RHEOLOGICAL PROPERTIES OF ARTERIES UNDER NORMAL AND EXPERIMENTAL HYPERTENSION CONDITIONS*-/M. G. SHARMA Department of Engineering Science and Mechanics and
T. M. HOLLIS Department
of Biology, The Pennsylvania State University. University Park. PA, U.S.A.
Abstract-Rheological properties of rabbit aortas in eitro were studied under control and hypertensive conditions by subjecting small segments of the aorta to uniaxial stress relaxation tests for various constant strain values ranging from 0.2 to 1.5 and stress-strain tests for deformation rates ranging from 8 to 4Omm/min. An examination of experimental results indicates that below a strain value E,,, which depends upon the location of aortic segment within the aorta, the arterial tissue essentially displays pure elastic response. For strains above this strain value, the tissue displays time dependent response. Based upon a nonlinear viscoelasticity theory, a constitutive relation involving six material constants for the characterization of the observed stress relaxation behavior has been developed. Results indicate that the material constants for normotensive and hypertensive aortas differ significantly. It is found that the effect of hypertension is to increase the in vitro stiffness of the aortas. The paper suggests the observed increase of stiffness to an increase in collagen fiber density and water absorption in the arterial wall.
INTRODUCTION
Heart and artery diseases currently account for more than 50% of the total deaths in the United States
(Newman, 1971), occurring most frequently in individuals who are in excess of 40 yr of age. Through many epidemiological studies, it is well established that atherosclerosis and arteriosclerosis are important contributing factors in over 80% of these deaths. It is likewise well estabbshed that hypertension potentiates or aggravates the severity of vascular disease, presumably both by pressure-induced vascular injury and compensatory changes in the arterial wall which represent structural adaptation to these elevated stresses. These adaptive changes alter the overall hemodynamic properties of the arterial system, and undoubtedly are related in some way to the preferential localization of atheroma and other arterial lesions (Robertson and Khairallah, 1973). Recent studies (Gow, 1970; O’Rourke. 1970) have indicated that arterial elasticity is affected in the early phases of hypertension. For instance, Gow (1970) and G’Rourke (1970) have reported marked increase in elastic modulus and decrease in arterial distensibility between normotensive and hypertensive dogs investigated under in uiao conditions. It appears that the observed increase in modulus and decrease in distensibility may be the result of elastic nonlinearity of the wall or the induced morphological changes that may occur under persistent hypertension. These mor-
phological changes may be attributed to degeneration, fibrosis and calcification. From
elastin in
CI‘LW
studies alone it is difficult to determine whether the observed changes in distensibility of arteries under hypertension are due to either of the above mechanisms or a combination of both. The present study is concerned with the iti ritro examination of rheological properties of rabbit aortas obtained from both normo- and hypertensive rabbits. Specifically, emphasis has been placed on an examination of local rheological properties from uniaxial stress relaxation and stress-strain experiments on isolated segments obtained from the ascending and two regions of the descending thoracic aorta, i.e. from the first through third intercostal arteries and in a region 1 cm anterior to the diaphragm. In particular, the variation of these rheological properties with two values of time durations of experimental hypertension have been studied, and mathematical expressions for characterization of observed nonlinear rheological response of the arteries have been developed. MATERIALS AND METHODS
Twenty-seven Dutch-belted rabbits, 1.5-2.0 kg body weight were anesthetized with Ketaset (50mg, 1 M) after having been premeditated with 5 mg acepromazine (1 M). Following a midline laparotomy and employing aseptic technique, viscera were reflected, the abdominal aorta was exposed and then subjected to a coarctation of approx. 507; between origins of the renal arteries by initially measuring the outside aortic diameter, then reducing this dia* Received 3 Nooember 1975. meter through use of sterile, 2-O silk thread. In control t The paper was presented in the 26th International animals the aorta was exposed and manipulated (sham opPhysiological Congress, 20-26 October, 1974, Delhi, India. eration). The incision was closed with 3 suture layers, i.e. 293
M. G.
294
SHARMA and
an interrupted layer of chromic 4-0 sutures to join the abdominal musculature. a subcutaneous continuous chromic 4-O suture layer. and finally interrupted skin sutures using 2-O silk thread. Animals were then given 30 units penicillin (IM). Animals were recovered and subsequently maintained in individual stainless steel cages under controlled environmental conditions (12 hr photoperiod, 2o’C. 8 air changes/24 hr). Prior to administration of anesthesia blood pressure was measured via arteriopuncture using a 23 gauge needle in the central ear artery. This needle was connected via a polyethylene cannula to a Statham 23 AC pressure transducer. and mean pressures were recorded using a Grass model 7 polygraph (Grass Instrument Corp., Quincy. Mass.). Animals were sacrificed by stunning at 2 and 4 weeks following abdominal aottic coarctation. Prior to sacrifice blood pressure was again measured in the central ear artery. Following sacrifice, the heart and entire thoracic aorta was removed, and perfused with phosphate buffered saline (PBS) to remove adhering blood. Three 0.5 cm thick sections were then removed from the aorta in the following regions: (1) ascending aorta, beginning at the left ventricular-aorta junction; (2) upper descending thoracic aorta. beginning with the first intercostal artery; (3) lower descending thoracic aorta, beginning 1.5 cm proximal to the diaphragm. The three segments of the aorta were excised with great care to avoid any branch connections. The heart was then opened, blotted to remove excess buffer, and heart weight of each animal was then noted. Additional aortic samples were obtained for analysis of aortic water content through the determination of wet weight. dry weight differences. AI1 rheological studies were completed within 2 hr following sacrifice. RHEOLOGICAL EXAMINATION OF AORTIC SEGMENTS
All rheological studies were performed using a Tensilon II (Toy0 Measuring Instrument Company.
Inside Head
T. M. HOLLIS
Japan) Universal testing apparatus that is specially suited for conducting rheological studies on biological tissues. With this apparatus the stress-strain behavior at selected rates ranging from 2 to 40mm/min may
be studied. In addition. the apparatus can be used to study the stress relaxation behavior at various values of constant strains. The apparatus is provided with a strip chart recorder for recording the experimental data. Throughout the duration of each experiment, a tissue segment under test was immersed in a plexiglass tank filled with PBS (pH 7.4, 37°C) through which a 95%0,:5%COz gas mixture was bubbled. A schematic diagram of the apparatus is shown in Fig. 1. Before starting any stress-strain and relaxation tests the tissue specimens were preconditioned by subjecting them to several load and unload cycles until the area of the resulting hysteresis loops did not vary appreciably. The magnitudes of load imposed in the preconditioning load cycles did not exceed the maximum loads used in stress-strain and relaxation tests. As the intent of this paper is to report any changes in mechanical properties of the arteries with hypertension, only mechanical properties of the tissues in the tangential direction have been evaluated for this purpose. PHYSIOLOGICAL MEASUREMENTS
The mean heart to body weight ratios (HW/BW) were (21.6 + S.E. 2) x 10m4, (22.8 f S.E. 17) x low4 and (22.9 + S.E. 6) x 10-O for control animals and those of the 2- and Cweek post-surgical groups respectively. Differences with respect to control are significant (p < 0.05) in each case. The blood pressures of 104/94, 120/111 and 133/119 (mm Hg) for each of the above groups indicate a prehypertensive condition, i.e. a state in which blood pressure is elevated but has not yet reached the defined limits for a fully established hypertensive state (Rojo-Ortega and Genest, 1968; Bolitho and Hollis, 1975) prevailing during this time. STRESS RELAXATION EXPERIMENTS
Fig. 1. A schematic diagram of test apparatus.
