Assam University Journal of Science &
Technology: Physical Sciences and Technology
Vol. 4 Number II
7-16,2009
Steady Flow of Blood through a Catheterized Tapered Artery with Stenosis:
A Theoretical Model
Devajyoti Biswas' and Uday Shankar Chakraborty2
Department of Mathematics
Assam University, Silchar-7880 11, India
E-mail: [email protected]
Abstract
The paper deals with a steady laminar flow ofblood through an annulus, enclosed between an arterial stenosis,
developed along Cl tapering wall and a uniform catheter, .co-axial to it. In this work, body fluid blood is
assumed to behave like a Newtonian fluid. A velocity slip condition is employed at the catheterized wall with
different sizes ofstenosis and zero-slip at the catheter boundary is taken. Analytic expressions are obtainedfor
axial velocity,flow rate, wall shear stress and apparent viscosity. Their variations with differentflow parameters
are plotted in figures. The behaviour of these flow variables in this constricted annular region has been
discussed. It may be worthmentionating that velocity and flow rate will increase in one hand but wall shear
stress and apparent viscosity, will be lowered on the other, due to the introduction of an axial slip. The effects
oftapering and stenosis are considered. Physiological implications ofthis theoretical modelling to bloodflow
situations are also included in brief
Keywords and phrases: Blood flow, catheterized tapered vessel, constriction, annulus, Velocity slip, Newtonian
fluid.
Introduction
The study of blood flow through tapered tubes is
important not only for an understanding of the
flow behaviour of the marvellous body fluid blood
in arteries, but also for the design of prosthetic
blood vessels (How and Black, 1987). Also, the
word stenosis or atherosclerosis is a generic
medical term which means narrowing of any body
passage, tube or orifice (Young, 1979). As stenosis
literally means the formation of atherosclerosis
plaques in the lumen of an artery, so, its presence
at different sites, along with its subsequent and
severe growth on the artery wall, results in serious
circulatory disorders (Young, 1979; Caro, 1973;
Fry, 1973). Further, atherosclerosis lesion or
plaque is a common form of cardiovascular
disease that severely influences the human health.
The partial occlusion of arteries due to stenotic
obstruction, is one of the most frequent anomalies
in blood flow (Liu et.al, 2004; Poltem et.al, 2006)
and this occlusion due to plaque formation, is
reported as the leading cause of mortality in
developed countries (Caro et.al, 1971). It has been
found that the initiation and localization of
stenosis, is closely related to the local
hemodynamic factors (such as wall shear stress,
pressure-flow relationship etc.). Although, there
remains uncertainty with regard to the exact
hemodynamic factors responsible for the initiation
of the arteriosclerosis, it has been demonstrated
that the development of stenosis is strongly related
to the characteristics of blood flow in arteries (Liu
et.al, 2004). Further, there is a good amount of
evidence that hydrodynamic factors play an
important role in the formation, development and
progression of an arterial stenosis (Caro et.al,
1971; Young and Tsai, 1973).
Sometimes for many useful medical and clinical
purposes, artificial catheters are inserted in
stenosed arteries. The pressure-flow relationship
alters considerably due to the presence of a
-7-
/
flexible catheter in the constricted region of an junction, a side branch, or other sudden change
artery, as this will lead to further enhancement of in flow geometry (Berger and Jou, 2000). There
the impedance or frictional resistance to flow as is no doubt that tapering in arteries, is a significant
well as. will alter the pressure distribution aspect of mammalian arterial system and the
(Jayaraman and Tewari, 1995; Biswas and formation of stenosis along the tapered wall, may
Bhatacharjee, 2003). Study of blood flow in the alter the flow situation to a great extent. In view
annular region of a mammalian artery, has already of the above considerations, we are interested to
drawn a good deal of interest and attention, among study the steady annular flow of blood through a
the researchers, in view of its potential catheterized tapered vessel with stenosis. In this
physiological significance and a lot of clinical case, blood is taken as a Newtonian fluid and an
applications. It may be worth mentioning that in axial slip in velocity, is introduced at the stenotic
certain flow situations, physiological fluid blood wall of the tapered artery.
may behave as a Newtonian fluid (Schlitching, Mathematical Formulations.
