The effect of age on the wall stiffness of the human thoracic aorta: A large deformation “anisotropic” elastic analysis

J. theor Biol. (1976) 59, 467-484

The Effect of Age on the Wall Stiffness of the Human Thoracic Aorta: A Large Deformation ‘LAnisotropic” Elastic Analysis I. MIRSKY

Department of Medicine, Division of Mathematical Biology, Harvard Medical School and Peter Bent Brigham Hospital, Boston, Mass. 02 115, U.S.A. AND

R. F. JANZ Data Systems Analysis Section, The Aerospace Corporation, Los Angeles, California, 90009, U.S.A. (Received 26 March 1975, and in revisedform 6 October 1975) Assuming a cylindrical geometry for the thoracic aorta, passiveelastic stiffness-stressrelations have heen obtained on the basis of a large deformation “anisotropic” elastic theory. Employing pressure-volume data obtained previously by Bader (1967)from human thoracic aortas of various age groups (29-85 yr), it is shown that the stiffnessof the aortic wall material increaseswith age, a result in qualitative agreementwith Bader’s studies. Although elastic stiffnessof the arterial wall increases with age, the large deformation theory, in contrast to the linear theory proposed by Bader, indicates that lumen volumes increasein such a manner so as to maintain constant operating stiffnesslevels with age at 100 mmHg pressure.Failure of the aorta to maintain constant level of elastic stiffnessmay be one causeof essentialhypertension.

1. Introduction In recent years there has been an increasedinterest by physiologists, clinicians and pathologists in the elastic properties of blood vessels. Furthermore, these studies appear to establish the fact that arterial walls display anisotropic elastic behavior, i.e. the elastic moduli of the vessel wall material vary in the different directions (Fenn, 1957; Patel, Janicki & Carew, 1969; Pate1 & Janicki, 1970; Vaishnav, Young, Janicki & Patel, 1972; Dobrin & Doyle, 1970; Doyle & Dobrin, 1971, 1973). As indicated by these investigators and by Anliker, Histand & Ogden (1968), reliable techniques are now 467

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available for the determination of these elastic constants. Since the problems of wave propagation through arteries and those of wall stress are intimately related to the elasticity of the vessel walls, it is important that an appropriate anisotropic elastic theory be developed in association with these experimental studies. With the exception of the studies by Simon, Kobayashi, Strandness & Wiederhielm, 1971, 1972; Vaishnav et al., 1972; Doyle & Dobrin, 1973; Mirsky, 1973a; Tanaka & Fung, 1974, most investigations related to the elastic properties of arteries have been based on the classical linear theory of elasticity. In particular, we refer to those studies relating to the anisotropic elastic behavior of arteries (Lambossy & Miiller, 1954; Lambossy, 1967; Atabek, 1968; Pate1 et al., 1969; Mirsky, 1967a, 19673, 1973b). The large deformation analyses of Simon et al., 1972 and Vaishnav et al., 1972, which consider the arteries to be anisotropic, are rather tedious and do not lend themselves readily to clinical applications. Furthermore, the assumption of transverse isotropy made by Simon et af. (1972) is in disagreement with the recent studies by Doyle et al., 1973. The purpose of the present study is to (a) develop simple expressions for the circumferential stress component in cylindrical elastic arteries based on a large deformation “anisotropic” elastic theory and (b) assess the effect of age on wall stiffness of arteries employing data obtained by Bader (1967).

2. Definitions It is appropriate to define some of the terms employed in this study since there is often confusion in the literature particularly in relation to wall stiffness of arteries. Stress Denoted here by 0 may be defined as force per unit cross-sectional area of a material and essentially is a measure of the intensity of forces. The units commonly employed are dyn/cm’ or g/cm’. Strain Associated with the term stress is the quantity strain denoted by E. It is a dimensionless quantity and represents a change in length with respect to a reference length. This reference length should be the length corresponding to the zero state of stress and often this length is replaced by a preloaded length which may lead to errors in analysis. In the present discussion we define an increment of strain to be de = dR/R, i.e. an instantaneous change in radius

AGE

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469

with respect to the instantaneous radius R. This is consistent with the natural given by E = log (R/R,) where R, is the zero stress radius.

strain definition

Elastic stiffness E

May be defined in qualitative terms as a measure of the resistance of an elastic material to deformation. Mathematically this is expressed as da/d&, i.e. an instantaneous change in stress with respect to an instantaneous change in strain. For biological materials, elastic stiffness is an increasing function of the stress, however, for most metallic materials, elastic stiffness is independent of the stress level and is termed the Young’s modulus.

