An evolution of pulse speed in arteries

Bderin

ofMarhemarica/Biology,

Vol. 58, No. 1, pp. 129-140, Elsevier Science

0 1996 Soaety

for Mathematical

009X3240/96

1996 Inc.

Biology

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0092-8240(95hMl310-M

AN EVOLUTION n

OF PULSE SPEED IN ARTERIES

HiLMjI DEMiRAY* Marmara Research Centre, Research In$itute for Basic Sciences, Department of Mathematics, P.O. Box 21, Gebze-Kocaeli Turkey

In this work, treating the artery as a thick-walled cylindrical shell made of an incompressible, isotropic and elastic solid, utilizing the large deformation theory and the stress-strain relation proposed by Demiray (1976b, Trans. ASME Ser. E, J. Appl. Mech., 98, 194-197), an explicit expression for the pulse speed is obtained and the effect of lumen pressure and the axial stretch on wave speed is discussed. Numerical results indicate that the wave speed increases with lumen pressure but decreases with the axial stretch. The results of the present model are compared with our previous work (Demiray, 1988, J. Biomech. 21,55-58) on the same subject.

NOMENCLATURE Cauchy stress tensor Hydrostatic pressure Lumen pressure Finger deformation tensor Deformation gradient Reciprocal metric tensor of the spatial frame Reciprocal metric tensor of the material frame Strain energy density function Basic invariants of Finger deformation tensor Stretch ratio in the axial direction Stretch ratio in circumferential direction Stretch ratio on the inner surface Stretch ratio on the outer surface Deformed inner radius Deformed outer radius *Also, Istanbul Technical University, Faculty of Sciences and Letters, Department Sciences, Maslak 80626, Istanbul, Turkey.

of Engineering

129

130

Pf A c ‘MK

H. DEMiRAY

Undeformed inner radius Undeformed outer radius Radial stress component Circumferential stress component Axial stress component Incremental pressure modulus Young’s modulus Mass density of fluid Vessel lumen area Speed of propagation Moens-Korteweg speed

1. Introduction. Wave measurement techniques are efficiently used, without damaging the specimen, to determine the mechanical and geometrical characteristics of the material under consideration, In the past, several theoretical and experimental studies have been conducted to determine pulse speeds in distensible tubes containing inviscid or viscous fluid. As is well known (Fung, 1985), the speed of the wave changes with material and geometrical characteristics of arteries, as well as the frequency of waves. Moreover, considering that for a healthy human being the mean intramural pressure is about 100 mm Hg and for in vim conditions the axial stretch is about 1.5, one may assume that during the blood flow, the blood vessels are subjected to large initial static deformations (c.f. Mirsky, 1973; Demiray, 1976a; Fung et al., 1979). These large initial deformations might affect the speed, transmission coefficient and other characteristics of the wave (Atabek and Lew, 1966; Rachev, 1979; Demiray and Ercengiz, 1991). In order to see how the wave speed changes with large initial deformation, as well as material and geometrical characteristics of the arteries, one should know the explicit expression of the wave speed. The majority of theoretical studies employed either small deformation theories (Womersley, 1957; Kuiken, 1984) or treated the arteries as thinwalled membranes (Atabek and Lew, 1966; Rachev, 1979; Demiray, 1992). Thin shell theories are applicable when the ratio of thickness to mean radius of the shell is less than l/20. However, for most arteries this ratio varies between l/4 - l/6. This consideration clearly indicates that the membrane or thin shell theories cannot be successfully used in the analysis of arterial mechanics. Moreover, the arteries are subjected to large initial static deformation. Therefore, in assessing the speed of propagation, these characteristics of the arteries should be taken into account. This study aims to give an explicit expression for the pulse wave speed as a function of large initial deformation as well as material and geometrical

EVALUATION OF PULSE SPEED IN ARTERIES

131

characteristics of the artery. For this purpose, the artery will be treated as a thick-walled cylindrical shell undergoing large deformation, both in the radial and axial directions. Employing the strain energy function proposed for soft biological tissues by Demiray (1976b), the explicit expression of the wave speed is obtained as a function of the above-mentioned parameters. The numerical results indicate that the pulse speed increases with lumen pressure and decreases with axial stretch. The result of the present model is also compared with our previous work on the same subject (Demiray, 1988). 2. Theoretical Preliminaries. Let us consider an elastic body, with volume I/ and surface S, occupying the region B, in three-dimensional physical space. Upon application of external forces, the body moves to a new configuration B, through the deformation x = x(X). The deformation must be the solution of Cauchy’s equations of equilibrium, tk’;

k = 0,

(1)

where tk’ is the Cauchy stress tensor and the index following the semicolon is used to denote the covariant differentiation of the corresponding tensor field. The summation convention applies on repeated indices. For a homogeneous, incompressible and isotropic elastic material the stress-strain relation may be expressed as (Eringen, 1962) tk’ = Pgkl +