1979). The laminar flow of a Newtonian fluid
through an annulus has been dealt in (Yuan, 1969). Flow Geometry and Co-ordinate system.
The annular blood flow in a catheterized stenosed -The schematic diagram of the flow geometry
uniform artery has been investigated in (Jayaraman corresponding to the steady flow of an
and Tewari, 1995; Biswas and Bhatacharjee, incompressible Newtonian fluid in an annular
2003). Recently, blood flow in tapered vessels, space between a constricted tapered tube of
with the presence of a stenosis, is modelled in (Liu undisturbed radius Ro and a co-axial uniform
et.al,2004).
circular pipe of radius KRo (K«I), is shown in
In recent years, theoretical, numerical and Fig.l.
experimental investigations (How and Black,
r~d~C:1:----'----------------1987; Young, 1979; Liu et.al, 2004; Poltem et.al,
---------..
R.
U(n
___C)
2006; Fung, 1981; Biswas, 2000), have been
performed to analyze blood flow in arteries. The
__::::::_=:_..,._ z
complex geometry of arteries (viz. bending,
"',-, KR.
bifurcating, branching, tapering, discrete etc.), is
also an important factor which obviously affects
the local hemodynamics (Guyton, 1970; Puniyani
1: =: : :=: : :=: : :=: : :=: : : ;: :!: : = = ~= = c: : : a{h: : : e: : : !": : : J~_: : :
~.
and Niimi, 1998). The relationship between flow
in the arteries, particularly the wall shear stresses,
and the sites where atherosclerosis develops has
motivated much of the research on arterial flow
in recent decades. It is now well accepted that the
sites where shear stresses are low, or change
rapidly in time or space, that are most vulnerable.
These conditions are likely to prevail at places
where the vessel is curved; bifurcated; has a
R(z) =
Fig. 1. Schematic diagram of a catheterized
tapered artery with stenosis
The tapered vessel segment having an axially
symmetric stenosis is mathematically modeled as
a rigid tube with a circular section and a catheter
is co-axial to it. The geometry of the constricted
tapered artery (Fig. 1. ) is mathematically modeled
as (Liu et.al, 2004).
Ru -m(z+d)- 8SCOsa(1+COSil"Z/
{ Ru -m(z+d)
2
)
/zo
(2.1)
where R(z) denotes the radius of the tapered
for the tapered artery,
arterial segment in the stenotic region, Ro is the
m{= tan a) represents the slope of
the tapered vessel. Let (r, e, z) be the system of
constant radius of the straight artery in the nonstenotic region, a is the angle of tapering,
88 cos a is the length of the stenosis at a length d
Zo
is the half length of the
stenosis and
co-ordinates, used to analyze the flow field in the
-8-
geometry as stated above, where r and () are
along the radial and circumferential directions and,
z-axis is taken along the axis of the artery.
The governing equation of a steady flow in this
annulus, as obtained from the equations (2.2-2.4)
above, is given by
"Governing Equations
Conditions.
C
and
Boundary
Consider a steady and laminar flow of blood in
an axi-symmetric catheterized tapered tube with
a stenosis.If we consider the blood flow in the
annular region between the constricted tapered
arterial segment and a co-axial catheter, to be onedimensional (I-D), the equations of motion in
(r, (), z) system (Schlitching, 1979) are written as
8p =0
8r
k~ ~ r ~ R{z) ,
(2.5)
r 8r 8r
p
where C = (- d ) is the pressure gradient.
dz
In order to study the annular flow of blood,
between the stenotic wall and a catheterized
tapered artery, we shall consider the following
case:
Boundary Conditions: A slip condition at the
stenotic wall and a no-slip condition at the catheter
wall, have been employed in the present case,
(2.2)
(2.3)
+ f-l ~(au) = 0,
u(r) = Us at
u(r)=O at
r= R(Z)}
r=kRo ,
(2.6)
where Us is the axial velocity slip at the stenotic
wall (Biswas, 2000)
(2.4)
where
U
=u{r) denotes the axial blood velocity,
f-l is the viscosity of blood and, p the pressure.
