Boundary effects on the streaming flow around a bubble located at the velocity antinode of a standing wave.

This study uses the singular perturbation method to analyze the streaming flow around a pulsating bubble at the velocity antinode of a standing wave. The bubble radially and laterally oscillates with small nondimensional amplitudes of ε` and ε, respectively. The momentum equation is expanded using ε. The frequency parameter M, which is the ratio of the bubble radius to the viscous length, is included in the expanded equations as OM-1. Four boundary conditions are solved: non-pulsating and pulsating assuming no-slip and shear-free boundaries. For the non-pulsating bubble, the streaming is on the order of OM-1 for the shear-free boundary. The flow has a quadrupole pattern, with direction from the equator to the poles. However, for the non-pulsating bubble with the no-slip boundary, the flow pattern is from the poles to the equator and the direction reverses after a critical value of M=13.3. When bubble pulsation is introduced, the intensity of the streaming increases and is proportional to M. The flow pattern is dipole with a direction from the south to the north pole for the shear-free boundary. For the non-slip boundary, the flow is quadrupole for small values of M and varies with the phase shift ϕ. As M increases, the flow intensifies and becomes dipole. For both cases, the maximum velocity is at the phase shift angle ϕ=135° and M=10.

Mathematical Model of Macromolecular Drug Transport in a Partially Liquefied Vitreous Humor.

The purpose of this study is to investigate the effect of partial liquefaction (due to ageing) of the vitreous humor on the transport of ocular drugs. In our model, the gel part of the vitreous is treated as a Darcy-type porous medium. A spherical region within the porous part of vitreous is in a liquid state which, for computational purposes, is also treated as a porous medium but with a much higher permeability. Using the finite element method, a time-dependent, three-dimensional model has been developed to computationally simulate (using the Petrov-Galerkin method) the transport of intravitreally injected macromolecules where both convection and diffusion are present. From a fluid physics and transport phenomena perspective, the results show many interesting features. For pressure-driven flow across the vitreous, the flow streamlines converge into the liquefied region as the flow seeks the fastest path of travel. Furthermore, as expected, with increased level of liquefaction, the overall flow rate increases for a given pressure drop. We have quantified this effect for various geometrical considerations. The flow convergence into the liquefied region has important implication for convective transport. One effect is the clear diversion of the drug as it reaches the liquefied region. In some instances, the entry point of the drug in the retinal region gets slightly shifted due to liquefaction. While the model has many approximations and assumptions, the focus is illustrating the effect of liquefaction as one of the building blocks toward a fully comprehensive model.

In Vivo Experimental and Analytical Studies for Bevacizumab Diffusion Coefficient Measurement in the Rabbit Vitreous Humor.

In order to measure the effective diffusion coefficient D of Bevacizumab (Avastin, Genentech) in the vitreous humor, a new technique is developed based on the "contour method" and in vivo optical coherence tomography measurements. After injection of Bevacizumab-fluorescein conjugated compound solution into the rabbit eye, the contours of drug concentration distribution at the subsurface of injection were tracked over time. The 2D contours were extrapolated to 3D contours using reasonable assumptions and a numerically integrated analytical model was developed for the theoretical contours for the irregularly shaped drug distribution in the experimental result. By floating the diffusion coefficient, different theoretical contours were constructed and the least-squares best fit to the experimental contours was performed at each time point to get the best fit solution. The approach generated consistent diffusion coefficient values based on the experiments on four rabbit eyes over a period of 3 h each, which gave D = 1.2 ± 0.6 × 10 - 6 cm 2 / s , and the corresponding theoretical contours matched well with the experimental contours. The quantitative measurement of concentration using optical coherence tomography and fluorescein labeling gives a new approach for the "noncontact" in vivo drug distribution measurement within vitreous.

Analytical and Computational Modeling of Sustained-Release Drug Implants in the Vitreous Humor.

Sustained ocular drug delivery systems are necessary for patients needing regular drug therapy since frequent injection is painful, undesirable, and risky. One type of sustained-release systems includes pellets loaded with the drug, encapsulated in a porous shell that can be injected into the vitreous humor. There the released drug diffuses while the physiological flow of water provides the convective transport. The fluid flow within the vitreous is described by Darcy's equations for the analytical model and Brinkman flow for the computational analysis while the drug transport is given by the classical convection-diffusion equation. Since the timescale for the drug depletion is quite large, for the analytical model, we consider the exterior surrounding the capsule to be quasi-steady and the interior is time dependent. In the vitreous, the fluid-flow process is relatively slow, and meaningful results can be obtained for small Peclet number whereby a perturbation analysis is possible. For an isolated capsule, with approximately uniform flow in the far field around it, the mass-transfer problem requires singular perturbation with inner and outer matching. The computational model, besides accommodating the ocular geometry, allows for a fully time-dependent mass-concentration solution and also admits moderate Peclet numbers. As expected, the release rate diminishes with time as the drug depletion lowers the driving potential. The predictive results are sufficient general for a range of capsule permeability values and are useful for the design of the sustained-release microspheres as to the requisite permeability for specific drugs.

MEASUREMENT OF THE HYDRAULIC CONDUCTIVITY OF THE VITREOUS HUMOR.

