Hunter Moss^{1}, Emilie T. McKinnon^{2}, Andreana Benitez^{2}, Dorothea D. Jenkins^{3}, G. Russel Glenn^{4}, Jens H. Jensen^{5}, and Joseph A. Helpern^{5}

Using fiber ball imaging (FBI), we isolate the intra-axonal compartment of the white matter with strong diffusion weighting in order to suppress the extra-axonal water fraction and then calculate its associated fractional anisotropy (FA). We call this parameter the fractional anisotropy axonal (FAA) in contrast to the conventional FA, and we compare these two measures in three subjects: a healthy young adult, a cognitively intact older adult with severe white matter abnormalities, and a neonate with acute hypoxic ischemic injury. Our results indicate that FAA reveals diffusion anisotropy that is not apparent using the conventional FA.

FBI determines spherical
harmonic expansion coefficients for the fODF from high angular resolution diffusion imaging
(HARDI) data using a single b-value shell.^{1} The b-value should be
about 4000 s/mm^{2} or larger in order to adequately suppress signal
from extra-axonal water,^{3} which is neglected in the analysis. More
precisely, one obtains the fODF, $$$ F( \bf n) $$$, for a direction $$$ \bf n $$$ as

$$ F({\bf n}) = \sum_{l=0}^{\infty}\sum_{m=-l}^{l} \it c_{l}^{m}\it Y_{l}^{m} \tt (\theta,\phi), $$

where $$$ \it Y_{l}^{m} $$$ are the spherical harmonics, $$$ (\theta,\phi) $$$ are the spherical angles for $$$ {\bf n} $$$, and $$$ \it c_{l}^{m} $$$ are the complex-valued expansion coefficients; because of reflection symmetry, $$$ c_{2l+1}^{m} = 0 $$$. The diffusion tensor, $$$ {\bf D}_a $$$, for the intra-axonal compartment is related to $$$ \it c_{l}^{m} $$$ by

$$ {\bf D}_a = \frac{D_a}{c_{0}^{0} \sqrt{30}} \begin{bmatrix} \frac{\sqrt{30}}{3}c_{0}^{0}-\frac{\sqrt{6}}{3}c_{2}^{0}+c_{2}^{2}+c_{2}^{-2} & ic_{2}^{2}-ic_{2}^{-2} & -c_{2}^{1}+c_{2}^{-1} \\ ic_{2}^{2}-ic_{2}^{-2} & \frac{\sqrt{30}}{3}c_{0}^{0}-\frac{\sqrt{6}}{3}c_{2}^{0}-c_{2}^{2}-c_{2}^{-2} & -ic_{2}^{1}-ic_{2}^{-1} \\ -c_{2}^{1}+c_{2}^{-1} & -ic_{2}^{1}-ic_{2}^{-1} & \frac{\sqrt{30}}{3}c_{0}^{0}+\frac{2\sqrt{6}}{3}c_{2}^{0} \end{bmatrix}. $$

Following from this, the FAA can be written as

$$ FAA = \sqrt{\frac{3\sum_{m=-2}^{2}|c_{2}^{m}|^{2}}{5|c_{0}^{0}|^{2}+2\sum_{m=-2}^{2}|c_{2}^{m}|^{2}}}. $$

To test this expression for
FAA, we acquired HARDI data at b = 6000 s/mm^{2} with 128 diffusion encoding
directions (adults) and 64 diffusion encoding directions (neonate) using a 3T
Siemens Prisma scanner. We also acquired diffusional kurtosis imaging (DKI)^{4}
datasets with 64 directions and b = 0, 1000, and 2000 s/mm^{2} in order to quantify the FA. The HARDI
acquisition had 3 mm isotropic voxels with TE = 110 ms (adults) and TE = 98 ms (neonate),
while the DKI acquisition had 3 mm isotropic voxels with TE = 110 ms (adults) and
2.5 mm isotropic voxels with TE = 85 ms (neonate). FAA and FA maps were
produced with in-house Matlab scripts and diffusional kurtosis estimator (DKE),^{5}
respectively. Regions of interest (ROIs) sampled from the frontal WM (fWM), the
posterior limb of the internal capsule (PLIC), and the splenium of the corpus
callosum (sCC) were manually drawn using MRIcron^{6} for each map.

As seen in Figure 1, the FAA (lower row) has substantially higher values in most WM regions in comparison to the FA (upper row). This suggests intra-axonal water diffusion to be more anisotropic than for the full tissue, as might be expected. Furthermore, FAA detects sizeable diffusion anisotropy in brain regions where the FA is low. This is most apparent for the older adult (Figure 1, panels B and E) and the neonate (Figure 1, panels C and F).

Table 1 shows mean ± standard deviation values for FA, FAA, and FAA/FA in three ROIs for each subject. In every case, FAA is greater than FA. The FAA/FA ratio varies across regions from 1.41 to 2.48 in the fWM, 1.18 to 2.08 in the PLIC, and 1.09 to 1.61 in the sCC.

The Litwin Foundation (PI: Helpern)

NIH/NIA K23AG044434 (PI: Benitez)

Rare Disease Foundation (PI: Benitez)

Medical University of South Carolina Neuroscience Institute Grant (PI: Jenkins)

T32DC0014435 (Trainee: McKinnon)

T32GM008716 (Trainee: Glenn)

1. Jensen, J.H., G. Russell Glenn, and J.A. Helpern, Fiber ball imaging. Neuroimage, 2016. 124(Pt A): p. 824-33.

2. Tuch, D.S., Q-ball imaging. Magn Reson Med, 2004. 52(6): p. 1358-72.

3. McKinnon, E.T., et al., Dependence on b-value of the direction-averaged diffusion-weighted imaging signal in brain. Magn Reson Imaging, 2017. 36: p. 121-127.

4. Jensen, J.H., et al., Diffusional kurtosis imaging: the quantification of non-gaussian water diffusion by means of magnetic resonance imaging. Magn Reson Med, 2005. 53(6): p. 1432-40.

5. Tabesh, A., et al., Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging. Magn Reson Med, 2011. 65(3): p. 823-36.

6. Rorden, C. and M. Brett, Stereotaxic display of brain lesions. Behav Neurol, 2000. 12(4): p. 191-200.

Figure 1. Comparison
of DKI-derived FA (upper row) with FBI-derived FAA (lower row) for approximately
similar anatomical axial slices in a young healthy adult (A, D), a cognitively
intact older adult with severe WM abnormalities (B, E), and a neonate with HIE
(C, F). The FAA has substantially higher values even in WM regions where the FA
is close to zero, possibly indicating better discrimination of axonal
microstructure. These differences are most apparent for the older adult and the
neonate.

Table 1. Mean
values (± standard deviations) for FA, FAA, and the FAA/FA ratio in manually
drawn ROIs. The ROIs correspond to the frontal WM (fWM), the posterior limb of
the internal capsule (PLIC), and the splenium of the corpus callosum (sCC). In all
of the ROIs, the FAA is larger than the FA. The FAA/FA ratio does show a strong
regional variation and is greatest in the fWM. The relatively smaller ratios
for the sCC may be due in part to high diffusion anisotropy for the
extra-axonal water confined to densely packed, unidirectional axonal fiber
bundles.