In the stress relaxation experiments ring segments from the upper, middle, and lower thoracic regions were subjected to various constant values of deformation, and the resulting true stresses necessary to maintain the deformations constant were evaluated. Stress relaxation behavior of control, two-week hypertensive and four-week hypertensive rabbit aortas were studied. At least nine animal aortas were tested under each category mentioned above. From the experimental data, typical stress relaxation curves for control, two-week and four-week hypertensive animals were obtained and shown plotted in Figs. 2-4. In these curves the strain values selected correspond to those that may occur in the aortic wail for the magnitudes of blood pressure under in uioo conditions. The true
Rheological
properties
‘95
of arteries
LEGEND
I
d
Upper Thorooc Aorta f. Control 0 2 Weeks Hypertensrve 0 4 Weeks Hypertensive Ring Specimens e = Tensile Strom -Experimental ---Theoretical
T
70
t
CC
Ring Specrmens c = Tensrle Strain -Experrmentol --Theoreticoi
\E’
P To6 b
\
=’
z" ?
30
5 E ‘- 20
iT f ---
i-f-f-i
--r=I.JS
0
IO
0
1
20
I 60 t fsed
I
40
I 80
I
100
I I20
stress in the relaxation test was obtained by dividing the load for a given stretch ratio i. by the instantaneous cross-sectional area as follows: P
Pi.
A
A’
I t%
I 80
I IO0
I I20
!
STRESSSTRAIF; EXPERIMENTS AT VARIOUS STRAIN RATES
Fig. 2. Variation of true tensile stress with time for rabbit aorta in stress relaxation experiment.
a=-_=-
I
40
Fig. 3. Variation of true tensile stress with time for rabbit aorta in stress relaxation experiment.
F
I
1
20
(1I
where d = true stress (kg/mm’) .A = instantaneous area of cross-section of the specimen A’ = original area of cross-section of the specimen i. = l/lo = axial stretch ratio P = applied force (kg) I = length ofthe specimen corresponding to load P lo = original length. In arriving at equation (11, we have assumed that the aortas are incompressible (Carew et al.. 1968). In each experiment the original gage length I,, was evaluated by ascertaining the distance between the pins supporting the specimen, just when the load began to register in the .X-Yrecorder of the test apparatus at the beginning of the test. The original area of cross-section for each arterial segment was evaluated by measuring its width and thickness at various locations circumferentially by a micrometer and finding the average of all the measurements. The expression for nonlinear strain used in the reduction of data is as follows (Treloar. 1958):
Ring specimens of thoracic aortas of control, twoweek hypertensive and four-week hypertensive animals were subjected to monotonically increasing tensile load at various constant crosshead sneeds. namelv 8. 10. 20 and 40 mm/min. Figures 5-7’represent the stress-strain curves obtained for control, two-week
LEGEN 80
Lower Thorocic Aorta A Control 0 2 Weeks Hypertensive 0 4 Weeks Hypertensive Ring Specimens c = Tensile Strain -Experimental ---Theoretical
t
I
0
I 20
I 40
I 80
I 80
I IO0
I I20
t fsed L
where E = tensile strain.
Fig. 4. Variation of true tensile stress with time for rabbit aorta in stress relaxation experiment.
M. G. SHARMAand T. M. HOLLIS THEORETICAL INTERPRETATION I EGFNQ 0 Upper
Thomctc
0 Middle Thoracic b Lower
An examination of the stress relaxation data indicated that the relaxation modulus representing the ratio of observed stress to imposed strain not only varied with time but also was found to be a function of the magnitude of strain. This suggests that the thoracic aorta of rabbits displays nonlinear viscoelastic behavior. A method of representing the nonlinear viscoelastic behavior has been suggested by Green and Rivlin (Green and Rivlin. 1957). According to this theory, the stress necessary to produce a deformation at time t, depends upon all previous values of the strain rates to which the material has been subjected. That is, the stress may be assumed to be a functional of history of strain as follows:
Aorto Aorto
Thomcic
Aorta
contra/ Deformation Rote (mm/min..l
a(r) = F
[
F
I
I
1 ;=--x
,
where F represents a functional t’ = past time t = present time. TENSILE
STRAIN
c
Fig. 5. True tensile stress vs tensile strain curves for rabbit aorta. hypertensive
and
Since the variation
four-week
hypertensive
animals.
of stress-strain
curves with crosshead speeds was found to be small compared to the scatter in the test data, only mean stress-strain curves with statistical variation were drawn for each part of the aorta examined, i.e. the upper, middle, and lower aortic regions.
u
When the functional F is linear, it can be represented by the Boltzmann Superposition integral that is the basis of the linear viscoelasticity theory (Flugge, 1967). This superposition integral can be written as
Thoroclc
Aorta
14
Thomcic
0 Middb
Thorocic Aorto
Aorta
A Lower
Thorocic Awta
De formotion Rotalmm./min..) T \a’20 3 b :Kx)
2.0
2.4
Fig. 6. True tensile stress vs tensile strain curves for rabbit aorta.
0
a
6 0
IO
20
f
0
08 1.6 I.2 TENSILE STRAIN l
0 Upper
4 Weeks Hypertensive
on Rote lmm./min.l
0.4
&f$
E(t - t’) dt' 0 where E(t) = the relaxation modulus for the material that is a function of time only. When the functional F in equation (1) is nonlinear and continuous, Green
Middle Thomcic &HO r
s I
a(t) =
0
0.4
0.8 TENSILE
1.2 I.6 STRAIN
2.0
2.4
l
Fig. 7. True tensile stress vs tensile strain curves for rabbit aorta.
297
Rheological properties of arteries and Rivlin (1957) have shown that the functional can be represented by an infinite series of multiple integrals as follows:
t -
t2,
t -
d+3) x -ddt, dt,
W,) -
ts)
dtI
W2) -
dt,
dt, dt,,
(5)
where El, E2, E3 . . . are Kernel functions. In equation (5) the integrand of the first term can be understood to be representing the contribution of the strain increment &(tI) to the final stress. The integrand of the second term can be interpreted as representing the joint contribution of the strain increment &(t,) and de(t2) to the final stress. For a strain history corresponding to stress relaxation as given below e(t) = M(t),
(6)
where the unit step function If(t)=
0
=1
t o,
the stress can be obtained by substituting equations (6) and (5) and evaluating the integrals as follows:
modulus by dividing the stress with strain as follows: E(1.C) = fi = E,(r) + E2(f. f)E E + E,(t. t. rk’ +
E4k
(7) we can obtain
0
0.2
0.4
Fig. 8. Variation
tk3.
(8)
E(E) = E, + E,c for E I lo, (9) where E(c) is the elastic modulus in tension corresponding to the observed nonlinear elastic response below the critical strain lO and El and E2 are the material constants. Beyond the magnitude of strain co, the observed viscoelastic response was represented by adding the fourth term in the series [equation (S)] as follows: Eke) = E, + E+ + E4(t)(c - cd3
for
E 2 lo, (10)
+ Eg(t. t, t, t)E4. (7) equation
t.
