Dr. Daniel Riahi, Dr. Ranadhir Roy, Samuel Cavazos

University of Texas-Pan American34th Annual Texas Differential Equations

Conference

Modeling and computation of blood flow resistance of an atherosclerotic artery with

multiple stenoses

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.

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.

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.

Mathematical Model

0

*

0

.

( ) .

.

.

.

.

.

R radius of artery

R z radius of stenoses

thickness of stenoses

L length of stenoses

d length from origin

r Radial axis

z axial axis

Formulae

0

,mh R

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.

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

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.

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

the impedance.

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

Graph of impedance against e.

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

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

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

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

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

Graph of the axial velocity with a different e value.

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

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.

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.