Modeling the Blink Cycle and Lipid Dynamics in the Tear Film

在这个项目中，我们将通过一系列数学模型研究人类泪液膜的动态，其中泪液膜的各个方面越来越具有挑战性。 该层级将包括具有和不具有表面活性剂效应、具有表面张力和在眼表面上具有滑动的膜表面。 该模型是基于润滑理论和产生的高阶非线性偏微分方程将使用分析和计算方法求解。 将结果与泪膜厚度的体内测量进行密切比较。 目前，一个有限差分空间离散化与向后微分公式软件数值求解常微分代数方程（Daspk）被用来解决我们的模型中最简单的;两个新的计算方法也将开发。 一种新的方法将使用径向基函数作为该方法的基础。 另一个将修改一个移动重叠网格方法涉及的软件包序曲，插值边界拟合有限体积网格在域的边缘和一个直线网格在域的内部。截至1998年，多达1000万美国人需要使用人工泪液制剂，干眼综合征（DES）占眼科医生就诊的很高比例。 最近的评论呼吁更好地了解泪膜，以改善DES的治疗和产品。 该项目将为人类泪膜开发新的数学模型，并将开发新的计算方法来解决这些问题。 计算方法将与劳伦斯利弗莫尔国家实验室的数学家和特拉华州大学的教授和学生共同开发。 该结果将与俄亥俄州州立大学视光学学院进行的泪膜厚度动态的体内测量进行密切比较。 这些结果的影响将是双重的。 首先，我们将为验光师和眼科医生提供对泪膜的新的定量了解;我们希望这将有助于改善DES的治疗。 第二，计算方法将广泛用于解决移动非矩形几何中的困难的时变问题。 在实现这些目标，我们将培养博士生在数学建模和计算数学的多学科应用。

In this project, we will study the dynamics of the human tear film via a hierarchy of mathematical models with increasingly challenging aspects of the tear film. The hierarchy will include film surfaces with and without surfactant effects, with surface tension, and with slip on the eye surface. The models are based on lubrication theory and the resulting high-order nonlinear partial differential equations will be solved using analytical and computational methods. The results will be closely compared with in vivo measurements of the tear film thickness. Currently, a finite-difference spatial discretization together with backward-differentiation-formula software for numerically solving ordinary differential-algebraic equations (Daspk) is being used to solve the simplest of our models; two new computational approaches will be developed as well. One new approach will use radial basis functions as a basis for the method. The other will modify a moving overset grid method involving the software package Overture, which interpolates between boundary fitted finite volume grids at the edges of the domain and a rectilinear grid in the interior of the domain. As of 1998, up to 10 million Americans required the use of artificial tear preparations and dry eye syndrome (DES) accounts for a high percentage of visits to ophthalmologists. Recent reviews on the subject call for a better understanding of the tear film to improve treatments and products for DES. This project will develop new mathematical models for the human tear film and will develop new computational methods to solve these problems. The computational methods will be developed jointly with mathematicians at Lawrence Livermore National Laboratory and University of Delaware professors and students. The results will be closely compared with in vivo measurements of tear film thickness dynamics made in the College of Optometry at the Ohio State University. The impacts of these results will be twofold. First, we will provide new quantitative understanding of the tear film for optometrists and ophthalmologists; we hope it will help improve treatment for DES. Second, the computational methods will be broadly useful in solving difficult time-varying problems in moving non-rectangular geometries. In accomplishing these goals, we will train PhD students in multidisciplinary applications of mathematical modeling and computational mathematics.