PROBLEMS VESSELS

OF

THE

DYNAMIC

BEHAVIOR

OF BLOOD

AS S H E L L S

A. S. Vol'mir

and

M. S . G e r s h t e i n

UDC 678:624.074.4.001

P r o b l e m s of the dynamic behavior of blood v e s s e l s as deformable shells with a viscous fluid flowing in t h e m a r e investigated. A multicoat elastic shell in which an active ( m u s cular) coat is distinguished is proposed as a model of the v e s s e l . P r o b l e m s that can be solved on the basis of the model a r e d i s c u s s e d .

P r o b l e m s of the d y n a m i c s of blood v e s s e l s r e g a r d e d as deformable shells have already been disc u s s e d repeatedly in the literature; reviews of s o m e investigations along this line a r e given in the authors' a r t i c l e [1] and in [2-4]. Here we will attempt to p r o p o s e a model of a multicoat shell which with some approximation can d e s c r i b e the behavior of real v e s s e l s and to outline ways of solving individual p r o b l e m s . A model of the r e c t i l i n e a r portion of a v e s s e l regarded as an elastic shell with a fluid flowing in it s e r v e s for studying the propagation of a pulse wave and the behavior of a r t e r i e s u n d e r v a r i o u s dynamic loads. In addition, the c o n s t r u c t i o n of such a model is n e c e s s a r y for m a t h e m a t i c a l and e l e c t r i c a l analog modeling of the closed blood circulation s y s t e m as a whole [5]. Apparently the f i r s t study where the blood flow was considered f r o m the viewpoint of mechanics b e long to E u l e r . The propagation velocity of pulse waves in v e s s e l s was determined by Young: c o = (Eh/ 2Rpf) 1/2, where E is the modulus of elasticity of the m a t e r i a l of the v a s c u l a r wall; R and h are the radius of c u r v a t u r e and thickness of the wall; p f is the density of the fluid. L a t e r this relation was obtained by v a r i o u s methods by Weber, Resal, Moens, and Korteweg. In the physiological literature it is called the M o e n s - K o r t e w e g f o r m u l a o r the Resal formula. The p r o b l e m of the wavelike motion of blood in an e l a s tic v e s s e l with c o n s i d e r a t i o n of the v i s c o s i t y of the fluid was studied in [6]. A large p a r t of the studies on the propagation of the pulse wave is based on linearized t h e o r i e s . will name s o m e a s s u m p t i o o s usually made in linearization.

We

Since the a v e r a g e velocity of the blood flow (of the o r d e r of 25 c m / s e c in the aorta) is s m a l l in c o m p a r i s o n with the p r o p a g a t i o n velocity of the pulse wave (from 500 to 1200 c m / s e e ) , the nonlinear t e r m s in the equations of motion for the fluid a r e neglected. It is known that at low s h e a r velocities the blood behaves as a non-Newtonian fluid [7]. However, in large v e s s e l s the flow velocity is such that the v i s c o s i t y can be considered constant, f o r all p r a c t i c a l p u r p o s e s . It is a s s u m e d that the radial displacements of the a r t e r i a l walls during pulsation are so s m a l l {about 1/20 of t h e i r a v e r a g e diameter) that there is no need to take into account g e o m e t r i c nonlinearity. Finally, the assumption of linear elasticity of the m a t e r i a l of the v a s c u l a r wall is introduced. In a n u m b e r of studies this m a t e r i a l is considered to be linear v i s c o e l a s t i c [8]. Let us c o n s i d e r the s t r u c t u r e of blood v e s s e l s . The wails of the v e s s e l s consist of s e v e r a l with different p r o p e r t i e s (Fig. 1). The inner coat 1 consists mainly of elastin and is lined on the with quite smooth cells. The middle coat 2 is f o r m e d f r o m smooth m u s c u l a r f i b e r s , The d e g r e e velopment of the m u s c u l a r coat is different in different v e s s e l s . The n u m b e r of m u s c u l a r fibers

coats inside of dedecreases

N. E. Zhukovskii M i l i t a r y - A i r Academy. T r a n s l a t e d f r o m Mekhanika P o l i m e r o v , No. 2, pp. 373-379, M a r c h - A p r i l , 1970. Original article submitted July 10, 1969.

© 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy o[ this article is available from the publisher for $15.00.

