Rheologiea Aela
Rheol Acta 23:548-555 (1984)
An analysis of the flow of blood through thick-walled vessels, considering the effect of tethering J. C. Misra and K. Roy Choudhury (sen.) Department of Mathematics, Indian Institute of Technology, Kharagpur (India)
Abstract: A theoretical analysis of the flow of blood through the blood vessel is presented. Both the radial and longitudinal tetherings are accounted for. By taking the orthotropicity of the wall tissues into consideration, blood is treated as a Newtonian, viscous and incompressible fluid. The applicability of the analysis is illustrated through the numerical computations of the derived analytical expressions by using experimentally determined values of the material parameters and the effect of tethering on the phase velocities of blood flow is thereby quantified. Key words." Hemodynamic flow, tethering effect, orthotropicity, phase velocity, transmission coefficient 1. Introduction
For the sake of proper diagnosis of cardiovascular diseases which are caused by circulatory disorders of the body, the mechanical behaviour of the circulatory system must be thoroughly known. That is why in recent years the study of the problem of h e m o d y n a m i c flow in arteries has become a subject of intensive research both to medical workers and to scientists of other disciplines. In an expository paper, various studies on the role of biological fluids in the formation of vascular diseases was critically reviewed by N e r e m and Cornhill [1]. The constitutive relations for soft tissues, in general, together with an extensive bibliography of previous investigations in this area was presented in an excellent manner by Fung [2]. The mechanical properties of blood vessels were described by Bergel [3]. An introduction to blood flood problems may be found in Anliker [4] and Kenner [5]. References to earlier studies made until the beginning of the last decade in the respective areas have also been cited in their articles. The papers of Bonis and Ribreau [6], Brunn [7], Haut et al. [8], Chato [9], Kenner and Wieczorek [10], Holenstein et al. [11], Hecht etal. [12] and Rudinger [13] are some notable recent theoretical as well as experimental studies of various aspects of the mechanics of bio-fluids in general, and blood flow as well as vascular deformation, in particular. The longitudinal tethering force exerted by the tissues surrounding the arterial wall seems to be incorporated in an analytical study in a systematic manner by Womersley [14]. The analysis is based on the assumption that the nonlinear convective acceleration terms of the Navier-Stokes equations do not affect the flow behaviour appreciably. Chang and 962
Atabek [15], as well as Atabek et al. [16] made extensive studies on the effect of inlet lengths on the flow behaviour of blood while the effects of initial stresses on the wave propagation through arteries were studied by several investigators, e.g. Atabek and Lew [17], Mirsky [18] and Anliker et al. [19]. The effect of initial stress together with that of surrounding tissues on the pulse wave propagation in arteries was considered by Atabek [20]. A nonlinear analysis of pulsatile blood flow in arteries was made by Ling et al. [21]. An investigation of the problem of steady laminar blood flow through uniformly tapering tubes of circular cross-section was conducted by Walawender et al. [22], the tapering angle being considered to be small. Fich and Welkowitz [23] considered a model of the aorta which was tapered in area, included elastic constants and was essentially reflectionless. It proved to be a good representation for the calculation of pressure and flow wave propagation in the aorta. Analog and digital data techniques were also developed by Welkowitz et al. [24] for calculating various parameters of the aorta. Excellent reviews on previous studies (experimental as well as theoretical) on hemodynamic flow through blood vessels may be found in Noordergraaf [25] and McDonald [26]. In a recent study, Rachev [27] considered an analytical model by taking into consideration the effect of muscle contraction on pulse wave propagation for canine middle descending thoracic aorta and also for humans of different age groups [28]. Many of the above mentioned analytical studies were based on the assumption of linearity of vascular tissues; in quite a few of these the wall material was treated as linearly elastic, while in some the material was considered to be linearly viscoelastic. In the analysis presented here for the p r o b l e m of wave propagation of blood, an attempt is made to incorporate both the radial and longitudinal tethering effects, by paying due attention to the d a m p i n g material behaviour of arterial wall tissues, as per previous
Misra and Choudhury, An analysis of the flow of blood through thick-walled vessels, considering the effect of tethering experimental observations (cf. Stacy [29], Wiederhielm [30] and Fung et al. [31]). Also in conformity to the experimental observation of Patel et al. [32] that the arterial walls have material symmetry with respect to the cylindrical configuration, orthotropicity of wall tissues has been taken into account here. It may be mentioned here that these mechanical properties of wall tissues were taken into account in our earlier studies as well (cf. Misra [33] and Misra et al. [34-371). By computing the derived analytical expressions and using the experimentally determined values [38] for the non-dimensional parameters, the effect of tethering on the phase velocities as well as the transmission coefficient of the propagating waves through arteries is quantified. The computed results are presented graphically.