An examination of the cross plots of the relaxation modulus against strain for various constant values of time (see a typical plot in Fig. 8) indicates that below a strain value of Ed, the relaxation modulus representing the ratio of stress to strain is independent of time. This means that no viscoelastic effect is displayed by the material below a strain value of co. Beyond the value of co, a spread in relaxation modulus versus strain curves results indicates a viscoelastic response. From the cross plots of relaxation modulus data such as that shown in Fig. 8 and using a standard curve fitting procedure, it was found that the first two terms and the fourth term in the series represented by equation (8) were sufficient to describe the observed stress relaxation behavior, with the first two terms representing the time independent behavior below the critical strain of co and the fourth term representing the time dependent behavior as follows:
a(t) = E,(t)r + E&, t)E2 + E3(f, t, t)?
From
f.
the relaxation
0.6
where E(t. E) is the nonlinear relaxation modulus in tension, Edt) is the time dependent modulus representing the observed rheological response beyond the
0.8 1.0 SJRAlN c
of relaxation
modulus
1.2
with strain
L4
for rabbit
1.6
aorta.
I.8
2.0
M. G. SHARMAand T. M. HOLLIS
298
Table 1. Mechanical constants representing the rheological behavior of rabbit aortas E, X 1o-3 kg/mm2 ( -+SE.)
Ez x lo-’
0.46 + 0.01
(k SE.)
Ed x 1o-3 kg/mm’ (k SE.)
E, x 10-3 kg/mm’ (fS.E.)
(* i.E.,
4.70 It 0.6
7.84 + 0.25
3.75 f 0.12
8.76 k 0.52
19.8 + 2.2
0.27 f 0.02
3.37 * 0.45
7.4 f 0.40
2.03 5 0.20
5.72 * 0.35
40.8 k 4.5
0.24 f 0.02
5.27 k 0.80
+ 0.80
1.21 + 0.14
5.46 f 0.65
54.1 + 6.8
0.28 + 0.03
7.20 + 1.16
10.51 + 1.69
4.62 * 1.69
9.78 f 2.76
21.8 i 8.7
0.24 f 0.04
7.85 k 1.21
11.51 + 1.77
2.80 + 0.42
8.46 + 1.12
24.9 + 4.2
0.37 f 0.05
6.89 f 1.8 . 10.82 + 1.76
5.36 f 1.16
10.67 k 1.26
23.7 + 6.5
0.46 f 0.14
3.75 f 0.55
12.15 + 1.9
4.99 f 0.92
9.8 + 1.61
18.7 _+4.5
0.31 f 0.04
7.67 f 2.2
14.56 f 1.7
2.51 & 0.36
5.72 f 1.03
38.2 + 4
3.63 - 0.88
11.27 It 2.05
18.3 f 2
(&, Upper thoracic aorta Mid thoracic aorta g u Lower thoracic aorta 2 Upper thoracic aorta -Y ‘;; g Mid thoracic aorta A$ $ x Lower thoracic c aorta H
a Upper thoracic aorta % ‘@ % ; y Mid thoracic aorta 4% 2 Lower thoracic aorta
0.43 f 0.18
7.7 + 2.70
kg/mm’
7.53 If: 0.87
Expression for relaxation modulus function for rabbit aortas E(t,e) = E, + E+ + (l&e-%) SE. = standard error. strain. cc. lt, is the critical strain beyond which material displays the rheological properties.
the
An examination of the data indicated that the time dependent modulus conforms to the following mathematical relation. (11) where Ed = decay modulus associated with the relaxation
process equilibrium modulus r = relaxation time.
EC =
Table 1 shows the value of the material constants for aortas from the control, two-week and four-week hypertensive animals. Each value represents a mean of at least 9 animals. DlSCUSSlON
OF RESULTS
An examination of stress relaxation data presented in Figs. 24 indicates that the stress for each region of the thoracic aorta reaches the equilibrium value within a duration of about two minutes. The equilibrium stress for each segment of the aorta is about 90-93x of its initial value. However, the initial and equilibrium stresses are functions of imposed strain values in a relaxation test and depend upon the location of the portion of the segment within the aorta as well as the duration of mechanical stress imposed by the elevated pressure. Figures S-7 representing the stress-strain curves for both normotensive and hypertensive rabbit aortas show that the variation of strain rate has very little influence on these curves. This indicates that no rheological response is displayed by rabbit aortas in short-time stress-strain experiments for the range of imposed strain rates.
+ E,)(E - G$.
Figures 5-7 also show that the stiffness of the aortic segment depends upon its location in the thoracic aorta. For instance, stiffness of the thoracic aorta increases from the upper thoracic region to the lower thoracic region. This observation is in agreement with the companion studies conducted by other investigators (Bergel, 1961; Pate1 and Vaisnav, 1972). In order to obtain a constitutive relation for the description of the observed rheological properties, the relaxation modulus was evaluated from the experimental data. The variation of this modulus with the imposed strain (see Fig. 8) indicates that up to a certain value of strain co, the arterial wall displays essentially a perfect elastic behavior. Beyond this strain value which depends both upon the location of arterial segment in the thoracic aorta and the duration of elevated blood pressure, the tissue displays the time-dependent rheological response. Based upon the nonlinear viscoelasticity theory and the experimental data on relaxation modulus variation with strain (Fig. 8). a constitutive relation for the description of rheological properties of rabbit aortas was developed and is given by equation (10). In order to determine how well the developed constitutive relation describes the observed stress relaxation behavior, equation (10) in conjunction with material constants given in Table 1 were used to predict the stress relaxation variation with time for various strain values. The predicted stress relaxation curves were compared with the experimentally observed curves as shown in Figs. 24. These figures indicate the predicted stress relaxation curves fit very well with the experimental curves and the discrepancy between the observed and evaluated values is not more than 6%. Table 1 shows that the six material constants are needed for the description of the rheological behavior of the thoracic aorta. The physical significance of these constants may be
Rheological properties of arteries Table 2. Mechanical stress-strain Tensile strain d No. rabbits Average weight (kg) Average pressure (mm Hg) Heart wt./body wt. Upper thoracic aorta True tensile stress u (+ S.E.) x lo-’ kg/mm2 Mid thoracic aorta True tensile stress u ( f SE.) x 10V3 kg/mm2
Lower thoracic aorta True tensile stress c (+ S.E.) x 10e3 kg/mm’
299
values for normotensive and hypertensive rabbit aortas
Control
2-week hypertensive
4-week hypertensive
9 4.5 + 0.126 104194 0.00216 + 2 x 1O-4
9 4.5 + 0.126 120/111 0.00228 + 1.7 x 1O-3
9 4.5 * 0.126 1331119 0.00229 + 6 x 1O-4
0.62 0.90 1.20 1.50
5.5 * 11.5 f 21.5 + 32.0 k
0.1 0.5 1.0 2.0
6.0 f 0.1 13.0 * 0.2 24.0 + 1.0 38.0 &- 1.5
11.0 + 0.1 20.5 & 1.5 33.0 + 2.0 51.5 f 2.0
0.62 0.90 1.20
7.0 * 0.1 13.5 * 1.0 24.5 + 1.5
13.0 * 0.3 22.0 * 1.0 31.0 * 4.0
13.5 + 0.1 23.0 + 0.5 38.5 + 2.5
1.50
47.0 f 2.0
57.0 * 5
0.62 0.90 1.20 1.50
8.5 f 17.0 f 30.0 f 47.0 +
18.0 f 32.0 + 52.0 k 82.0 +
0.2 1.0 2.0 4.0
3.0 3.5 6.0 7.0
64 + 4.0 16.5 k 30.5 * 60 + 132 *
0.1 2.0 4.0 20
True tensile stress 0 = (tensile force)/(instantaneous area). SE. = standard error. Tensile strain E = (1’ - 1)/2. I = Stretch ratio. explained as follows. The constants El and E2 describe the nonlinear elastic properties of the arterial tissue. The constant E, may represent the elastic response of mostly elastin in the arterial wall with the collagen fibers being in the coiled state and not contributing very much in resisting the mechanical stress. With the increase in strain, some of the collagen fibers in the arterial tissue reach their unstretched length (Burton, 1962; Bader and Kapel, 1958) and contribute more in resisting the stress along with elastin. This type of behavior which is represented by both the constants El and EZ, presumably continues until a critical strain value E,, is reached. Beyond this strain value the arterial tissue displays a viscoelastic response. The constants Ed and E, and 7 in Table 1 are representative of these rheological properties. This rheological behavior may be attributed to the interfacial slip occurring between the stretched collagen and the elastin substance surrounding it (Sharma, 1974). The relaxation time 7, which represents the time necessary for the stress to decrease (l/e)* times its original value in a relaxation test, indicates the duration of viscoelastic transition (Ferry, 1961) that is usually observed to occur in all rheological materials. The relaxation process for the arterial tissue involves the reduction of the modulus value from its high initial value E(O,e) to a relatively lower equilibrium value E( X, E). This reduction is a gradual process that is part of a viscoelastic transition. The initial * Where r is the Naperian base. * The stiffness is defined as the ratio of stress to strain.