Solutions
The general integral of equation (2.5) is obtained
as
Cr2
u(r)+-=B+Alnr,
4J1
kRn :5: r:5:R(z),
(2.7)
where A and B are constants of integration. As a result of applying condition (2.6) in equation (2.7),
the expression for velocity function becomes
u(r)~~[(kR,)'-r21+ It~7d)[US+~~R(Z))2-(kRo)2}J
4f-l
In R(z)
4f-l
.,
kRo 5.r5.R(z), (2.8)
kRo
The rate of mass flow Q for the steady, incompressible and laminar flow is determined by
integrating the quantity
R(z)
Q = 2;r
f ru {r )dr ,
(2.9)
r=kRo
which becomes, with the help of the equation (2.8),
-9-
Expression for wall shear stress, defined by
'iR(z)
ou(r)]
or
,
= - P--
(2.11)
r=R(z)
becomes with the help of the equation (2.8)
~
C
TR(z)
=2"R(z)-
R(z) [
In(R(Z))
.
Us
C S')2
+ 4pl\R(z)
-(kRo)
2}]
(2.12)
kRo
Apparent viscosity can be computed with the formula
ne(R(z)t
Po =
8Q
(2.13)
'
where Q has the representation in equation (2.10) •
Thus with the help of equation (2.10), we get,
_
fl. -
8u s
{..
1-(~)) 1
~(R(Z))2 1- 21n(R~)) +-;; 1-
(kRu)'
R(z)
-
H~Jl'r
In( :~))
(2.14)
A second representation ofthe above flow variables can be obtained as fo.1lows:
Velocity function:
(2.15)
Flow rate:
Q ='ii
S
[2(R(Z)) (R(z»)
-(fR~)2]+ (R(z)t -(kRot _ KR(z))
21n(~~)
_(kRo)2
In( ~~))
t
(2.16)
Wall shear stress:
(2.17)
Apparent viscosity:
(2.18)
- 10 -
_
r
r=-,
'(R(z)
Ro
CRo
'( R(z)
=--
'(0
=-2
Jia
(2.19)
fda
=Ji
'(0
Results and Discussions.
Steady laminar flow of a Newtonian fluid (blood)
,through an annulus of a catheterized tapered vessel
'with a constriction is presented in Fig.I. The
Analytical expressions for velocity, flow rate, wall
shear stress and apparent viscosity are given by
equations (2.8-2.14). It is observed that velocity
is a function of shear viscosity Ji , pressure
(iv)
When R(z) ;.t Ro ' kRo = 0, 'o.s ;.t 0,
a = 0 = m it provides I-D flow in a
stenosed uniform artery.
(v)
When R(z) ;.t Ro ' kRo = 0, Os ;.t 0, a ;.t ;.t
m, it results in I-D flow through a tapered
gradient C , radius of the tapered arterial segment
(vi).
°
artery with constriction.
When R(z) ;.tRo' kRo = 0, Os ;.to, a;.tO ;.t m
it represents the present model with wall
slip or no-slip cases.
R(z), catheter radius kRo' slip velocity Us and r,
the radial coordinate. The non-dimensional forms
ofthe aforesaid flow variables are represented in
the equations (2.15-2.19), and the variations of
these flow variables are shown in figures (Fig.2Fig.II). Further, the radius R(z), (as taken in
equation 2.1) depends on Ro , the radius of the
uniform artery; a , the tapering angle, zo' the half
length of the stenosis,
os' the maximum height
of the stenosis at an axial distance Z = d , and m,
the slope of the tapered tube. The present model
includes the following cases, viz.
(i)
When
a
R(z) = Ro , kRo
= 0,
Os
Axial Velocity Profile.
Velocity expression (in equation 2.8) is a linear
function of C, f.l, Us but a non-linear function of,
KRo R(z). A comparison of velocity profiles in
the annular region of a steady state tapered artery
in the presence of a catheter co-axial to it, for slip
and no-slip cases at the stenosis location and the
flow boundary is shown in Fig.2. , a comparison
of velocity profiles for different tapering angles
is shown in Fig.3 .and a comparison of velocity
profiles for different sizes of stenosis is
represented in FigA. Wherefrom, the following
observations that can be made are:
=0,
=0 = m , it reduces to Poiseuille model
Z= -Zo
- - - cs=o,o"
--C,=OOI
--Cs=O
with velocity slip (us> 0) or no-slip
(us
(ii)
= 0) at wall.
When R(z) = Ro , kRo
a
0,3
:;t:
0,
Os = 0,
= 0 = m it yields to
0,2
an annular flow in
a catheterized uniform artery, without
constriction.
