Dr. Daniel Riahi, Dr. Ranadhir Roy, Samuel Cavazos University of Texas-Pan American

1. Modeling and computation of blood flow resistance of an atherosclerotic artery with multiple stenoses Dr. Daniel Riahi, Dr. Ranadhir Roy, Samuel Cavazos University of Texas-Pan American 34th Annual Texas Differential Equations Conference

2. Introduction • Atherosclerosis is a circulatory disease which causes lesions and plaques in blood vessels, preventing a sufficient amount of blood from reaching the distal bed. Plaques which contain calcium may initiate the formation of blood clots, also inhibiting blood from reaching the distal bed. Plaques that form in the coronary arteries may lead to heart attack, and clots in brain vessels may lead to a stroke. The most common formation sites for these clots are the coronary arteries, the branching of the subclavian and common carotids in the aortic arch, the division of the common carotid to internal and external carotids, the renal arterial branching in the descending aorta, and in the ileofemoral divisions of the descending aorta.

3. Purpose • There are two cases which we will consider for this project. The first case we considered is when the blood flow is consistent, or steady, while the second case is when the blood flow is unsteady. In this presentation, we consider only the effects of the stenoses with the steady blood flow. We began our research by using computational methods and modeling. The resistance (or impedance) of the blood flow is determined by the relationship between the blood flow and the pressure drop, and the distribution of pressure and the shearing stress through the stenoses.

4. Objective • Derive the governing non-axisymmetric and unsteady equations and the boundary conditions for the mathematical models of the flow of blood for non-Newtonian fluid cases in the artery. • Carry out analysis and derive expressions for pressure drop, impedance, shear stress distribution and variations at the artery wall, at the stenosis throats and critical height. • Develop the computer program which uses numerical methods to estimate quantitative effects of various parameters involved on the results of the analysis.

6. Mathematical Model

7. Formulae

8. This is the graph of the volume pressure flow. The graph shows how the pressure gradient (Volume Flow Rate) is consistent until it enters the stenoses at z = 0.5, where it begins to drop.

9. Pressure Gradient with e = 0.04. The decrease in e also decreased the total Volume Flow Rate.

10. The graph shows the shear stress at the walls of the artery. As expected, the stress on the artery walls increases in the stenoses with the greatest stress at the stenoses throats.

11. Here is the graph of the impedance (Flow Resistance) against g. As g increases, so does the impedance.

12. The change from e=0.1 to e=0.04 also decreases the impedance.

13. Graph of impedance against e.

14. Graph of shear stress against g. Again, the stress increases as g increases.

15. Shear Stress with e = 0.4 at the critical heights of the stenoses.

16. Graph of shear stress at the stenoses critical heights against e.

17. Graph of Axial Velocity at critical throats against the radius r of the artery.

18. Graph of Axial Velocity against z. The axial velocity decreases as the value of r increases.

19. Graph of the axial velocity with a different e value.

20. Shown above is the graph of the axial velocity with respect to the radius.

21. Results • The flow rate decreases as the blood flows through the stenoses. • As the size of the stenoses increases, the pressure gradeint increases. • The shear stress on the artery walls increases at the stenoses, reaching it’s maximum value at the stenoses critical point. • As the stenoses increases, so does the impedance and shear stress. • As the radius of the artery increases, the axial velocity decreases. • Axial velocity decreases as the radius increases.

22. Conclusion • We have developed the mathematical models for non-axisymmetric equations and the boundary conditions for of the flow of blood for Newtonian fluid in the artery. • We have carried out analysis and derived expressions for pressure drop. • Developed the computer program using numerical methods to estimate quantitative effects of various parameters involved on the results of the analysis. • Analytic expression have been developed for the thickness of the peripheral layer. Slip and core viscosity was obtained in terms of measure quantities (flow rate), centerline velocity, pressure gradient. • Computed the results and data for the dependent variables for realistic parameter regimes for the case of human arteries and found the effect and the roles played by the stenoses on the blood flow.