Computational Fluid Dynamics for Coronary Diseases: Analysis of Different Murray's Law Based Boundary Conditions Open Access

Abstract

Coronary Disease is becoming the one of the most serious threat toward human health. In most cases, atherosclerosis is responsible for the coronary artery stenosis, which further leads to coronary diseases. Investigating the abnormal blood flow pattern in coronary artery provides a way to inspect the existence of coronary artery stenosis. In this paper, we will first study the use of Murray's law in computational fluid dynamics and specify how to construct the Murray's law based boundary condition.

Further, we will introduce the general form of Navier-Stokes equation, which offers a mathematical model for fluid flow and related phenomena. In this research, the blood flow is our main interest. Several simplifications on Navier-Stokes equation allow us to depict blood flow in the vessel properly. Such modification leaves us the steady three-dimensional Navier-Stokes equation for incompressible Newtonian viscous fluid. The weak formulation of the modified Navier-Stokes equation is then introduced, which will be utilized in the computational fluid dynamic procedure to approximate the solution. In order to solve the Navier-Stokes equation with FreeFem++, the finite element method is chosen for space discretization and Picard iteration scheme is selected to linearize the problem. The previous steps together provide the algorithm to solve for pressure and velocity of blood.

With the solutions of Navier-Stokes equations with different Murray's law based boundary conditions, we calculated and compared the pressure drop and flow split. The results showed the choosing exponent between 2.25 to 2.75 in Murray's law will not significantly affect the Navier-Stokes solutions.

Table of Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Coronary Arteries Disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Atherosclerosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Computational fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Previous Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Objective and Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Space discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Linearization Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

2.4.1 Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2 Implemented Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

2.5 Streamline Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4. Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

About this Honors Thesis

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School	Emory College
Department	Mathematics and Computer Science
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics
Keyword	Navier-Stokes Murray's law
Committee Chair / Thesis Advisor	Veneziani, Alessandro, Emory University
Committee Members	Huang, Hao , Emory University Saadein, M Reza, Emory University

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