Models for Tear Film Structure, Dynamics and Parameter Identification

2008年，近500万50岁以上的美国人患有中度至重度干眼症，人们认为泪膜功能障碍在这种情况下起着主要作用。该项目将开发数学模型来解释泪膜的部分功能，主要研究人员将与领先的验光师密切合作，以实现这一目标。该项目的一个主要目的是使用数学模型来确定泪膜失效的情况，这通常被称为泪液破裂。计算机生成的模拟泪液破裂模型的答案可以用来确定无法通过实验测量的局部数量；这些数量对于理解泪液破裂背后的机制和某些干眼症的原因至关重要。另一个主要推力将使用两层流体模型来更好地理解该层的功能，比目前已知的要好得多。脂质层漂浮在水层之上；脂质层是一种比水更复杂的流体，水层可以被认为像水一样起作用。新的、更复杂的漂浮脂质层模型将被开发出来：脂质层将被视为所谓的液晶。我们将一起求解两层的模型，并将其与实验结果进行比较。我们期望解释一些实验中在眼睛上看到的脂质层模式，我们将解决新的数学问题来获得这些解释。这两个项目的重点将推进我们对泪膜和干眼症的理解，以及推进应用数学。该奖项将通过研究为研究生提供培训。pi将研究可能应用于人眼泪膜的薄流体膜模型的动力学。两层膜有一个浮在水层上的油性脂层。越来越多的证据表明，在许多情况下，浮脂层是液晶。然而，对于这种两层薄膜，基本上没有推导或求解的模型。在该项目的第一个主要推力中，我们建议开发漂浮在水面上的向列和近晶A液晶的基本模型。在该项目的第二个主要目标中，我们还通过一系列问题开发了泪膜模型的参数识别方法，这些问题最终将导致识别泪膜中观察到的生理泪液破裂的物理过程。该项目将借鉴薄液膜和液晶文献，求解这些薄膜的新模型，并证明参数识别方法对计算产生的噪声数据的有效性。该项目还将包括验光师，他们将根据观察到的数据检验理论。pi将结合建模，分析和计算方法，以及我们合作者的实验数据，以促进对悬浮在牛顿流体（水）上的液晶组成的两层膜的理解。推导这些复杂的模型并在静止和运动域上求解它们是一项重大的数学挑战。例如，pi不知道任何扩展流动薄膜模型的近晶A液晶；我们将推导和求解自由和浮动伸展层的模型。应用参数识别方法来确定膜模型层次的噪声测试数据的参数，将使这些复杂流体系统的参数识别成为可能。除了在复杂流体动力学方面取得重大成果外，该项目的结果将使眼表研究人员能够更好地了解薄膜动力学和泪膜的功能。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In 2008, nearly 5 million Americans age 50 or older suffered from moderate to severe dry eye, and it is thought that dysfunction of the tear film plays a major role in this condition. This project will develop mathematical models that explain how parts of the tear film function, and the principal investigators will work closely with leading optometrists 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 often called tear breakup. Computer-generated answers to models that simulate tear breakup can be used to determine local quantities in that cannot be measured experimentally; 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 layer 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 so-called liquid crystal. We will solve models for the two layers together and compare them to experimental results. We expect to explain some lipid layer patterns seen experimentally on the eye, and we will solve new mathematical problems to obtain these explanations. Both thrusts of the project will advance our understanding of the tear film and dry eye, as well as advance applied mathematics. The award will provide graduate student training through research.The PIs will investigate the dynamics of thin fluid film models that may be applied to the tear film on the human eye. The two-layer film there has an oily lipid layer floating atop an aqueous layer. There is growing evidence that the floating lipid layer is a liquid crystal in many instances. However, there are essentially no models derived or solved for such two-layer films. In the first main thrust of the project, we propose to develop fundamental models for nematic and smectic A liquid crystals floating on water. In the second main thrust of the project, we also develop parameter identification methods for tear film models through a sequence of problems that will ultimately lead to identifying physical processes responsible for observed physiological tear breakup in the tear film. The project will draw upon the thin liquid film and liquid crystal literature, solve new models for these films, and prove the effectiveness of parameter identification methods on noisy data generated computationally. The project will also involve optometrists to test the theories on observed data. The PIs will combine modeling, analytical and computational approaches, together with experimental data from our collaborators, to advance understanding of two-layer films composed of liquid crystals floating on Newtonian fluids (water). Deriving these complex models and solving them on stationary and moving domains are substantial mathematical challenges. For example, the PIs are unaware of any extensional flow thin film models for smectic A liquid crystals; we will derive and solve models for both free and floating extensional layers. Applying parameter identification approaches to determine parameters on noisy test data for a hierarchy of film models will enable identification of parameters in these complex fluid systems. Beyond significant results in dynamics of complex fluids, the results of the project will enable ocular surface researchers to better understand thin film dynamics and the function of the tear film.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.