Some measure of the appropriateness of the effective diffusivity model can be gleaned from the data in Figure 5. The close agreement between the experimental drug concentration trace and the computational one is evidence that MDV3100 the complex phenomena underlying ultrasound-enhanced drug transport can be captured in the model with one critical parameter (porosity) and two less important parameters (thickness and delay time). Presumably, if more parameters were required or a multiscale approach was necessary, close agreement in both the pre-ultrasound and post-ultrasound regimes would not be attained. An estimate of the agreement between the experiments and computations, and hence of the appropriateness of the model, is provided by the optimization algorithm. The algorithm yields not only a best guess for the porosity, but an estimate of the RMS error between experiments and the effective diffusivity model.
The large uncertainty in the porosity ratio for an intensity of 2 W/cm2 is due to the single trial for which the ratio is 12.5. This ratio, along with the ratios for all of the other trials, are illustrated in Figure 6. Ignoring the singularly large value, the mean porosity ratio for the 2 W/cm2 intensity is 3.2, with a standard deviation of 0.8. With or without the singularly large value, a one-tailed t-test yields the conclusion that the porosity ratio is significantly larger than one (p < 0.05). The standard deviations as a percentage of the mean, ignoring the single large value, are less than 30% for the four intensities. At the higher pressures used in the study, ultrasound-induced cavitation potentially played a role in changing the porosity of the epithelial layer. Although the peak negative pressure is likely too low to initiate cavitation (mechanical index < 0.5 in all cases) within the cornea, the cornea was adjacent to a liquid mixture containing tap (non-degassed) water combined with sodium fluorescein. As reported by Atchley et al. (1988), the cavitation threshold in water at a frequency of 0.98 MHz is on the order of 400 kPa, comparable to the higher pressures in this study. (In the study of Atchley et al., the water was degassed, but contained latex particles.) Using the same model transducer as the one from our study, and a passive cavitation detector, Castellanos et al. (2017) detected noticeable energy (about 10 dB above noise level) at frequencies equal to half and 1.5 times the fundamental frequency (800 kHz), during 1 W/cm2 exposures. The presence of energy at these frequencies is a strong indicator that inertial cavitation was present. Morphologically, the increase in porosity may be due to a transient increase in intracellular spacing within the epithelial layer, caused by oscillation of cavitation-induced bubbles near the cornea surface, or to the microjets arising from the collapse of the bubbles (Paliwal and Mitragotri 2006). The small reduction in epithelial thickness is possibly due to the removal of cells from the surface layers of the epithelium (Zderic et al. 2004a).
Although the pressure across the corneal surface was assumed to be uniform, it was seen in Figure 2 that the pressure decreased by about a factor of 2 over the radius of the orifice. The effect of the variable pressure distribution on the variability of the porosity distribution was estimated in the following manner. Given that the pressure varies over the scale of the orifice radius, about 6 mm, and the thickness of the epithelium is less than one-hundredth of that, we assume that the pressure field is locally uniform, that is, the epithelium at any radial location responds to the pressure of the ultrasound beam at that location. To model the increase in porosity with pressure, we hypothesized a local enhancement of porosity on pressure of the formwhere p0 is the threshold pressure below which no porosity enhancement occurs. That pressure and the scaling constant b were determined by first integrating this function for LE across the entire corneal surface, to find the total enhancement TE: