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Pulsatile flow of blood with periodic body acceleration
1~. J. Engng Sci. Vol. 29, No. 1, pp. 113-121, 1991 Printed in Great Britain. All rights reserved
PULSATILE
FLOW OF BLOOD WITH PERIODIC ACCELERATION P. CHATURANI
Department
0020-7225/91 $3.00+ 0.00 Copyright @ 1991Pergamon Press plc
of Mathematics,
BODY
and V. PALANISAMY
Indian Institute of Technology, Powai, Bombay-400 076, India
Abstract-Pulsatile flow of blood through a rigid tube has been studied under the influence of body acceleration. With the help of finite Hankel and Laplace transforms, analytic expressions for axial velocity, fluid acceleration, wall shear and instantaneous volume flow rate have been obtained. It is of interest to note that these solutions can be used for all the feasible values of pulsatile and body acceleration Reynolds numbers Re, and Re,. This is in contrast with the existing results where different approximate solutions have to be used for different ranges of Re, and Re,. Using physiological data, the following qualitative and quantitative results have been obtained. The amplitude of the instantaneous volume flow rate, for flows with body acceleration, decreases shaprly as the tube radius decreases (from aorta to arteriole). This variation of amplitude is very slow for flows with no body acceleration. Another interesting result is the maximum of the axial velocity and fluid acceleration shifts from the tube axis to the vicinity of the tube wall as the tube diameter increases. The variation of the amplitude of wall shear with tube diameter (aorta to coronary) is less for flows with body acceleration than that of flows with no body acceleration. The phase lag between pressure gradient and flow rate changes sharply with tube diameter in narrow tubes, it varies asymptotically in wide tubes. The obtained results are qualitatively in good agreement with existing theoretical observations. Quantitatively, they differ from the other theoretical results (19 to 3000%). The difference in the results of the analyses decreases as the tube diameter increases (arteriole to aorta).
1. INTRODUCTION The human body may be subjected to body acceleration in many situations e.g. helicopter crew members, astronauts, jet pilots and crew while taking off and landing, jackhammer operators while working, athletes and sportsmen for their sudden movements. Prolonged exposures to body acceleration leads to many health problems, e.g. headache, abdominal pain and loss of vision etc. [l, 21. It is possible that dangerous combinations of amplitude and frequency of body acceleration and blood flows may be responsible for such health problems. It is, therefore, desirable to set a standard for short and long term exposures of human beings to such an acceleration. It is also possible that a proper understanding of interactions of body acceleration with blood flow may lead to a therapeutic use of controlled body acceleration [3, 41, to the development of new diagnostic tools and to better design of protective pads. The present investigation is a step in this direction. Due to physiological importance of body acceleration, many mathematical models have been proposed for blood flow with body acceleration [5-91. Sud and Sekhon [8] have considered the pulsatile flow of blood through a rigid circular tube with body acceleration by assuming blood as a Newtonian fluid. They have obtained solutions for axial velocity, fluid acceleration, wall shear etc., in terms of Bessel’s functions with complex arguments. To compute flow variables from these expressions is very difficult, if not impossible, for general values of Re, and Reb, even for professional mathematicians, leave aside non-professional mathematicians, in particular medical scientists. In their analysis, two limiting cases have been considered and their approximate solutions have been obtained. There are two more limiting cases for which approximate solutions can be found. Besides having four approximations for various limiting values of Re, and Re,, intermediate values of Re, and Re, will pose serious problems for computation. In this investigation, solutions have been obtained which are free from the above mentioned difficulty of computing.
2.
FORMULATION
OF
THE
PROBLEM
Let us consider the pulsatile flow of blood through a rigid cylindrical tube with body acceleration. Blood is assumed to be Newtonian and incompressible fluid. Flow as axially 113 ES 29:1-H
114
P. CHATURANIand V. PALANISAMY
symmetric, pulsatile and fully developed. given by
The pressure gradient and body acceleration
-$4D+A,cosw,r
(g) are (1)
g = a0 cos(oi$ + $)
(2)
Where wP = 2nf, and fP is pulse frequency, A0 and Al are pressure gradient amplitude of oscillatory part, a, is the amplitude of body acceleration, o,, = acceleration frequency, 9 is phase angle of g with respect to the heart action and t is time. Under the above mentioned assumptions, the equation of motion for coordinates (r, 8, z) can be written as [8]
of steady flow and 2~rfb and fb is body (pressure gradient) flow in cylindrical
dU
~~=pa,cos(w,r+~)+A,+A,cosw,r+~~
where u is velocity in axial direction, p and p are the density and viscosity of blood. The initial and boundary conditions are [8] u(r
0)
=
2
(ii)
(R2- r2)(Ao+Ad 4p
at r = R (Tube radius)
u(r, t) = 0
u(r, t) is finite at r = 0.
(iii)
(4)
3. ANALYSIS
The consecutive applications of finite Hankel and Laplace transforms on partial differential equation (3) and initial and boundary conditions (4) leads to an algebraic equation of which solution can be easily found. The inversion of which gives the final solution as A,[vk? cos o,f + wp sin ~,t] [v2k; + co’,] + pa,[vk:
cos(t&t + #) + t%bsin(e.&t + @)] [v*k; + co;]
-e
A A_
-vkfr
[
+ pu,[vk:
(Ao+A,)+
AIvk;
vk:
[v2k; + w;]
vk;2
cos c#J+ o,, sin f$] [v2k: + o;]
11
(5)
where v = ,u/p, k, = ~i/R and 5i are zeros of Jo. The expression of u consists of four terms. The first and second terms correspond to steady and oscillatory parts of pressure gradient, the third term represents body acceleration and the last is transient term. As t * 00, the transient term approaches to zero and we get the desired solution for pulsatile pressure gradient and body acceleration as
(6) where s = _l_+ c[ET cos w,t + a’sin ~,t]
m41 + p[‘$ COS(Wbt + @)+ 0’ sin(e&t ]f4 + P”1
E?
w R2 (u2=A=Re V
P’
154 +
/j2=$=Rebr
E =
+ $)]
A,IAo,
p=pa, Ao .
