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.

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.

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.

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.

White matter degradation near cerebral microbleeds is associated with cognitive change after mild traumatic brain injury.

To explore how cerebral microbleeds (CMBs) accompanying mild traumatic brain injury (mTBI) reflect white matter (WM) degradation and cognitive decline, magnetic resonance images were acquired from 62 mTBI adults (imaged ∼7 days and ∼6 months post-injury) and 203 matched healthy controls. On average, mTBI participants had a count of 2.7 ± 2.6 traumatic CMBs in WM, located 6.1 ± 4.4 mm from cortex. At ∼6-month follow-up, 97% of CMBs were associated with significant reductions (34% ± 11%, q < 0.05) in the fractional anisotropy of WM streamlines within ∼1 cm of CMB locations. Male sex and older age were significant risk factors for larger reductions (q < 0.05). For CMBs in the corpus callosum, cingulum bundle, inferior and middle longitudinal fasciculi, fractional anisotropy changes were significantly and positively associated with changes in cognitive functions mediated by these structures (q < 0.05). Our findings distinguish traumatic from non-traumatic CMBs by virtue of surrounding WM alterations and challenge the assumption that traumatic CMBs are neurocognitively silent. Thus, mTBI with CMB findings can be described as a clinical endophenotype warranting longitudinal cognitive assessment.

Survey of the year 2008: applications of isothermal titration calorimetry.

Isothermal titration calorimetry (ITC) is a fast, accurate and label-free method for measuring the thermodynamics and binding affinities of molecular associations in solution. Because the method will measure any reaction that results in a heat change, it is applicable to many different fields of research from biomolecular science, to drug design and materials engineering, and can be used to measure binding events between essentially any type of biological or chemical ligand. ITC is the only method that can directly measure binding energetics including Gibbs free energy, enthalpy, entropy and heat capacity changes. Not only binding thermodynamics but also catalytic reactions, conformational rearrangements, changes in protonation and molecular dissociations can be readily quantified by performing only a small number of ITC experiments. In this review, we highlight some of the particularly interesting reports from 2008 employing ITC, with a particular focus on protein interactions with other proteins, nucleic acids, lipids and drugs. As is tradition in these reviews we have not attempted a comprehensive analysis of all 500 papers using ITC, but emphasize those reports that particularly captured our interest and that included more thorough discussions we consider exemplify the power of the technique and might serve to inspire other users.

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).

Nucleation of protein crystals under the influence of solution shear flow.

Several recent theories and simulations have predicted that shear flow could enhance, or, conversely, suppress the nucleation of crystals from solution. Such modulations would offer a pathway for nucleation control and provide a novel explanation for numerous mysteries in nucleation research. For experimental tests of the effects of shear flow on protein crystal nucleation, we found that if a protein solution droplet of approximately 5 microL (2-3 mm diameter at base) is held on a hydrophobic substrate in an enclosed environment and in a quasi-uniform constant electric field of 2 to 6 kV cm(-1), a rotational flow with a maximum rate at the droplet top of approximately 10 microm s(-1) is induced. The shear rate varies from 10(-3) to 10(-1) s(-1). The likely mechanism of the rotational flow involves adsorption of the protein and amphiphylic buffer molecules on the air-water interface and their redistribution in the electric field, leading to nonuniform surface tension of the droplet and surface tension-driven flow. Observations of the number of nucleated crystals in 24- and 72-h experiments with the proteins ferritin, apoferritin, and lysozyme revealed that the crystals are typically nucleated at a certain radius of the droplet, that is, at a preferred shear rate. Variations of the rotational flow velocity resulted in suppression or enhancement of the total number of nucleated crystals of ferritin and apoferritin, while all solution flow rates were found to enhance lysozyme crystal nucleation. These observations show that shear flow may strongly affect nucleation, and that for some systems, an optimal flow velocity, leading to fastest nucleation, exists. Comparison with the predictions of theories and simulations suggest that the formation of ordered nuclei in a "normal" protein solution cannot be affected by such low shear rates. We conclude that the flow acts by helping or suppressing the formation of ordered nuclei within mesoscopic metastable dense liquid clusters. Such clusters were recently shown to exist in protein solutions and to constitute the first step in the nucleation mechanism of many protein and nonproteinsystems.

Nucleation of insulin crystals in a wide continuous supersaturation gradient.

Modifying the classical double pulse technique, by using a supersaturation gradient along an insulin solution contained in a glass capillary tube, we found conditions appropriate for the direct measurement of nucleation parameters. The nucleation time lag has been measured. Data for the number of crystal nuclei versus the nucleation time were obtained for this hormone. Insulin was chosen as a model protein because of the availability of solubility data in the literature. A comparison with the results for hen-egg-white lysozyme, HEWL was performed.

Effects of buoyancy-driven convection on nucleation and growth of protein crystals.

Protein crystallization has been studied in presence or absence of buoyancy-driven convection. Gravity-driven flow was created, or suppressed, in protein solutions by means of vertically directed density gradients that were caused by generating suitable temperature gradients. The presence of enhanced mixing was demonstrated directly by experiments with crustacyanin, a blue-colored protein, and other materials. Combined with the vertical tube position the enhanced convection has two main effects. First, it reduces the number of nucleated hen-egg-white lysozyme (HEWL) crystals, as compared with those in a horizontal capillary. By enabling better nutrition from the protein in the solution, convection results in growth of fewer larger HEWL crystals. Second, we observe that due to convection, trypsin crystals grow faster. Suppression of convection, achieved by decreasing solution density upward in the capillary, can to some extent mimic conditions of growth in microgravity. Thus, impurity supply, which may have a detrimental effect on crystal quality, was avoided.

Instability of protein drops via applied electric field: mathematical and experimental aspects.

Drops (5-15 microL) consisting of a protein solution readily crystallize and could provide an opportunity for a simultaneous examination of their thermodynamic and kinetic properties at various sizes. These drops experienced different pressures and therefore different surface tensions. Starting from the expression for the interface traction between protein fluid and silicon medium (with different dielectric constants), we have derived an equation accounting the influence of the electric field strength on the geometry of a protein drop. If the field strength increases, the lysozyme drop between two electrodes elongates and some crystals nucleate on the cathode side. In this situation numerous factors besides the intensity of the electric field--such as the solution composition, the charge and size of the protein molecule, the purity of the protein substance, and the consistency of bubbles of water--can have a significant effect on the crystallization rate and location.