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

高度狭窄的动脉在生理条件下可能塌陷。塌陷可能导致疲劳加速，斑块纤维帽破裂进入血流，随后导致下游小动脉阻塞。这会直接导致心脏病发作和中风。研究者和他的同事使用数学、计算和实验方法来研究这一坍塌过程。由于塌缩管的形状是未知的和不断变化的，因此难以处理，因此数学模型主要局限于一维模型。提出了一个具有自由移动边界的三维非线性粘性数学模型，并提出了一种新的数值求解方法。数值方法采用长波渐近展开式的解作为数值初始条件，采用边界迭代法求未知的运动边界、流速和压力。通过实验确定了弹性管模型的压力-面积关系（管律），并在实验室装置中量化了管坍塌的确切压力-流动条件。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。验证的模型及其数值解为进一步研究坍塌过程奠定了基础。获得的结果将有助于早期发现和预防狭窄，确定斑块帽破裂的原因，以及量化可能发生塌陷的生理条件。虽然在计算和实验中使用了颈动脉的生理尺寸和相关参数范围，但本项目中开发的方法将在涉及自由移动边界的广泛应用中有用。动脉中聚集的斑块会使其变窄；这种狭窄叫做狭窄。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，这会导致动脉塌陷。然后，斑块可能在血液中破裂，随后阻塞下游的小动脉。这会直接导致心脏病发作和中风。研究人员使用数学、计算和实验方法研究这一坍塌过程。先前的模型主要局限于一维模型，因为难以处理形状未知且不断变化的坍塌管。该项目开发了一个三维数学模型，用于解释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。其中一名研究人员在实验室装置中进行了实验，以确定弹性管模型的压力-面积关系（管律）和管坍塌的确切压力-流动条件。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早期发现和预防狭窄，确定牙菌斑帽破裂的原因，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在涉及自由移动边界的其他应用中有用。

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.