dr. daniel riahi, dr. ranadhir roy, samuel cavazos university of texas-pan american 34 th annual...
TRANSCRIPT
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
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.
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.
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.