Blood is considered here as an incompressible, viscous and Newtonian fluid and its flow as laminar. In a wave propagation problem such as the one considered in the present paper, since the velocity components of the fluid flow are small compared to the phase velocity, the convective acceleration terms in the NavierStokes equations will be disregarded. Thus the motion of the flow of blood will be characterized by the linearized equations [02Ul
0ul Ot
Owl 0~- --
1 0b/1
U1
02U1~
\ ~
T
II0
if'l= W1 '
r
z
=__
~:__
a fl0
bl0
/5= '
0{2_
a
ia(D C
P ~f bl2 '
~a
-
tUo
t=__
u0
a
- ifl (say)
(2)
represent dimensionless quantities. The solutions of the eqs. (1) in their dimensionless form are given by
0--7- -{- OZ2 ]
• (i/??) e
'
-I-fl2 )
,2,,
W l = - [ i f l a A l J o ( i f l f ) + ( i V • ic~2uOa V +~2 )/ ( / • Jo i
i°~2u°a ) ] i~2(?-P'] v +f12 ? e ~ ~/,
--ia2c° AlJo(iflr) e '~= t-V# ,
(3)
V (1)
together with the continuity equation OUl F-u l ' OWl
0r
~ j = U1
fi-
1 0p Of ~r ~-v/--~'-r2+ r 0r r 2 +-~5-z2) 1 Op 4_v(O2w1+ 1 Owl 02Wlt Of 0 ~
For the sake of convenience in the analysis that follows, let us make the equations dimensionless. Let u0 be a quantity having the dimension of velocity and let a and b be respectively the inner and outer radii of the blood vessel wall under consideration. Then
U1 = - i f l [ A i J l ( i f l r ) + A 2 J l ( i / i ~ 2 u ° a
2. Equations characterizing the flow of blood
549
r + ---~-z = 0 '
in which ul, Wl, are respectively the radial and longitudinal velocities of the blood and p is the hydrostatic pressure, Ofthe density, and v the kinematic viscosity. Let us consider the solutions of these equations in the form ul = U1 (r) e wl = WI (r) e
where co represents the angular frequency and c the complex propagation velocity.
A1 and A2 being arbitrary constants. The forces acting on the inner surface of the vessel wall are due to the pressure and friction of the blood. The radial and longitudinal components of these forces are given by cry i~=1=-/5+ 2v 001 [
uoa 07 ~=1
_[oe/l _ iflU1 ]
crf I~=1= uoa [ 0?
~=1 "
3. Motion of the vessel wall In the present study, a straight thick-walled circular cylindrical segment of blood vessel is considered. Let the segment be uniform and of a constant thickness. In conformity to the results of experimental investigations mentioned in section 1, orthotropicity and viscoelasticity of wall tissues are also considered, and the analysis is restricted to the linear theory. The study pertains to
550
Rheologica Acta, Vol. 23, No. 5 (1984)
a case in which the motion of the vessel wall is axisymmetric and the wall displacements are small, whilst the wavelengths of oscillations are very large in comparison to the radius of the wall. With the above assumptions for the vascular wall material the constitutive relations are taken in the form ~Trr = Cll err + C12 eoo q- ci3 ezz, (700 = C12 err -[- C22 e00 q- c23 ezz, Gzz = C13 err -I- C23 eo0 --k c33 ezz,
(4)
Grz = C44 erz ,
tained. To make the equations dimensionless let us introduce the following non-dimensional quantities. u W ~ = - -Z, ? = - -/", ~ = --, ~=--, a a a a = "" ~ij co.=c~jJ2 and ~/j= (7) ~u~ 0u02' Let the non-dimensional displacements for the harmonically travelling waves be assumed to be of the form
where a 0 denotes the components of the stress-tensor and e U those of the strain tensor for the wall deformation. The deformation being assumed to be axisymmetric, one can set
.
