Measurements on the Elasticity and Damping
of Isolated Aortic Strips of the Dog
By
RICHARD W. LAWTON,
M.D.
The ex|3erimentiil data obtained on the viscoelastic behavior of isolated aortic strips of the dog
consist of: (]) the resonant or natural frequency of forced or free longitudinal vibration; (2) the
logarithmic decrement. From these data, static and dynamic moduli and the time constants for
the specimens are computed. Variations of these values with frequency, temperature and elongation are presented. The experimental observations are not adequately explained by a simple
mechanical analogue. An alternative explanation is offered based on studies of dynamic elasticity
in high polymers.
Downloaded from http://circres.ahajournals.org/ by guest on June 14, 2017
B
ECAUSE T H E AORTA functions in a
dynamic fashion, the experimental
evaluation of its mechanical properties
by dynamic methods is highly pertinent.
However, the normal distending force for the
aorta is complex. The characteristic shape of
the pulse wave results from a large number of
sinusoidal harmonically related waves of
varying amplitudes and frequencies. The rapid
pressure fluctuations in early systole are represented by high frequency components; in late
diastole low frequencies prevail so that the
distension of the segment is nearly static. In
addition, the static volume-pressure curve for
the aorta is nonlinear so that variations in the
mean pressure within it may be expected to
change its mechanical properties as a whole.
Thus, the elastic and viscous behavior of the
aorta during life may vary in a complex
fashion throughout the cardiac cycle and from
cycle to cycle.
The experimental data and the analysis that
follows is an attempt to simplify the complex
problem of the mechanical behavior of the
aorta under dynamic conditions. The measurements consist of: (1) the resonant or natural
frequency obtained during forced or free vibration of isolated aortic strips of the dog, (2)
the logarithmic decrement of the freely vibrat-
ing specimen, (3) the static elongation for a
given load and, (4) the temperature. The
amplitude of vibration was usually held to less
than 5 per cent of the specimen length. At these
amplitudes the length-tension relation for
aortic strips is assumed to be linear.
METHODS
The apparatus for the study of free and forced
longitudinal vibrations in isolated aortic strips is
shown diagrammatically in figure 1. Data on 30
strips from 15 animals were studied. The strips were
cut in the long axis from the opened aorta and
measured approximately 1 X 5 cms. Specimens were
sometimes used fresh but often were stored overnight
or longer in Ringers solution in the icebox. The thickness was measured with a stage micrometer. The
strip was suspended between spring clamps; the
upper clamp was held in the chuck of a phonograph
cutter head which was set in vibration by a low
frequency oscillator-amplifier combination. The
amplitude of vibration of the lower end of the
specimen was detected by means of a light-shielded
photovoltaic cell (Photovolt #S90) and a dc light
source. The low frequency spectrum (0-25 c.p.s.)
was explored, and the resonant frequency, that is,
the frequency for maximum amplitude, determined
for each applied weight by observation of the signal
on an oscilloscope. Measurements of the logarithmic
decrement were carried out by photographing the
slow-sweep oscilloscope trace of the decline in
amplitude during free vibration. Arrangement was
made to modulate the trace with a 60-cycle timing
signal in order to improve the accuracy of the
measurement of frequency during free vibration.
In general, the amplitude of vibration of the low
end of the strip was held to below five per cent of
the specimen length. At these levels, little or no
frequency variation occurred during free vibration.
From the Aviation Medical Acceleration Laboratory, U. S. Naval Development Center, Johnsville and Department of Physiology, University of
Pennsylvania School of Medicine, Philadelphia, Pa.
This investigation was supported in part by a research grant from the National Heart Institute, U. S.
Public Health Service.
Received for publication February 28, 1955.