322

4

2

3

Fig. 1

Fig. 2

with a d e c r e a s e of the c a l i b e r of the v e s s e l . The o u t e r coat 3 is f o r m e d by the m o s t r i g i d , collagenous f i b e r s with an a d m i x t u r e of e l a s t i n . E x p e r i m e n t s c a r r i e d out with a r t e r i a l s e g m e n t s [9, 10] showed that the e l a s t i c p r o p e r t i e s of the w a l l on the whole depend c o n s i d e r a b l y on the load - i n t e r n a l p r e s s u r e and longitudinal f o r c e s . The m a t e r i a l of the v a s c u l a r wall cannot be c o n s i d e r e d i n c o m p r e s s i b l e , as was done in a n u m b e r of s t u d i e s . As a model of a v e s s e l we m u s t take a c y l i n d r i c a l s h e l l made of an o r t h o t r o p i c n o n l i n e a r l y e l a s t i c c o m p r e s s i b l e m a t erial. The d e s c r i b e d e x p e r i m e n t s w e r e c a r r i e d out with a r t e r i a l s e g m e n t s kept s e v e r a l d a y s in p h y s i o l o g i c a l s o l u t i o n . The e x p e r i m e n t s did not p e r m i t a q u a n t i t a t i v e e v a l u a t i o n of the effect of c o n t r a c t i o n of the m u s c u l a r coat of the v e s s e l u n d e r the effect of n e u r o g e n i c and h u m o r a I f a c t o r s on i t s m e c h a n i c a l b e h a v i o r . H o w e v e r , it is known [11] t h a t such effect p l a y s a quite s u b s t a n t i a l r o l e in r e g u l a t i n g p e r i p h e r a l c i r c u l a t i o n . H e r e we p r o p o s e d in c o n s t r u c t i n g the m o d e l to d i s t i n g u i s h an a c t i v e - m u s c u l a r - coat whose c o n t r a c t i o n c r e a t e s the i n i t i a l f o r c e in the s h e l l and changes the w a l l t h i c k n e s s . A d e s c r i p t i o n of the m e c h a n i c a l b e h a v i o r of c o n t r a c t i l e m u s c u l a r t i s s u e u n d e r the joint e f f e c t of e x c i t a t i o n and m e c h a n i c a l {static) loads was given by L. V. N i k i t i n . * In e x a m i n i n g t h e d y n a m i c p r o b l e m we w i l l c o n s i d e r that the w a l l of the v e s s e l r e p r e s e n t s a c y l i n d r i c a l s h e l l c o n s i s t i n g of i n n e r and o u t e r e l a s t i c c o a t s and a m i d d l e m u s c u l a r coat. The o r t h o t r o p i c h o m o g e n eous c o a t s of v a r i o u s t h i c k n e s s c o n s i s t of m a t e r i a l s having d i f f e r e n t e l a s t i c p r o p e r t i e s and a r e connected along c o m m o n s u r f a c e s o The n o n l i n e a r i t y of the e l a s t i c p r o p e r t i e s of the w a l l m a t e r i a l in the p h y s i o l o g i c a l r a n g e has c o m p a r a t i v e l y little e f f e c t on the b e h a v i o r of the v e s s e l d u r i n g p r o p a g a t i o n of the p u l s e wave. This n o n l i n e a r i t y m u s t be t a k e n into account when s e l e c t i n g the e l a s t i c c o n s t a n t s Ei and ~ij in r e l a t i o n to the l e v e l of i n t e r n a l p r e s s u r e and longitudinal t e n s i o n of the v e s s e l and to the d e g r e e of c o n t r a c t i o n of the m u s c u l a r coat. We will r e l a t e the c o o r d i n a t e s y s t e m x, y, z with the r e f e r e n c e s u r f a c e of the s h e l l (Fig. 2). The c o a t s of the s h e l l a r e n u m b e r e d f r o m 1 to 3, 1 c o r r e s p o n d i n g to the i n n e r c o a t . The t h i c k n e s s of the i - t h c o a t is denoted by h i and the c o o r d i n a t e s of the i n s i d e and o u t s i d e b o u n d a r y of the c o a t a r e denoted by z i and z i ÷ l , r e s p e c t i v e l y . The t o t a l t h i c k n e s s of the s h e l l f r o m z 1 to z 4 is equal to h. The c o n t a c t conditions b e t w e e n the c o a t s c o n s i s t s in the e q u a l i t y of the d i s p l a c e m e n t s and t a n g e n t i a l f o r c e s of the c o a t s in p l a n e s xz and yz on the b o u n d a r y s u r f a c e s and in the a b s e n c e of m u t u a l p r e s s u r e of the c o a t s . We w i l l b e guided h e n c e f o r t h by the K i r c h h o f f - L o v e h y p o t h e s e s in which t h e s e conditions a r e m e t a u t o m a t i c a l l y . The s t r a i n s of the s h e l l e l e m e n t s a r e e x p r e s s e d with c o n s i d e r a t i o n of finite r a d i a l d i s p l a c e m e n t s b y m e a n s of the following r e l a t i o n s [12]: •

au

a2w

1 / 8w

c)2w

(1) 8u

Ov

ag

ox

~ = ~.-- +-q-.-

--2z

a2w

aw

c~w

a---~gg 4 (ix

ag

*The solution of the p r o b l e m of e q u i l i b r i u m of a b l o o d v e s s e l on the b a s i s of the i n d i c a t e d m o d e l was given in his r e p o r t at the s y m p o s i u m on m a t h e m a t i c a l m o d e l s of blood c i r c u l a t i o n {Moscow, 1969).