-- U2 (r) e '=' t - 7 , .
-
fl~
fiz
(s)
aro = azo = O . The moduli c~j= ~0-+ i gij are complex functions of frequency, the real parts ~0 denote the elastic moduli and the imaginary parts g~j the viscous moduli. As reported by Bulanowski [39], the viscous parameter gij is nearly 15 per cent of the elastic parameter ~ij. Thus any change in the viscous parameter will have a negligible effect on the mechanical behaviour of the blood vessel in comparison to that in the elastic parameter. This observation justifies taking the ratio of the viscous and elastic parts of the modulus as a constant throughout the analysis. For harmonic waves, the stresses and strains along with the displacement components may be expressed as
02 and fizzbeing functions of f only. Substituting the eqs. (7) and the relations (8) in the equations of motion (6), we obtain
f2 d2 G
d G + G (al F2 - m2) + bl ?2 dW2
_ d2 fiz2 dfiz2 . r - -d+ -2- ~ -?+ J F_ Wd2
C44fl 2
Cll
the components of the displacement and the stress are complex and functions of r; the superscript v refers to quantities for the vessel wall. If u and w denote the radial and tangential displacements, the strain-displacement relations are err =
8u/Sr,
eoo = u/r and e= = 8 w / S z .
(5)
The equations of motion for the axisymmetric deformation of the vessel wall may be expressed as Oarr
dU2 d7 = 0
T + e U2 +
(9)
with Q~4y2
where
= o,
Cl]
C1'3-'[-C44 ,
C13--C23 cl = - i f l - - ,
bl = -
d=
Cll
e =--i/~( ~44q- ~23] C44
\
] '
i f l ~
Cll
--ifl
f=-
C44
(C13"]-a44),
c3--~3/~2q- Ov2°~4 C44
C44
'
m2 = L'22
( l O)
The solutions of the eqs. (8) obtained by employing the Frobenius method are found to be of the form
1
8arz + r
(or
-
o00) =
Uz=(BI+B21nT) ~ a ~ l ) n Y 2 n + l + B 2 ~ 8azz 1 8 arz 8z ~-Tarz+-~-r =qwtt'
n=0
b2nr 2n+l
n=l
(6)
wheref, denotes 82f/St 2. On substitution of (4) and (5) in (6), a displacement formulation of the equations of motion may be ob-
n=0
n=0
fiz2 = Bill (7) + B2f2 (T) + B3fs (F) + B4f4 (r),
(11)
Misra and Choudhury, An analysis of the flow of blood through thick-walled vessels, considering the effect of tethering BI, B2, B3, B4 being arbitrary constants and
( el)
(2n + 1) 2 n=0
a ~ (s = 1, 1, _+m) are connected through the recurrence relations
m2a~1272,,
a2[s+l)2(s )
2n q--b7 bl ~(1) ~2n+2 ~2n r
al
m2q
+a0 a l ( s + l ) 2 s - l +
+f(s 2
-b,d f2 (?) = ~'~ b2n [ ( 2 n
-l+g =0, [(2n+s+4)2-m2]azn+4
(2n+s+3) 2 2n+s+l+
a~J2{(2n +
m 2) s + l +
?2n
+ 5) 2 - m 2]
al ~ b2n r2n+2 bl n=l ( 2n + cl + + In ~
551
+ a(2n+s+3) 2 1) 2 - m 2}
cl
2n+s+5+
{(2n+s+2)2-m 2}
+f 2n+s+3+
2n+ oo
-bid
2n+s+l+
2n+s+3+
2n +-~-i + 2 oo
f2n
- Z [(2n+l)2-m2]a~12 ( n=0 oo al Z /
cl]2 2n +-~71] bl
a~d2f2n+2 ---~,--\2
• (2n+s+3-~)]
+alf
a2n+2
2n+s+3+
a2~=0,
(12) [ aa2n ]
while b 2 . = [ as ]s=l'
_ ~
_2_n____+ (2 1) ~(1) =2~
72n+m-I
( 2 n + m + ~ l - - 5)
n=0 ,-,(m) ~-2n+m+ 1 y2n '_______
2n+m+l+
4. Boundary conditions The unknown constants A1, A2, B1, B2, B3 and B4 involved in the eqs. (3) and (1 1) can be determined by using the boundary conditions of the problem, which may be stated as follows: a~ (i) / d l = - ~
f4 (?) = ~ a~-nm)[(2n -
m) 2 - m 2]
n=0
at
on
?=1
i.e. the radial velocity of blood is equal to that of the arterial wall at the inner surface of the wall.