Pulsatile flow of blood with periodic body acceleration
The instantaneous
115
volume flow rate Q defined as Q = 2nlR ru dr
(7)
gives
Q=
(8)
The physiological important quantities like, wall shear and fluid acceleration,
can be written as
=-2A,RzS r=R
I
where
s, =
~a*[ - g: sin o,t + LY* cos wpt] ]Ef + a’1 + Pj3*[ - Zj:sin( uhf + #) + /3’ cos( 0,c + #)] K4 +
4. RESULTS
B”1
AND DISCUSSION
Velocity profiles for different values of the parameters (Table 1) have been computed from equation (6). It is observed that the effects of body acceleration depend on tube diameter. In small diameter tubes (coronary artery) body acceleration influences the velocity near the axis more than near the wall, magnitude increase is observed which is maximum near the axis. As the tube diameter increases, the influence of body acceleration shifts from the axis to the vicinity of the wall (f(tw,) = 0). As a matter of fact, for wide tubes there is almost no influence near the axis (aorta, Fig. 1) but near the wall, velocities rise sharply and the maximum of the profile shifts towards the wall as the body acceleration amplitude increases. A comparison of the present analysis with Sud and Sekhon [8] analysis shows that the difference between the two analyses increases as the tube diameter decreases. For some reason, they did not observe this shift of maximum towards the wall. Computation of velocity from their analysis also confirms similar change in velocity profile. For coronary artery, axial velocity variation with f for various parameters is shown in Fig. 2. The results of present analysis are compared with the corresponding observations of Sud and Sekhon [8] analysis. It is observed that velocity for their analysis is higher than present analysis at i = 0. As the cycle progress, the velocity becomes less than the present analysis value and then retain its starting behaviour (in the final part of the cycle). An increment in body acceleration frequency (fb) leads to a decrease in velocity. Because of this (increment in fb) velocity never becomes negative for any value of f A shows a comparison of velocity profiles, for flows with and without body acceleration, significant difference (Fig. 1). For wide tubes, like aorta, velocity profiles have fluctuations which are shown in Fig. 1. Table 1. Steady flow rate and pressure arteries [8]
gradient
for different
Blood vessels
Radius (em)
Flow rate Q. (cm3 s-‘)
Pressure gradi$nt A, (dyne em )
Aorta Femoral Carotid Coronary Arteriole
1.0 0.5 0.4 0.15 0.008
71.67 19.63 12.57 3.41 0.00008
7.30 32.00 50.00 698.65 2000.00
P. CHATURANI
116
and V. PALANISAMY
#, q0,f,,=fb=1.2 -----
-.-
00-o A
“0’9
o.
Aorta
,
= 0.2A,
=0
... a0 = g I
I
A,
-a,=0
Femoral
coronary
.
00 = a I
Fig. 1. Variation of velocity profiles for different arteries.
In Fig. 3, variation of fluid acceleration with tube radius and i has been shown for different values of parameters. It is of interest to note that fluid acceleration is identically zero in Sud and Sekhon [8] analysis for all narrow tubes (coronary, arteriole, etc.), whereas in the present analysis, it is not so. The non-dimensionalization and approximation used in their analysis is such that when fP =fb and f = 0 or integer multiple of n, fluid acceleration is zero. The factor body acceleration frequency plays an important role in fluid acceleration. It increases the amplitude of fluid acceleration as fb increases from 1.2 to 2.4. It is observed that the amplitude of fluid acceleration for flows with body acceleration is several times more than that of without 0
fp=
fb=
1.2,
A,=
0 2A,,
( Sud
1
-
fb=1.2,
----~----
fb= 2 4
fp= 1 2,
,
fp=
? 2
A,= , A,=
a,,= 8
fb=
12
,f,=2.4,A,=02Ao,
----
fb=
12
,f,=
g
.
f,=
12
,fp=1.2,A,=0.2A,,,
, a0 = g
x
g C6 I
Sekhon
0.28,, 0 2A.
o.
q
)
.#J = 45
1.2
o,=g oo=g
,Ar=05A0,
a,=0
,f,=12,fp=12,A,=02Aora0=g
.o
0.6
0.2
r E
O 1
.o
0.6 06 0.4 02 f-l -736
-04
0
04
06
1.2
1.6
0
04 u x
0.6
lo-“(cm/s
12
0
04
06
12
1
Fig. 2. Variation of velocity profiles for coronary artery with various parameters.
