Modeling Tear Film Dynamics

泪膜的扩散和动力学将用一个层次的计算模型来研究。起始点是由两个偏微分方程（PDEs）控制的单个液体层，一个用于泪膜厚度，另一个用于泪膜渗透压。项目进展到具有额外pde的模型，包括表面活性剂输运和多液体层。更详细地了解蒸发和脂质层动力学，特别是应用于撕裂膜破裂和渗透压变化，将从这项工作中出现。在这些问题的计算方法中，将使用两种方法。对于眼睛形状的域，通过劳伦斯利弗莫尔国家实验室的Overture框架的overset网格方法提供了管理复杂边界所需的许多软件功能；特拉华已经开发了一些专门针对泪膜问题的解决方案。对于更简单的几何图形，研究人员使用谱和自适应径向基函数（RBF）方法来利用一些问题中快速变化的小区域附近的无网格方法。这些方法的成功应用为理解泪膜动力学带来了新的能力。对泪膜动力学的新认识将使许多人受益。截至1998年，多达1000万美国人需要使用人工泪液制剂；近500万50岁以上的美国人患有中度至重度干眼症。该项目改进了公认的数学模型，以包括更多重要的生理效应，特别是渗透压，某些盐和糖在泪膜中的组合浓度。医生怀疑渗透压在干眼症的发展中起着至关重要的作用，该模型对泪膜和渗透压动力学产生了新的见解。该项目得益于与俄亥俄州立大学验光师的密切合作。这种重要的合作使研究团队能够理解和解释计算生成的结果，并与通过实验获得的图像和视频进行密切比较。计算方法得益于与劳伦斯利弗莫尔国家实验室的持续合作。结果的传播在数学家和他们在应用数学界和生物医学界的验光师合作者之间联合进行；研究结果在数学、流体力学和眼睛相关领域的重大科学会议上发表。作为实现这些目标的一部分，研究生接受数学建模和计算数学的多学科应用培训。

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 thedevelopment 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 inmathematics, 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.