modulus and the equilibrium modulus can be obtained by substituting t = 0 and t = x in equations (10) and (11) as follows: E(0.e) = El + E2c + (Ed + E,)(E - coj3 E(x,E)
= E, + E,e + E,(E - E,,)~.
(12)
(13)
Equations (12) and (13) indicate that the arterial wall behaves elastically before the beginning and at the end of the viscoelastic transitions. That the influence of continued exposure of the thoracic aorta to hypertension is to alter its in vitro stiffness properties,* can be seen from the results in Table 2, where the stress values for constant values of strains for control, two-week and four-week hypertensive animals have been shown. The stress values were obtained from stress-strain curves shown in Figs. 5-7 and represent the short-time behavior. Table 2 indicates that the stress necessary to produce a given strain increases with the duration of hypeztension for upper thoracic and mid thoracic aortas. The increase of stress between two- and four-week hypertension for upper thoracic aortas is much larger than that for mid thoracic aortas. In the case of lower thoracic aortas there is actually a decrease in stress between two- to four-week hypertension. However, the stress necessary to produce the strain is higher for two-week hypertensive aortas than that for the control. Figures 2-4 representing the stress relaxation behavior also indicate the same trend as mentioned above, although for upper thoracic aorta the mean stress relaxation versus time curve for two-week hypertensive animals gives higher stress values than
M. G. SHARMAand T. M. HOLLIS
300
that for four-week hypertensive animals. The above observations from stress relaxation and short-time stress-strain curves seem to indicate that rabbit aortas under chronic hypertension display higher stiffness than that under control conditions. Two possible explanations to the observed increase in stiffness in rabbit aortas with hypertension may be given as follows. The elevated blood pressure that occurs under induced hypertension leads to a higher hoop (circumferential) stress. This higher stress may produce strains exceeding the critical strain E,, [see equation (lo)], resulting in a viscoelastic flow which may weaken the aortic wall in resisting the mechanical stresses. To compensate for the reduced strength of the aortic wall, the collagen fiber density in the wall increases which results in the observed increase in stiffness of the aorta with hypertension. Recent findings (Fry, 1972) indicate that the increased stretch of the aorta that occurs as a result of elevated blood pressure may lead to an increased permeability of the wall. This may promote increased transport of certain fluids from the blood stream. An estimate of water content in the aortas of control and hypertensive rabbits has indicated a 10% increase in the absorption of water in the aortic wall of hypertensive animals. Considering a mathematical equation (Briggs, 1966) for the tensile modulus of a material with spherical inclusions as follows, we can show that an increase in modulus of the aortic wall occurs as a result of water absorption. E _,]+A?
1
1-v
v
24-5v’ [
Eo
(14)
where tensile modulus of the aortic tissue with pores filled with water E. = tensile modulus of the aortic tissue V = volume content of the water v = Poisson’s ratio of the tissue E =
As the aortic tissue can be considered pressible (Carew et aI., 19%
as incom-
the PoissOn’s ratio v may
be assumed to be 0.5. Substitutina this value for v in equation (14), we obtain the following expression for the ratio of moduli. E
-
Eo
= 1 + 2.5 V.
(15)
Equation (15) shows that for a 10% increase in the volume fraction of the water content, the elastic modulus of the tissue increases by 25%. Although the applicability of equation (14) for the prediction of the increase in elastic modulus observed in hypertensive rabbit aortas needs to be ascertained, we introduce equation (14) to advance our thesis that an increase in the water absorption in the aortic wall may be in fact one of the causes of the observed increase in the stiffness (or elastic conditions.
modulus)
under
hypertensive
In conclusion, our experimental data on rheological properties of rabbit aortas in vitro indicate a significant increase in elastic stiffness (or modulus) as a result of hypertension. We believe that this increase in modulus is a consequence of the morphological changes occurring in the aortic wall leading to the influx of water and the possible increase in the collagen fiber density. In oitro rheological experiments are ideal for studying the morphological changes in the aortic wall resulting from hypertension as the observed changes in rheological properties due to hypertension studied from in Lliw experiments may be due to a combination of nonlinear response of the wall as well as the morphological changes. Acknowledgemenrs-This investigation was supported in part by the National Science Foundation Grant GK-41484X and by the Biomedical Sciences Support Grant RR07082 from the General Research Support Branch of the National Institutes of Health. REFERENCES Bader, H. and Kapal, E. (1958) Altersvetinderungen der Aortern elastizatat Gerontol. 2. 253-265. Bergel, D. H. (1961) Static elastic properties of the arterial wall. J. Physiol. Lond. 156, 445-457. Bolitho, G. A. and Hollis, T. M. (1975) Aortic histamine synthesis in experimental and neurogenic hypertension. Proc. Sot. Exp. Biol. Med. 148, 118%1192. Briggs, W. D. (1966) In Composife Moteriafs (Edited by Holliday, L.), p. 40. Elsevier, Amsterdam. Burton, A. C. (1962) In Handbook of Physiology (Edited by Hamilton, W. F. and Dow, P.). American Physiological Society, Washington, D.C. Carew, T. E., Vaishnav, R. N. and Patel, D. J. (1968) Circ. Res. 23, 61-68. Ferry, J. D. (1961) &coelasric Properfies of Polymers. Wiley, New York. Fliigge, W. (1967) Hscoelascicity. Blaisdell, New York. Fry, D. L. (1972) Responses of arterial wall to certain physical factors. In Atherogenesis: Initiating Factors. pp. 93-125. Ciba Foundation Symposium. Elsevier, Amsterdam. Gow, B. S. (1970) Viscoelastic properties of conduit arteries. Circ. Res. Suppl. II, 26-27; 113-122. Green, A. E. and Rivlin, R. S. (1957) Mechanics of non-
linear material with memory. Arch. Ration. Me& Anal. 1, 1-21. Newman, E. V. (1971) Arteriosclerosis. A report by the National Heart and Lung Institute Task Force on Arteriosclerosis, National Institute of Health, DHEW Pub. No. (NIH) 72-219. O’Rourke, M. F. (1970) Arterial hemodynamics in hypertension. Circ. Res. Suppl. II, 26-27; 123-133. Patel, D. J. and Vaisnav, R. N. (1972) The rheology of large blood vessels. In Cardiovascular Fluid Dynamics (Edited by Bergel, D. H.), Vol. 2, p. 1. Robertson, A. L., Jr. and Khairallah P. A. (1973) Arterial endothelial permeability and vascular disease, the “trap door” effect. Exp. Molec. Path. 18. 241-260. Rojo-Ortega, J. M. and Genest, J. (1968) A method for production of experimental hypertension in rats. Can. J. Physiol. Phmmacol. 46, 833-834.