An isotropic elastic material properties in all directions.

is one that possesses the same material

Transverse isotropy

If the elastic properties of a material are isotropic in two dimensions, it is said to be transversely isotropic. Anisotropy

The elastic properties of an anisotropic material vary with the direction and, in the case of an orthotropic material, the elastic properties differ in the three mutually perpendicular directions. Arterial walls exhibit orthotropic elasticity. 3. Theoretical Considerations We are particularly interested in the evaluation of arterial wall stresses and the subsequent assessment of arterial wall stiffness. Thus a number of assumptions are required for the present analysis : (I) Assume that the descending thoracic aorta may be considered as a cylindrical tube of circular cross-section and uniform thickness. Although the cross-section may not be circular at very low pressures, it does assume a circular shape with uniform thickness at the physiological intravascular pressures (Bergel, 1960). (2) The aortic tissue is homogeneous, incompressible, elastic and anisotropic. Histologically, Wolinsky & Glagov (1964, 1967) have shown that the artery has a reasonably uniform structure in the longitudinal and circumferential directions but not in the radial direction. Homogeneity also implies that no distinction is made between the contributions of the elastin and

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collagen (different constituents of the wall). However, it is important to note the studies of Roach & Burton (1957) who concluded that the resistance to stretch at low pressures was due almost entirely to elastin fibres, at physiological pressures it was due to both collagen and elastin fibres and at high pressures it was almost entirely due to collagen fibres. With regard to incompressibility, Carew, Vaishnav & Pate1 (1968) have definitively concluded that this is the case under physiological states of stress. These assumptions are discussed in more detail in a review article by Pate1 & Vaishnav (1972). (A)

PRESSURE-VOLUME

RELATIONS LARGE

AND

WALL

DEFORMATION

STRESS

COMPONENTS

BASED

ON

ANALYSES

Guided by the work of Blatz, Chu & Wayland (1969) in their studies on the mechanical behaviour of elastic animal tissue, and by Ko & Blatz (1964), the radial (cRR) and circumferential (a& stress components are assumed in the form bRR = cm

=

Q(R) W%>+QW

(1)

where Q(R) is a “hydrostatic pressure”, F(&) is as yet an undetermined function of the stretch ratios A,, 1 to be defined later and R is the deformed radial coordinate. The rationale for this representation follows from the facts that (a) the arterial wall behaves more as an anisotropic elastic material, (b) they uncouple the radial direction from the circumferential direction which is consistent with the fibrous structure of the artery and (c) mathematically, these expressions are simple to work with as will be observed in the following analysis. Assuming a cylindrical geometry for the thoracic aorta, the stress equation of equilibrium may be written as (Timoshenko 8z Goodier, 1951), do,,/dR Direct integration

+ (bRn - oee)lR = 0

(2)

of this equation yields CRR =

+RR

-%e)

WR+oRR@)

(3) =

IF@;)

dR/R-P

where crRR(a) = -P (lumen pressure) is the boundary condition on the endothelial surface R = a and use has been made of expressions (1). The remaining boundary condition bRR(b) = -PO on the adventitial surface

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R = b yields the following relation for the transmural

AP = P-P,,

The incompressibility ratios are defined as

= ;F(AI;)

1, = R/r

and

471

pressure AP given by

dR/R

(4)

P assumption requires that ,I,J,J

& = dRJdr,

AORTA

= 1 where the stretch 1 = I/&,

(3

(Green & Adkins, 1960). Note that cylindrical polar co-ordinates (R, 0, Z) in the deformed body are taken as the reference frame, lo is the undeformed length and I the deformed length. Thus the point (R, 0, 2) was initially at the point (r, 0, Z/I) of the undeformed body and r is a function of R only. Since I, = R/r, we obtain on differentiation, dl,=dR/r-(R/r2)dr=y(J--f: = &d;(‘-&/,I,)

ik) = &(I -,I$)

dR/R

(6)

employing the above incompressibility relation. Furthermore, the Iumen volume V, volume V, at zero pressure and the wall volume V,Vare related to the deformed and undeformed radii in the following manner: v = na21

v, = 7ca&

V+V,

v-0+ VW= xb&

= nb21

.