~2n-m-- ,
(ii) v~l=~-= (3/
oo a~-nm) 72n_m+ 1 al ~ . . . . . .
bl n=°(2n_ m + l + cl] \
b o = O , a n d b l , cl, d,f, a n d m
are given by the expressions in (50).
f3 (r) = ~ a~'~) [(2n + m) 2 - m 21 oo al S '
( n = 0 , 5 , 2 .... )
b,/
.
on
F=I
i.e. the axial velocities of blood and the wall are equal at the inner surface. (iii) ~f--a~r-V on
T=I
552
Rheologica Acta, Vol. 23, No. 5 (1984)
i.e, the continuity of the radial stress of the fluid and that of the wall at the inner surface. (iv) ~ £ = ~
on
~=1
i.e. the axial shearing stress of the fluid is the same as that of the wall at the inner surface. (v) The axial shearing stress on the outer boundary is equal to the longitudinal component of the force exerted by the surrounding vascular tissues. (vi) The radial stress on the outer boundary is equal to the radial component of the tethering force. The effect of the surrounding tissues in the longitudinal direction may be described through the use o f a model consisting of a spring, a dashpot and a lumped additional mass (cf. [38]). A schematic representation of the model is shown in figure 1. The radial tethering may also be illustrated in a similar fashion. Let kl, cl and k,., c~ denote respectively the spring and dashpot coefficients of the mechanical model for the longitudinal and radial tethering. If M~ be the additional mass, the tethering forces in the longitudinal and radial directions are respectively equal to
[ d2w+ cl -dw \ - ~ + kt w) ,
- IMa-jy
(M. d2bl2d+, G - -dbt ~+
k,u
).
(13)
Due to the lack of additional information regarding the parameters kr and c,, we take kr=kz
and
c~=ct
We shall further make use of the dimensionless quantities J~a = M a
ct
Cr = Ct = - - ,
oa '
/~ = / ~ =
OUo
kla
•
The above-mentioned boundary conditions ( i ) - ( v i ) yield
(-~1 =i°~2U2
on
?=1,
[/~71=
on
g = 1,
i 0~2 I/V2
0 -fi+
2 v Ol Uoa
] =C11"-~-1-~'12 d~: G -- ic13
~-'f2/?
on
O[--~r-r-i/?Ul =c44[ dr i/?~72 on dG
~ = 1,
r=l,
(15)
O2 r
= (~4_
i ~ 2 G + kO U= on T= b/a = ~ ( s a y ) ,
_ dP172
c44 --d-7-r - i /? O2 = ( ;fi~ ~ a - i o~2e , + k , ) W2 on ~ = a
where 0 = OJm.
(16)
From eqs. (3) and (11), the boundary conditions (15) and the non-dimensional quantities defined earlier, it is convenient to assume the dimension of u0 as u0 = v/a. Using the relations (3) and (11) in the boundary conditions, we get six equations involving six unknowns. Then from the condition for the existence of a nontrivial solution, a frequency equation is obtained in the form l au] = 0,
(17)
where aij ( i , j = 1, 2,..., 6) are the coefficients of a matrix, whose different elements are given i n the appendix, when according to one of our basic assumptions the radius of the vascular wall is taken to be small in comparison to the wave lengths of oscillation i.e.,
(14)
Qu 2
which implies that
2 +/?62 ~= e2, c~0
Jo (rio) ~ 1 and J1 (rio) ~/?0/2.
