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

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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.