(7)

Hence I,(b) = b/b, = [(V+ V,)JA(V, + v,p

= &lb

and

(8)

&(a) = a/a0 = [v/&]*

= A,,

and the integral expression (4) may therefore be rewritten in the form k3b (9) Since ,IBa, 1,, are functions of the volume V as defined by expression (8), expression (9) represents an integral form of the pressure-volume relationship. It is convenient now to assume an infinite series expression for the function F(,Ui) in the form

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where the constants a,, are determined from known pressure-volume data, (see Appendix A). If in particular, the pressure-volume relation is expressed in the exponential form Ap = A+B eC(“-“O) (11) it is shown in the Appendix A that the function F(#) by the expression F(Ui)

may be approximated

- A$[ - 2(A + B)/log, c1+ 2BCT/,(A1,2 - 1) exp {C VO(@ - 1)} +nfl

2B(aCV,)“(U,2

- l)“/{(l

-a”)(n - l)!)]

(12)

for N sufficiently large, where c1 = V&V, + VW)in the present analysis, the stretch ratio 1 is assumed to be constant at a given pressure level but may vary from one pressure level to the next. (B) UNIAXIAL

STIFFNESS-STRESS

Consider now the three dimensional equations (1) to be of the form ci=

RELATION

stress-stretch relations implied

Q(R)

pi = Fi(nniz) $- Q(R)

by

(13)

i=2,3

If we assume that the arterial wall material is under a state of uniaxial stress, then g1 = 0, g3 = 0 and the stress-stretch relations (13) reduce to u1 = Q(R) = 0 u2

= F2(U3 + Q(R) = F&A:)

u3 = F&@+Q(R)

= Fs(U:)

(14) = 0

where F,(U~) is defined by expression (12). Defining elastic stiffness on the basis of the natural strain definition, obtain

we

E = da/& = A, da/d&

= 1, dF/dl, = 2F+2(L12,2)2 2BCl/,{l+CV,,(112,2-1)) I

exp CV,(Ui-1)

+~~~2Bn(CV,cl)YM:-I)“-l/(1-cr”)(n-l)!]

(15)

where E = log Iz,. It is expression (15) which is plotted against the stress c in order to yield the uniaxial stiffness-stress relations for arteries of various age groups.

AGE (C)

EXPRESSIONS

EFFECT FOR

ON

WA1.L

CLRCUMFERENTIAL ON LINEAR

STIFFNESS STRESS

OF AND

473

AORTA

ELASTIC

STIFFNESS

BASED

THEORY

From classical linear theory (Timoshenko & Goodier, 19.51), the circumferential wall stress at radius r in cylindrical tubes subjected to an internal pressure p is given in the form ug = pa2( 1 + bZ/rZ)/(b’ - u2) (16) where II, b are the internal and external radii. In particular the circumferential wall stresses at the endothelial and adventitial surfaces respectively are CTi= p(fz2+ b2)/(b2 - a2) (17) u. = 2paZ/(62-a2) with o*-(To = p. Since the wall thickness radius ratio is relatively small at the high pressures, we obtain a simple relation for the lumen volume/wall volume ratio in the form V/V, = na21/n(b2 - a’)1 = u2/(b2 - a2) - a2/h(2u) = 42h (19) where h = b-u is the wall thickness. Bader (1967) in his analysis considered the wall stress at the epicardial surface since the pressoreceptors are mainly situated in this region, hence the circumferential wall stress co may be written as u,, = 2pu2/(b2 - u2) = 2pV//lv, - pa/h. G.91 Furthermore, an empirical relation was developed for the tangential elastic modulus (stiffness) Et in the form Et = [2.16V(dp/dV)+1.2p](u/tz). (21) This formula was based on a single value of Poisson’s ratio = 0.2 and a ratio = 1.2 for Et/E, where El is the longitudinal modulus. These assumptions will be discussed later on in the analysis. 4. Methods In his studies Bader (1967) obtained pressure-volume relations in 27 thoracic aortas taken from normotensive humans of different ages between 22 and 85 yr. Volume changes in the aortas were accomplished with a motor-driven 500 ml syringe. Four extension-release cycles were performed and the fourth extension curve was used to obtain the slopes AP/AV at 10 mmHg intervals. The lumen volume V, at zero transmural pressure was

I. MIRSKY

474

AND

R. F. JAN'Z

obtained by emptying the contents of the aorta into a graduated cylinder. The wall volume V,,,was measured by immersing the aorta in a known volume of water and noting the volume increase. Unfortunately, values of A were not given. Figure 1 shows typical pressure-volume relations of six of the 27 human thoracic aortas of different ages and these curves are employed in the present numerical analysis. 5. Numerical Results and Discussion For computational purposes, the pressure-volume data displayed in Fig. 1 were curve fitted to exponential functions in the form Ap = p-p, = A+B ec(v-vo) (22) where A, B, C are constants determined from a non-linear regression analysis. The relations for the various age groups, volumes V,, at zero transmural pressure and wall volumes VWare given in Table 1. Several points must be noted here in relation to Table 1. Firstly, as observed in Fig. 1, the pressure-volume curves are not purely exponential