1 6
20
2.4
Pulsatile flow of blood with periodic body acceleration
117
Coronary 0
f,=
1.2
,ft,=
1.2,A,= (Sud
0.2Ao,
oo=g
B Sokhon
C.63)
---------
fp=1.2,fp=l.2,A~=0.2Ao, f,= 1.2 ,f,=2.4,A,=0.2Ao,
ao=g oo=g
----
f,=1.2,fb=l.2,A,=0.SAo,
ao=g
-... .
fp’
2.4,
f,=
fbZ 1.2,A,‘0,2A0
1.2,f,=l.2,A,=0.2Ao,
,
f,,=fb=1.2,A,=0.2A0 ao=g
ao’g o,=O
Carotid
A
Femoral
0
Aorta
A
1 .o 0.0 0.6 0.4
0.6
0.2 0 -12
-8
-4
0
4
8
12
-6
-4
0
Fx IO-’
4
(cm/s2
6
0
2
4
6
S
10
12
1
Fig. 3. Variation of fluid acceleration profiles for different arteries.
body acceleration. Further, in wider arteries, body acceleration influences the fluid acceleration near the vicinity of the wall. As the tube diameter increases, maximum value of fluid acceleration shifts towards the wall. Variation of wall shear stress with various parameters has been studied and shown in Figs 4 and 5. For a fixed steady state pressure gradient Ao, the maximum value of wall shear increases as the tube diameter decreases. However, for a constant average flow rate Q,, wall shear maximum decreases as the tube diameter decreases. For a constant A0 (constant Q,), as the tube diameter decreases (increases), the duration of r, in the positive region in a cycle increases. It may be noticed that situations like A0 and Q, constant are useful for theoretical studies, in cardiovascular system, flow rate, pressure gradient and tube diameter change simultaneously. Also it will be of interest to compare the values of r,, obtained from the present analysis with those of Sud and Sekhon [8]. The wall shear for coronary, carotid, femoral and aorta have been computed from the two analyses and shown in Fig. 5. Though the curves of the two analyses are quire close to each other, showing qualitatively good argreement between the two analyses, the quantitative difference between the two analyses is quite significant, in particular for coronary artery, for example in the first part of the cycle, the difference between the two analyses varies from 0 to 70%, in the later part of the cycle, it varies O-50%. A casual look at graph does not indicate this. The most interesting result is the variation of wall shear amplitude with tube diameter in cardiovascular system under prevailing physiological conditions (Table 2). As the tube diameter decreases, from aorta to coronary, amplitude of wall shear increases many fold from 0.09 to 8.48 (present analysis) in absence of body acceleration, for cases with body acceleration, these variations are considerably lower. Since the variations are of order of lOOtl%, it was decided to study the similar variation for Sud and Sekhon [8] analysis. It has been found that the variation in their analysis is also of the same order (Table 2). However, a further decrease in the tube diameter leads to a decrease in the amplitude of r,. At this stage, it may be emphasized that the amplitude of t, is not a function of tube diameter alone, prevailing values of pressure gradient, flow rate etc., are also contributing. The body acceleration frequency change leads to some interesting results for wide tubes, the amplitude
1.2
fp=1.2,
f,=
i
I
Al=0.2A0
ao=q,A,=0.5Ao
, aO=O,
I
for coronary
NE 9
0
60
I
120
I
I
240
with different
160
parameters
300
360
artery.
-1.2
-1
-1
-1.
-0.8
-0.E
c
-0.6
3-o I-
‘0 t x
z-0.2
F >
-0.E
-0.4
-0.2
a
7
A,=
variation
1 120
.+ = 180
f&=90
+=o #$z45
1.2,
I
f,=
60
A
.
---
f,=
(b) Wall shear
.o
4
0
0.2
1.2
1.2,
Q, A,=0.2Ao
02
f,=
.
f,*2.4,ao=g,Al-0.2Ao
, ao=
#=O
0.4
f,=
0
fb=1.2,
1.2
fp=
f,=
0.64,
fp=1.2,ao=q,A1=0.2Ao
fb=24,
)
f,=1.2,f,=12,00=g,A,=0.2Ao
Fig. 4. (a) Watt shear variation
2
x
‘0
lu
z
e
NE 9
x
.. . . . .
---
-
0.4
(0)
I
T
240
with phase
160
1
0.2Ao,ao=
angle
P
1
1 360
for coronary
300
artery.
Pulsatile flow of blood with periodic
119
body acceleration
na. “._
+=O,fp=
fb=l.2,
A,=0.2A,,oo=g Sud Et Sekhon
0.6
.
Coronary
A
Aorta
Present -.-
analysis Aorta Femoral
---
0.2
C63
Carotid .
Coronary
x
Arteriole
0 “E \” e A -0.2 z! N b vx -0.4 3 -0.6
-0
6
- 1.0
-1.2
Fig. 5. Variation
of wall shear for different arteries.
decreases with the increase of frequency (Table 2), the changes are of the order of 50%. However, for narrow tubes, initially it increase, reaches its maximum and then starts decreasing but the decrease is slow. The phase lag between the pressure gradient and wall shear decreases as the tube diameter decreases (Table 2). The expression for instantaneous volume flow rate is given by equation (8) which consists of three parts, the first and second parts are due to steady and oscillatory pressure gradients and the third part is due to body acceleration; it is a linear function of the pressure gradient and body acceleration. As the amplitude of body acceleration increases, the flow rate amplitude increases. Under the prevailing physiological conditions, the amplitude of flow rate in different arteries changes with the amplitude of the body acceleration in the following manner Amplitude of flow rate in aorta> amplitude amplitude of flow rate in arterioles.