Sharma, M. G. (1974) Viscoelastic behavior of conduit arteries. Biorheology 11, 279. Treloar, L. R. G. (1958) The Physics of Rubber Elasticity. Oxford University Press, London.
T. M. HOLLIS Department
of Biology, The Pennsylvania State University. University Park. PA, U.S.A.
Abstract-Rheological properties of rabbit aortas in eitro were studied under control and hypertensive conditions by subjecting small segments of the aorta to uniaxial stress relaxation tests for various constant strain values ranging from 0.2 to 1.5 and stress-strain tests for deformation rates ranging from 8 to 4Omm/min. An examination of experimental results indicates that below a strain value E,,, which depends upon the location of aortic segment within the aorta, the arterial tissue essentially displays pure elastic response. For strains above this strain value, the tissue displays time dependent response. Based upon a nonlinear viscoelasticity theory, a constitutive relation involving six material constants for the characterization of the observed stress relaxation behavior has been developed. Results indicate that the material constants for normotensive and hypertensive aortas differ significantly. It is found that the effect of hypertension is to increase the in vitro stiffness of the aortas. The paper suggests the observed increase of stiffness to an increase in collagen fiber density and water absorption in the arterial wall.
INTRODUCTION
Heart and artery diseases currently account for more than 50% of the total deaths in the United States
(Newman, 1971), occurring most frequently in individuals who are in excess of 40 yr of age. Through many epidemiological studies, it is well established that atherosclerosis and arteriosclerosis are important contributing factors in over 80% of these deaths. It is likewise well estabbshed that hypertension potentiates or aggravates the severity of vascular disease, presumably both by pressure-induced vascular injury and compensatory changes in the arterial wall which represent structural adaptation to these elevated stresses. These adaptive changes alter the overall hemodynamic properties of the arterial system, and undoubtedly are related in some way to the preferential localization of atheroma and other arterial lesions (Robertson and Khairallah, 1973). Recent studies (Gow, 1970; O’Rourke. 1970) have indicated that arterial elasticity is affected in the early phases of hypertension. For instance, Gow (1970) and G’Rourke (1970) have reported marked increase in elastic modulus and decrease in arterial distensibility between normotensive and hypertensive dogs investigated under in uiao conditions. It appears that the observed increase in modulus and decrease in distensibility may be the result of elastic nonlinearity of the wall or the induced morphological changes that may occur under persistent hypertension. These mor-
phological changes may be attributed to degeneration, fibrosis and calcification. From
elastin in
CI‘LW
studies alone it is difficult to determine whether the observed changes in distensibility of arteries under hypertension are due to either of the above mechanisms or a combination of both. The present study is concerned with the iti ritro examination of rheological properties of rabbit aortas obtained from both normo- and hypertensive rabbits. Specifically, emphasis has been placed on an examination of local rheological properties from uniaxial stress relaxation and stress-strain experiments on isolated segments obtained from the ascending and two regions of the descending thoracic aorta, i.e. from the first through third intercostal arteries and in a region 1 cm anterior to the diaphragm. In particular, the variation of these rheological properties with two values of time durations of experimental hypertension have been studied, and mathematical expressions for characterization of observed nonlinear rheological response of the arteries have been developed. MATERIALS AND METHODS
Twenty-seven Dutch-belted rabbits, 1.5-2.0 kg body weight were anesthetized with Ketaset (50mg, 1 M) after having been premeditated with 5 mg acepromazine (1 M). Following a midline laparotomy and employing aseptic technique, viscera were reflected, the abdominal aorta was exposed and then subjected to a coarctation of approx. 507; between origins of the renal arteries by initially measuring the outside aortic diameter, then reducing this dia* Received 3 Nooember 1975. meter through use of sterile, 2-O silk thread. In control t The paper was presented in the 26th International animals the aorta was exposed and manipulated (sham opPhysiological Congress, 20-26 October, 1974, Delhi, India. eration). The incision was closed with 3 suture layers, i.e. 293
M. G.
294
SHARMA and
an interrupted layer of chromic 4-0 sutures to join the abdominal musculature. a subcutaneous continuous chromic 4-O suture layer. and finally interrupted skin sutures using 2-O silk thread. Animals were then given 30 units penicillin (IM). Animals were recovered and subsequently maintained in individual stainless steel cages under controlled environmental conditions (12 hr photoperiod, 2o’C. 8 air changes/24 hr). Prior to administration of anesthesia blood pressure was measured via arteriopuncture using a 23 gauge needle in the central ear artery. This needle was connected via a polyethylene cannula to a Statham 23 AC pressure transducer. and mean pressures were recorded using a Grass model 7 polygraph (Grass Instrument Corp., Quincy. Mass.). Animals were sacrificed by stunning at 2 and 4 weeks following abdominal aottic coarctation. Prior to sacrifice blood pressure was again measured in the central ear artery. Following sacrifice, the heart and entire thoracic aorta was removed, and perfused with phosphate buffered saline (PBS) to remove adhering blood. Three 0.5 cm thick sections were then removed from the aorta in the following regions: (1) ascending aorta, beginning at the left ventricular-aorta junction; (2) upper descending thoracic aorta. beginning with the first intercostal artery; (3) lower descending thoracic aorta, beginning 1.5 cm proximal to the diaphragm. The three segments of the aorta were excised with great care to avoid any branch connections. The heart was then opened, blotted to remove excess buffer, and heart weight of each animal was then noted. Additional aortic samples were obtained for analysis of aortic water content through the determination of wet weight. dry weight differences. AI1 rheological studies were completed within 2 hr following sacrifice. RHEOLOGICAL EXAMINATION OF AORTIC SEGMENTS
All rheological studies were performed using a Tensilon II (Toy0 Measuring Instrument Company.
Inside Head
T. M. HOLLIS
Japan) Universal testing apparatus that is specially suited for conducting rheological studies on biological tissues. With this apparatus the stress-strain behavior at selected rates ranging from 2 to 40mm/min may
be studied. In addition. the apparatus can be used to study the stress relaxation behavior at various values of constant strains. The apparatus is provided with a strip chart recorder for recording the experimental data. Throughout the duration of each experiment, a tissue segment under test was immersed in a plexiglass tank filled with PBS (pH 7.4, 37°C) through which a 95%0,:5%COz gas mixture was bubbled. A schematic diagram of the apparatus is shown in Fig. 1. Before starting any stress-strain and relaxation tests the tissue specimens were preconditioned by subjecting them to several load and unload cycles until the area of the resulting hysteresis loops did not vary appreciably. The magnitudes of load imposed in the preconditioning load cycles did not exceed the maximum loads used in stress-strain and relaxation tests. As the intent of this paper is to report any changes in mechanical properties of the arteries with hypertension, only mechanical properties of the tissues in the tangential direction have been evaluated for this purpose. PHYSIOLOGICAL MEASUREMENTS
The mean heart to body weight ratios (HW/BW) were (21.6 + S.E. 2) x 10m4, (22.8 f S.E. 17) x low4 and (22.9 + S.E. 6) x 10-O for control animals and those of the 2- and Cweek post-surgical groups respectively. Differences with respect to control are significant (p < 0.05) in each case. The blood pressures of 104/94, 120/111 and 133/119 (mm Hg) for each of the above groups indicate a prehypertensive condition, i.e. a state in which blood pressure is elevated but has not yet reached the defined limits for a fully established hypertensive state (Rojo-Ortega and Genest, 1968; Bolitho and Hollis, 1975) prevailing during this time. STRESS RELAXATION EXPERIMENTS
Fig. 1. A schematic diagram of test apparatus.