/l////11//////1.. In(act ire
v///////~//////
C, [ - - ~ - -
5. Numerical results and discussion
tissues From the eqs. (2) we have
Kt //.//7/A Fig. 1. A schematic representation of the longitudinal tethering effect
/? = co~c,
in which c is the complex wave propagation velocity. Writing c = c l + ic=, the complex quantity /? can be written as p = p~ + i & ,
Misra and Choudhury, An analysis of the flow of blood through thick-walled vessels, considering the effect of tethering
The values of the non-dimensional constraint parameters (14) were reported by Patel et al. [38]. Atabek [20] used the same data for his calculation. The respective values are
where fll and f12 are both real and are given by acoc I
ae)c2
d÷
= d +
Thus for a fixed value of e, the frequency equation iai/I = 0
is a transcendental equation involving two unknowns cl and c2. After calculating the values of cl and c2, the phase velocity
d+c Cl
~1=0,
10 6 ,
kl=0.
Also for the present calculations, the following values are taken for other non-dimensional quantities involved in the above analysis (cf. [18, 39]). 1 4 '
v,.z=0.3,
ff'i = Ei [ l ''~ i ~-i] where Ei = ~ia2 and /~i =/~ia2
QvY2
Qvv2 "
Moreover, let us choose a non-dimensional reference velocity V0defined through (6 P0 l//~0 V 20
-
1) '
so that the non-dimensional phase velocity can be introduced as
V2(6-1).
e
1"0
,',¢/
Ez Eo
] -
3 '
Vzo=O.1,
vo~=0.4,
III
1
c44=--fEo,
6=1.15,
0=0.96
and /~k =/~1 (1 + 0.15i) (VOr, Vzo, V,.z denote Poisson ratios). On the basis of the present analysis, the effects of the tethering forces were quantified by using the above mentioned data through numerical computations. The effect of tethering on the phase velocity has been displayed in figure 2, while the effect of tethering on the transmission coefficient is illustrated in figure 3. In the figures, the solid lines represent the results for the tethered vessel, While the dashed lines represent the corresponding results for a free vessel (when the effect of tethering is ignored). The numerical values of the non-dimensional phase velocities computed by Atabek [20], who considered the vessel wall to be thin-walled and elastic, are also presented in figure 2. A comparison of these results with those obtained here illustrates the effect of the damping material behaviour of the vessel wall and its thickness on the phase velocity of the wave propagation.
s-I
./
. S
/~-
TETHERED VESSEL . . . .
0"~
I II
02
~ta=0.15,
kl = 1.6 x
C2
can be computed. The Young's moduli Ej exhibit the same form as that of % so that
,>%°
cl = 550,
Eo
Cl
0"8
3~ra = 0.7,
Er
and the transmission coefficient ~-2n
553
F R E E VESSEL ;
OUR RESULT
:
RESULT
OF A T A B E K
{ 19&8)
I
q
I
[
I
L
A
2
3
~
5
6
7
8
Fig. 2. Effect of tethering on the non-dimensional phase velocity (Vff Vo)
554
Rheologica Acta, Vol. 23, No. 5 (1984)
Appendix
0.B
The calculated expressions for the various elements of the matrix [a•] given by eq. (17) are as follows: a l l = - - ~ f12-,
a12 = flOJl (~0)
a13 = - i~2 ~ a~1) , n=0
~ b2n,
a14 = - i ~ 2 ~'~
/0"
ERED VESSEL --
.
.
.
.