of flow rate in coronary
artery >
The amplitude of flow rate decreases, as the body acceleration frequency increases. However, in very narrow tubes like arteriole, amplitude initially increases with fb, reaches its maximum Table2. variationof amplitude
and phase lag of wall shear (dyne cm-*) for different arteries for ,p = 0, A,=0.2A,,f,=1.2 Amplitude
Blood vessels
&=0.6 aa=g
&=1.2 ao=g
&=2.4 a,=g
Aorta Femoral Coronary Arteriole Carotid
81.52 82.34 71.07 4.27 82.64
67.19 67.76 70.10 5.65 67.60
46.21 47.90 51.63 4.88 48.10
Phase lag a,=0
0.09 0.42 8.48 1.58 0.65
Sud and Sekhon [8] a,=0
fb = 0.6 %=K
fb = 1.2 aO=g
fb = 2.4 ao=g
0.10 0.43 8.41 1.60 0.67
360 360 360 260 360
230 220 210 180 220
290 290 290 100 7W
P. CHATURANI and V. PALANISAMY
120
Table
3. Variation
of amplitude
and phase lag of flow rate (cm’s_‘) Q =O, &= 1.2, a,=& A, =O.?A,
for different
Amplitude Blood vessels Aorta Femoral Carotid Coronary Arteriole Table 4. Comparison
fb = 1.2
fb = 2.4
408.36 101.91 65.33 4.68 0.43 x 1o-4
369.21 84.03 51.31 4.61 0.58 x 10 -.I
189.69 44.61 21.12 3.13 0.5 x 1om4
fh=0.6
fb=1.2
160 140 130 40 0
Max
a,, = 0 Present Sud and Sekhon
[8]
72.20 72.31
rh=g Present Sud and Sekhon
[8]
440.88 522.29
Femoral Min
Max
71.15 71.03 -297.54 -378.95
20.15 20.40 103.66 144.80
Coronary Min
19.11 18.86 -64.40 -105.54
Max
fb=2.4
80 80 70 40 0
of instantaneous volume flow rate (cm3 s-‘) maximum and minimum Sod and Sekhon [8] for $I = 0, fp = fb = 1.2, A, = 0.2A,,
Analysis
for
Phase lag
fb=0.6
Aorta
arteries
40 40 40 30 0 for different
arteries
with
Arteriole Min
Max
Min
4.02 5.71
2.92 1.23
O.O#l 0.0204
O.WOO6 -0.02023
8.08 22.25
-1.14 - 15.31
0.00014 0.07269
o.oooo3 -0.07253
and then decreases (Table 3). The maximum and minimum of flow rate for flows with and without body acceleration are calculated and compared with corresponding values of Sud and Sekhon [8] analysis (Table 4). For flow with body acceleration, the obtained extreme values are lower than those predicted by Sud and Sekhon [8] analysis. Difference between the values obtained from the two analyses is significant and it increases as the tube diameter decreases (19-3000%). Further, Table 4 shows that difference between the extreme values of flow rate for flows, with and without body acceleration, in wide tubes is more differentiable than those in narrow tubes. As the frequency of the body acceleration increases, phase lag between the flow rate and pressure gradient decreases (Table 3). Further, it shows that for small diameter tubes, phase lag varies sharply as tube diameter increases, for wide tubes it varies asymptotically. 5.
CONCLUSIONS
The present analysis has a distinct advantage over the analysis of Sud and Sekhon [8]. Here, we have to use only one expression for computing the values of each flow variables u, F, r,, Q for required values of Re, and Re,, whereas in Sud and Sekhon [8] analysis, depending on the values of Re, and Re,, one has to choose the expresson. Qualitatively both analyses tend to predict similar trends, quantitatively differences varies from 19 to 3000% depending on the flow region. The difference between the two analyses decreases as the tube diameter increases. Using physiological data, the following observations have been made. The position of maximum of axial velocity and fluid acceleration is tube diameter dependent. It shifts from axis to the vicinity of the wall as tube diameter increases from arteriole to aorta. Variation of amplitude of wall shear for flow with body acceleration is slower than tht of flow with no body acceleration. For flow variables like wall shear and instantaneous volume flow rate, body acceleration effect is more in large arteries (aorta) than smaller ones (arteriole). The phase lag changes sharply in smaller arteries and asymptotically in larger arteries. In the present analysis, it has been assumed that the vessel wall is rigid, flow is fully developed and blood is a Newtonian fluid. In reality, blood vessells (wide arteries) are elastic, blood behaves like non-Newtonian fluid (narrow arteries) and the flow is developing rather than developed because of blood vessels frequent bifurcations. Here we have used same body acceleration for all blood vessels, in reality, an appropriate component of body acceleration should be used, which depends on the angle between the body acceleration and the flow direction. The present analysis could be improved with the incorporation of the above mentioned factors.
Pulsatile flow of blood with periodic body acceleration Acknowfedgemenr-This
121
work was supported by CSIR Grant No. 25(20)/EMR-II.
