The spreading and dynamics of the tear film will be studied with a hierarchy of computational models. The starting point is a single liquid layer governed by two partial differential equations (PDEs), one for the tear film thickness and one for the tear film osmolarity. The project progresses to models with additional PDEs involving surfactant transport and multiple liquid layers as well. More detailed understanding of evaporation and lipid layer dynamics, especially as applied to tear film breakup and osmolarity variation, will emerge from this work. Two methods will be used in the computational approaches to these problems. For eye-shaped domains, an overset grid method via the Overture framework from Lawrence Livermore National Laboratory provides much of the software capability needed to manage complex boundaries; some solvers specific to the tear film problems have been and are developed at Delaware. For simpler geometries, the investigators use spectral and adaptive radial basis function (RBF) methods to take advantage of a grid free approach near small regions of rapid variation in some of the problems. Successful application of these methods bring new capabilities for understanding tear film dynamics.
New understanding of tear film dynamics would benefit a large number of people. As of 1998, up to ten million Americans required use of artificial tear preparations; nearly 5 million Americans age 50 or older suffer from moderate to severe dry eye symptoms. This project improves accepted mathematical models to include more physiologically important effects, particularly osmolarity, the combined concentration of certain salts and sugars in the tear film. Osmolarity is suspected by physicians of being crucial in the development of dry eye, and the model yields new insights into tear film and osmolarity dynamics. The project benefits from synergy in the intensive collaboration with optometrists at the Ohio State University. This vital collaboration allows the research team to understand and interpret the computationally-generated results and make close comparisons with images and videos obtained through experiments. The computational approaches benefit from continued collaboration with Lawrence Livermore National Laboratory. Dissemination of results are joint with between the mathematicians and their optometrist collaborators in both the applied-mathematical and biomedical communities; results are presented at major major scientific meetings in mathematics, fluid-dynamics, and eye-related fields. As part of reaching these goals, graduate students are trained in multidisciplinary applications of mathematical modeling and computational mathematics.
This grant developed mathematical models for the tear film on the front of the eye and used computers to solve them. In some cases, new methods for the computer were developed to solve those (partial differential equation) problems, and in some cases those methods were in a software package (the Overture framework) that can solve a wide variety of problems. The mathematical models incorporated fluid flow in very thin films (under 5 millionths of a meter thick), evaporation of water to the environment, transport of salts inside the tear film and osmosis from the ocular surface, as well as supply and drainage of tears around the eyelid margins. The tear film models predicted the osmolarity (saltiness) of the tear film all over the front of the eye; this is a key variable in the diagnosis and development of dry eye disease. Currently, it is not possible to experimentally measure osmolarity all over the eye surface because most of the tear film is too thin; it can only be measured near the outer canthus (corner) of the eye in the relatively thick meniscus at the eyelid margin. To our knowledge, our prediction is the best quantitative indication of the osmolarity. The highest osmolarity is expected to occur in a thin region near the eyelid margin over the cornea. Local spots of excessive tear film thinning and break up may develop similarly high levels of osmolarity. These high levels of osmolarity are easily enough to cause irritation to the eye surface. We also created mathematical models for heat transfer, ocular surface temperature and blinking. The models could predict temperature on the ocular surface as well as inside the front (anterior chamber) of the eye. We used a relatively simple model for the underlying eye (a rectangle), but we could recover observed ocular surface temperatures easily with reasonable parameter choices. We also found dynamics for the temperature inside the eye that have not yet been measured: small regions that are cooler than the surrounding tissue may be briefly trapped inside the eye during and after a blink. Subsequent cooling, once the eye is open, then eliminates these isolated interior spots of low temperature. Finally, we broke new ground by developing mathematical models of the fluorescent intensity in widely used tear film visualization methods involving fluorescein. In the latter experiments either a pre-made strip or a solution is used to instill fluorescein into the tear film. Subsequently a blue light is shown onto the eye and the aqueous part of the tear film glows green. Dark regions of may appear over time in the tear films of most subjects, indicating that tear film break up has occurred, and if it allowed to continue, its extent. The fluorescence observation is most often a qualitative one that is used to assess tear film stability. Depending on the concentration of fluorescein used, different observations may be made and in some cases one may make quantitative observation of the tear film thickness. We developed a mathematical model that quantified what happens over a wide range of fluorescein concentration during tear film thinning. The results from the model clarified in which regimes fluorescence imaging may be used for which purposes. We then went on to develop a mathematical model that showed the fluorescent intensity all over the open eye, in order to facilitate close comparison with experimental results. We also developed mathematical models for small localized regions of fast thinning and tear film break up, and found that the fluorescent imaging gave a visualization of the break up region that was smaller than the actual thickness indicated in many situations. This result was something of a surprise and is of great help to interpret experimental images in a quantitative manner. We also found that the fluid flow inside the tear film had a much more pronounced effect on the fluorescein concentration distribution than it did on the osmolarity distribution; this is because the fluorescein diffuses at about 1/4 of the rate of salt (sodium chloride in particular). We expect the models to be of great utility in understanding commonly used imaging methods for the tear film. The mathematical models were developed in close collaboration with optometrists and extensive use was made of their in vivo experimental data for both model development and evaluation of the results. Beyond the usual conferences and journals for the mathematical and physical sciences, we have written papers for optometrists and ophthalmologists, and we have gone to their conferences to present posters on our work. The results appear to be well-received, and have the potential to affect basic science and clinical methods in optometry and ophthalmology. Two undergraduate students from this project went to optometry school; others went to mathematics graduate school or industry. Two PhDs were awarded for the models studied here.