Experiments were performed in moist air; the
specimen was suspended within a tubular lucite
403
Circulation Rrtearck, Volume III, Juli IMS
404
ELASTICITY AND DAMPING OF AORTIC STRIPS
Downloaded from http://circres.ahajournals.org/ by guest on June 14, 2017
temperature-controlled moist chamber, fitted on one
side with an optically flat window through which
length measurements of the specimen were made by
means of a precision cathetometer. The distance
between two ink dots placed in the middle section
of the strip was determined. This procedure reduced
end effects on the measurements. Suitably stable air
temperatures and high humidity were obtained by
bubbling air through a reflux condenser containing
water of known temperature which opened into one
side of the chamber. At high humidity, fogging was
prevented by thermal radiation from a thermostatically controlled soldering iron directed at the
area through which length measurements were
made. At equilibrium, temperatures were stable to
within approximately ±0.5 C. Both air temperature
and specimen temperature were measured by copper-constantan thermocouples. When a wide range
of temperatures was required (0-60 C), experiments
were performed in a large walk-in chamber (KoldHold) where a high humidity could be maintained.
The experimental procedure was as follows:
before beginning any stress-strain experiment, the
strip was stretched rapidly by hand to 50-75 per
cent of its initial length and then released. Several
cycles of such stretching and recoil enhanced the reversibility of subsequent stress-strain measurements,
as has been noted by other investigators.1 • ! Initially,
the largest weight to be used was hung on the specimen which was allowed to elongate to constant
length or until the rate of elongation became insignificant. The specimen was then progressively
unloaded. Before measuring the length for a given
load, the resonant frequency was determined and
the logarithmic decline during free vibration photographed.
The specimen then was allowed one to two minutes
to reach equilibrium length and temperature. After
all the weights were removed, the specimen was
progressively loaded again and the resonant frequency and the logarithmic decrement determined
as before. Finally, the specimen was completely
unloaded and the unstretched length determined.
EQUATIONS FOR ANALYSIS OF THE DATA
The experimental data may be expressed in terms,
of a simple model composed of a parallel elastic and
viscous component. For a mass of m grams hanging
from an elastic element of constant G dynes/cm,
and a parallel viscous element of damping constant
b dynes sec./cm., the frequency of simple harmonic;
motion is given by Page3 as
G/m - (6/2m)J
(1)
where co» = 2TT times /, the natural frequency in
c.p.s. The logarithmic decrement is
5
(2)
"» 1T&/W1O).
For the case of forced vibration the resonant frequency cor is
wr! = G/m - (b/m)5
(3)
Ordinarily when the damping is small the last term
of equations 1 and 3 may be neglected so that
Wa1 = a)r* = G/m. In the calculation of G for aortic
strips the damping term has been omitted. Thisleads to an error in G of only one per cent or less.
While G and b are characteristic of the model of
the particular specimen under test, the modulusof elasticity E, and the coefficient of internal viscosity 7, representative of the tissue as a whole, are,
according to Ballou and Smith,-4
E = (L/A)G dynes/cm.1,
(4>
7 = (L/A)b dynes sec./cm.2
(5)
and,
where L = the stretched sample length in cm., and
A •» the stretched cross-sectional area, in cm.1"
The relaxation time or time constant r is
T
™ y/E
— b/G sec.
(6)
The relaxation time T is the time required for the
stress to relax to 1/e of its initial value. For small
damping 5 = TTGOT and the phase angle <f> between the
displacement and the force according to Nolle' is
given approximately by
Tan <f> — 5/v =
FIG. 1. Diagram of apparatus. Specimen suspended from cutter head (C.H.) supporting weight
(WT.) within moist chamber. OS, low frequency
oscillator; AM, amplifier; SC, oscilloscope;
PRE-AM., oscilloscope preamplifier; PC, photovoltaic cell; D.C.L.T., dc light source; T, thermocouple; CATH., cathetometer.
GOT
(7)
For these experiments on aortic strips <f> was usually
less than 5 degrees.
RESULTS
Effects of Temperature. In figure 2, at a constant load of 37.5 grams, the values for the
resonant frequency GO, the logarithmic decre-
405
RICHARD W. LAWTON
Downloaded from http://circres.ahajournals.org/ by guest on June 14, 2017
25
35
40
TEMP
45
CO
55
Fici. 2. Effects of temperature on resonant frequency (u), logarithmic decrement (&), resting length
(Lo), stretched length (/>), and ratio L/La(a). Ordinates for L and Lo = dot separation in mms. The
resonant frequency u «• 2rf where / = frequency
in c.p.s.