REFERENCES [l] E. P. HIATT, J. P. MEECHAN and GALAMBOS, Publication 901, NAS-NRC, Washington, D.C. (1969). [2] R. R. BURTON, S. D. LEVERETT J r and E. D. MICHAELSOW, Aerospace Med. 46, 1251 (1974). (31 A. C. ARNTZENIUS, J. D. LAIRD, A. NOORDERGRAFF, P. D. VERDOUW and P. H. HUISMAN, Bibl. Cardiol. 29, 1 (1972). [4] P. D. VERDOUW, A. NOORDERGRAFF, A. C. ARNTZENIUS and P. H. HUISMAN, Bibl. Cardiol. 31,57 (1973). [5] L. E. HOOKS, R. M. NEREM and T. J. BENSON, Inr. J. Engng Sci. 10,989 (1972). [6] S. PARVATHAMMA and R. DEVANATHAN, Bull. Math. Biol. 45, 721 (1983). [7] V. K. SUD and G. S. SEKHON, Bull. Math. Biol. 46, 937 (1984). [8] V. K. SUD, and G. S. SEKHON, BUN. Math. Biol. 47, 35 (1985). [9] J. C. MISRA and B. K. SAHU, Comput. Math. Appl. 16, 993 (1988). (Received and accepted 10 July 1990)
PULSATILE
FLOW OF BLOOD WITH PERIODIC ACCELERATION P. CHATURANI
Department
0020-7225/91 $3.00+ 0.00 Copyright @ 1991Pergamon Press plc
of Mathematics,
BODY
and V. PALANISAMY
Indian Institute of Technology, Powai, Bombay-400 076, India
Abstract-Pulsatile flow of blood through a rigid tube has been studied under the influence of body acceleration. With the help of finite Hankel and Laplace transforms, analytic expressions for axial velocity, fluid acceleration, wall shear and instantaneous volume flow rate have been obtained. It is of interest to note that these solutions can be used for all the feasible values of pulsatile and body acceleration Reynolds numbers Re, and Re,. This is in contrast with the existing results where different approximate solutions have to be used for different ranges of Re, and Re,. Using physiological data, the following qualitative and quantitative results have been obtained. The amplitude of the instantaneous volume flow rate, for flows with body acceleration, decreases shaprly as the tube radius decreases (from aorta to arteriole). This variation of amplitude is very slow for flows with no body acceleration. Another interesting result is the maximum of the axial velocity and fluid acceleration shifts from the tube axis to the vicinity of the tube wall as the tube diameter increases. The variation of the amplitude of wall shear with tube diameter (aorta to coronary) is less for flows with body acceleration than that of flows with no body acceleration. The phase lag between pressure gradient and flow rate changes sharply with tube diameter in narrow tubes, it varies asymptotically in wide tubes. The obtained results are qualitatively in good agreement with existing theoretical observations. Quantitatively, they differ from the other theoretical results (19 to 3000%). The difference in the results of the analyses decreases as the tube diameter increases (arteriole to aorta).
1. INTRODUCTION The human body may be subjected to body acceleration in many situations e.g. helicopter crew members, astronauts, jet pilots and crew while taking off and landing, jackhammer operators while working, athletes and sportsmen for their sudden movements. Prolonged exposures to body acceleration leads to many health problems, e.g. headache, abdominal pain and loss of vision etc. [l, 21. It is possible that dangerous combinations of amplitude and frequency of body acceleration and blood flows may be responsible for such health problems. It is, therefore, desirable to set a standard for short and long term exposures of human beings to such an acceleration. It is also possible that a proper understanding of interactions of body acceleration with blood flow may lead to a therapeutic use of controlled body acceleration [3, 41, to the development of new diagnostic tools and to better design of protective pads. The present investigation is a step in this direction. Due to physiological importance of body acceleration, many mathematical models have been proposed for blood flow with body acceleration [5-91. Sud and Sekhon [8] have considered the pulsatile flow of blood through a rigid circular tube with body acceleration by assuming blood as a Newtonian fluid. They have obtained solutions for axial velocity, fluid acceleration, wall shear etc., in terms of Bessel’s functions with complex arguments. To compute flow variables from these expressions is very difficult, if not impossible, for general values of Re, and Reb, even for professional mathematicians, leave aside non-professional mathematicians, in particular medical scientists. In their analysis, two limiting cases have been considered and their approximate solutions have been obtained. There are two more limiting cases for which approximate solutions can be found. Besides having four approximations for various limiting values of Re, and Re,, intermediate values of Re, and Re, will pose serious problems for computation. In this investigation, solutions have been obtained which are free from the above mentioned difficulty of computing.
2.
FORMULATION
OF
THE
PROBLEM
Let us consider the pulsatile flow of blood through a rigid cylindrical tube with body acceleration. Blood is assumed to be Newtonian and incompressible fluid. Flow as axially 113 ES 29:1-H
114
P. CHATURANIand V. PALANISAMY
symmetric, pulsatile and fully developed. given by
The pressure gradient and body acceleration
-$4D+A,cosw,r
(g) are (1)
g = a0 cos(oi$ + $)
(2)
Where wP = 2nf, and fP is pulse frequency, A0 and Al are pressure gradient amplitude of oscillatory part, a, is the amplitude of body acceleration, o,, = acceleration frequency, 9 is phase angle of g with respect to the heart action and t is time. Under the above mentioned assumptions, the equation of motion for coordinates (r, 8, z) can be written as [8]
of steady flow and 2~rfb and fb is body (pressure gradient) flow in cylindrical
dU
~~=pa,cos(w,r+~)+A,+A,cosw,r+~~
where u is velocity in axial direction, p and p are the density and viscosity of blood. The initial and boundary conditions are [8] u(r
0)
=
2
(ii)
(R2- r2)(Ao+Ad 4p
at r = R (Tube radius)
u(r, t) = 0
u(r, t) is finite at r = 0.
(iii)
(4)
3. ANALYSIS
The consecutive applications of finite Hankel and Laplace transforms on partial differential equation (3) and initial and boundary conditions (4) leads to an algebraic equation of which solution can be easily found. The inversion of which gives the final solution as A,[vk? cos o,f + wp sin ~,t] [v2k; + co’,] + pa,[vk:
cos(t&t + #) + t%bsin(e.&t + @)] [v*k; + co;]
-e
A A_
-vkfr
[
+ pu,[vk:
(Ao+A,)+
AIvk;
vk:
[v2k; + w;]
vk;2
cos c#J+ o,, sin f$] [v2k: + o;]
11
(5)
where v = ,u/p, k, = ~i/R and 5i are zeros of Jo. The expression of u consists of four terms. The first and second terms correspond to steady and oscillatory parts of pressure gradient, the third term represents body acceleration and the last is transient term. As t * 00, the transient term approaches to zero and we get the desired solution for pulsatile pressure gradient and body acceleration as
(6) where s = _l_+ c[ET cos w,t + a’sin ~,t]
m41 + p[‘$ COS(Wbt + @)+ 0’ sin(e&t ]f4 + P”1
E?
w R2 (u2=A=Re V
P’
154 +
/j2=$=Rebr
E =
+ $)]
A,IAo,
p=pa, Ao .