In the stress relaxation experiments ring segments from the upper, middle, and lower thoracic regions were subjected to various constant values of deformation, and the resulting true stresses necessary to maintain the deformations constant were evaluated. Stress relaxation behavior of control, two-week hypertensive and four-week hypertensive rabbit aortas were studied. At least nine animal aortas were tested under each category mentioned above. From the experimental data, typical stress relaxation curves for control, two-week and four-week hypertensive animals were obtained and shown plotted in Figs. 2-4. In these curves the strain values selected correspond to those that may occur in the aortic wail for the magnitudes of blood pressure under in uioo conditions. The true
Rheological
properties
‘95
of arteries
LEGEND
I
d
Upper Thorooc Aorta f. Control 0 2 Weeks Hypertensrve 0 4 Weeks Hypertensive Ring Specimens e = Tensile Strom -Experimental ---Theoretical
T
70
t
CC
Ring Specrmens c = Tensrle Strain -Experrmentol --Theoreticoi
\E’
P To6 b
\
=’
z" ?
30
5 E ‘- 20
iT f ---
i-f-f-i
--r=I.JS
0
IO
0
1
20
I 60 t fsed
I
40
I 80
I
100
I I20
stress in the relaxation test was obtained by dividing the load for a given stretch ratio i. by the instantaneous cross-sectional area as follows: P
Pi.
A
A’
I t%
I 80
I IO0
I I20
!
STRESSSTRAIF; EXPERIMENTS AT VARIOUS STRAIN RATES
Fig. 2. Variation of true tensile stress with time for rabbit aorta in stress relaxation experiment.
a=-_=-
I
40
Fig. 3. Variation of true tensile stress with time for rabbit aorta in stress relaxation experiment.
F
I
1
20
(1I
where d = true stress (kg/mm’) .A = instantaneous area of cross-section of the specimen A’ = original area of cross-section of the specimen i. = l/lo = axial stretch ratio P = applied force (kg) I = length ofthe specimen corresponding to load P lo = original length. In arriving at equation (11, we have assumed that the aortas are incompressible (Carew et al.. 1968). In each experiment the original gage length I,, was evaluated by ascertaining the distance between the pins supporting the specimen, just when the load began to register in the .X-Yrecorder of the test apparatus at the beginning of the test. The original area of cross-section for each arterial segment was evaluated by measuring its width and thickness at various locations circumferentially by a micrometer and finding the average of all the measurements. The expression for nonlinear strain used in the reduction of data is as follows (Treloar. 1958):
Ring specimens of thoracic aortas of control, twoweek hypertensive and four-week hypertensive animals were subjected to monotonically increasing tensile load at various constant crosshead sneeds. namelv 8. 10. 20 and 40 mm/min. Figures 5-7’represent the stress-strain curves obtained for control, two-week
LEGEN 80
Lower Thorocic Aorta A Control 0 2 Weeks Hypertensive 0 4 Weeks Hypertensive Ring Specimens c = Tensile Strain -Experimental ---Theoretical
t
I
0
I 20
I 40
I 80
I 80
I IO0
I I20
t fsed L
where E = tensile strain.
Fig. 4. Variation of true tensile stress with time for rabbit aorta in stress relaxation experiment.
M. G. SHARMAand T. M. HOLLIS THEORETICAL INTERPRETATION I EGFNQ 0 Upper
Thomctc
0 Middle Thoracic b Lower
An examination of the stress relaxation data indicated that the relaxation modulus representing the ratio of observed stress to imposed strain not only varied with time but also was found to be a function of the magnitude of strain. This suggests that the thoracic aorta of rabbits displays nonlinear viscoelastic behavior. A method of representing the nonlinear viscoelastic behavior has been suggested by Green and Rivlin (Green and Rivlin. 1957). According to this theory, the stress necessary to produce a deformation at time t, depends upon all previous values of the strain rates to which the material has been subjected. That is, the stress may be assumed to be a functional of history of strain as follows:
Aorto Aorto
Thomcic
Aorta
contra/ Deformation Rote (mm/min..l
a(r) = F
[
F
I
I
1 ;=--x
,
where F represents a functional t’ = past time t = present time. TENSILE
STRAIN
c
Fig. 5. True tensile stress vs tensile strain curves for rabbit aorta. hypertensive
and
Since the variation
four-week
hypertensive
animals.
of stress-strain
curves with crosshead speeds was found to be small compared to the scatter in the test data, only mean stress-strain curves with statistical variation were drawn for each part of the aorta examined, i.e. the upper, middle, and lower aortic regions.
u
When the functional F is linear, it can be represented by the Boltzmann Superposition integral that is the basis of the linear viscoelasticity theory (Flugge, 1967). This superposition integral can be written as
Thoroclc
Aorta
14
Thomcic
0 Middb
Thorocic Aorto
Aorta
A Lower
Thorocic Awta
De formotion Rotalmm./min..) T \a’20 3 b :Kx)
2.0
2.4
Fig. 6. True tensile stress vs tensile strain curves for rabbit aorta.
0
a
6 0
IO
20
f
0
08 1.6 I.2 TENSILE STRAIN l
0 Upper
4 Weeks Hypertensive
on Rote lmm./min.l
0.4
&f$
E(t - t’) dt' 0 where E(t) = the relaxation modulus for the material that is a function of time only. When the functional F in equation (1) is nonlinear and continuous, Green
Middle Thomcic &HO r
s I
a(t) =
0
0.4
0.8 TENSILE
1.2 I.6 STRAIN
2.0
2.4
l
Fig. 7. True tensile stress vs tensile strain curves for rabbit aorta.
297
Rheological properties of arteries and Rivlin (1957) have shown that the functional can be represented by an infinite series of multiple integrals as follows:
t -
t2,
t -
d+3) x -ddt, dt,
W,) -
ts)
dtI
W2) -
dt,
dt, dt,,
(5)
where El, E2, E3 . . . are Kernel functions. In equation (5) the integrand of the first term can be understood to be representing the contribution of the strain increment &(tI) to the final stress. The integrand of the second term can be interpreted as representing the joint contribution of the strain increment &(t,) and de(t2) to the final stress. For a strain history corresponding to stress relaxation as given below e(t) = M(t),
(6)
where the unit step function If(t)=
0
=1
t o,
the stress can be obtained by substituting equations (6) and (5) and evaluating the integrals as follows:
modulus by dividing the stress with strain as follows: E(1.C) = fi = E,(r) + E2(f. f)E E + E,(t. t. rk’ +
E4k
(7) we can obtain
0
0.2
0.4
Fig. 8. Variation
tk3.
(8)
E(E) = E, + E,c for E I lo, (9) where E(c) is the elastic modulus in tension corresponding to the observed nonlinear elastic response below the critical strain lO and El and E2 are the material constants. Beyond the magnitude of strain co, the observed viscoelastic response was represented by adding the fourth term in the series [equation (S)] as follows: Eke) = E, + E+ + E4(t)(c - cd3
for
E 2 lo, (10)
+ Eg(t. t, t, t)E4. (7) equation
t.