FREE
a15 = - i ~ 2
~ ai m),
n=l
n=0
a16= - i~2 ~ a~-.ml , a21 = - flo, a22= flo~oJo(%),
VESSEL
n=O
2
3
4
.5 oL
G
7
8
'-'-'--"
a23 = - i~2fl (1),
a24 = - i~2f2(1),
a26 = - i ~ 2 f 4 ( ~ ) ,
a31 = ~ [ ~ o2 - ~o2],
a25 = - i~2f3(1),
a3z = - 2 [floJ1(%) + ~oc%Jo(%)],
Fig. 3. Effect of tethering on the transmission coefficient
a33=--?u [ ~ (2n+l)atl)J +712 ~ aSl)n-ifl?13fl(1), n=0
From our results presented in figure 2 it appears that the phase velocity first increases with an increase in frequency and then attains nearly a constant value; of course, the frequency at which this constant value is attained for a tethered vessel is different from that in the case of a f r e e vessel. One also observes from this figure that the combined effect of tethering in the longitudinal and radial directions has a marked effect on the phase velocity (particularly for values o f the non-dimensional frequency less than 5). O f course, this effect is not very pronounced in the results of Atabek [20], who considered only longitudinal tethering. In table 1 a few of our computed results are presented, together with the corresponding values obtained by Bulanowski and Yeh [39], who considered the material of the vessel wall to be transversally isotropic and ignored the effect of radial tethering. Thus it appears that the effect of the degree of anisotropicity on the non-dimensional phase velocity of the waves of the first kind is not very pronounced for large values of ~, at least when the analysis is made under the purview of a linear theory. This observation is in conformity to the conclusion made by Atabek [20]. One may further note from figure 3 that the transmission of waves of the first kind is significantly affected by vascular tethering. This also agrees well with the observation of Atabek [20].
n=0
a34=--c11[n=0 ~ a~1)+ ~ b2n(2n+l)]-El2 ~ b2n n=l n=l + iflg13f2 (1), a 3 5 = - [ ~u n=0~'atnn)(2n+m)+c12n=0~a~m)--iflc13f3(1)] ' a36=-[c11 ~a[-nm'(2n--m)+c12 n=0
a41 = ~f13,
~,a~nm)--iflOl3f4(l)J, n=0
a42 = _ 0~2j1(0~0),
a43 = - 044f~(l) - i a45 = -- c44
(1) -- ifl
a~=-044
1)-i
,
a
a44 = -- c44f~(1),
a~r~) , a~-~m) , a51=0, a52=0,
a,3 = e. ~ (2n + 1) at~ 62" + 012 ~ ai~ 62~- iPO13fl (6) n=0
n=0
-- [Ma ~4- io~20r- kr] ~ a~l)n62"+1 , n=0
a ~ = ~v ln6 ~ at~ (2n +1) 62"+ e. n=0
+011
n=0
at~ 62" + ~12 ln6
b2.(2n+l)62"
n= 1
a~32"+
n=l
1
b2.62"
-- ifle13A (6) -- (J~a C(4 -- i~2er- L) oo
"[ln6n~=o a~62n+l+ n=l~b2n62n+l] ' Table 1. Values of non-dimensional phase velocities (V1/Vo)
a55=~'11 ~aSm)(2n+m)62n+m-1+O12 n=O
c~
Our results
Results of Bulanowski and Yeh [39]
~ aSm)62n+m-I n=O
-- iflO13f3 (6) -- (ff'la ~4_ i Ct2 er_ kr) ~ ai m) 62n+m , n=O
Tethered
Free
Tethered
Free
0.87 1.03
0.95 0.98
1.02 1.1
1.00 1.01
oO
6/56= 011 ~ 6/~-nm) (2 n - m) 62n-m-I + 012 Z a~nm) 62. . . . 1 n=0
5 I0
oO
n=0
-- iflc13f4 (6) -- (/~a ~4 -- i~20r-- ]('r) ~ a~nrn)~2n-m , n=0
Misra and Choudhury, An analysis of the flow of blood through thick-walled vessels, considering the effect of tethering a61 = 0,
a62= 0,
-- (J~fa ~ 4 -
a64= e44
iO;2el --
~ ) fl (03,
(03 - ifl log 5 }"~at~ 52n+1 n=0
_
( # o ~ 4 _ i~2e~ _ fi0f2 (03,
a6s = e44
(03 -
i/~ ~
at~') a 2n+'n
.=0
-- (-/~a 0~4- ic~2el
-
]Q)f3 (03,
a66 = c44 f~ (03 - ifl ~'~ a~n m) 62n-m n=0 -
(Ma c~4- i~2et - fi/)f4 (03,
d in which f ' (c0 means ~ x f ( X ) x=~ and all the summations are taken over n.
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