Pulsatile flow of blood with periodic body acceleration
The instantaneous
115
volume flow rate Q defined as Q = 2nlR ru dr
(7)
gives
Q=
(8)
The physiological important quantities like, wall shear and fluid acceleration,
can be written as
=-2A,RzS r=R
I
where
s, =
~a*[ - g: sin o,t + LY* cos wpt] ]Ef + a’1 + Pj3*[ - Zj:sin( uhf + #) + /3’ cos( 0,c + #)] K4 +
4. RESULTS
B”1
AND DISCUSSION
Velocity profiles for different values of the parameters (Table 1) have been computed from equation (6). It is observed that the effects of body acceleration depend on tube diameter. In small diameter tubes (coronary artery) body acceleration influences the velocity near the axis more than near the wall, magnitude increase is observed which is maximum near the axis. As the tube diameter increases, the influence of body acceleration shifts from the axis to the vicinity of the wall (f(tw,) = 0). As a matter of fact, for wide tubes there is almost no influence near the axis (aorta, Fig. 1) but near the wall, velocities rise sharply and the maximum of the profile shifts towards the wall as the body acceleration amplitude increases. A comparison of the present analysis with Sud and Sekhon [8] analysis shows that the difference between the two analyses increases as the tube diameter decreases. For some reason, they did not observe this shift of maximum towards the wall. Computation of velocity from their analysis also confirms similar change in velocity profile. For coronary artery, axial velocity variation with f for various parameters is shown in Fig. 2. The results of present analysis are compared with the corresponding observations of Sud and Sekhon [8] analysis. It is observed that velocity for their analysis is higher than present analysis at i = 0. As the cycle progress, the velocity becomes less than the present analysis value and then retain its starting behaviour (in the final part of the cycle). An increment in body acceleration frequency (fb) leads to a decrease in velocity. Because of this (increment in fb) velocity never becomes negative for any value of f A shows a comparison of velocity profiles, for flows with and without body acceleration, significant difference (Fig. 1). For wide tubes, like aorta, velocity profiles have fluctuations which are shown in Fig. 1. Table 1. Steady flow rate and pressure arteries [8]
gradient
for different
Blood vessels
Radius (em)
Flow rate Q. (cm3 s-‘)
Pressure gradi$nt A, (dyne em )
Aorta Femoral Carotid Coronary Arteriole
1.0 0.5 0.4 0.15 0.008
71.67 19.63 12.57 3.41 0.00008
7.30 32.00 50.00 698.65 2000.00
P. CHATURANI
116
and V. PALANISAMY
#, q0,f,,=fb=1.2 -----
-.-
00-o A
“0’9
o.
Aorta
,
= 0.2A,
=0
... a0 = g I
I
A,
-a,=0
Femoral
coronary
.
00 = a I
Fig. 1. Variation of velocity profiles for different arteries.
In Fig. 3, variation of fluid acceleration with tube radius and i has been shown for different values of parameters. It is of interest to note that fluid acceleration is identically zero in Sud and Sekhon [8] analysis for all narrow tubes (coronary, arteriole, etc.), whereas in the present analysis, it is not so. The non-dimensionalization and approximation used in their analysis is such that when fP =fb and f = 0 or integer multiple of n, fluid acceleration is zero. The factor body acceleration frequency plays an important role in fluid acceleration. It increases the amplitude of fluid acceleration as fb increases from 1.2 to 2.4. It is observed that the amplitude of fluid acceleration for flows with body acceleration is several times more than that of without 0
fp=
fb=
1.2,
A,=
0 2A,,
( Sud
1
-
fb=1.2,
----~----
fb= 2 4
fp= 1 2,
,
fp=
? 2
A,= , A,=
a,,= 8
fb=
12
,f,=2.4,A,=02Ao,
----
fb=
12
,f,=
g
.
f,=
12
,fp=1.2,A,=0.2A,,,
, a0 = g
x
g C6 I
Sekhon
0.28,, 0 2A.
o.
q
)
.#J = 45
1.2
o,=g oo=g
,Ar=05A0,
a,=0
,f,=12,fp=12,A,=02Aora0=g
.o
0.6
0.2
r E
O 1
.o
0.6 06 0.4 02 f-l -736
-04
0
04
06
1.2
1.6
0
04 u x
0.6
lo-“(cm/s
12
0
04
06
12
1
Fig. 2. Variation of velocity profiles for coronary artery with various parameters.