An examination of the cross plots of the relaxation modulus against strain for various constant values of time (see a typical plot in Fig. 8) indicates that below a strain value of Ed, the relaxation modulus representing the ratio of stress to strain is independent of time. This means that no viscoelastic effect is displayed by the material below a strain value of co. Beyond the value of co, a spread in relaxation modulus versus strain curves results indicates a viscoelastic response. From the cross plots of relaxation modulus data such as that shown in Fig. 8 and using a standard curve fitting procedure, it was found that the first two terms and the fourth term in the series represented by equation (8) were sufficient to describe the observed stress relaxation behavior, with the first two terms representing the time independent behavior below the critical strain of co and the fourth term representing the time dependent behavior as follows:
a(t) = E,(t)r + E&, t)E2 + E3(f, t, t)?
From
f.
the relaxation
0.6
where E(t. E) is the nonlinear relaxation modulus in tension, Edt) is the time dependent modulus representing the observed rheological response beyond the
0.8 1.0 SJRAlN c
of relaxation
modulus
1.2
with strain
L4
for rabbit
1.6
aorta.
I.8
2.0
M. G. SHARMAand T. M. HOLLIS
298
Table 1. Mechanical constants representing the rheological behavior of rabbit aortas E, X 1o-3 kg/mm2 ( -+SE.)
Ez x lo-’
0.46 + 0.01
(k SE.)
Ed x 1o-3 kg/mm’ (k SE.)
E, x 10-3 kg/mm’ (fS.E.)
(* i.E.,
4.70 It 0.6
7.84 + 0.25
3.75 f 0.12
8.76 k 0.52
19.8 + 2.2
0.27 f 0.02
3.37 * 0.45
7.4 f 0.40
2.03 5 0.20
5.72 * 0.35
40.8 k 4.5
0.24 f 0.02
5.27 k 0.80
+ 0.80
1.21 + 0.14
5.46 f 0.65
54.1 + 6.8
0.28 + 0.03
7.20 + 1.16
10.51 + 1.69
4.62 * 1.69
9.78 f 2.76
21.8 i 8.7
0.24 f 0.04
7.85 k 1.21
11.51 + 1.77
2.80 + 0.42
8.46 + 1.12
24.9 + 4.2
0.37 f 0.05
6.89 f 1.8 . 10.82 + 1.76
5.36 f 1.16
10.67 k 1.26
23.7 + 6.5
0.46 f 0.14
3.75 f 0.55
12.15 + 1.9
4.99 f 0.92
9.8 + 1.61
18.7 _+4.5
0.31 f 0.04
7.67 f 2.2
14.56 f 1.7
2.51 & 0.36
5.72 f 1.03
38.2 + 4
3.63 - 0.88
11.27 It 2.05
18.3 f 2
(&, Upper thoracic aorta Mid thoracic aorta g u Lower thoracic aorta 2 Upper thoracic aorta -Y ‘;; g Mid thoracic aorta A$ $ x Lower thoracic c aorta H
a Upper thoracic aorta % ‘@ % ; y Mid thoracic aorta 4% 2 Lower thoracic aorta
0.43 f 0.18
7.7 + 2.70
kg/mm’
7.53 If: 0.87
Expression for relaxation modulus function for rabbit aortas E(t,e) = E, + E+ + (l&e-%) SE. = standard error. strain. cc. lt, is the critical strain beyond which material displays the rheological properties.
the
An examination of the data indicated that the time dependent modulus conforms to the following mathematical relation. (11) where Ed = decay modulus associated with the relaxation
process equilibrium modulus r = relaxation time.
EC =
Table 1 shows the value of the material constants for aortas from the control, two-week and four-week hypertensive animals. Each value represents a mean of at least 9 animals. DlSCUSSlON
OF RESULTS
An examination of stress relaxation data presented in Figs. 24 indicates that the stress for each region of the thoracic aorta reaches the equilibrium value within a duration of about two minutes. The equilibrium stress for each segment of the aorta is about 90-93x of its initial value. However, the initial and equilibrium stresses are functions of imposed strain values in a relaxation test and depend upon the location of the portion of the segment within the aorta as well as the duration of mechanical stress imposed by the elevated pressure. Figures S-7 representing the stress-strain curves for both normotensive and hypertensive rabbit aortas show that the variation of strain rate has very little influence on these curves. This indicates that no rheological response is displayed by rabbit aortas in short-time stress-strain experiments for the range of imposed strain rates.
+ E,)(E - G$.
Figures 5-7 also show that the stiffness of the aortic segment depends upon its location in the thoracic aorta. For instance, stiffness of the thoracic aorta increases from the upper thoracic region to the lower thoracic region. This observation is in agreement with the companion studies conducted by other investigators (Bergel, 1961; Pate1 and Vaisnav, 1972). In order to obtain a constitutive relation for the description of the observed rheological properties, the relaxation modulus was evaluated from the experimental data. The variation of this modulus with the imposed strain (see Fig. 8) indicates that up to a certain value of strain co, the arterial wall displays essentially a perfect elastic behavior. Beyond this strain value which depends both upon the location of arterial segment in the thoracic aorta and the duration of elevated blood pressure, the tissue displays the time-dependent rheological response. Based upon the nonlinear viscoelasticity theory and the experimental data on relaxation modulus variation with strain (Fig. 8). a constitutive relation for the description of rheological properties of rabbit aortas was developed and is given by equation (10). In order to determine how well the developed constitutive relation describes the observed stress relaxation behavior, equation (10) in conjunction with material constants given in Table 1 were used to predict the stress relaxation variation with time for various strain values. The predicted stress relaxation curves were compared with the experimentally observed curves as shown in Figs. 24. These figures indicate the predicted stress relaxation curves fit very well with the experimental curves and the discrepancy between the observed and evaluated values is not more than 6%. Table 1 shows that the six material constants are needed for the description of the rheological behavior of the thoracic aorta. The physical significance of these constants may be
Rheological properties of arteries Table 2. Mechanical stress-strain Tensile strain d No. rabbits Average weight (kg) Average pressure (mm Hg) Heart wt./body wt. Upper thoracic aorta True tensile stress u (+ S.E.) x lo-’ kg/mm2 Mid thoracic aorta True tensile stress u ( f SE.) x 10V3 kg/mm2
Lower thoracic aorta True tensile stress c (+ S.E.) x 10e3 kg/mm’
299
values for normotensive and hypertensive rabbit aortas
Control
2-week hypertensive
4-week hypertensive
9 4.5 + 0.126 104194 0.00216 + 2 x 1O-4
9 4.5 + 0.126 120/111 0.00228 + 1.7 x 1O-3
9 4.5 * 0.126 1331119 0.00229 + 6 x 1O-4
0.62 0.90 1.20 1.50
5.5 * 11.5 f 21.5 + 32.0 k
0.1 0.5 1.0 2.0
6.0 f 0.1 13.0 * 0.2 24.0 + 1.0 38.0 &- 1.5
11.0 + 0.1 20.5 & 1.5 33.0 + 2.0 51.5 f 2.0
0.62 0.90 1.20
7.0 * 0.1 13.5 * 1.0 24.5 + 1.5
13.0 * 0.3 22.0 * 1.0 31.0 * 4.0
13.5 + 0.1 23.0 + 0.5 38.5 + 2.5
1.50
47.0 f 2.0
57.0 * 5
0.62 0.90 1.20 1.50
8.5 f 17.0 f 30.0 f 47.0 +
18.0 f 32.0 + 52.0 k 82.0 +
0.2 1.0 2.0 4.0
3.0 3.5 6.0 7.0
64 + 4.0 16.5 k 30.5 * 60 + 132 *
0.1 2.0 4.0 20
True tensile stress 0 = (tensile force)/(instantaneous area). SE. = standard error. Tensile strain E = (1’ - 1)/2. I = Stretch ratio. explained as follows. The constants El and E2 describe the nonlinear elastic properties of the arterial tissue. The constant E, may represent the elastic response of mostly elastin in the arterial wall with the collagen fibers being in the coiled state and not contributing very much in resisting the mechanical stress. With the increase in strain, some of the collagen fibers in the arterial tissue reach their unstretched length (Burton, 1962; Bader and Kapel, 1958) and contribute more in resisting the stress along with elastin. This type of behavior which is represented by both the constants El and EZ, presumably continues until a critical strain value E,, is reached. Beyond this strain value the arterial tissue displays a viscoelastic response. The constants Ed and E, and 7 in Table 1 are representative of these rheological properties. This rheological behavior may be attributed to the interfacial slip occurring between the stretched collagen and the elastin substance surrounding it (Sharma, 1974). The relaxation time 7, which represents the time necessary for the stress to decrease (l/e)* times its original value in a relaxation test, indicates the duration of viscoelastic transition (Ferry, 1961) that is usually observed to occur in all rheological materials. The relaxation process for the arterial tissue involves the reduction of the modulus value from its high initial value E(O,e) to a relatively lower equilibrium value E( X, E). This reduction is a gradual process that is part of a viscoelastic transition. The initial * Where r is the Naperian base. * The stiffness is defined as the ratio of stress to strain.
