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37
Residual Strain in Rat Left Ventricle
Jeffrey H. Omens and Yuan-Cheng Fung
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Residual stress in an organ is defined as the stress that remains when all external loads are
removed. Residual stress has generally been ignored in published papers on left ventricular
wall stress. To take residual stress into account in the analysis of stress distributions in a
beating heart, one must first measure the residual strain in the no-load state of the heart.
Residual strains in equatorial cross-sectional rings (2-3 mm thick) of five potassium-arrested
rat left ventricles were measured. The effects of friction and external loading were reduced by
submersing the specimen in fluid, and a hypothermic, hyperkalemic arresting solution
containing nifedipine and EGTA was used to delay the onset of ischemic contracture. Stainless
steel microspheres (60-100 gm) were lightly imbedded on the surface of the slices, and the
coordinates of the microspheres were digitized from photographs taken before and after a
radial cut was made through the left ventricular free wall. Two-dimensional strains computed
from the deformation of a slice after one radial cut were defined as the residual strains in that
slice. It was found that the distributions of the principal residual stretch ratios were
asymmetric with respect to the radial cut: in areas where substantial transmural strain
gradients existed, the distributions of strain components were different on the two sides of the
radial cut. A second radial cut produced deformations significantly smaller than those
produced from the first radial cut. Hence, a slice with one radial cut may be considered stress
free. Our results show that the circumferential residual strain was negative in the endocardial
region (stretch ratios were significantly less than 1.00, p<0.001 for five slices), while those in
the epicardial region were either positive or negative with a much smaller magnitude. This
residual strain distribution suggests that, in general, there is compressive circumferential
residual stress at the inner layers of the ventricle. A residual stress distribution of this nature
may reduce the endocardial peak in circumferential tensile stress at end diastole predicted by
analytical models using nonlinear material properties. (Circulation Research 1990;66:37-45)
nalysis and use of residual stress, defined as the
stress in a body when all external loads are
removed, have been part of mechanical engineering for well over a century.' 2 Only in the past few
years have these ideas been applied to biological
tissues. Several recent analyses of the arteries have
taken the initial stress into account,3'4 resulting in
decreased transmural circumferential stress gradients when the vessel is inflated. Without residual
stress, tensile circumferential stress at physiological
arterial pressures can be up to 10 times as large at the
inner wall compared with the outer wall.5
An accurate biomechanical analysis of stress and
strain must use the zero-stress state as a reference
state. Once the stress-free state is defined, residual
A
From the Department of Applied Mechanics and Engineering
Sciences (Bioengineering), University of California San Diego, La
Jolla, California.
Supported in part by the National Science Foundation grant
EE85-18559, and by the National Institutes of Health grant
HL-07089.
Address for correspondence: Jeffrey Omens, UCSD R-012, La
Jolla, CA 92093.
Received September 15, 1988; accepted July 24, 1989.
stress and externally loaded stress states may be
analyzed. Assuming that residual stress exists, the
first step in a stress analysis is the quantification of
residual strain, or the strain when a body is deformed
from the stress-free state to the no-load state. Once
the residual strain is known, then an appropriate
constitutive law, referred to the stress-free state, can
be employed to calculate the stress. The aim of the
present study is to quantitatively describe the twodimensional residual strains in an equatorial slice of
the rat left ventricle.
At the present time, no experimental measurement of residual strain in the myocardium exists.
Most stress analyses assume the unloaded left ventricle to be stress free. Critical reviews of some of
these models have been published by Huisman et al,6
Yin,7'8 and Pelle et al.9 Many of these mathematical
models show large stress concentrations at the
endocardium when the ventricle is passively inflated.
This effect is more pronounced when nonlinear constitutive properties are used.7"0 If a compressive
residual stress were present at the inner layers of the
ventricular wall, then end-diastolic endocardial stress
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Circulation Research Vol 66, No 1, January 1990
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concentrations will be reduced. Hence, the enddiastolic circumferential stress may be more uniform
across the left ventricular wall thickness.
Our initial experiments on left ventricular residual
stress used whole hearts with a meridional slit entirely
through the left ventricular wall.11,12 These initial
experiments showed the existence of residual stress,
as well as a distinctive asymmetrical deformation
pattern of the left ventricle after the slit was cut. In
the mean time, an arresting protocol and cardioplegic solution were developed, which deterred the
effects of ischemia while preserving the diastolic
mechanical properties of the tissue. To more accurately quantitate the residual strain, latitudinal slices
of the heart were made first, followed by a radial cut,
whereby the surface strain could be directly measured. These strains are only a small sampling of the
many residual strain components that are present in
the left ventricle. Even with the limited results for
residual strain obtained so far, general characteristics
of residual stress may be surmised.
