Hong-Hsi Lee^{1}, Els Fieremans^{1}, and Dmitry S. Novikov^{1}

We explore axial diffusivity dependence on both diffusion time and gradient pulse width in major white matter tracts. This allows us to differentiate between two possible arrangements of restrictions (e.g. beads) along fibers: (1) short-range disorder, or (2) “hyperuniform” disorder (arrangement qualitatively closer to periodic). Unexpectedly, model prediction for hyperuniform disorder is more consistent with our data than for short-range disorder. If conformed histologically in human or animal studies, this would mean that restrictions along axons are not “purely” randomly distributed but rather spatially correlated − perhaps, for optimizing physiological constraints.

**Short-range
disorder** is characterized by a finite correlation length $$$l_c$$$,
beyond which the statistics of restrictions is Poissonian (uncorrelated) (**Fig.1**,
blue). Irrespective of the microscopic nature of restrictions (e.g. beads), short-range
disorder in their placement yields the following functional form for the transverse or axial diffusivity [2,3]:
$$D_\parallel^{\rm sr}\simeq\,D_\infty+\frac{A^{\rm
sr}}{\Delta-\delta/3}\cdot\left[2\sqrt{\Delta}-\frac{16}{15}\sqrt{\delta}\right],\quad(1)$$
where $$$D_\infty$$$ is the bulk diffusivity at $$$\Delta\to\infty$$$, and
$$$A^{\rm sr}$$$ is the strength of restrictions
($$$\mu$$$m$$$^2$$$/ms$$$^{-1/2}$$$). When $$$\Delta\gg\delta$$$,
$$$D_\parallel^{\rm sr}$$$ scales with ~$$$1/\sqrt{\Delta}$$$. The correlation
length is estimated by $$$l_c^\parallel\simeq 2A^{\rm sr}\sqrt{\pi/D_\infty}$$$
[1].

The $$$\sim 1/\sqrt{\Delta}$$$ diffusivity scaling, observed in GM with OGSE [3,5], is consistent with short-range disorder in the histological observations [4] of varicosities along neurites in gray matter. In contrast, the varicosity distribution along axons in WM tracts has not been characterized.

**Hyperuniform
disorder** [3,6] implies a much more, though not perfectly, ordered
arrangement (**Fig.1**, green). An example is a “shuffled lattice”, i.e. individual
restrictions displaced from their ideal periodic positions by independent
random shifts drawn from a distribution with finite variance $$$\sigma_a^2$$$. While
local snapshots of disordered configurations look deceptively similar to
short-range disorder (**Fig.1**), the functional form of the axial diffusivity is
markedly different [2,3]:
$$D_\parallel^{\rm hu}\simeq\,D_\infty+\frac{A^{\rm
hu}}{\Delta-\delta/3}\cdot\left[\frac{16}{3}\cdot\frac{1}{\sqrt{\delta}}\right],\quad\quad\,\,\,(2)$$
where $$$A^{\rm hu}$$$ is the strength of restrictions
($$$\mu$$$m$$$^2\cdot$$$ms$$$^{1/2}$$$). When $$$\Delta\gg\delta$$$,
$$$D_\parallel^{\rm hu}$$$ scales as $$$\sim 1/(\Delta\sqrt{\delta})$$$, which
is qualitatively different from $$$D_\parallel^{\rm sr}$$$ in Eq.(1). In this
case, the relevant length scale is the variance-to-mean-spacing ratio $$$\sigma_a^2/\bar{a}=
4\sqrt{\pi}\cdot A^{\rm hu}\cdot D_\infty^{5/2}$$$ [3].

Diffusion measurements were performed on five healthy subjects (1 males/ 4 females, 24-44 y/o) using a 3T Siemens Prisma scanner with a 64-channel head coil. We scanned each volunteer twice for about 40 min in total using monopolar pulsed gradient spin echo sequences for different combinations of ($$$\Delta,\delta$$$). Other parameters were three $$$b$$$=0 images and $$$b$$$=500 s/mm$$$^2$$$ images along 30 diffusion gradient directions,TE/TR=150/5000 ms, voxel size of (2.7 mm)$$$^3$$$, and FOV=(221 mm)$$$^2$$$.

In **scan
1**, we varied $$$\delta$$$=4.7-49 ms and fixed $$$\Delta$$$=55 ms; in **scan 2**, we fixed $$$\delta$$$=15
ms and varied $$$\Delta$$$=21-100 ms. A series of WM regions-of-interest (ROIs)
were created by thresholding the fractional anisotropy (FA) map at 0.3-0.7 (FA
thresholded ROI); another series were created based on the JHU DTI WM atlases [7]
(anatomical WM ROI). The axial diffusivity $$$D_\parallel$$$ was averaged over
each ROI.

[1] Fieremans, E., et al. NI 129,414-427 (2016).

[2] Lee, H.H., et al. Proc ISMRM 23, 2777 (2015).

[3] Novikov, D.S., et al. PNAS 111,5088-5093 (2014).

[4] Shepherd, G.M.G., et al. PNAS 99,6340-6345 (2002).

[5] Does, M.D., et al. MRM, 49(2),206-215 (2003).

[6] Torquato, S. & Stillinger, F.H., PR E 68,041113 (2003).

[7] Mori, S., et al. MRI atlas of human white matter. Elsevier (2005).