@ck’f

'PBk',

(2)

where P is the hydrostatic pressure to be determined from the field equations and the boundary conditions, gk’ is the reciprocal metric tensor of the spatial frame ,I?, ckl is the Finger deformation tensor and other quantities are defined by ck’ = GKLF,kF;, Bk’

E

F+

axk dXK

1,~~’ - c;crnl,

Here GKL is the reciprocal metric tensor of the material frame XK, IS is the strain energy density function and I,, I, and I3 = 1 (incompressibility) are the basic invariants defined by I, = c$,

I,

=

;B;,

I3 = det(cF) = 1.

(4)

132

H. DEMiRAY

In order to obtain the deformation field completely, one must know the functional form of the strain energy density function. As a first approximation for the characterization of mechanical behavior of the arterial wall material, we proposed the following type of strain energy function (Demiray, 1976b):

Z=

A {exp[ a(12 -

3)] - l},

(5)

where (Y and /3 are two material constants to be determined mental measurements. Introducing (5) into (2) we get

from experi-

tkl = Pgkl + /3 exp[ a(lz - 3)] Bkl.

(6)

Insertion of (6) into (1) and the incompressibility condition give four equations for unknowns xk and P. These equations may be solved under properly posed boundary conditions to determine the mechanical field completely. 3. Extension and Inflation of an Artery. In this section we shall consider a cylindrical artery subjected to an axial stretch A,, lumen pressure Pi and zero pressure from the outer surface. Employing the incompressibility condition, the symmetrical deformation in the cylindrical polar co-ordinates may be given by

r=(R2/h,+C)1’2, e=o,

z=A,Z,

(7)

where C is an integration constant to be determined from the boundary conditions and (R, 0, Z) and (r, 13, z) are the cylindrical co-ordinates of a material point before and after deformation. Introducing (7) into the expression of Finger deformation tensor ckf and Bkz in (3) we have the physical components of Bk’ as B” =

Bk[=o

1

1

(k#Z)

1 B@@=h2h2+e z A2’ z and

I,=*iAt+

Bf”=h2h2+ e z

$

+ $, e

where A, is the stretch ratio in the circumferential

e

direction.

-

1

Ai

(8)

EVALUATION OF PULSE SPEED IN ARTERIES

Introducing (8) into equation (6) the physical components stress tensor become

133

of the Cauchy

(9) with 1 $A,2+-+--3

1

A;

A;

.

(10)

The stress components must satisfy the equilibrium equations given in the cylindrical polar co-ordinates as

dt,, --&-

1

+

-(t,,-t&J

I

=o,

P= P(r).

(11)

Introducing (9) and (10) into (11) and integrating the result with respect to Y we obtain

(12) Here we have employed the differential relation

(13)

where D is another integration constant to be determined from the boundary condition. Since the outer surface of the artery is free of stress, from (12) we have

(14)

134

H. DEMiRAY

Then the radial stress component

takes the form

dt.

(15)

On the other hand, since the radial stress on the inner surface is -Pi, from (15) we get (16) Here the superscripts i and o stand for the values of a quantity on the inner and outer surfaces, respectively. Equation (16) relates the deformation of the artery to the inner pressure Pi and the axial stretch h,. If the inner pressure-radii relation is known experimentally, by comparing theoretical results (16) with the experimental measurements, the material constants (Yand p can be determined numerically so as to obtain the best fit between the experiment and the theoretical model. For that purpose we introduce

where II is the number of experimental measurements performed on the specimen, P$&) is the kth measurement of inner pressure corresponding to the kth measurement of the inner radius rik) and P.(k) is the I(theo) theoretical inner pressure calculated from (16) by using the inner and outer radii obtained in kth measurements. The coefficients (Y and j3 can be determined from (17) so as to make the left side minimum. Simon et al. (1972) conducted experiments on a canine abdominal artery and measured the inner pressure and inner radius relations listed in Table 1 for A, = 1.53. Using the least squares method, for the best fit of theoretical model to experimental measurements, the values of material coeffiTable 1. Pressure-radii relation for an artery at zero stress Ri = 0.31 x 10e2 m, R, = 0.38 x lo-’ m EXP pi (Pa) 3,350 6,670 10,000 13,340 20,000 26,680

ri X lo-*

Cm)

roX lo-’