It is known that biological tissues can adapt to
external stress and strain. Bone remodeling is an
example that has been well documented.13 It is
thought that cardiac hypertrophy and growth develop
in response to changes in diastolic and systolic wall
stress.14 Our general hypothesis is that growth and
change affect the stress and strain in a tissue, more
specifically the residual stress and strain. The zerostress state of an organ displays the natural shapes
and sizes of the cellular and extracellular structures
of the organ in the undeformed condition. If the cells
or extracellular matrix remodel by hypertrophy, hyperplasia, or resorption, it is necessary to describe the
change in shape and size of the cells and matrix; a
simple way to do this is to describe the change of the
zero-stress state. Hence, the changes in zero-stress
state, or equivalently the residual stress and strain,
reflect tissue remodeling.
Materials and Methods
Isolated Heart Preparation
Five adult male (200-300 g) Sprague-Dawley rats
were anesthetized with pentobarbital (50 mg/kg)
intraperitoneally. A tracheostomy was performed,
and the animal was ventilated with 70%02-30% N2
After 10 minutes of ventilation, the heart was
arrested by clamping the ascending aorta and injecting
an iced, heparinized (10,000 units/l) hyperkalemic
Krebs-Henseleit solution directly into the left ventricle
through the apex. The perfusate contained (in g/l)
NaCl 5.40, Na2SO4 0.22, NaC6H11O7 2.08, KCl 1.87,
MgCl2. 6H20 0.23, and dextrose 1.0 and was bubbled
with 95% 02-5% CO2 gas. Also added to the perfusate
were the calcium channel blocker nifedipine (2x10-4
g/l) and the calcium chelator EGTA (10 mM/l). The
arrested hearts were rapidly removed, and the ascending aorta was cannulated for coronary perfusion at
20°-23° C for 2-3 minutes using the hyperkalemic
solution with approximately 50 mm Hg pressure.
gas.
B
A
FIGURE 1. A: Schematic representation of the left ventricle,
showing equatorial slice. This extracted slice, when placed in
the holding tank, will be in the no-load condition. B: Same
slice with a radial cut directly opposite the right ventricle (not
shown). This configuration is referred to as the stress-free state.
Equatorial Slice and Radial Cut
In defining the residual strain, the stress-free state
was used as the reference state. Since cutting the
tissue causes the surface traction* acting on the cut
surface to vanish, the stress-free state can be revealed
by cutting the tissue in enough places to relieve all of
the residual stress components. If a thin slice of
tissue is obtained from the ventricle, then the stress
components perpendicular and tangential to the cut
surface of the slice become zero, and the state of
stress in the thin slice is that of the so-called "generalized plane-stress" state, which is two-dimensional.
Slicing of the rat heart was a simple and fast
procedure. The arrested, isolated heart was secured
with a pin at the apex. Holding the aortic cannula at
the other end, a single-edged razor blade was used to
slice the heart perpendicular to its axis in the equatorial region (Figure 1A). The slices were 2-3 mm
thick and were always analyzed with the base end
facing up.
A preliminary study was performed using epicardial markers that showed that this slicing technique
did not produce any measurable deformation in the
circumferential direction. In other words, within the
experimental accuracy, the shape of the equatorial
slice was the same as its configuration while part of
the intact ventricle.
After slightly drying the surface of the slice with an
absorbent cloth, stainless steel microspheres (60-100
gm diameter) were sprinkled on the slice and lightly
pressed into the tissue. The illuminated microspheres
served as a grid for strain field measurements. With
the microspheres in place, the slice was moved to a
small dish containing the same perfusate as previously described at room temperature. The slice was
completely submerged in the fluid, and any loose
microspheres were rinsed from the surface. The slice
assumed its natural shape resting on a plastic pedestal. The surrounding fluid counteracted the gravita*Surface traction, or stress vector, represents the force per unit
area acting on the surface.
Omens and Fung Residual Strain in Rat Left Ventricle
39
FIGURE 2. Equatorial slice of a rat heart
showing imbedded microspheres. The photo~
graph on the left is before radial cutting
(no-load state) and that on the right is after
radial cutting (stress-free state).
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1 cm
tional and frictional forces, leaving the slice in the
no-load state. After a photograph was taken, the free
wall of the left ventricle was cut radially (Figure 1B)
at a location opposite to the right ventricle and
photographed within 30 seconds of radial cutting. It
was imperative that the photograph of the radially
cut configuration be taken as quickly as possible so
that extraneous deformations resulting from contracture were not recorded. Photographs of an equatorial
slice before and after radial cutting are shown in
Figure 2. After the radial cut was made, the slice was
considered to be in the "stress-free" state.
three planar points is homogeneous (constant).