(iii)
When
R(z) = Ro ,kRo
:;t:
0, 8s
'*
0,1
0,
a = 0 = m it results to an annular flow in
a catheterized uniform artery, with a
constriction.
)
Fig.2. Variation of axial velocity with tube
radius for different slip velocities.
- 11 -
- -
Z=Zo
-:-'1iId
:\1odet"ate
- - Severe
Z=-Zo
-~
0.5
r
- _ -(1=0.
-0=1·
.,
0..4
Z=o.
--a=13o.
Z=O
t:z 0..3
LZ
0..3
0..2
0..2
0.1
0..1
0.1
0..2
0..4
0.3
0.5
r
0.6
0..7
0..8
US
------i>
)
0.7
0.8
o.S
Fig.3. Variation of axial velocity with tube
radius for different tapering angles.
Fig.4. Var.iation of axial velocity with axial
distance for different stenosis sizes
(i)
(vi) Also, with an increase in stenosis height or
for a reduction in annular area, velocity
diminishes for slip and no-slip cases.
In this annular region, profiles indicate a nonparabolic trend for both slip and no-slip
cases.
Flow rate
(ii) As. r,adial~o-ordinate ..!...- increases in full
Ro
scale from KRo to R(z), velocity increases
Ro
The expression for flow rate as given in equation
Ro
rapidly to a greater value, wherefrom, it
gradually decreases to a lower value at or
near the vessel wall.
(iii) Velocity is maximum at maximum crosssectional area at z
= -zo
(at the initiation
of the stenosis), a higher value at z = Zo (at
the termination of the stenosis) and,
minimum at minimum cross-sectional area,
z = 0 (at the throat of the stenosis).
(2.10), is a function of Us ,KRo , R(z), m, 8s and
Q
against
Zo ~ Z ~ Zo
for slip
z. The variations in the rate of flow
axial distance z in full scale -
or no-slip at the flow boundary are presented in
Fig.5 , variation of flow rate against the axial
distance for various tapering angles is presented
in Fig.6 and It can be noticed from the figures
that variation of flow rate against the axial distance
for different stenosis sizes is shown in Fig. 7.
(iv) As expected, velocity increases with slip
employed at the stenotic wall. Its values are
higher for flows with slip (us > 0) than those
1:s=0_05 1;s=o
with no-slip case. As slip moves upward,
velocity attains a higher and more higher
value, in case of this constriction.
0.2
Us=O.Ol
0.1
(v) Profiles indicate that increasing of tapering
angle of the artery decreases the axial
velocity of blood.
.,
Fig.5. Variation of Flow rate with axial
distance for different slip velocities.
- 12 -
- - - <Mild
--~odera.t~
-Severe
/
0=0
""--~//'
/
Q
0.3
0.2
11'
,
_
Z
2
3
)
-,
-2
-3
·4
4
Fig.6. Variation of Flow rate with axial distance
for different tapering angles.
Fig.7. Variation of Flow rate with axial
distance for different stenosis sizes.
(i)
are ·shown in Fig.9 and for different sizes of
stenosis are shown in Fig.l0. The shear stress
profiles clearly indicate that
Flow rate changes with axial locations in
non-uniform annular region.
(ii) It is minimum at the minimum annular region
_9
(i.e. at the throat ofthe stenosis) and attains
the maximum flow rate at the initiation of
the stenosis. It has a higher value at the
termination ofthe stenosis. In the two equal
annular
regions
-Zo
-1:
0.7
:s; z:S; 0
and 0 :s; Z :s; Zo ,Q increases from the smallest
magnitude at the throat to the greater values
at the initiation or termination of the stenosis.
(iii) In all axial locations, Q computed with a
wall slip is more than that obtained with zeroslip at the boundary. As slip increases in
magnitude, Q acquires higher and higher
magnitudes.
(iv)
-4
-3
-2
Fig.8. Variation of wall shear stress with axial
distance for different slip velocities
Q obtained in the artery with tapering seen
to be lower than its value, found in the
annular flow model without tapering. Also
increase in tapering angle further reduces the
volume flow rate.
(v) Flow rate decreases with the increase in
stenosis size.
