of axonal fiber bundles impacts the diffusion anisotropy of the
extra-axonal space.