Mathematical Sciences: Mathematical and Experimental Studies of Blood Flow in Collapsible Carotid Arteries with Stenoses

具有高度狭窄的动脉可能在生理条件下塌陷。 塌陷可能导致加速疲劳和斑块纤维帽破裂进入血流，随后阻塞下游的小动脉。 会直接导致心脏病和中风。 研究人员和他的同事使用数学，计算和实验方法来研究这个崩溃过程。 数学模型主要局限于一维模型，因为很难处理塌陷管，其形状是未知的和不断变化的。 建立了一个三维非线性粘性流体运动数学模型，并提出了一种新的数值求解方法。 数值方法使用长波渐近展开的解决方案作为数值初始条件和边界迭代方法来找到未知的移动边界和流速和压力。 进行实验来定义的压力-面积关系（管法）的弹性管模型和量化的确切的压力-流量条件管崩溃在实验室设置。 实验室实验和体内测量为数学模型的制定和验证提供了数据。 验证模型连同其数值解形成的基础上，进一步调查的崩溃过程。 所获得的结果将有助于狭窄的早期检测和预防，确定斑块帽破裂的原因，并量化可能发生塌陷的生理条件。 虽然在计算和实验中使用的颈动脉的生理尺寸和相关的参数范围，在这个项目中开发的方法将是有用的，在广泛的应用中，自由移动的边界涉及。 动脉中聚集的斑块会使动脉变窄，这种变窄称为狭窄。 就像水在狭窄的通道中流动一样，血液在狭窄处流动得更快。 在某些情况下，这可能会导致动脉塌陷。 然后，斑块可能会在血液中破裂，随后阻塞下游的小动脉。 会直接导致心脏病和中风。 研究人员使用数学，计算和实验方法研究这种崩溃过程。 以前的模型主要局限于一维模型，因为在处理塌陷管，其形状是未知的和不断变化的困难。 该项目开发了一个三维数学模型，该模型考虑了血流和塌陷动脉形状的变化，以及一种新的数值方法来求解该模型。 其中一名研究人员进行实验，以确定弹性管模型的压力-面积关系（管定律）和实验室装置中管塌陷的确切压力-流量条件。 实验室实验和体内测量为数学模型的制定和验证提供了数据。 获得的结果将有助于狭窄的早期检测和预防、确定斑块帽腐败的原因以及可能发生塌陷的生理条件。 在这个项目中开发的数值方法将是有用的，在其他应用中，自由移动边界涉及。

Arteries with high grade stenoses may collapse under physiological conditions. The collapse may lead to accelerated fatigue and rupture of the fibrous cap of the plaque into the bloodstream with subsequent blockage of small arteries downstream. That can lead directly to heart attacks and strokes. The investigator and his colleague use mathematical, computational and experimental methods to study this collapsing process. Mathematical models have been limited primarily to one-dimensional models because of the difficulty in handling the collapsing tube, whose shape is unknown and changing. A three-dimensional nonlinear viscous mathematical model with free moving boundaries is introduced and a novel numerical method is developed to solve the model. The numerical method uses solutions of the longwave asymptotic expansions as the numerical initial condition and a boundary-iterative method to find the unknown moving boundary and the flow velocity and pressure. Experiments are performed to define the pressure-area relationship (tube law) for the elastic tube models and to quantify the exact pressure-flow conditions for tube collapse in a laboratory set-up. The laboratory experiments and in vivo measurements provide data for the formulation and verification of the mathematical model. The validated model together with its numerical solutions form a basis for further investigations of the collapsing process. Results obtained will be useful for early detection and prevention of stenoses, identifying the causes of rupture of plaque caps, and quantifying physiological conditions under which collapse may occur. While physiological dimensions of the carotid arteries and related parameter ranges are used in the computations and experiments, the methods developed in this project will be useful in a broad range of applications where free moving boundaries are involved. Plaque collecting in an artery can narrow it; this narrowing is called a stenosis. Just like wat er flowing in a narrowed channel, the blood will flow faster past a stenosis. In certain conditions, this can cause the artery to collapse. Then the plaque may break up in the bloodstream and subsequently block small arteries downstream. That can lead directly to heart attacks and strokes. The investigators study this collapsing process, using mathematical, computational and experimental methods. Previous models have been limited primarily to one-dimensional models because of the difficulty in handling the collapsing tube, whose shape is unknown and changing. This project develops a three-dimensional mathematical model that accounts for the blood flow and the change in shape of the collapsing artery, and a novel numerical method to solve the model. One of the investigators conducts experiments to determine the pressure-area relationship (tube law) for the elastic tube models and the exact pressure-flow conditions for tube collapse in a laboratory set-up. The laboratory experiments and in vivo measurements provide data for the formulation and verification of the mathematical model. Results obtained will be useful for early detection and prevention of stenoses, identifying the causes of pupture of plaque caps, and physiological conditions under which collapse may occur. The numerical methods developed in this project will be useful in other applications where free moving boundaries are involved.