Models for Tear Film Structure, Dynamics and Parameter Identification

2008年，近500万50岁或以上的美国人患有中重度干眼，泪膜功能障碍被认为是导致这种情况的主要原因。这个项目将开发数学模型，解释泪膜的某些部分是如何发挥作用的，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作为一种方法来识别泪膜失效的条件，这通常被称为泪膜破裂。计算机生成的模拟泪水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水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，这比目前已知的要好得多。脂层漂浮在水层之上；脂层是一种比水更复杂的流体，可以假设水层的作用类似水。新的、更复杂的浮动类脂层模型将被开发出来：类脂层将被视为所谓的液晶。我们将一起求解这两个层的模型，并将它们与实验结果进行比较。我们希望解释一些实验中在眼睛上看到的脂层模式，我们将解决新的数学问题来获得这些解释。该项目的两个重点都将促进我们对泪膜和干眼症的理解，以及推进应用数学。该奖项将通过研究提供研究生培训。PI将研究可能应用于人眼泪膜的薄膜流体模型的动力学。那里的两层膜在水层上漂浮着一层油性脂层。越来越多的证据表明，浮动类脂层在许多情况下是液晶。然而，对于这样的双层薄膜，基本上还没有得到或求解的模型。在该项目的第一个主要推力中，我们建议建立漂浮在水面上的向列相和近晶A液晶的基本模型。在项目的第二个主旨中，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性撕裂的物理过程。该项目将借鉴薄膜和液晶的文献，解决这些薄膜的新模型，并证明参数识别方法在计算产生的噪声数据上的有效性。该项目还将邀请验光师在观测数据上测试理论。PI将结合建模、分析和计算方法，以及来自我们合作者的实验数据，以促进对漂浮在牛顿流体(水)上的液晶组成的两层膜的理解。推导出这些复杂的模型，并在静止和运动的区域上求解它们是相当大的数学挑战。例如，PI不知道近晶A液晶的任何伸展流动薄膜模型；我们将推导和求解自由伸展层和浮动伸展层的模型。应用参数识别方法来确定一系列薄膜模型的噪声测试数据上的参数，将使识别这些复杂流体系统中的参数成为可能。除了在复杂流体的动力学方面取得的重大成果外，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NSF的法定使命，并通过使用基金会的智力优势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.