1 6
20
2.4
Pulsatile flow of blood with periodic body acceleration
117
Coronary 0
f,=
1.2
,ft,=
1.2,A,= (Sud
0.2Ao,
oo=g
B Sokhon
C.63)
---------
fp=1.2,fp=l.2,A~=0.2Ao, f,= 1.2 ,f,=2.4,A,=0.2Ao,
ao=g oo=g
----
f,=1.2,fb=l.2,A,=0.SAo,
ao=g
-... .
fp’
2.4,
f,=
fbZ 1.2,A,‘0,2A0
1.2,f,=l.2,A,=0.2Ao,
,
f,,=fb=1.2,A,=0.2A0 ao=g
ao’g o,=O
Carotid
A
Femoral
0
Aorta
A
1 .o 0.0 0.6 0.4
0.6
0.2 0 -12
-8
-4
0
4
8
12
-6
-4
0
Fx IO-’
4
(cm/s2
6
0
2
4
6
S
10
12
1
Fig. 3. Variation of fluid acceleration profiles for different arteries.
body acceleration. Further, in wider arteries, body acceleration influences the fluid acceleration near the vicinity of the wall. As the tube diameter increases, maximum value of fluid acceleration shifts towards the wall. Variation of wall shear stress with various parameters has been studied and shown in Figs 4 and 5. For a fixed steady state pressure gradient Ao, the maximum value of wall shear increases as the tube diameter decreases. However, for a constant average flow rate Q,, wall shear maximum decreases as the tube diameter decreases. For a constant A0 (constant Q,), as the tube diameter decreases (increases), the duration of r, in the positive region in a cycle increases. It may be noticed that situations like A0 and Q, constant are useful for theoretical studies, in cardiovascular system, flow rate, pressure gradient and tube diameter change simultaneously. Also it will be of interest to compare the values of r,, obtained from the present analysis with those of Sud and Sekhon [8]. The wall shear for coronary, carotid, femoral and aorta have been computed from the two analyses and shown in Fig. 5. Though the curves of the two analyses are quire close to each other, showing qualitatively good argreement between the two analyses, the quantitative difference between the two analyses is quite significant, in particular for coronary artery, for example in the first part of the cycle, the difference between the two analyses varies from 0 to 70%, in the later part of the cycle, it varies O-50%. A casual look at graph does not indicate this. The most interesting result is the variation of wall shear amplitude with tube diameter in cardiovascular system under prevailing physiological conditions (Table 2). As the tube diameter decreases, from aorta to coronary, amplitude of wall shear increases many fold from 0.09 to 8.48 (present analysis) in absence of body acceleration, for cases with body acceleration, these variations are considerably lower. Since the variations are of order of lOOtl%, it was decided to study the similar variation for Sud and Sekhon [8] analysis. It has been found that the variation in their analysis is also of the same order (Table 2). However, a further decrease in the tube diameter leads to a decrease in the amplitude of r,. At this stage, it may be emphasized that the amplitude of t, is not a function of tube diameter alone, prevailing values of pressure gradient, flow rate etc., are also contributing. The body acceleration frequency change leads to some interesting results for wide tubes, the amplitude
1.2
fp=1.2,
f,=
i
I
Al=0.2A0
ao=q,A,=0.5Ao
, aO=O,
I
for coronary
NE 9
0
60
I
120
I
I
240
with different
160
parameters
300
360
artery.
-1.2
-1
-1
-1.
-0.8
-0.E
c
-0.6
3-o I-
‘0 t x
z-0.2
F >
-0.E
-0.4
-0.2
a
7
A,=
variation
1 120
.+ = 180
f&=90
+=o #$z45
1.2,
I
f,=
60
A
.
---
f,=
(b) Wall shear
.o
4
0
0.2
1.2
1.2,
Q, A,=0.2Ao
02
f,=
.
f,*2.4,ao=g,Al-0.2Ao
, ao=
#=O
0.4
f,=
0
fb=1.2,
1.2
fp=
f,=
0.64,
fp=1.2,ao=q,A1=0.2Ao
fb=24,
)
f,=1.2,f,=12,00=g,A,=0.2Ao
Fig. 4. (a) Watt shear variation
2
x
‘0
lu
z
e
NE 9
x
.. . . . .
---
-
0.4
(0)
I
T
240
with phase
160
1
0.2Ao,ao=
angle
P
1
1 360
for coronary
300
artery.
Pulsatile flow of blood with periodic
119
body acceleration
na. “._
+=O,fp=
fb=l.2,
A,=0.2A,,oo=g Sud Et Sekhon
0.6
.
Coronary
A
Aorta
Present -.-
analysis Aorta Femoral
---
0.2
C63
Carotid .
Coronary
x
Arteriole
0 “E \” e A -0.2 z! N b vx -0.4 3 -0.6
-0
6
- 1.0
-1.2
Fig. 5. Variation
of wall shear for different arteries.
decreases with the increase of frequency (Table 2), the changes are of the order of 50%. However, for narrow tubes, initially it increase, reaches its maximum and then starts decreasing but the decrease is slow. The phase lag between the pressure gradient and wall shear decreases as the tube diameter decreases (Table 2). The expression for instantaneous volume flow rate is given by equation (8) which consists of three parts, the first and second parts are due to steady and oscillatory pressure gradients and the third part is due to body acceleration; it is a linear function of the pressure gradient and body acceleration. As the amplitude of body acceleration increases, the flow rate amplitude increases. Under the prevailing physiological conditions, the amplitude of flow rate in different arteries changes with the amplitude of the body acceleration in the following manner Amplitude of flow rate in aorta> amplitude amplitude of flow rate in arterioles.
