A tunable permeability phantom for validating time-dependent diffusion models
Antonios Papaioannou1, Dmitry Novikov1, and Els Fieremans1

1Center For Biomedical Imaging, New York University Medical Center, New York, NY, United States


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.


Well-characterized phantoms that mimic the salient characteristics of microstructure in biological tissues (e.g., the brain and muscle) are instrumental for testing and validation of diffusion MRI. Phantom development so far has been mainly focused on anisotropic phantoms, either consisting of hollow1 or plain2 fibers, used for sequence testing, tractography validation3 and testing of biophysical models for axon diameter mapping4 and extracellular5,6 diffusion. However, molecular transport in the presence of permeable membranes is ubiquitous in living tissues, and a well-characterized permeable phantom is still lacking. Here we introduce a phantom made of permeable porous films that can be tuned in terms of its microstructural characteristics, including pore size, pore density and permeability. We test the phantom by performing time-dependent diffusion experiments on three different MRI and NMR systems (Bruker, Siemens and TecMag), and comparing against theory.


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 denoised7 and the signal was averaged within the ROI followed by a DKI8,9 (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 diffusivity10. Additional NMR experiments were performed on the setup of reference10 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 coefficient11


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 using11,

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

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 brain12, and peaks at approximately 6$$$\tau_r$$$.


The present work demonstrates the time-dependence of diffusivity and kurtosis of a custom made permeable phantom on clinical and preclinical systems. Our results reveal a characteristic power-law scaling (1) of the diffusivity with time, agreeing with the theory, as well as a non-monotonic time-dependence of kurtosis. The microstructural features of this phantom are well-characterized with a permeability that can be tuned, making it a promising candidate for testing and validating the effect of exchange on various MRI contrasts including diffusion.


The authors acknowledge Thorsten Feiweier from Siemens Healthcare. Jelle Veraart and Gregory Lemberskiy for useful discussions about the data and Jerzy Walczyk for machining some of the phantom parts. Research was supported by the NIH/NINDS award R01NS088040 and by NIH/NCI 5P30CA016087 and NIH P41 EB017183.


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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 reference10.

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.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)