The NSF has awarded a grant to Profs. Richard Braun (PI) and Tobin Driscoll (Co-PI) to study mathematical models for tear film dynamics. The grant is for $325,000 over three years beginning July 1, 2019, and is co-funded from the Division of Mathematical Sciences' Applied Mathematics program and Life Science Ventures. This grant continues a string of grants from the NSF in this area for the PIs and their group.
Million American age 50 or older suffer from moderate to severe dry eye syndrome (DES); this multifactorial disease can cause discomfort, inconvenience, and damage to the ocular surface. It is thought that dysfunction of the tear film plays a major role in this condition. The PIs will develop mathematical models that explain how parts of the tear film function, and they will work closely with leading optometrists at Indiana University and the Ohio State University to achieve this goal.
One main thrust of the project will use mathematical models as a way to identify the conditions when the tear film fails, which is called tear breakup in many cases. Computer-generated answers to partial differential equation models that simulate tear breakup can be used to determine local quantities in the tear film that cannot be measured experimentally with current technologies. An example is the tear film osmolarity (or saltiness), a variable which is thought to be crucial in the development of DES. Solving the models becomes part of a bigger problem that is often called a parameter identification problem. These quantities are crucial to understanding the mechanisms behind tear breakup and the causes for some instances of dry eye.
The other main thrust will use two-layer fluid models to understand the function of the layers much better than is currently known. The lipid layer floats atop an aqueous layer; the lipid layer is a more complex fluid than water, and the aqueous layer can be assumed to act like water. New, more complex models for the floating lipid layer will be developed: the lipid layer will be treated as a liquid crystal. The liquid crystal could be either nematic or smectic-A, for which fundamental models in some instances are already known. Those models will be adapted and extended for this new project. The group will solve models for the two layers together, and compare them to experimental results. They expect to explain some lipid layer patterns seen experimentally in vivo, and they will solve new mathematical problems to obtain these explanations.
The project thrusts described above will make heavy use of imaging data for the tear film. The eye is such a sensitive system that non-invasive imaging methods are required to obtain useful data. However, those imaging methods often don't directly provide the quantities of interest (e.g., the osmolarity). The mathematical models thus provide a crucial function to estimate those important quantities. Both thrusts of the project will advance understanding of the tear film and DES, as well as advance applied mathematics.
As is common for these grants, graduate and undergraduate students will be supported with stipends and travel expenses to present their work. The PIs will mentor the students in these activities, receive limited summer salary and also travel to conferences. Collaboration with optometrists and fellow mathematicians will largely occur via meetings over the internet.
The group is funded through the Center for Applications of Mathematics to Medicine (CAMM). Further information about the group's work can be found at the CAMM website at mathandmedicine.org. The site includes short videos discussing selected results at the CAMM YouTube channel. CAMM includes a variety of biomedical projects beyond the tear film and ocular surface, including work on colon cancer, atherosclerosis, osteoporosis, and more, and descriptions of those project can be found at the above sites.