Antonios Papaioannou^{1}, Dmitry Novikov^{1}, and Els Fieremans^{1}

Well characterized phantoms are essential for time-dependent diffusion model validations. Here we introduce a phantom made of permeable barriers with highly tunable micro-structural characteristics such as pore size, pore density and permeability and perform time-dependent diffusion measurements using three NMR and MRI systems. Our experimental results agree with the theory of time-dependent diffusion in disordered systems, making the phantom suitable for model validations in clinical and preclinical systems.

Phantom: Porous films (Sterlitech-CO) made of polycarbonate were used to create a permeable phantom. The films were circular with a diameter of 27$$$mm$$$, 6$$$\,\mu\,m$$$ thick, had pores $$$0.15\,\mu\,m$$$ in diameter and a density of 5 pores/$$$\mu\,m^2$$$. A cylindrical tube was designed to hold the films together (Fig.1a), with two pistons on either side of the cylinder for controlling the distance between the films. The phantom was constructed by stacking 90 films along $$$z$$$ with an average distance between them $$$\bar{a}=30\,\mu\,m$$$, then filled with water, sonicated and depressurized for microbubble removal before performing MR experiments. In addition, the tube was embedded in a larger cylinder filled with water to enhance the signal when scanning on the Siemens-Prisma (Fig.1c). Note that for the NMR experiments $$$\bar{a}=12\mu\,m$$$.

MR experiments
were performed on three systems: a preclinical imaging system (7T Bruker-Biospec), a clinical system (3T Siemens-Prisma) and an NMR system (4.2T
TecMag-Apollo). For the diffusion measurements a STEAM sequence was used.
Below are the experimental parameters for each system. Biospec: number of
directions$$$\,=\,124$$$, $$$b=0.2,\,0.6,\,1.0,\,1.5,\,2.0\,ms/\mu\,m^2$$$,
$$$\Delta=11-3000\,ms$$$, $$$\delta$$$=2.1$$$\,ms$$$, TE=19$$$\,ms$$$,
TR=6$$$\,s$$$, Res=0.5x0.5x2.0$$$\,mm^3$$$. Prisma: number of directions$$$\,=\,124$$$, $$$b=\,0.5,\,1.0,\,1.2,\,1.5\,ms/\mu\,m^2$$$, $$$\Delta=40-1900\,ms$$$,
$$$\delta$$$=10$$$\,ms$$$, TE=93$$$\,ms$$$, TR=8$$$\,s$$$,
Res=2.0x2.0x1.5$$$\,mm^3$$$. Apollo: $$$b=\,0.1,\,0.13,\,0.17,\,0.22,\,0.26,\,0.32\,ms/\mu\,m^2$$$,
$$$\Delta=1.6-3500\,ms$$$, $$$\delta$$$=120$$$\,\mu\,s$$$, TE=15$$$\,\mu\,s$$$, TR=10$$$\,s$$$. The diffusion images were denoised^{7} and the
signal was averaged within the ROI followed by a DKI^{8,9} (Diffusion
Kurtosis Imaging) weighted linear least squares fit. For the NMR experiments, the b-value was chosen such that $$$bD_0<1$$$ to avoid higher order cumulants and a least squares fit was performed to obtain the diffusivity^{10}. Additional NMR experiments were performed on the setup of reference^{10} using phantoms of varying pore size from $$$0.15-0.65\,\mu\,m$$$.

The presence of
microstructure in complex media manifests itself as power-law tail in the time-dependent
diffusion coefficient^{11}

$$D(t)\simeq\,D_\infty+c\,t^{-\vartheta},\quad(1)$$

for times $$$t>>\tau_r$$$ much exceeding the residence time $$$\tau_r\equiv\bar{a}/2\kappa$$$ between adjacent membranes, where $$$\kappa$$$ is the membrane permeability, and $$$D_{\infty}$$$ is the diffusivity at $$$t\rightarrow\infty$$$.

In the case of short-range disorder (similar to
Poissonian disorder), the exponent $$$\vartheta=1/2$$$ in Eq.(1). Here, we experimentally estimate the
permeability of our membranes using^{11},

$$\kappa = \frac{D_0}{\bar{a}\zeta}\,,\quad\zeta=\frac{D_0}{D_{\infty}} - 1,\quad(2)$$

and show the validity of scaling (1).

Fig. 2 shows the diffusivity time dependence in the phantom along ($$$z$$$) and transverse ($$$x-y$$$) to the porous films. The inset of Fig. 2 highlights the diffusivities normalized with $$$D_0$$$ at the experimental temperature. A characteristic time-dependence was observed along the $$$z$$$ direction whereas in directions $$$x-y$$$, the diffusivity had no time-dependence. Since the permeable films were placed in random positions along the cylindrical tube (Fig. 1a), a time-dependence according to Eq.(1) is expected for the diffusivity. Indeed, by performing a three degrees of freedom least squares fit, and normalizing with $$$D_{\infty}$$$, a power law of $$$\vartheta=0.62$$$, $$$\vartheta=0.41$$$ and $$$\vartheta=0.59$$$ was observed, in agreement with Eq.(1). The permeability of the phantom can be estimated from Eq.(2) using the $$$D_0$$$ at the experimental temperature and $$$D_{\infty}$$$ from the fit. For the NMR system $$$\kappa\simeq\,0.04\,\mu\,m/ms$$$ whereas for Biospec and Prisma MRI systems $$$\kappa\simeq\,0.03\,\mu\,m/ms$$$ and $$$\kappa\simeq\,0.025\,\mu\,m/ms$$$ respectively. Fig. 3 shows the permeability-values of phantoms made with films of different pore size.

Fig. 4 shows the time dependence of the kurtosis
for the scans performed on the MRI systems. Kurtosis along the $$$z$$$
direction showed a non-monotonic time-dependence, similar to what was observed in the brain^{12}, and peaks at approximately 6$$$\tau_r$$$.

1. Komlosh, M. E., et al. "Anisotropic phantom to calibrate high-q diffusion MRI methods." Journal of Magnetic Resonance 275 (2017): 19-28.

2. Lemberskiy, Gregory, et al. "Validation of surface‐to‐volume ratio measurements derived from oscillating gradient spin echo on a clinical scanner using anisotropic fiber phantoms." NMR in Biomedicine 30.5 (2017).

3. Fillard, Pierre, et al. "Quantitative evaluation of 10 tractography algorithms on a realistic diffusion MR phantom." Neuroimage 56.1 (2011): 220-234.

4. Komlosh, Michal E., et al. "Pore diameter mapping using double pulsed-field gradient MRI and its validation using a novel glass capillary array phantom." Journal of magnetic resonance 208.1 (2011): 128-135.

5. Fieremans, Els, et al. "Simulation and experimental verification of the diffusion in an anisotropic fiber phantom." Journal of Magnetic Resonance 190.2 (2008): 189-199.

6. Burcaw, Lauren M., Els Fieremans, and Dmitry S. Novikov. "Mesoscopic structure of neuronal tracts from time-dependent diffusion." NeuroImage 114 (2015): 18-37.

7. Veraart, Jelle, et al. "Denoising of diffusion MRI using random matrix theory." NeuroImage 142 (2016): 394-406.

8. Veraart, Jelle, et al. "Weighted linear least squares estimation of diffusion MRI parameters: strengths, limitations, and pitfalls." NeuroImage 81 (2013): 335-346.

9. Jensen, J. H., Helpern, J. A., Ramani, A., Lu, H., & Kaczynski, K. (2005). Diffusional kurtosis imaging: The quantification of non‐gaussian water diffusion by means of magnetic resonance imaging. Magnetic Resonance in Medicine, 53(6), 1432-1440.

10. Papaioannou, A., Novikov, D. S., Fieremans, E., & Boutis, G. S. (2016). Diffusion distinguishes between structural universality classes of disordered media. arXiv preprint arXiv:1607.08639.

11. Novikov, D. S., Jensen, J. H., Helpern, J. A., & Fieremans, E. (2014). Revealing mesoscopic structural universality with diffusion. Proceedings of the National Academy of Sciences, 111(14), 5088-5093.

12. Pyatigorskaya, N., Bihan, D., Reynaud, O., & Ciobanu, L. (2014). Relationship between the diffusion time and the diffusion MRI signal observed at 17.2 tesla in the healthy rat brain cortex. Magnetic resonance in medicine, 72(2), 492-500.

a) Schematic of the phantom. A cylindrical tube
(blue) made out of Teflon, with a length of 125 mm and an inner diameter of 27
mm holds the films together. The permeable films (dark blue) were stacked along
$$$z$$$ at random positions as shown above. The two pistons on top and bottom
(gray) were positioned such that the average distance between the permeable
barriers $$$\bar{a}\simeq30 \mu m$$$. b) Signal (b=0) from the water inside the
films acquired with Bruker BioSpin. c) Signal (b=0) from the water inside the
films as well as the surrounding unrestricted water acquired with Prisma.

Normalized diffusivities along $$$z$$$ plotted
with respect to $$$t/\tau_r$$$ exemplify the short-range nature of the phantom
used in this work. The long time scaling of the diffusivity resembles a
power-law with a dynamical exponent $$$\vartheta\simeq 0.5$$$ in agreement with
the theory (Eq.(1)). Inset: Diffusivities
normalized with the $$$D_0$$$ at the corresponding temperature; the solid
squares represent the average diffusivity along directions $$$x-y$$$ and solid
circles diffusivities along $$$z$$$ (through the permeable films).

Estimated permeabilities of the phantoms with
respect to pore size using experimental data acquired as described in reference^{10}.

Estimated kurtosis along $$$z$$$ shows a
non-monotonic behavior for the diffusion times investigated in this work and
showed a peak at approximately $$$\sim6\tau_r$$$. On the other hand kurtosis along directions $$$x-y$$$ was flat and closer to zero.