The hydraulic conductivity of the vitreous humor has been measured for the bovine eye. The experiment was carried out by placing it within upright cylindrical chamber, open at both ends, and letting its liquid content drain out of the bottom opening by gravity, through a 20µm nylon mesh filter. Additional negative pressure was provided at the exit by a hanging drainage tube. The diminishing vitreous volume was measured in terms of the height in the chamber and recorded as a function of time. The reduction in the vitreous liquid content also caused the hydraulic conductivity to reduce and this parameter was quantified on the basis of previously-developed theories of fibrous porous media that have been very well established. A theoretical model with a fully analytical expression for the vitreous volume undergoing drainage was developed and used as a least-squares best fit to deliver the initial hydraulic conductivity value of K 0/µ=(7.8 ± 3.1) × 10-12 m2 (Pa-s). The measurements were made with the hyaloid membrane intact and therefore represents an effective conductivity for the entire system, including possible variations within the vitreous.

Diffusive Transport in the Vitreous Humor: Experimental and Analytical Studies.

In relation to intravitreal drug delivery, predictive mathematical models for drug transport are being developed, and to effectively implement these for retinal delivery, the information on biophysical properties of various ocular tissues is fundamentally important. It is therefore necessary to accurately measure the diffusion coefficient of drugs and drug surrogates in the vitreous humor. In this review, we present the studies conducted by various researchers on such measurements over the last several decades. These include imaging techniques (fluorescence and magnetic resonance imaging (MRI)) that make use of introducing a contrast agent or a labeled drug into the vitreous and tracking its diffusive movement at various time points. A predictive model for the same initial conditions when matched with the experimental measurements provides the diffusion coefficient, leading to results for various molecules ranging in size from approximately 0.1 to 160 kDa. For real drugs, the effectiveness of this system depends on the successful labeling of the drugs with suitable contrast agents such as fluorescein and gadolinium or manganese so that fluorescence or MR imagining could be conducted. Besides this technique, some work has been carried out using the diffusion apparatus for measuring permeation of a drug across an excised vitreous body from a donor chamber to the receptor by sampling assays from the chambers at various time intervals. This has the advantage of not requiring labeling but is otherwise more disruptive to the vitreous. Some success with nanoparticles has been achieved using dynamic light scattering (DLS), and presently, radioactive labeling is being explored.

Mass diffusion coefficient measurement for vitreous humor using FEM and MRI.

In early studies, the 'contour method' for determining the diffusion coefficient of the vitreous humor was developed. This technique relied on careful injection of an MRI contrast agent (surrogate drug) into the vitreous humor of fresh bovine eyes, and tracking the contours of the contrast agent in time. In addition, an analytical solution was developed for the theoretical contours built on point source model for the injected surrogate drug. The match between theoretical and experimental contours as a least square fit, while floating the diffusion coefficient, led to the value of the diffusion coefficient. This method had its limitation that the initial injection of the surrogate had to be spherical or ellipsoidal because of the analytical result based on the point-source model. With a new finite element model for the analysis in this study, the technique is much less restrictive and handles irregular shapes of the initial bolus. The fresh bovine eyes were used for drug diffusion study in the vitreous and three contrast agents of different molecular masses: gadolinium-diethylenetriaminepentaacetic acid (Gd-DTPA, 938 Da), non-ionic gadoteridol (Prohance, 559 Da), and bovine albumin conjugated with gadolinium (Galbumin, 74 kDa) were used as drug surrogates to visualize the diffusion process by MRI. The 3D finite element model was developed to determine the diffusion coefficients of these surrogates with the images from MRI. This method can be used for other types of bioporous media provided the concentration profile can be visualized (by methods such as MRI or fluorescence).

Noncontact thermophysical property measurement by levitation of a thin liquid disk.

The purpose of the current research program is to develop techniques for noncontact measurement of thermophysical properties of highly viscous liquids. The application would be for undercooled liquids that remain liquid even below the freezing point when suspended without a container. The approach being used here consists of carrying out thermocapillary flow and temperature measurements in a horizontally levitated, laser-heated thin glycerin disk. In a levitated state, the disk is flattened by an intense acoustic field. Such a disk has the advantage of a relatively low gravitational potential over the thickness, thus mitigating the buoyancy effects, and helping isolate the thermocapillary-driven flows. For the purpose of predicting the thermal properties from these measurements, it is necessary to develop a theoretical model of the thermal processes. Such a model has been developed, and, on the basis of the observed shape, the thickness is taken to be a minimum at the center with a gentle parabolic profile at both the top and the bottom surfaces. This minimum thickness is much smaller than the radius of disk drop and the ratio of thickness to radius becomes much less than unity. It is heated by laser beam in normal direction to the edge. A general three-dimensional momentum equation is transformed into a two-variable vorticity equation. For the highly viscous liquid, a few millimeters in size, Stokes equations adequately describe the flow. Additional approximations are made by considering average flow properties over the disk thickness in a manner similar to lubrication theory. In the same way, the three-dimensional energy equation is averaged over the disk thickness. With convection boundary condition at the surfaces, we integrate a general three-dimensional energy equation to get an averaged two-dimensional energy equation that has convection terms, conduction terms, and additional source terms corresponding to a Biot number. A finite-difference numerical approach is used to solve these steady-state governing equations in the cylindrical coordinate system. The calculations yield the temperature distribution and the thermally driven flow field. These results have been used to formulate a model that, in conjunction with experiments, has enabled the development of a method for the noncontact thermophysical property measurement of liquids.