FIG. 1. Pressure-volume relations of human thoracic aortas which are typical for various age groups. (Reproduced from the studies of Bader, 1967.) TABLE

1

Values of V,, V, and pressure-volumeconstantsA, B, C for various age groups

Age Cvd 29 40 50 60 75 85

A (mHg) 54.7 43.8 62.5 37.1 10.5 5.5

B (JnIazl

C (ml-l)

0.308 O-178 0.0772 0.0354 0.838 0.896

0.086 0.0852 0.0768 0.0817 0.0653 0.0559

VO

(ml)

26.9 41 *o 60.9 91 .o 167.9 194.9

VW(ml> 9.3 13.4 21.2 32-s 54.0 70.3

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475

over the entire range of pressure and therefore expression (22) should not be employed in the low pressure range at least for the age groups 29-60 yr. Since values for the wall volume VW were not available in the paper by Bader (1967), average values were evaluated from the relation VW/V = (b2 - u*)/a* = h(2a+ /?)/a* = (2h/u+h*/a*) (23) on the basis of values of u/h at pressures of 0, 100 and 200 mmHg (Fig. 2 of Bader, 1967). On the basis of this data, stress-volume stretch and stiffness-stress relations for the six age groups have been obtained and are shown plotted in Figs 2 and 3. Over the physiological ranges of pressure it was necessary to employ only four to five terms in the series expressions (12) and ( 15). The

IO 9 8

3-

2-

I-

Volume

stretch

ratio

LA;

- I

FIG. 2. Stress-volume stretch relations for aortas of different age groups. Note that at a given stress level above 1 kg/cm2, the volume stretch (“strain”) decreases with increase in age.

476

I. MIRSKY

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i

Stress o (kg/cm’) FICL 3. Stiffness-stress relations for various age groups. In general the relation between stiffness(E) and stress(a) is linear and the slope of these linear relations representsthe stiffnessconstant for the arterial wall material (E = ka+c). It can be observedhere that the arterial wall stiffnessincreaseswith age from 29-85 yr, there being no significant differencebetween the 75 yr and 85 yr age groups.

slopes of these stiffness-stress relations describe the degree of stiffness of the arterial wall material and increases with age, i.e. for a given stress, stiffness increases with age, a resuIt in agreement with that obtained from Bader’s studies. Generally, these stiffness-stress relations are approximately linear at the higher stress levels and may be written in the form E = ka + c where k denotes the stiffness constant and is given in Table 2 for the various age groups. It should be noted that Bader obtained qualitatively similar stiffness-stress relations employing the simplified formulae (20) and (21). Figure 4 displays the circumferential stress distribution through the wail thickness for various age groups at a lumen pressure of 100 mmHg. These curves were obtained from expressions (1) where use has been made of the

AGE EFFECT

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477

TABLE 2

St$ness constantsk for each age group AgeCY~) kt 29 40 50 60 75 85

-

17.6 24.7 28.0 33.2 38.4 38.5

-FNote that k is the slope of the linear portion of the stiffness-stress curve which is represented in the form E = ka+c and is evaIuated on the basis of equations (12) and (15). Furthermore, there is no statistically significant difference between the 75 and 85 yr age groups.

boundary conditions Q(a) = -P and Q(b) = 0. With the exception of the age group (50 yr) there is a progressive decrease of both endothelial and adventitial stresses with age. Although this result is also in qualitative agreement with that of Bader, there are marked gradients of stress in going from the endothelial to the adventitial surfaces. This latter result is not obtained with the linear theory of elasticity which is shown plotted in Fig. 4 for the 75 yr age group and displays a relatively constant stress through the wall thickness. These sharp gradients of stress have also been noted in studies by other investigators who have employed large deformation analyses (Simon et al., 1972; Doyle et al., 1973; Mirsky, 1973a). In their studies with carotid arteries, Doyle et al. (1973) noted that the elastic lamellae were close together near the lumen and widely spaced near the adventitia and conjectured that this development may have resulted from the circumferential stress distribution. Finally, Fig. 4 indicates that midwall stresses are not markedly different from the mean stresses which are given by the expression IJ,,, = pa/h.

