By Lina Sorg
More than 140 million people around the world wear contact lenses, and 45 million of these users are in the U.S. The very first contact lenses were composed of glass, though the material has naturally evolved with time to improve both comfort and ocular health. Modern-day lenses are made of soft hydrogel and adapt to changes in curvature and eye shape. Yet despite the popularity of contact lenses, nearly 22 percent of wearers in developed countries eventually decrease their frequency or stop utilizing their lenses altogether. Established users cite discomfort as the primary reason behind this decision.
During a contributed presentation at the 2022 SIAM Conference on the Life Sciences, which is currently taking place in Pittsburgh, Pa., in conjunction with the 2022 SIAM Annual Meeting, Lucia Carichino of Rochester Institute of Technology (RIT) introduced a mathematical model that explores the effect of eye deformation on contact lens comfort. Her model is the first of its type to account for the eye’s deformability.
For the last three years, Carichino has been combining mathematical modeling with experimental data to understand the interaction between the contact lens and the eye. She and her collaborators at RIT—Kara Maki and David Ross—have been working with several manufacturers of contact lenses to expedite the empirical design process with the goal of improving lens design and comfort level. For example, the material in question changes the properties of contact lenses, and the thickness and shape depend on lens type. Patients with astigmatism might wear toric lenses, which are thicker on the bottom outer rim (the thickness in standard, more spherical lenses does not change as drastically). Researchers must account for these and other factors when working with contact lens models. As such, the material, thickness, and shape of the lens serve as the design parameters.
Figure 1. A schematic representation of the reference frame of the eye and contact lens models, coupled via the suction pressure \(p\). Image courtesy of Lucia Carichino.
Carichino considers two key elements within her project: (i) How the lens will conform to the eye, and (ii) how the eye—which is not a rigid material—will deform with a lens on top of it. All existing math models of contact lenses assume that the eye is rigid. “Imaging putting a contact lens on a statue,” Carichino said. “We want to take into consideration the fact that the eye can deform, and the lens can push or pull on it.”
For the sake of this study, Carichino ignores the surrounding tear film fluid and instead simply models the contact lens, the eye itself, and the interaction between the two. She assumes that the lens is axially symmetric, takes the shape of the eye upon insertion, and transforms from an undeformed to deformed configuration when on the eye. First, Carichino derives the radial tension and Hoop tension, both of which depend on the eye shape, lens thickness, and material parameter (i.e., squishiness of the lens). She then notes the balance of forces as a second-order ordinary differential equation (ODE) before rewriting it as a system of first-order ODEs. A shooting method solves the resulting system.
“If we balance the vertical forces, the downward component of the radial tension balances the upward component of the pressure,” Carichino said. This outcome results in suction pressure \((p)\) under the lenses, which prevents them from falling out (see Figure 1). “This pressure has different signs, positive or negative,” she continued. “In some areas it pushes on the eye and in some areas it lifts the eye up.”
Carichino assumes axial symmetry for the eye, which she models as a homogeneous, isotropic, linear elastic material (she acknowledges that doing so is a significant assumption because the eye consists of many different materials). The nonlinear suction pressure \(p\) couples the lens model to the eye model with the help of an iterative numerical algorithm that handles the change in curvature on the domain (see Figure 2). Ultimately, the lens and eye apply pressure and force in opposite directions to yield the balance of normal stresses.
Figure 2. Iterative numerical algorithm that couples the eye and contact lens models. Image courtesy of Lucia Carichino.
Given an average eye and lens shape with balanced suction pressure, Carichino computes the corresponding displacement in the radial and vertical direction. The limbus—the junction of the cornea and sclera in the eye—marks the point of curvature where the lens is pulling/lifting the eye across the deformation. “We can actually show this deformation in the full domain,” Carichino said.
Carichino’s preliminary results suggest that the eye actually does not deform much in the presence of contact lenses. “The lens is pushing and pulling on the eye, but the deformation is very small — less than one percent of the corneal epithelium thickness,” she said. “The eye deformation pretty much does not influence the suction pressure of the lens. What’s really important is the difference in curvature between the lens and the eye.”
In the future, Carichino hopes to continually make improvements to her model so researchers can apply it to different scenarios. Specifically, she wants to see how a change in lens thickness affects the model output. She also intends to expand the model to explore a more realistic interior domain of the eye and account for the effects of interocular pressure.
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Lina Sorg is the managing editor of SIAM News. |