Mathematical Sciences: Mathematical and Experimental Studiesof Pulsatile Flow in Free Moving Elastic Tubes

研究人员开发了锥形自由运动弹性管内粘性流动的二维(轴对称)非线性数学模型、求解该模型的数值方法以及在实验模型中测量管壁运动的超声波方法。该模型旨在表示有狭窄和动脉粥样硬化的大动脉中的血液流动。给出了具有局部硬化、加厚和闭塞的自由运动弹性管内流动的详细局部化数值解。在实验室中利用弹性管进行了相应的实验。将数值计算结果与实验结果进行了比较，并对数学模型进行了进一步修正。这些结果还与已发表的活体测量结果进行了比较，以验证模型和技术的有效性。该项目所获得的数值结果和开发的实验技术将有助于更好地了解大动脉中的血液流动，并可能在未来具有临床价值。众所周知，心血管疾病是美国发病率和死亡率的主要原因。特定的疾病通常是动脉粥样硬化、狭窄和动脉瘤，由于诊断技术的不足，这些疾病往往没有被发现。本研究旨在提供一种有效的改进的大动脉血流数学模型，一种高效的数学模拟的数值方法，以及一种准确的无创超声测量技术。这些可能对血管疾病的早期诊断非常有用，因此有可能在未来拯救许多人的生命。本课题发展的数值方法同样适用于其他许多自由运动边界问题，对应用数学具有重要意义。

The investigators develop a two-dimensional (axisymmetric) nonlinear mathematical model of viscous flow in tapered free moving elastic tubes, a numerical method to solve the model, and an ultrasonic method to measure the movement of vessel walls in an experimental model. The model is intended to represent blood flow in large arteries with stenoses and atherosclerosis. Detailed localized numerical solution for the flow in free moving elastic tubes with local hardening, thickening and occlusions is obtained. Corresponding experiments are made in the lab using elastic tubes. The numerical and experimental results are compared and the mathematical model may be further refined. Those results are also compared with published results of in vivo measurements to verify the validity of the models and techniques. The numerical results obtained and experimental techniques developed in this project will lead to better understanding of blood flow in large arteries and may be of clinical value in the future. As is well-known, cardiovascular diseases are the leading causes of morbidity and mortality in the United States. Specific diseases are typically atherosclerosis, stenoses, and aneurysms that often remain undetected due to inadequate diagnostic techniques. This research aims to provide a valid improved mathematical model for blood flow in large arteries, an efficient numerical method for mathematical simulations, and an accurate noninvasive ultrasound measurement technique. These could be very useful for the early diagnosis of blood vessel diseases and therefore have the potential to save many lives in the future. The numerical method developed in this project is also applicable to many other free moving boundary problems and is itself of significance to applied mathematics.