A Mathematical Model for Ocular Tear and Solute Balance

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Abstract
In this paper we develop a mathematical model that can predict the steady-state tear film thickness and the dynamic tear film thickness and the solute concentration after instillation of a solute-laden fluid in the eye.The mathematical model developed in this paper is based on a balance between the inflow and outflow of tears into the tear film. It incorporates a tear drainage model and a model that relates the tear film thickness to the meniscus radius of curvature. To predict the solute concentrations, the tear balance is coupled with the solute balance. The differential equations for the unsteady balances are solved numerically.The model predicts that the tear film thickness depends on a number of physiological factors, such as rates of tear production and evaporation, geometry and modulus of the canaliculi, and surface tension and viscosity of tears, and varies from about 3 to 15 microm. The model also predicts that the drainage time for an instilled volume of 15 microl is 1283 s. Additionally, the time required for the tracer concentration to decay to 1% of the value immediately after instillation of a drug-laden 40 microl drop is about 2480 s. Similarly, the time for intensity decay for a radioactive tracer after 25 microl instillation is about 1566 s. Also, the model predicts that the fraction of the instilled drug that reaches the cornea is about 1.3% for topical application of timolol.The predicted results agree reasonably with the reported experimental results, at least qualitatively. The model developed here can serve as a useful tool to develop a more precise understanding of various issues related to tears and also evaluate the effect of various parameters on the tear volume.

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