Research Article Open Access
Volume 3 • Issue 4 • 1000127
J Phys Chem Biophys
ISSN: 2161-0398 JPCB, an open access journal
Open Access Research article
Kol and Woafo, J Phys Chem Biophys 2013, 3:4
DOI: 10.4172/2161-0398.1000127
Keywords: Blood flow; Stented vessel; Navier-stokes equations
Introduction
Since cardiovascular diseases are one of the major causes of
mortality in the world, research works on the nature of these pathologies
and the techniques used for the treatment have become a priority to the
entire scientific community. To treat these vascular diseases, such as
atherosclerosis, aneurysms and stenoses, the endovascular technique
is the more comfortable technique used. It consists of a prosthesis
insertion into the diseased region, using a catheter [1].
But, the implantation of the prosthesis modifies the vascular wall,
dragging a modification of the flow in the vessel. Tis modification of
the flow can alter the wall, either at the interfaces or either on the stent
itself. Tese, therefore, lead to the fractures of the stent, the ruptures
of suture or holes in the stent coat [2]. It also appear complications
as displacement of the prosthesis in relation to their initial position
(migration) due to the widening of the collar of an aneurysm, or to a
over sizing too important of the prosthesis [3,4]. Furthermore, some
studies show that the rigidity of the prosthesis lead to its migration
[5]. We can note between other of the undesirable phenomena, as the
endofuites that are due to the persistence of the blood flux in outside of
the prosthesis, and in the aneurysmal bac, for example [4].
Much of the works done on stents are through numerical
investigations [6-9]. Tortoriello and Pedrizzetti [10] examined the
effects of stent implantation using an axisymmetric 2-D numerical
fluid-solid model. Tis analysis of pulsatile flow revealed that the
compliance mismatch and overexpansion caused by the stent both
enhanced the flow disruption in the stented region, thus reducing to
a minimum, and causing rapid variations in flow near the stent ends.
But, they disregard the effects of the nonlinearity on the wall.
In this study, we can extends the analysis of the Chakravarty and
Mandal model [11], by introducing, and analyzing the effects of the
spatial variation of the tube radius, and wall rigidity, as well as those
of localized deformation, such as aneurysms, stenoses, atherosclerosis
and prostheses. We focus on the deformation of the wall and the
nonlinearity coefcient of elasticity.
Te rest of the paper is therefore outlined as follows. In section 2,
one can briefly present the physical and mathematical model. In section
3, one makes a mathematical development. We can give a quantitative
*Corresponding author: Guy Richard Kol, School of Geology, Mining and Mineral
Processing, University of Ngaound´er´e, P.O. Box 454, Ngaound´er´e, Cameroon,
Tel: 237-9961-9187; E-mail: kolguyr@yahoo.fr
Received May 01, 2013; Accepted September 27, 2013; Published September
30, 2013
Citation: Kol GR, Woafo P (2013) Semi-analytical Study of Blood Flow through
a Prosthesis Inserted in an Affected Blood Vessel. J Phys Chem Biophys 3: 127.
doi:10.4172/2161-0398.1000127
Copyright: © 2013 Kol GR, et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
Semi-analytical Study of Blood Flow through a Prosthesis Inserted in an
Affected Blood Vessel
Guy Richard Kol
1,2*
and Paul Woafo
2
1
School of Geology, Mining and Mineral Processing, University of Ngaound´er´e, P.O. Box 454, Ngaound´er´e, Cameroon
2
Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812,
Yaound´e, Cameroon
discussion following the results obtained from numerical simulation in
section 4, and the last section is devoted to some concluding remarks.
Physical and Mathematical Formulation
Parameters of modeling
Te model in consideration here is a long narrow elastic tube filled
with an incompressible, viscous New- tonian fluid. Terefore, for the
modeling of this flow, one can introduce the equation presenting the
mass conservation of the fluid accompanied with the Naviers-Stokes
set of equations, taking into consideration the nonlinear coefcient of
elasticity of the wall, and the inertial effect as
Mass conservation
0
U U W
r r z
∂ ∂
+ + =
∂ ∂
(1)
Linear momentum conservation
2 2
2 2
1 1 W W W P W W W
U W
t r z f z f r r
r z
µ
ρ ρ
∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + =− + + +
∂ ∂ ∂ ∂ ∂
∂ ∂
(2)
2 2
2 2 2
1 1 U U U P U U U U
U W
t r z f r f r r
r r z
µ
ρ ρ
∂ ∂ ∂ ∂ ∂ ∂ ∂
+ + =− + + − +
∂ ∂ ∂ ∂ ∂
∂ ∂
(3)
where W and U are, respectively, the axial and radial fluid velocity, P
is the pressure in the vessel, ρ
f
is the fluid density and µ is the dynamic
viscosity of the fluid.
For the wall dynamics, using the second law of Newton on a
portion of the vessel wall, one can obtain the relation, which can be
Abstract
In this work, a two dimensional model of the flow is considered, with focus on effects of the nonlinearity coeffcient of
elasticity, variation of the radius and the Young modulus. The model described characterizes the blood flow consecutively
in aneurysms, stenoses and prostheses. We obtain in the three cases that the increase in the coeffcient of nonlinearity
decreases the axial fluid velocity, and weakly influence the radial velocity. The velocity of the flow remains parabolic,
decrease in the aneurys-mal bag, and increase in the stenosis when the severity of blood vessels diseases varies. We
found that aneurysms of small widths present high peaks of wall shear stress, and so predisposes to the formation of
thrombus. Finally, we determine the maximum value of elasticity that helps to enhance the performance of prosthesis.
Journal of Physical Chemistry &
Biophysics
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ISSN: 2161-0398