Cm)

Theo. Pi (Pa)

0.348 0.396 0.425 0.442 0.467 0.485

0.401 0.445 0.473 0.485 0.520 0.524

3,275 6,580 9,938 13,830 19,886 25,970

Deviation (%o) -1.7 -1.4 -0.6 +3.6 -0.6 - 2.7

EVALUATION OF PULSE SPEED IN ARTERIES

135

cients were determined as (Y= 0.82 and j3 = 10,100 Pa. Employing these numerical coefficients in (161, the theoretical values of pressure are calculated and listed in the fourth column of the table. The deviation between the experiment and the model is given in the fifth column of the table. The result reveals that the maximum deviation between theory and experiments is about 3.6 percent, which seems to be a fairly good approximation. 4. Speed of the Pulses in Arteries. In this section we consider an infinitely long, straight and homogeneous tube filled with an incompressible inviscid fluid. For a one-dimensional model, the non-linear governing equations describing the flow of the fluid are the conservation of mass and the balance of linear momentum, given by

CJA &Awl dz t+-=

0

(18)

1 dPi dw dw z+wdz+--=o, Pf

(19)

d.2

where A denotes the internal area of the tube, w is the velocity in the axial direction and Pi(z, t)is the fluid pressure (or the inner pressure of the tube). These equations involve three unknowns A, w and Piand, therefore, a third equation which relates the inner cross sectional area of the tube to the inner pressure is needed. Since the fluid is inviscid, the inner pressure is the only factor that changes the inner radius. Thus in general one can write

Pi=Pi(A) or

A =A(Pi).

(20)

Here Pi depends on the axial co-ordinate z and time t through dependence on A(z,t). Introducing (20) into (19) we get

the

c2 dA

dW

dW

atfW

A dz dz+--=O*

Here c is known as the local speed of the propagation

AdP. AAPi c2=___2~___ pf aA Pf 4.4*

(21) and is defined by (22)

Having determined the inner radius as a function of the inner pressure (see equation (1611,we can calculate the speed of propagation by using equation

136

H. DEMiRAY

(22). For that purpose we consider the change in internal radius due to change in the lumen pressure Pi. Giving small increments to he and Ai, i.e. A: = he + AAL and A: = AZ+ AA:, where A’, is the value of the circumferential stretch corresponding to lumen pressure Pi’ = Pi + APi, from (16) we have

I I

APi=p

F(A;)AA;.

.

(23)

In obtaining equation (23) from (16) we utilized the mean-value theorem for integration. Using the global incompressibility condition and the definition of circumferential stretch, i.e.

AA;=A;--

hrll ‘0

r,,Ar, = ri Ari,

(24)

’

equation (23) becomes

Introducing

the incremental

pressure modulus Er

and noting the relation Ari/ri = AA/2A,

as

equation (25) can be written as

APi=E~AA/2A.

(27)

Hence the pulse velocity (22) may be expressed by c2 = Ep/2pf.

(28)

EVALUATION OF PULSE SPEED IN ARTERIES

137

If the initial deformation vanishes, i.e. AL= h,O= A, = 1, the incremental pressure modulus reduces to ,

(29)

where Ri and R, are the undeformed values of the inner and outer radii, respectively. Recalling the relation between the classical Young’s modulus E and the coefficient p ( /3 = E/3; Demiray and Antar, 1994) from equations (28) and (29), we have (30)

This expression is exactly the same as that given by Bergel (1961) for an incompressible elastic tube. Moreover, if the arterial wall is very thin, then equation (30) reduces to 2_2EH 4 c - T = &; ’ 3pfR

&=EH 2PfR’

(31)

where R is the midradius, H is the thickness of the arterial wall and cMMK is the Moens-Korteweg wave speed. For numerical analysis it is convenient to introduce the non-dimensional propagation velocity u and pressure pi as