Wall Shear Stress
Expression for wall shear stress (in equation 2.12)
depends on, us' R(z), m, . Its variations (obtained
from equation 2.17) versus axial locations of z
for in the annular region for slip or no-slip cases,
are shown in Fig.S, for different tapering angles
-4
-3
-2
2
-0.1/
Z --7
·0.2
Fig.9. Variation of wall shear stress with axial
distance for different tapering angles
- 13 -
Apparent Viscosity
0..9
i
---Mild
0.8
-+---+- ~oderate
Apparent viscosity
-Severe
Jio is computed with the help
of equation 2.14.and its variations against axial
location z in (-zo' zo) for the annular region in
case of mild, moderate, severe stenoses and for
slip or no-slip cases are shown in Fig. 11.
0..7
0.6
0..5
_D"~(I!
--u~
.,
·2
·3
. . "
.
Fig.tO. Variation of Wall shear stress with axial
distance for different stenosis sizes
Il:Jii
1.'5
1
,~,
(i)
It decreases with an employment of slip
condition at the flow boundary and, as slip
increases,
(ii)
TR(z)
TR(z)
decreases further.
reaches a maximum magnitude at a
minimum area of annular region and rapidly
decreases in the diverging constricted region.
However, this maximum value obtained with
a slip is lower than that computed with zeroslip at the boundary.
(iii) As axial location z increases from z = -zo to
z
=
0,
TR(Z)
increases from the minimum
value at mouth of the constricted annular
region to the maximum one at the throat of
the annulus. On the remaining equal portion,
it decreases from the maximum value at the
mouth of the constriction (z=O) to a lower
one at the other end of the annular stenotic
region (z = zo)' However, throughout this
axial variation, shear stress is lowered as a
result of employing slip at the flow boundary.
(iv) . The effect of tapering on wall shear stress
reveals that the tapering of the artery does
not change the flow pattern, but only changes
the values.
(v) The wall shear stress increases with the
increase of height of the stenosis.
.2,
Fig.H. Variation of apparent viscosity with
axial distance for slip and no-slip cases
It is observed from the profile that
(i)
Jio
attains the greatest. magnitude at the
minimum constricted annular area (at z=O) and
the lower values at the maximum annular area (at
z = -zo or at z = zo ).
(ii) As expected,
Jio
decreases with slip at the
flow boundary and its magnitude so obtained with
slip is lower than that value obtained with nowall slip.
(iii)
Jio increases with the increase in height of
the stenosis for slip and no-slip cases.
Conclusion
In the present case, an annular flow model of
blood (behaving as a Newtonian fluid), inside the
region of a catheterized tapered artery, has been
presented with the presence of a constriction at
the vessel wall. Analytical expressions for
velocity, flow rate, wall shear stress and apparent
viscosity, have been obtained. It is of interest to
note that Poiseuille flow, one-dimensional (I-D)
- 14 -
flow through tapered tubes, one-dimensional (1D) flow in a constricted tapered arteries, onedimensional (1-0) flow in a catheterized uniform
tub.es, one-dimensional (1-0) annular flow in a
catheterized uniform artery with stenosis of
Newtonian fluids with velocity slip or no-slip
cases at wall, are considered in the present model
as its special cases. As expected, velocity increases
with slip and it increases further with the increasing
values of slip while it decreases with an increase
of height ofthe stenosis and tapering angle of the
artery. Flow rates increases with slip and, it attains
the greater magnitudes at either end of the
constricted annular region and the lowest value at
the throat of the stenosis. Increase in stenosis size
decreases the flow rate. Wall shear stress attains
its maximum magnitude at the throat of the
stenosis and lower magnitudes at either end of the
stenosis. Apparent viscosity reduces with slip but
its trend remains the same with the variation of
tapering angle. Also, the magnitudes of velocity
and flow rate are found to be smaller in the present
model than those in non-tapering catheterized
annulus with stenosis. Further, the peak value of
wall shear stress in this case is seen to be bigger
than that peak value obtained in the constricted
annulus without tapering. But this peak value is
found to be lowered as the employment of a wallslip. The present model indicates that the
introduction of a velocity slip at wall, flow rate
can be accelerated on one hand and resistance to
flow can be retarded on the other. Also, separation
phenomena and back flow regime are indicated
in the post stenotic region. However, separation
in velocity occurs earlier in zero-slip case than
that with a velocity slip. The present model shows
that the tapering of an artery does not alter the
flow pattern, but only changes the values. It also
indicates that height of the stenosis is more
important factor influencing the annular blood
flow than tapering of a blood vessel. As high wall
shear stress in tapered arteries may not only
damage the vessel wall but also, it may activate
platelets, cause platelet aggregation and finally
result in formation of a thrombus, leading to
fatality, wall shear stress needs to be investigated.
