Scott Swanson^{1}, Dariya Malyarenko^{1}, and Thomas Chenevert^{1}

A set of materials based on nanostructured lamellar vesicles with restricted diffusion compartments is constructed to achieve tunable diffusion kurtosis behavior. The observed apparent diffusion (D_{app}) and kurtosis (K_{app}) model parameters span the range of values found in vivo. Effect of vesicle population, size, and porosity is studied for estimated diffusion parameters. These nanostructured systems provide an ideal platform for a diffusion kurtosis phantom used to validate quantitative imaging protocols and results.

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Fig. 1. Samples with uni- and multi-lamellar vesicles are composed of regions of relatively "free" or unrestricted water molecules (P1 pool and diffusion constant D1) and of impeded or restricted waters (P2 pool and effective diffusion constant D2) where P1 + P2 = 100%. Because the mean free path of diffusion, about 20 microns at 100 ms, is much greater than the size of the vesicles, increases in D2 arise from more porous vesicle walls allowing the restricted diffusion to mix with the free diffusion. Scale bar is 1 micron.

Fig 2. (A) Log normalized MR signal intensity as a function of b value for CA-BTAC samples between 0.5 and 5% (w/w). Data are fit to a biexponential decay. (B) Estimated diffusion constant D1 of unrestricted waters as a function of CA-BTAC concentration. (C) Estimated percentage of sample (P2) that is restricted by lamellar vesicles as a function of CA-BTAC concentration. Estimated errors of the fitting in B and C are within markers.

Fig 3. (A) Region of CA-BTAC diffusion data appropriate for DKI analysis. Data are fitted to the Kurtosis model (shown as solid line). (B) Estimated apparent diffusion constant, D_{app}, as a function of CA-BTAC concentration. (C) Estimated kurtosis constant, K_{app}, as a function of CA-BTAC concentration. Dapp is inversely related to CA-BTAC concentration and Kapp is nearly linear, allowing one to create a sample with any desired kurtosis value between 0.76 and 4.5 (Table 1).

Fig 4. (A) Diffusion in 1% CA-BTAC and DEC-CTAB. More porous DEC-CTAB results in a larger effective diffusion constant D2. Other parameters (D1 and P2) are similar. (B) DKI analysis and fitted results. Changing the porosity of vesicles has a minor effect on D_{app} and significant effect on K_{app}. (C) Diffusion in 1.6% (w/w) CA-SDS vesicles. Sonicating vesicles reduces size and restricted volume, leading to nearly free diffusion in the sonicated sample. Changing the ratio of alcohol (CA) to surfactant (SDS) changes the porosity of the sample and affects kurtosis values.

Table 1. Diffusion parameters estimated for the 10 lamellar vesicle samples using a biexponential model and a kurtosis model. For the biexponential model, S/S(0) = P1 exp(-*b* D1) + P2 exp(-*b* D2) and for the kurtosis model ln(S/S(0)) = -*b* D_{app} + 1/6 K_{app} (*b* D_{app})^{2}