ment 5, the resting length Lo, and the extended
length L for a typical isolated dog aortic strip
are shown as a function of temperature. The
relative change in length, L/Lo or a, also is
computed. Variations in resonant frequency
are observed between 20 and 40 C where there
is a general trend towards a lower frequency
with higher ternperatures. In other experiments, at temperatures below 15 and above
45 C, a sharp rise in frequency occurred. Although the logarithmic decrement seems to
parallel the resonant frequency between 26
and 38 C in figure 2, the relationship is not
•exact. The calculated spring constant, G,
has fallen about 13 per cent over this temperature interval. Although the magnitude of the
viscous term in equation (3) is reduced by 47
per cent over this same temperature range, its
contribution to the calculated value of the
spring constant at 38 C is only 0.10 per cent
while at 2G C it is 0.26 per cent. This substantial change in the viscous term thus exerts
a negligible effect on the value of the spring
constant as measured by resonant frequency
determinations under these experimental
conditions and another explanation for the frequency change must be sought. At temperatures below 15 C and above 45 C the logarithmic decrement rises markedly. Changes in the
stretched and unstretched length of the specimen with temperature always occur. As a
general rule, however, the ratio L/Lo is
relatively independent of temperature in the
physiological range as shown in the diagram
so that it has been used to present the data in
the subsequent sections.
Dijnaviic and Static Moduli. In figure 3,
values for the elastic moduli for a typical
isolated aortic strip are shown under static
(E.) and dynamic conditions (Ed) at constant
temperature. Because both E. and EH are
characteristic of the specimen only at a given
elongation the stretched length (L) and the
stretched cross-sectional area (A) must be
used in the calculations. Previous investigations8 have established that elongation of
isolated aortic strips is isovolumetric. Accordingly, the unstretched length Lo and
unstretched cross-sectional area Ao are converted to their stretched counterparts by the
relations L = LQCX, and A = A0/a. The
equations for the static and dynamic moduli for
figure 3 are then given by:
AAL
L
A
dynes/cm.2;
(a. - \)Ao
GLo
dynes/cm.4
(8)
(9)
RELATIVE LENGTH (L/Lo)
FIG. 3. Comparison of static (E.) and dynamic
(Ed) moduli at various elongations for typical aortic
strip of the dog.
406
ELASTICITY AND DAMPING OF AORTIC STRIPS
TABLE 1.—Effect of Temperature on the Static (E,)
ami Dynamic (Ed) Moduli, the Relative Length (a),
the Product (or) and the Resilience (R)
Temp F grama
26.7
37.7
50.2
50.2
Liu
1.280
1.287
E, x io« Ed x 10"
2.94
2.89
6.47
5.76
0.044
0.036
0.69
0.79
Downloaded from http://circres.ahajournals.org/ by guest on June 14, 2017
Over the entire range of elongations the
dynamic moduli are higher than the static
values. A similar elevation is found in the case
of many high polymers studied by dynamic
methods.7 Both static and dynamic moduli
may have minimum values at small elongations. The curvature and the dependence on
elongation under dynamic conditions is much
more pronounced than under static conditions.
In table 1, 15 pairs of observations on 8
strips from 4 dogs are given illustrating the
effects of temperature on the static and dynamic moduli. To obtain these data the procedure
employed was as follows: Observations were
first made at room temperature (average value
26.7 C). The temperature then was raised
(average value 37.7 C) and, after equilibration
for approximately one hour at the new temperature, the observations were repeated. Only
two temperatures were used. Values given in
the table are averages and include a range of
loads from 12.5 to 107.5 Gm. Xote that
despite a 10 C change in temperature, the
average relative length a is nearly constant so
that the average E, is also approximately
constant. The dynamic modulus of elasticity
Ed, however, decreases with a rise in temperature.
Viscous Behavior. The relaxation time is independent of the specimen dimensions and is,
therefore, experimentally a reliable measure of
the viscous properties of the specimen (equation 6). A number of quantities may be derived which are primarily dependent on the
relaxation time. For instance, the fraction of
the vibration energy persisting in the second
of two successive free oscillations, the so-called
dynamic work recovery or resilience,-4 is given
by
R = e —2 r u T
(10)
Values for WT and R are given in table 1. The
resilience is a measure of the magnitude of the
hysteresis loop and is temperature dependent.