However slip reduces the wall shear stress and
hence damaging of diseased vessel wall, can be
lowered. Therefore as a recourse, slip at the vessel
wall can be exploited in lessening the damage to
the inner artery wall as well as improving the
normal functioning of an occluded artery. Thus
the present model may be used as a tool for
accelerating the flow rate, reducing the resistance
to flow and restoring the normal functioning of
the diseased arterial wall. Exploration is in
progress to understand the functioning of flow
parameters in two-dimensional (2D) motion of
blood in arteries with both radial and axial velocity
slip at the stenotic wall.
References
1.. Berger, S.A.; Jou, L-D. (2000). Flows in Stenotic
Vessels. Annual Review of Fluid Mechanics.
32:347-384
2. Biswas,D.(2000). Blood Flow Models: A
Comparative Study, Mittal Publications, New Delhi
3. Biswas,D.; Bhattacharjee,A.(2003). "Blood Flow
in Annular Region of a Catheterized Stenosed
Artery". Proc. 30,h NC-FMFP. 104-110
4. Caro, C.G. (1973). "Transport of Material Between
Blood and Wall in Arteries, In Atherogenesis:
Initiating Factors", Ciba Foundation Symposium,
Elsevier, Experta Medica, North Holland,
Amstardam,12:127-164
5. Caro, C.G.; Fit-Gerald,J.M.; ScHroter, R.C. (1971).
"Atheroma and Arterial Wall Shear: Observation,
Correlation and Proposal of a Shear Dependent
Mass Transfer of Atherogenesis".Proc. Roy. Soc.
B, 177: 109-159
6. Fung,Y.C.(1981).Biomechanics: Mechanical
Properties of Living Tissues, Springer-Verlag, New
York Inc.
7. Fry, D.L. (1973). "Responses of the Arterial Wall
to Certain Physical Factors, In Atherogenesis:
Initiating Factors", Ciba Foundation Symposium,
Elsevier, Experta Medica, North Holland,
Amstardam, 12:.93-125
8. Guyton, A.C.(1970). Text Book of Medical
Physiology, Igaku Shoin International, Philadelphia
9. How, T.V.; Black, R.A. (1987). Pressure loses in
Non-Newtonian Flow Through Rigid wall Tapered
Tubes. Biorheology. 24: 337-351
10. Jayaraman, G.; Tewari, K.(1995). Flow in
Catheterized curved Artery, Medical and
Biological Engineering and Computing (IFBME).
33(5):720-724
- 15 -
11. Liu,G.T.; Wang, X.J.; Ai,B.Q.; Liu,G.L.(2004),
Numerical Study of Pulsating Flow Trough- a
Tapered Artery with Stenosis. Chinese Journal of
Physics. 42 (4-I): 401-409
12.Poltem,D.; Wiwatanapataphee,B.; Wu,Y.H.;
Lenbury, Y.(2006). A Numerical Study Of NonNewtonian Blood Flow in Stenosed Coronary
Artery Bypass with Graft. ANZIAM J. 47(EMAC
2005): 277-291
13. Puniyani,R.R.; Niimi,H.(1998). Applied Clinical
Hemorheology, Quest Publications, Mumbai
14. Schlitching, H. (1979). Boundary Layer Theory,
Mc-Graw-Hill, New York
15. Young, D.F.; Tsai,F. Y. (1973). Flow Characteristics
in Models of Arterial Stenoisis-II Unsteady Flow.
J. Biomecanics. 6, 547-561
16. Young, D.F. (1979). Fluid Mechanics of Arterial
Stenosis. J.Biomech.Engng., Trans. ASME. 101:
157-175
17. Yuan, S.W. (1969). Foundations of Fluid
Mechanics, Prentice-Hall of India Pvt. Ltd., New
Delhi
- 16 -