In Table 3 the stiffnesses and mean stresses at levels of strain corresponding to pressures of 100 mmHg are presented and compared with those obtained by Bader (1967). The agreement between the linear theory and large deformation theories appears to be remarkable in view of the fact that the Bader analysis has some limitations, namely (a) classical linear theory cannot adequately describe the elastic behaviour of materials undergoing deformations of the order of 50-100%; (b) a single value for Poisson’s ratio is not consistent with anisotropic elasticity theory which may require up to six Poisson ratios (Vaishnav et a/., 1972). Furthermore, the value of O-2 chosen T.B. 31

478

1. MlRSKY

AND

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R.

thickness

F.

JANZ

ratio

FIG. 4. Circumferential wall stressdistribution for various age groups at 100 mmHg pressure. Note that in contrast to the linear theory the stress gradients through the wall

thickness are rather marked whenevaluated on the basis of the large deformation theory. Furthermore, the midwall stresses are close to the average stresses which decrease with an increase in age. - - - -, Linear theory; -, finite theory.

by Bader has not been validated by other investigators (Carew et al., 1968) who indicated that it should be close to 0.5 for incompressible materials; (c) longitudinal (EJ and circumferential (Et) elastic moduli have been shown by Pate1 et al. (1969) to vary with the stretch ratios i10and 3, and that the ratio EJE, is neither 1.2 or a constant. It is quite possible that this close agreement between the two theories is coincidental as shown in Appendix B where an alternative expression for elastic stiffness has been developed. The expression developed there is in good agreement with the one obtained by Bader and is based on average

TABLE 3

Elastic stiffness at 100 mmHg pressurebased011 Bader theory and large deformation theory

W

Mean stress? u ig/cm2)

29 40 50 60 75 85

2550 2270 1880 1590 1270 1140

Age

Elastic stiffnessj (g/cm”) Bader theory Present theory 21,300 28.200 18,800 34,200 39,900 37,600

21,500 27,000 18,500 33,500 41,000 39,500

j’ The mean stress CTis calculated from the formula a=pu/h

=p/(b/a-1)

=p/[(FF)+-l]

$ The elastic stiffness based on the Bader theory is evaluated from expression (21) and values based on the present large deformation theory are taken directly from the stiffnessstress (E-o) curves plotted in Fig. 3.

stress and endothelial strain. However, as WCshall now observe the agreement ends here. Bader endeavoured to show that the increase in stiffness with age at a pressure of 100 mmHg is compensated by an increase of the lumen volume of the aorta so as to maintain a constant low level for dP/dV which he incorrectly termed “volume elasticity”. Based on the assumption that dP/dV remains constant between 20 and 60 yr (Kapal & Bader, 1958), he showed using equation (21) that a IOO-yr old aorta has to have a lumen volume of approximately 1 litre in order to work properly as a buffering chamber. This result is obviously unreal and his theory appears to break down at ages greater than 50 yr. Figure 5 shows the actual plot of lumen volume versus age at a pressure of 100 mmHg and comparisons are made with those obtained on the basis of the linear and finite theories assuming the elastic stiffness to remain at a constant value of 21,300 g/cm2 (i.e. the value at 29 yr of age). Since a comparison is being made between aortas of different size the use of elastic stiffness is more appropriate than the quantity dP/dV employed by Bader. For the linear theory the volumes were calculated in the following manner: (1) Stresses corresponding to an elastic stiffness of 21,300 g/cm2 were obtained from the various stiffness-stress curves (not shown here). (2) With these stresses, the lumen volume was calculated from the formula (20) in the form V = aVJ2p with p = 100 mmHg. 31’

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JANZ

_ ~_-__---.-----

300

-..---. s

250

50

-L--!---l-_l-~I0

IO

20

30

40

50

60

70

80

!

Age (yr)

FIG. 5. Volume-age relations at 100 mmHg pressure as obtained from both the linear and large deformation theories assuming a constant elastic stiffness level of 21,300 g/cm”. Except for the 50 yr age group, there are wide discrepancies in volumes based on the linear theory (0) as opposed to the close agreement obtained with the large deformation theory (0). -, true volume.