For length-tension experiments its value is
equal to the work area below the retraction
curve (work recovery) divided by the work area
below the elongation curve (work input).
A general relation between w and r appears
to exist in the case of aortic strips over the
limited frequency range of our data. This suggests that cor might be treated as a characteristic constant for the tissue under these experimental conditions, as has been noted for
some polymer systems by other investigators.4
In figure 4 data from two dogs are presented.
At constant temperature (28 C) four static
stress-strain cycles were performed on a single
aortic strip from each dog. Values for a> and r
were determined at various points along the
curves. At the end of each stress-strain cycle
approximately 1 cm. was cut from the end of
the strip. This alteration in length produced a
somewhat wider range of frequencies. The
average value for ur is 0.077 for these experiments and the solid line in the diagram roughly
approximates the experimental points. At this
temperature the resilience is thus approxi-
0 004.
0003
0.00
20
30
40
50
60
70
8 0
9 0
FREOUENCY (CJ)
FIG. 4. Plot of resonant frequency (u) vs. relaxation time (T). Solid line equals theproduutur = 0.077.
For explanation see text.
RICHARD W. LAWTON
Downloaded from http://circres.ahajournals.org/ by guest on June 14, 2017
12 13 14 15 16 17
RELATIVE LENGTH [ L / L o )
Fia. 5. Variations of the resonant frequency (u),
tlie logarithmic decrement (S), the spring constant
(0), and the "coefficients of hysteresis" UT and u6
with elongation. Note that the ordinate for UT has
been expanded as compared to that for S, although
I0
6/r
II
— UT.
mutely 0.62. Given cor constant and because
ub/G = cor, one might expect the quantity
col) to follow quite closely the form of the
curve for the spring constant, G. This appears
to be only approximately the case for many
high polymers7 and also for aortic strips as
indicated in figure 5. The value of cor goes
through a minimum. This elongation, where
viscous loases are a minimum, represents the
area of maximum energy transfer. In the intact
dog maximum energy transfer in the aorta
appears to take place in the range of normal
mean aortic pressure according to the experiments of Peterson.8 "Whether identical
elongations of the aorta are involved requires
further study. Preliminary calculations indicate that this is the case.
DISCUSSION
A departure from physiological conditions
has been made in this study and certain pre-
•±07
cautions taken in order to simplify analysis
of the viscoelastic behavior of the aorta. The
preliminary stretching procedure brings about
reversible static stress-strain behavior and,
in addition, reduces the magnitude of the
viscous losses so that the contribution to the
calculated dynamic modulus by this factor is
small (equations 1 and 3). This latter effect
may be related in part to the disappearance of
active smooth muscle participation in the
elastic response of fresh specimens. Such losses
as are present may be related primarily to the
fibrous structures in the specimen.
When precautions concerning the nature of
the vibration and the state of the vessel are
taken, and the data are analyzed by a simple
mechanical analogue, the so-called "elastic
constant" of the system varies continuously
with elongation under both static and dynamic
conditions. The elastic behavior of isolated
aortic strips, therefore, cannot be considered
as linear except over a small range of elongation.
The increase of the elastic moduli under
dynamic conditions and the decrease in the
dynamic moduli with a rise in temperature are
not readily explained. The contribution of the
viscous term to the value of the dynamic
modulus in both cases is small and conventional
concepts of damping (proportional to velocity)
apparently cannot account for the experimental
observations. A reasonable suggestion, however, has been made8 on the basis of studies of
the dynamic behavior of polymeric systems in
general. If the specimen is imagined as being
composed of a randomly intermeshed network
of molecular chains, the uncoiling process under
dynamic conditions might be impeded by
network entanglement. Because of these entanglement effects, portions of the molecular
chains do not participate in the elastic response; the chain as a whole behaves as though
it were shorter and, thus, of more limited extensibility. An elevated modulus results. As the
frequency changes (at constant temperature),
the percent of the network which responds,
changes, and accordingly, the time constant
for the system changes such that cor appears to
be approximately a constant (fig. 4). As the
temperature is increased, the thermal motions
408
ELASTICITY AND DAMPING OF AORTIC STRIPS
Downloaded from http://circres.ahajournals.org/ by guest on June 14, 2017
of the molecules increase, entanglement is
reduced, uncoiling is enhanced and the dynamic
modulus is reduced. The observed viscous behavior of the specimen may be interpreted,
then, as related only to that portion of the network participating in the responses and to vary
with the percent of the network involved.