323

T

Here R is the radius of c u r v a t u r e of the r e f e r e n c e surface; u, v, w are the displacements of its points.

7"

The s t r e s s e s a r e related with strains by the following expression: Exi

(rx ~=

.

. (e,~ i +~yx

1 -- ~txy~tyx ~

i

ey i ), • ] !

Evi

I1

(2)

btyx , x y Ti = Giy i .

H÷...

T÷...

T*...

}

H+...

The positive directions of the internal f o r c e s and m o ments in an element of the shell are shown in Fig. 3. The values of these quantities are determined in the following manner:

Fig. 3

3

Zi+ 1

N,,-h~o= 2

3

i~l

Zl

i=1

i=l 3

i~l

Zi+ 1

f c~idz; Nu--N~v= z~

(3)

zi

Zt+ 1

zl+ !

3

Zt

i=1 3

i=!

Zi

(4)

zi+ I

z~

Here Nx0 and Ny0 a r e the initial static f o r c e s acting in the shell. In e x p r e s s i o n s (3), (4) integration is p e r f o r m e d with r e s p e c t to each coat f r o m its inside s u r f a c e to the outside, and s u m m a t i o n is done o v e r all t h r e e coats. The equations of motion for the shell in f o r c e s and displacements have the f o r m : 3

ON,: OT ~ ~ - ~ Ox Oy I ( OMy R Oy

a2M~

OMx 02w O:u ~ T -~-Px . p/hi=0; Ox Oxz - 7 ~= OH) Ox

a2H

ONy Oy

OT

Ox

02U 2 Oihi:O; - ~ ~=1

a2M~ N,, ~-~x(

(5)

aw _ aw\ 3

0 i

Ow

,

O~v ~

°~w 2

pih~=O.

Here Px and Pz a r e components of the hydrodynamic forces acting on the v e s s e l . In examining the a x i s y m m e t r i c p r o b l e m of pulse wave propagation in a r t e r i e s Eqs. (5) can be s i m p li fled:

ONx Ox

OMx 02w 02u 2 p~h~=0; Ox Ox2 . + p ~ - ~ ~=~ g

Ox2

324

OX '~ ~ Ox I --

+Pz

~

i=l

p~hi:0.

(6)

Substituting into these equations the values of the internal f o r c e s (3), (4) and e x p r e s s i n g the s t r e s s e s in t e r m s of strains (2), we a r r i v e by m e a n s of e x p r e s s i o n s (1) at equations in d i s p l a c e m e n t s . The motion o f the bIood in the v e s s e l will be r e g a r d e d as a l a m i n a r a x i s y m m e t r i c flow of a Newtoninn fluid. We write out the N a v i e r - S t o k e s equations c o r r e s p o n d i n g to this assumption in cylindrical c o o r dinates x and y,

Oox Ot

__2£ Pl Ox

1

Ov~ at

1 dp pf

-

I 02v~

1 Ov~

I 02vr

10v~

-

+7-37-

02v~ \

|

/

o_77_ ! ; 02Vr

(7)

vr \ [

; IJ

and the equation of continuity

Op] Fpfo( Ov~ vr Ovx \ - ~ - r +.-7-+--~--xJ =0. Ot

(7a)

Here u is the kinematic v i s c o s i t y . It is a s s u m e d that the density of the fluid pf is a linear function of p r e s sure: pj = 9so'k

P--Po K

'

(8)

where K is the bulk modulus of the fluid in c o m p r e s s i o n .

aw

vr=-~;

au v~=--~

for r = R - z x .

(9)

The magnitude of the n o r m a l f o r c e s Px and Pz acting on the shell a r e determined by the formulas

t Ov~ av~ \ P~=OjV ~"~-~r +-~x l ;

Ovr pz=p--2plv Or "

(10)