Volume calculations based on large deformation theory required a more detailed analysis. (1) Stresses corresponding to an elastic stiffness of 21,300 g/cm2 were obtained from Fig. 3. (2) From Fig. 2, the corresponding values of (An,”- 1) and hence U”, were thus available. (3) Assuming &Is to be the geometric mean of AI& and A&&, i.e. nn,z = [(:
AGE

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481

age at least at the pressure level of 100 mmHg. Thus as Bader has suggested, failure of the aorta to maintain a constant level of stiffness may be one cause of essential hypertension. This behaviour is analogous to that occurring in some hypertrophied ventricles where the stiffness constant k is elevated but the operating diastolic stiffnesses remain within normal limits (Mirsky et al., 1973c). Finally, an interpretation of the present results require further confirmation in view of the fact that the experimental studies performed by Bader do not accurately simulate the in viva situation. In particular, Bergel (1960) has shown that sigmoidal pressure-volume relationships may be artifactual and arise from the fact that isolated segments of arteries are allowed to lengthen on inflation as opposed to the longitudinal tethering situation in viva.

APPENDIX A

Approximate Expression for the Function F(Mi) From equation (9) of the text, we obtain the transmural pressure AP in the integral form h3 AP = 1, F&l;) d&/&(U,2 - 1) (AlI Assuming F(#)

in the form F(aa;) = an; f qu; n:O

- 1)

the above integral reduces to L&l AP = j lZ& 2 Z,(#,>,-I dl, &lb n=O At%7 00 = 3 1, ,zo i&l,?; - I)“- ’ d(U,2 - 1) g &(A&

=; ”

1

l)“-’

1

d(U,2-- 1)

(A9

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Thus (A5) where

If the pressure-volume data is curve-fitted to an exponential function of the form Ap = A+B &(y-yO) = A+BnEo

[C(V-

Vo)]“/n! W)

= (A+B)+Bntl

[c(v-v,)‘l”~n!

we obtain by direct comparison of expressions (A5), (A6) the following values for the coefficients: i&J = -2(/i + B)/logp; a = ~OWO + G) b, = BC”/n ! (A7) n= 1,2,... iI, = 2B(CV,,)“/( 1 - a”)@ - 1) ! The coefficients Z, may be written in the alternative form a, = 2B(CV,)

[&+l-l]/(n-l)! W

= 2B(CVJ Hence the function F(I$)

[&+l]/(n

- I)!

reduces to

F(nn;) = n/l,” f ii”(nn; - 1) n=O

-t-2BCvo(nn,2-1) - nn;

“~,(c~o~-~(nd-l)“-l/(~-l)!]

- 2(A + @/log, a $ 2 2B(aCvo)“(nn,z - l)“/(l - cr”)(n - l)! II=1 (A9) +2BCVo(ll~ - 1) exp (CV,(n$ - l)} I for N sufficiently large.

AGE

EFFECT

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STIFFNESS

APPENDlX

OF

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483

B

An Alternative Expression for the Elastic Stiffness of Arteries A simplified expression for the elastic stiffness, which agrees very well with that given in the present analysis, is developed here on the basis of earlier studies by Frank (1920). In particular, we define the tangential elastic modulus Et in terms of an average stress c and endothelial strain E (which for thin-walled tubes is close to the average strain) as follows: E, = do/d&

(Bl)

where Q = pa/h and de = da/a. Here p is the lumen pressure, h is the wall thickness and “a” is the lumen radius. If the wall is relatively thin, the average stress may be rewritten in the approximate form IS = pa/h = fi = pa (b-a)

(b+a) -=------(b+a)

pa(2a +h) b*-a2

2pa2 b*-a2 032)

where b is the external radius, I is the length, V is the lumen volume and V,,, is the wall volume. Assuming the length 1 to remain constant at each pressure level, we may write V = na*l

and

dV = 2nal da = 2V da/a.

(B3)

Hence the elastic stiffness is given by E, = da/de = da d” = 2V da/dV I a = 2V(2p+2V

dp/dV)/V,,

--Za(l+F

g)

= (2p+2V$)(a/h)

(B4)

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This compares favorably with the expression

developed by Bader in his studies and quantitatively with the more appropriate expression (15) obtained in the present analysis. the

Supported in part by USPHS Grants HL 12711-6 and HL 14651-03 from National Heart and Lung Institute. REFERENCES

ANLIKER, M., HISTAND, M. B. & OGDEN, E. (1968). ATABEK, H. B. (1968). Biophys. J. 5,626. BADER, H. (1967). Circ&&on Res. 20,354. BERGEL, D. H. (1960). Z%e visco-elasfic properties

University of London.

Circulation Res. 23, 534. of the arterial

wall.

Ph.D.

Thesis,

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McGraw

J. 12,1008.