Sudhir Kumar Pathak^{1}, Vinod Jangir Kumar^{2}, Catherine Fissell ^{3}, Anthony Zuccolotto^{4}, and Walter Schneider^{5}

A Textile Anisotropic Brain Imaging Phantom incorporating textile water filled hollow fibers (taxons with inner/outer diameter 12/34 micron) is used to examine time-dependent diffusion. In this study, impermeable hollow tubes (taxons) with 12-micron diameter are used to test the relationship between axial and radial diffusivities with diffusion time (Δ) for a given taxon packing density. An inverse relationship of radial diffusivity with diffusion time (Δ) is established. A constant relationship of axial diffusivity with diffusion time is established. The dependence of these relationships on packing density is then tested and the radial diffusivity relationship is shown to vary with packing density.

Diffusion MRI (dMRI) can estimate tissue microstructure that depends upon the cellular length scale. Diffusion length can be related axonal diameter(s) and packing density. By manipulating either diffusion gradient (G) and/or diffusion time (Big Delta, Δ), the sensitivity of length scale can be studied using diffusion imaging (PGSE sequence^{4}). Equation (1) shows the relationship between diffusion time, gradient strength and b-value that can be used to vary diffusion time length. Investigation of these relationships to date^{2,3,5,6} has been limited by a need to confirm results with histological analysis. Empirical determination of the relationship between Δ and diffusion measurements such as radial and axial diffusivity in a sample with manufactured microstructure can validate and calibrate dMRI biophysical modeling such as mathematical models for restricted and hindered compartments.

$$b = \gamma^{2} G^{2} \delta^{2} \left( \Delta - \frac{\delta}{3} \right) $$

In this study diffusion time (Δ) was manipulated by making b-value constant and varying the diffusion gradient (G). Axial and radial diffusivity measurements are derived from the imaging data using DTI reconstruction^{9}. The relationship between these diffusivity measurements and diffusion time was tested with linear regression as described by equation:

$$ D(\Delta) = \beta_{0} + \frac{\beta_{1}}{\Delta} $$

The functional form of depends upon the structural exponents and effective spatial dimensionality (*d*) as proposed by Burcaw^{2 }and Novikov^{3}. The mathematical form of the time-dependence of axial (*d=1*) and radial (*d=2*) diffusivity is dependent on the structural disorder along the fibers. In this study, hollow impermeable fibers with constant radius are used to test the relationship (2). This suggests that axial diffusivity should be constant (no-time dependence) whereas radial diffusivity is dependent upon the packing density and should have the functional form described in equation (2).

Figure
1: A) Plot of Axial diffusivity w.r.t. to diffusion time. Coefficients of
variation for density are 19%, 36%, 21%, 23% respectively. B) Plot of Radial
diffusivity w.r.t. to
diffusion time. Plot shows an inverse relationship between diffusion time and
radial diffusivity. C) Plot of Axial diffusivity w.r.t. to diffusion time. Coefficients of
variation for density are 25%, 19.25%, 10.79%, 7.19% respectively. AD, RD and
FA are both estimated for a 10x10 mm^{2}
chamber with four different density patterns (12.5%, 25%, 50% and 100%).

Table
A: Regression coefficient for Axial Diffusivity with diffusion time. R2
and
associated p-values are reported for each regression coefficient. Variation of
R2 shows differences in packing density. 12.5%, 25% shows good fit vs higher
density (100%, 50%).

Table B: Regression coefficient for Radial Diffusivity with diffusion time. R2
and
associated p-values are reported for each regression coefficient.

Figure
3: A) SEM image of Taxons. B) SEM image of taxonal bundle with 40% packing
density. C) CAD model of actual phantom. D) Axial slice of T1 image. Density
cubes are shown in middle of the figure.