modulus and the equilibrium modulus can be obtained by substituting t = 0 and t = x in equations (10) and (11) as follows: E(0.e) = El + E2c + (Ed + E,)(E - coj3 E(x,E)
= E, + E,e + E,(E - E,,)~.
(12)
(13)
Equations (12) and (13) indicate that the arterial wall behaves elastically before the beginning and at the end of the viscoelastic transitions. That the influence of continued exposure of the thoracic aorta to hypertension is to alter its in vitro stiffness properties,* can be seen from the results in Table 2, where the stress values for constant values of strains for control, two-week and four-week hypertensive animals have been shown. The stress values were obtained from stress-strain curves shown in Figs. 5-7 and represent the short-time behavior. Table 2 indicates that the stress necessary to produce a given strain increases with the duration of hypeztension for upper thoracic and mid thoracic aortas. The increase of stress between two- and four-week hypertension for upper thoracic aortas is much larger than that for mid thoracic aortas. In the case of lower thoracic aortas there is actually a decrease in stress between two- to four-week hypertension. However, the stress necessary to produce the strain is higher for two-week hypertensive aortas than that for the control. Figures 2-4 representing the stress relaxation behavior also indicate the same trend as mentioned above, although for upper thoracic aorta the mean stress relaxation versus time curve for two-week hypertensive animals gives higher stress values than
M. G. SHARMAand T. M. HOLLIS
300
that for four-week hypertensive animals. The above observations from stress relaxation and short-time stress-strain curves seem to indicate that rabbit aortas under chronic hypertension display higher stiffness than that under control conditions. Two possible explanations to the observed increase in stiffness in rabbit aortas with hypertension may be given as follows. The elevated blood pressure that occurs under induced hypertension leads to a higher hoop (circumferential) stress. This higher stress may produce strains exceeding the critical strain E,, [see equation (lo)], resulting in a viscoelastic flow which may weaken the aortic wall in resisting the mechanical stresses. To compensate for the reduced strength of the aortic wall, the collagen fiber density in the wall increases which results in the observed increase in stiffness of the aorta with hypertension. Recent findings (Fry, 1972) indicate that the increased stretch of the aorta that occurs as a result of elevated blood pressure may lead to an increased permeability of the wall. This may promote increased transport of certain fluids from the blood stream. An estimate of water content in the aortas of control and hypertensive rabbits has indicated a 10% increase in the absorption of water in the aortic wall of hypertensive animals. Considering a mathematical equation (Briggs, 1966) for the tensile modulus of a material with spherical inclusions as follows, we can show that an increase in modulus of the aortic wall occurs as a result of water absorption. E _,]+A?
1
1-v
v
24-5v’ [
Eo
(14)
where tensile modulus of the aortic tissue with pores filled with water E. = tensile modulus of the aortic tissue V = volume content of the water v = Poisson’s ratio of the tissue E =
As the aortic tissue can be considered pressible (Carew et aI., 19%
as incom-
the PoissOn’s ratio v may
be assumed to be 0.5. Substitutina this value for v in equation (14), we obtain the following expression for the ratio of moduli. E
-
Eo
= 1 + 2.5 V.
(15)
Equation (15) shows that for a 10% increase in the volume fraction of the water content, the elastic modulus of the tissue increases by 25%. Although the applicability of equation (14) for the prediction of the increase in elastic modulus observed in hypertensive rabbit aortas needs to be ascertained, we introduce equation (14) to advance our thesis that an increase in the water absorption in the aortic wall may be in fact one of the causes of the observed increase in the stiffness (or elastic conditions.
modulus)
under
hypertensive
In conclusion, our experimental data on rheological properties of rabbit aortas in vitro indicate a significant increase in elastic stiffness (or modulus) as a result of hypertension. We believe that this increase in modulus is a consequence of the morphological changes occurring in the aortic wall leading to the influx of water and the possible increase in the collagen fiber density. In oitro rheological experiments are ideal for studying the morphological changes in the aortic wall resulting from hypertension as the observed changes in rheological properties due to hypertension studied from in Lliw experiments may be due to a combination of nonlinear response of the wall as well as the morphological changes. Acknowledgemenrs-This investigation was supported in part by the National Science Foundation Grant GK-41484X and by the Biomedical Sciences Support Grant RR07082 from the General Research Support Branch of the National Institutes of Health. REFERENCES Bader, H. and Kapal, E. (1958) Altersvetinderungen der Aortern elastizatat Gerontol. 2. 253-265. Bergel, D. H. (1961) Static elastic properties of the arterial wall. J. Physiol. Lond. 156, 445-457. Bolitho, G. A. and Hollis, T. M. (1975) Aortic histamine synthesis in experimental and neurogenic hypertension. Proc. Sot. Exp. Biol. Med. 148, 118%1192. Briggs, W. D. (1966) In Composife Moteriafs (Edited by Holliday, L.), p. 40. Elsevier, Amsterdam. Burton, A. C. (1962) In Handbook of Physiology (Edited by Hamilton, W. F. and Dow, P.). American Physiological Society, Washington, D.C. Carew, T. E., Vaishnav, R. N. and Patel, D. J. (1968) Circ. Res. 23, 61-68. Ferry, J. D. (1961) &coelasric Properfies of Polymers. Wiley, New York. Fliigge, W. (1967) Hscoelascicity. Blaisdell, New York. Fry, D. L. (1972) Responses of arterial wall to certain physical factors. In Atherogenesis: Initiating Factors. pp. 93-125. Ciba Foundation Symposium. Elsevier, Amsterdam. Gow, B. S. (1970) Viscoelastic properties of conduit arteries. Circ. Res. Suppl. II, 26-27; 113-122. Green, A. E. and Rivlin, R. S. (1957) Mechanics of non-
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