(32) Using the numerical values of the material coefficients obtained in the previous section, the numerical values of wave speed and lumen pressure are calculated and the results are depicted in Figs. 1 and 2. The numerical results indicate that the wave speed increases with increasing lumen pressure, while it decreases with increasing axial stretch ratio (Fig. 1). Qualitatively, similar results had been observed in our previous model (Demiray, 1988). For smaller axial stretch ratios, the present model gives lower wave speed as compared to our previous model (Demiray, 1988; Fig. 2). However, for larger axial stretch ratios, the wave speed of the present model is larger than the previous model. Moreover, as can be seen from Fig. 1, in the present model the wave speed is less sensitive to the changes in the axial

138

H. DEMiFUY

4.00

3.60

3.20

V 2.40 3

1.60

1.20

0.80?1111IIII,,,,IIIIIII,I,,III,II,IIII,,,II,IIIIII,,,,IIIIIIIII, 1.20 0.00 0.40 0.80

1.60

2.00

2.40

4

Figure 1. The variation

of pulse wave speed with inner pressure stretch ratio.

and axial

stretch ratio. Finally, we should remark that the wave speed we are dealing with here corresponds to slow or secondary wave speed in the terminology used in Demiray and Ercengiz (1991). As pointed out there, the speed of primary (or fast) wave increases with both lumen pressure and axial stretch ratio.

5. Conclusion. In this paper, utilizing the theory of finite elasticity and a particular type of strain energy density function, the speed of a pressure pulse wave propagating in an elastic tube subjected to a large lumen pressure and an axial stretch and filled with an inviscid fluid is studied and the effects of lumen pressure and axial stretch on wave speed are indicated. The numerical results reveal that the speed of propagation increases with inner pressure but decreases with axial stretch ratio (Fig. 1). This behavior is common to all soft biological tissues whose tangent modulus increases with deformation. However, for engineering materials this is not the case; that is, the speed decreases with intramural pressure (Atabek and Lew, 1966). The present results are qualitatively in agreement with the experi-

EVALUATION

OF PULSE SPEED IN ARTERIES

139

Figure 2. The variations of pulse wave speeds of the models with inner pressure _and axial stretch ratios.

mental results of Anliker et al. (1968) and the theoretical Rachev (1979) and Demiray and Ercengiz (1991).

derivation

of

This work was supported by the Turkish Academy of Sciences. LITERATURE Anliker M., M. B. Histand and E. Ogden. 1968. Dispersion and attenuation of small artificial pressure waves in the canine aorta. Circulation Res. 23, 539-551. Atabek, H. B. and H. S. Lew. 1966. Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube. Biophys. J. 7, 480-503. Bergel, D. H. 1961. The static elastic properties of arterial wall. J. Physiof. 156, 445-457. Demiray, H. 1976a. Some basic problems in biophysics. Bull. Math. Biof. 38, 701-711. Demiray, H. 1976b. Stresses in ventricular wall. Trans. ASME Ser. E. J. Appl. Mech. 98, 194-197.

Demiray, H. 1988. Pulse velocities in initial stressed arteries. J Biomech. 21, 55-58. Demiray, H. 1992. Wave propagation through a viscous fluid contained in a prestressed thin elastic tube. Znt. J. Eng. Sci. 30, 1607-1620. Demiray, H. and N. Antar. 1994. Harmonic waves in an elastic thin tube filled with a viscous fluid. Unpublished. Demiray, H. and A. Ercengiz. 1991. Wave propagation in a prestressed elastic tube filled with a viscous fluid. Znt. J. Eng. Sci. 29, 575-583. Eringen, A. C. 1962. Nonlinear Theory of Continuous Media. New York: McGraw-Hill.

140

H. DEMiRAY

Fung, Y. C. 1985. Biodynamics: Circulation. New York: Springer. Fung, Y. C., K. Fronek and P. Pattitucci. 1979. Pseudoelasticity of arteries and choice of its mathematical expression. Am. J. Physiol. 35, 626-631. Kuiken, G. D. C. 1984. Wave propagation in a thin walled liquid-filled initially stressed tube. J. Fluid. Mech. 141, 289-308. Mirsky, I. 1973. Ventricular and arterial wall stresses based on large deformation analysis. Biophys. J. 13, 1141-1159. Rachev, A. I. 1979. Effect of transmural pressure and muscular activity on pulse waves in arteries. Trans. ASME J. Biomech. Eng. 102, 119-123. Simon, B. R., A. S. Kobayashi, D. E. Stradness and C. A. Wiederhelm. 1972. Reevaluation of arterial constitutive laws. Circulation Res. 30, 491-500. Womersley, J. R. 1957. An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries. Technical Report TR 56-614, WADC.

Received Received 15 January 1995 Revised version accepted 28 May 1995