Even for relatively low frequencies of
vibration this model suggests that the untangling of the molecular chains during extension
of the specimen might be noticeably different
from their behavior during recoil. Presumably
these differences might account in part for the
hysteresis loops observed in biological tissues
such as the aorta. While the behavior of the
specimen during extension might be quite
variable, depending upon the manner of uncoiling the chains, the behavior during recoil
might be more reproducible, as appears to be
the case for aortic specimens.1'2 The dependence of the elastic modulus and the damping on
the driving frequency has physiological meaning in terms of the frequency spectrum of the
pressure pulse in the aorta during life. To the
extent that these data are applicable to the
living situation, it would appear that the high
frequency components present during early
systole should result in a larger modulus, a
small displacement and small damping. In late
diastole, where low frequencies predominate,
a smaller modulus, a larger displacement and
greater damping are to be expected.
CONCLUSIONS
Measurements of the static and dynamic
moduli and the damping in isolated aortic
strips of the dog indicate that changes in the
dynamic modulus with temperature and the
increase of the dynamic over the static moduli
cannot be accounted for on the basis of conventional viscous behavior. A qualitative explanation of these changes based on variable
entanglement of molecular chains in a polymer
model is offered.
ACKNOWLEDGEMENT
It is with pleasure that the author acknowledges
the skillful technical assistance of Mr. Herman
Sharma and the helpful criticism of Professor Allen
King of the Department of Physics, Dartmouth
College.
REFERENCES
1
REMINGTON, J. W., HAMILTON, W. F. AND DOW, P.:
Some difficulties involved in the prediction of the
stroke volume from the pulse wave velocity.
Am. J. Physiol. 144: 536, 1945.
'KRAFKA, J., JR.: Comparative study of histophysics of aorta. Am. J. Physiol. 125: 1, 1939.
PAGE, L.: Introduction to Theoretical Physics.
1
New York, Van Nostrand, 1935.
BALLOU, J. W. AND SMITH, J. C : Dynamic meas* urements of polymer physical properties. J.
Appl. Physics 20: 493, 1949.
6
NOLLE, A. W.: Dynamic mechanical proi>erties of
rubber-like materials. J. Polymer Sc. 5: 1, 1950.
6
LAWTON, R. W.: The thermoelastic behavior of
isolated aortic strips of the dog. Circulation Research 2: 344, 1954.
T
MOONEY, M. AND BLACK, S. A.: Elongation hysteresis of hevea and synthetic elastomers.
Canadian J. Res. 28F: S3, 1950.
8
PETERSON, L. H.: The dynamics of pulsatile blood
flow. Circulation Research 2: 127, 1954.
1
MEYER, K. H.: Natural and Synthetic High Polymers. Ed. 2, New York, Interscience, I960, p.
737.
Measurements on the Elasticity and Damping of Isolated Aortic Strips of the Dog
RICHARD W. LAWTON
Downloaded from http://circres.ahajournals.org/ by guest on June 14, 2017
Circ Res. 1955;3:403-408
doi: 10.1161/01.RES.3.4.403
Circulation Research is published by the American Heart Association, 7272 Greenville Avenue, Dallas, TX 75231
Copyright © 1955 American Heart Association, Inc. All rights reserved.
Print ISSN: 0009-7330. Online ISSN: 1524-4571
The online version of this article, along with updated information and services, is located on the
World Wide Web at:
http://circres.ahajournals.org/content/3/4/403
Permissions: Requests for permissions to reproduce figures, tables, or portions of articles originally published in
Circulation Research can be obtained via RightsLink, a service of the Copyright Clearance Center, not the
Editorial Office. Once the online version of the published article for which permission is being requested is
located, click Request Permissions in the middle column of the Web page under Services. Further information
about this process is available in the Permissions and Rights Question and Answer document.
Reprints: Information about reprints can be found online at:
http://www.lww.com/reprints
Subscriptions: Information about subscribing to Circulation Research is online at:
http://circres.ahajournals.org//subscriptions/