The Mathematics Behind Ophthalmic Drug Delivery and Contact Lenses

By Lina Sorg

The human tear film is a thin, complex fluid system in the eye that contributes to both ocular health and visual function. When one inserts a contact lens, the lens divides the tear film into a pre-lens and post-lens film. Stability of these two films is essential for comfortable and safe contact lens wear. But a lens’ effect on the tear film concerns more than just ocular comfort, and researchers are exploring the logistics of contact lenses as drug delivery systems.

Drug delivery in the eye is complex. Most ophthalmic drugs are delivered via eye drops, but a multitude of problems can arise because of the drops’ short residency time (due to blinking, drainage, or runoff). Those drops that do enter the eye often end up elsewhere, as up to 95 percent of an administered drug can be lost in the body. A patient must therefore administer eye drops in high and frequent doses for sustained drug delivery, which is often inconvenient.

Drug-eluting contact lenses are a promising therapeutic tool and an attractive alternative to high-frequency eye drops for the treatment of various chronic and acute eye disorders, such as glaucoma. However, the time scales of drug release rates and the distribution path throughout the eye complicate the process. During a minisymposium presentation at the 2020 SIAM Annual Meeting, which took place virtually last week, Daniel Anderson of George Mason University used mathematical models to investigate the feasibility of ophthalmic drug delivery via contact lenses. He is particularly interested in drug delivery to the cornea, since contact lenses are in close proximity to the corneal surface.

Previous researchers have also investigated this prospect. For example, a 2006 study by Chi-Chung Li and Anuj Chauhan examined diffusion in contact lenses and post-lens transport over the course of multiple blinks. Li and Chauhan modeled the pre-lens tear film with two different boundary conditions and determined that the efficiency of drug delivery to the cornea ranged from 20 to 95 percent (a very large window), depending primarily on the boundary condition. Other studies have experimented with hydrogel lenses, diffusive transport of drugs across the sclera (the white, outer layer of the eye), and the effects of blinking and contact lens motion within the eye.

Anderson began with a model that investigates the influence of specific solute absorption into the hydrogel polymer matrix. It treats hydrogels as a two-component mixture comprised of water volume fraction and polymer volume fraction. The model accounts for the water volume and polymer volume fractions of the hydrogel, the concentration of the solute that can diffuse through the water portion of the gel and the concentration that is absorbed in the hydrogel polymer, and the solute diffusion coefficient through the water phase. Experimental and theoretical work has indicated that a local equilibrium characterized by Henry’s law—which states that the amount of dissolved gas in a liquid is proportional to its partial pressure above that liquid—applies in this scenario. Under this assumption, the model reduces to a classical diffusion equation that includes an effective diffusion coefficient.

Next, Anderson presented a compartment model of the open-eye portion of an interblink (an interblink interval reflects the time between blinks). This model was based on student work at the University of Delaware’s 2019 Graduate Student Math Modeling Camp and inspired by Li and Chauhan’s earlier results. The accompanying set of governing equations describe a pre-lens dynamic, assume a uniform thickness film, and account for diffusion from the lens. Boundary conditions are present on the upper and lower surface of the contact lens, and Anderson computes the diffusion equation across the lens thickness. The post-lens film has a relatively simple equation for thickness.

A “blink reset” occurs after each blink. The contact lens concentration retains its current values during a blink, but the pre-lens film resets to its original thickness. Furthermore, each blink yields a completely fresh tear film and post-lens reset, both of which lead to some drug loss. Anderson employs a finite difference discretization of the contact lens diffusion equation and a method of lines coupled to ordinary differential equations in pre/post-lens tear films to address this problem.

He then revealed the model’s results. First, Anderson focused on drug concentration in the contact lens. His simulation involved 1,800 blinks with a five second interblink for 150 minutes. Approximately 10 percent of the drug remained in the contact lends after 75 minutes, and roughly one percent of the drug remained after the full 150 minutes. he next examined drug concentration in the pre- and post-lens tear film with the same blinking conditions. The highest concentration naturally occurs immediately, with the blink reset responsible for the subsequent decline. “There’s a gradual decline as the drug leaves the system in various ways,” Anderson said.

Finally, Anderson turned to drug delivery and loss. He compared three simulations: 1,800 blinks with a five second interblink, 900 blinks with a 10 second interblink, and 3,600 blinks with a 2.5 second interblink (see Figure 1). For each scenario he modeled the pre-lens drug loss, post-lens drug loss, and amount of drug delivered to the cornea. Roughly 30 percent of the drug reached the cornea and 60-70 percent was lost in the pre-lens film for the 1,800 blink scenario. While very little of the drug was lost in the post-lens film, this does not account for possible loss due to the translational motion of the contact lens itself. As one might suspect, more frequent blinking reduced drug delivery to the cornea, while less frequent blinking preserved more of the drug in the pre-lens film.

The compartment model is an efficient tool for the study of multiple blinks, as one can easily adapt it for variable blinking, examine preliminary cases of Gaussian or Poisson blink statistics, and use it to explore variations in initial drug distribution from contact lenses. “It’s nice to have a model that runs quickly and answers quickly,” Anderson said. However, the model does require a lot of assumptions, most significantly the supposition that the pre- and post-lens concentrations are uniform in drug distribution. Predictions are sensitive to this assumption. Compartment models also neglect drug loss due to the translational motion of the contact lens in the eye.

Given the importance of pre-lens tear film processes with respect to drug delivery via contact lenses, Anderson and his team decided to look more closely at the pre-lens tear film dynamics during blinking and contact lens wear. Anderson thus presented an idealized cross-sectional geometry of a tear film and contact lens to address the influence of contact lens motion on pre-lens tear film dynamics and drug distribution while blinking. He set up his model using Newton’s equations for the translational motion of a contact lens during a blink, which account for separate upper lid, lower lid, post-lens, pre-lens, and elastic forces. To create this model, Anderson adapted an existing model of the tear film over a blink by adding the effect of contact lens motion. The goal was to incorporate solute transport to better understand drug distribution during the pre-lens blink.

Ultimately, Anderson’s compartment model for drug delivery via contact lenses includes detailed dynamics in the pre-lens tear film and allows researchers to track a drug over many hours and multiple blink cycles. He also explored tear film dynamics and contact lens motion during blinking by focusing on pre-lens tear film forces. Anderson hopes to advance his study even further by incorporating solute transport in the pre-lens film during blinking and contact lens wear to better resolve drug transport during a blink and help inform compartment models.

Lina Sorg is the managing editor of SIAM News.