of flow rate in coronary
artery >
The amplitude of flow rate decreases, as the body acceleration frequency increases. However, in very narrow tubes like arteriole, amplitude initially increases with fb, reaches its maximum Table2. variationof amplitude
and phase lag of wall shear (dyne cm-*) for different arteries for ,p = 0, A,=0.2A,,f,=1.2 Amplitude
Blood vessels
&=0.6 aa=g
&=1.2 ao=g
&=2.4 a,=g
Aorta Femoral Coronary Arteriole Carotid
81.52 82.34 71.07 4.27 82.64
67.19 67.76 70.10 5.65 67.60
46.21 47.90 51.63 4.88 48.10
Phase lag a,=0
0.09 0.42 8.48 1.58 0.65
Sud and Sekhon [8] a,=0
fb = 0.6 %=K
fb = 1.2 aO=g
fb = 2.4 ao=g
0.10 0.43 8.41 1.60 0.67
360 360 360 260 360
230 220 210 180 220
290 290 290 100 7W
P. CHATURANI and V. PALANISAMY
120
Table
3. Variation
of amplitude
and phase lag of flow rate (cm’s_‘) Q =O, &= 1.2, a,=& A, =O.?A,
for different
Amplitude Blood vessels Aorta Femoral Carotid Coronary Arteriole Table 4. Comparison
fb = 1.2
fb = 2.4
408.36 101.91 65.33 4.68 0.43 x 1o-4
369.21 84.03 51.31 4.61 0.58 x 10 -.I
189.69 44.61 21.12 3.13 0.5 x 1om4
fh=0.6
fb=1.2
160 140 130 40 0
Max
a,, = 0 Present Sud and Sekhon
[8]
72.20 72.31
rh=g Present Sud and Sekhon
[8]
440.88 522.29
Femoral Min
Max
71.15 71.03 -297.54 -378.95
20.15 20.40 103.66 144.80
Coronary Min
19.11 18.86 -64.40 -105.54
Max
fb=2.4
80 80 70 40 0
of instantaneous volume flow rate (cm3 s-‘) maximum and minimum Sod and Sekhon [8] for $I = 0, fp = fb = 1.2, A, = 0.2A,,
Analysis
for
Phase lag
fb=0.6
Aorta
arteries
40 40 40 30 0 for different
arteries
with
Arteriole Min
Max
Min
4.02 5.71
2.92 1.23
O.O#l 0.0204
O.WOO6 -0.02023
8.08 22.25
-1.14 - 15.31
0.00014 0.07269
o.oooo3 -0.07253
and then decreases (Table 3). The maximum and minimum of flow rate for flows with and without body acceleration are calculated and compared with corresponding values of Sud and Sekhon [8] analysis (Table 4). For flow with body acceleration, the obtained extreme values are lower than those predicted by Sud and Sekhon [8] analysis. Difference between the values obtained from the two analyses is significant and it increases as the tube diameter decreases (19-3000%). Further, Table 4 shows that difference between the extreme values of flow rate for flows, with and without body acceleration, in wide tubes is more differentiable than those in narrow tubes. As the frequency of the body acceleration increases, phase lag between the flow rate and pressure gradient decreases (Table 3). Further, it shows that for small diameter tubes, phase lag varies sharply as tube diameter increases, for wide tubes it varies asymptotically. 5.
CONCLUSIONS
The present analysis has a distinct advantage over the analysis of Sud and Sekhon [8]. Here, we have to use only one expression for computing the values of each flow variables u, F, r,, Q for required values of Re, and Re,, whereas in Sud and Sekhon [8] analysis, depending on the values of Re, and Re,, one has to choose the expresson. Qualitatively both analyses tend to predict similar trends, quantitatively differences varies from 19 to 3000% depending on the flow region. The difference between the two analyses decreases as the tube diameter increases. Using physiological data, the following observations have been made. The position of maximum of axial velocity and fluid acceleration is tube diameter dependent. It shifts from axis to the vicinity of the wall as tube diameter increases from arteriole to aorta. Variation of amplitude of wall shear for flow with body acceleration is slower than tht of flow with no body acceleration. For flow variables like wall shear and instantaneous volume flow rate, body acceleration effect is more in large arteries (aorta) than smaller ones (arteriole). The phase lag changes sharply in smaller arteries and asymptotically in larger arteries. In the present analysis, it has been assumed that the vessel wall is rigid, flow is fully developed and blood is a Newtonian fluid. In reality, blood vessells (wide arteries) are elastic, blood behaves like non-Newtonian fluid (narrow arteries) and the flow is developing rather than developed because of blood vessels frequent bifurcations. Here we have used same body acceleration for all blood vessels, in reality, an appropriate component of body acceleration should be used, which depends on the angle between the body acceleration and the flow direction. The present analysis could be improved with the incorporation of the above mentioned factors.
Pulsatile flow of blood with periodic body acceleration Acknowfedgemenr-This
121
work was supported by CSIR Grant No. 25(20)/EMR-II.
REFERENCES [l] E. P. HIATT, J. P. MEECHAN and GALAMBOS, Publication 901, NAS-NRC, Washington, D.C. (1969). [2] R. R. BURTON, S. D. LEVERETT J r and E. D. MICHAELSOW, Aerospace Med. 46, 1251 (1974). (31 A. C. ARNTZENIUS, J. D. LAIRD, A. NOORDERGRAFF, P. D. VERDOUW and P. H. HUISMAN, Bibl. Cardiol. 29, 1 (1972). [4] P. D. VERDOUW, A. NOORDERGRAFF, A. C. ARNTZENIUS and P. H. HUISMAN, Bibl. Cardiol. 31,57 (1973). [5] L. E. HOOKS, R. M. NEREM and T. J. BENSON, Inr. J. Engng Sci. 10,989 (1972). [6] S. PARVATHAMMA and R. DEVANATHAN, Bull. Math. Biol. 45, 721 (1983). [7] V. K. SUD and G. S. SEKHON, Bull. Math. Biol. 46, 937 (1984). [8] V. K. SUD, and G. S. SEKHON, BUN. Math. Biol. 47, 35 (1985). [9] J. C. MISRA and B. K. SAHU, Comput. Math. Appl. 16, 993 (1988). (Received and accepted 10 July 1990)