The equations of motion of the shell (5) o r (6) and fluid (7) and (7a) with consideration of conditions (9) and relations (10) d e s c r i b e the dynamic b e h a v i o r of the blood v e s s e l . The model of the v e s s e l given h e r e can be reduced in c e r t a i n p a r t i c u l a r cases to the models known in the literature. Assuming the thickness of the coats to be equal and the values of the elastic constants E i and ~ij to be the same for all coats, we a r r i v e at the dynamic nonlinear equations for a o n e - c o a t shell. Such equations w e r e given for isotropic m a t e r i a l in [13]. If we d i s c a r d the nonlinear t e r m s in the equations of the orthotropic shell but take into account the effect of t r a n s v e r s e s h e a r , as was done, for example, in [14], we a r r i v e at the formulation used in [15] f o r solving the p r o b l e m of a pulse wave without c o n s i d e r a t i o n of large d i s p l a c e m e n t s . Linearizing the equations and considering the shell to be o n e - c o a t and o r t h o t r o p i c , we obtain the equation of motion in the f o r m given in [16]. Darther simplifications of the equations for the shell lead to a formulation in which the p r o b lem of pulse wave propagation was examined in [6, 17, etc.]. The simultaneous solution of the equations of motion of the shell and fluid can be accomplished for example, by the network method with the use of a digital computer, If we assign the pulsating p r e s s u r e at the s t a r t of the v e s s e l (there a r e extensive e x p e r i m e n t a l data on the c h a r a c t e r of change of p r e s s u r e ) , then by means of such a solution we can t r a c e the propagation of the pulse wave and find the displacements of the v a s c u l a r walls and change of p r e s s u r e along the v e s s e l ' s e n t i r e length. The indicated model can be used also when studying the c h a r a c t e r i s t i c s of blood c i r c u l a t i o n in the human o r g a n i s m subjected to the effect of long o r b r i e f (impact) g - f o r c e s . Such p r o b l e m s a r e of i n t e r e s t for space physiology [18]. In this c a s e t e r m s reflecting the effect of body f o r c e s must be introduced into the equations.

325

We will indicate still another p o s s i b i l i t y of u s i n g the p r o p o s e d model. It was shown in [19] that K o r o t k o v ' s sounds h e a r d when m e a s u r i n g the p r e s s u r e in the b r a c h i a l a r t e r y a r i s e as a result of loss of stability of the a r t e r y with fluid u n d e r the effect of external p r e s s u r e . The authors of the a r t i c l e conside r e d the v a s c u l a r wall homogeneous and isotropico An analysis of the p r o b l e m on the b a s i s of nonlinear equations of n o n a x i s y m m e t r i c motion of a multicoat shell (5) p e r m i t s giving a m o r e c o m p l e t e analysis of K o r o t k o v ' s phenomenon. A study of the effect of the initial p r e s s u r e and degree of c o n t r a c t i o n of the m u s c u l a r coat on the b e h a v i o r of v e s s e l s should be considered as one of the i m m e d i a t e t a s k s in examining all these p r o b l e m s . Such an a n a l y s i s should be combined with an e x p e r i m e n t a l investigation of the m e c h a n i c a l p r o p e r t i e s of v e s s e l s u n d e r static and dynamic loads with c o n s i d e r a t i o n of the d e g r e e of c o n t r a c t i o n of the smooth m u s c l e s of the v a s c u l a r wall. LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

326

CITED

A. V o l ' m i r and M. S. G e r s h t e i n , P r i k l . Mekhan., 5, 3 (1969). G. Rudinger, B i o m e d i c a l Fluid Symp., N. Y. (1966). R. Skalak, B i o m e c h a n i c s P r o c . Symp., N. Y. (1966), p. 20. E . O . Attinger, in: Advances in Biomedical Engineering and Medical P h y s i c s , N. Y. (1968), p. 1. M . L . E d e m s k i i , in: Physiology of C a r d i a c Election [in Russian], Kiev (1968), p. 28. I . S . G r o m e k a , P r o p a g a t i o n Velocity of Wavelike Motion of a Fluid in E l a s t i c Tubes [in Russian], Kazan' (1883). R . L . W h i t m o r e , Fourth Intern. Congr. on Rheology, Vol. I, N. Y. (1965), p. 57. R . H . Cox, B i o p h y s . J . , 8, 6 , 6 9 1 (1968). E . G . T i c k n e r and A. H. Sacks, Biorheology, 4, 4, 169 (1967). D . J . P a t e l and D. L. Fry, Circulation Res., 2_44, 1, 1 (1969). V . M . Khayutin, V a s o m o t o r Reflexes [in Russian], Moscow (1964). A . S . V o l ' m i r , Stability of D e f o r m a b l e S y s t e m s [in Russian], Moscow {1967). A . S . V o l ' m i r and M. S. G e r s h t e i n , Izv. AN A r m . SSR, Mekhanika, 1_99, 8 (1966). S . A . A m b a r t s u m a n , T h e o r y of Anisotropic Shells [in Russian], Moscow (1961). I. M i r s k y , Biophys. J . , 7, 2, 165 (1967). H . B . Atabek, Biophys. J., 8, 5 , 6 2 6 (1968). J . R . W o m e r s l e y , Phil. Mag., 4_~6,s . 7 , 3 7 3 , 199 (1955). V . V . P a r i n , R. M. B a e v s k i i , Yu. N. Volkov, and O. G. Gazenko, Space Cardiology [in Russian], Leningrad (1967). M. Anliker and K. Ro R a m a n , Int. J. Solids and S t r u c t u r e s , 2, 3 , 4 6 7 (1966).