Selected microsphere positions were digitized in
the no-load and stress-free states from the enlarged
photographs. The position of each microsphere was
digitized five times and averaged. The x-y coordinate data were used to estimate the strain components in each area defined by three selected points
forming a "triad." The finite Lagrangian strain
components were calculated from
Opening Angle
A simple quantitative measure that reflects the
residual strain distribution in a slice of the left ventricle is the opening angle (Figure 3C). Although the
opening angle cannot give the same description of
strain as the deformation analysis described in the
next section, opening angles may be incorporated into
analytic and continuum mechanics models for stress
and strain. The opening angle of a thin latitudinal slice
of the left ventricle was defined as the angle between
the two radial lines connecting the center of the
ventricular chamber and the centerlines of the walls at
the cut edges. The steps used to estimate the chamber
center and opening angle in the stress-free state are
outlined in Figures 3A and 3B.
where As and Aso are the lengths of each side of the
triad in the stress-free and no-load states, respectively, and Aai and Aaj are the respective x and y
components of Aso. To calculate the three independent strain components Eij, three linear equations
were written for the three sides of the triad, and
solved simultaneously.
The values of principal strains are of interest. By
solving an algebraic eigenvalue problem. the principal strains EB, EB and the directions of the mutually
orthogonal principal axes were readily calculated.
The principal stretch ratios were then calculated
Strain Calculation
The basis for our finite strain distributions is the
homogeneous strain analysis from two-dimensional
continuum mechanics theory.15 This method has been
previously used for calculation of myocardial twodimensional finite strains using ultrasonic crystals,'6
and has been extended for measuring myocardial
three-dimensional finite strains using radiopaque
myocardial implants.17,18 The basic assumption of
this analysis is that the strain in an area defined by
2
2
As2-As02= 1 > 2EilEaiaaj,
j=1 1=i
using
Ai=(2Ei+ 1)012,
i=1,2.
The principal stretch ratios, A1 and A2, were determined from principal strains, E1 and E2, and represent the ratios of the segmental length at zero stress
to that at no load. The principal stretch ratios are the
maximum and minimum of the stretch ratios of line
elements in all directions initiated from a single
point. In the principal directions there is no shear.
Note that from the equations given above, the
principal directions are referred to the no-load state.
When describing a stretch ratio or a strain, one must
state to which configuration the value refers to. Since
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Circulation Research Vol 66, No 1,
January 1990
,Cut
1.
W
CL
1251
CD50
'
-,,'
TIME (MINUTES)
2
A
B
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Anterior
C
FIGURE 3. Definition of the center of a stress-free slice and
opening angle. A: Drawing of a slice in the no-load state. The
chamber center (3) of this configuration is estimated by
constructing a diameter through the epicardial edge of the cut
(1) and the most distant epicardial point (2), crossing the right
ventricular chamber. The center is chosen to be a point along
this diameter midway between the two endocardial intersections, and segment (3-4) is the no-load radius. B: Same slice
in the stress-free state. The center of a line connecting the
epicardial corners of the two cut edges is found (line 1'-1"). A
diameter is constructed through the center of this line, intersecting the most distant epicardialpoint (2'). Using the radius
defined in the no-load state, the center of the stress-free
configuration (3') is chosen to be the same distance from the
septal endocardium along its diameter. Papillary muscles were
ignored in these measurements. C: Opening angle, defined as
the angle between the two radial lines connecting the center of
the ventricular chamber and the centerlines of the walls at the
cut edges. Note the asymmetry of the posterior and anterior
free walls.
our definition of residual strain is referred to the
stress-free state, each principal stretch ratio presented
in the following sections is simply the inverse of the
stretch ratio found from the previous equations.
Results
The present investigation is concerned with the
residual strains in equatorial slices of arrested myo-
FIGURE 4. Plot of opening angle vs. time for potassiumarrested equatorial slices using diffierent arresting protocols.
Circles are from hearts arrested after excision, then perfused
with the hyperkalemic perfsa te without nifedipine and EGTA.
Triangles are from hearts arrested in vivo, with nifedipine and
EGTA added to the perfusate. The first data points (zero time)
are acquired within 30 seconds of radial cutting. Hearts in the
firstgroup (circles) show signs of contracture within the first 30
seconds, as seen by the initial rise in the opening angle. Data
from three heart slices from each group are shown.
cardium without ischemic contracture. The cardioplegic solution described in "Materials and Methods"
section was formulated to maintain the tissue in the
arrested state without contracture. To demonstrate
the effectiveness of the perfusate and protocol, opening angles (as defined in "Materials and Methods"
section) of a few slices were examined over a period
of time. In the absence of ischemic contracture, the
opening angle is a function of the residual stress
distributions (among other properties such as geometry and material constants). As ischemic contracture
sets in, the forces generated by the muscle cells cause
the opening angle to increase. If a specimen with a
radial cut is allowed to reach full contracture (approximately 1 hour later), the opening angle approaches
1800, and the stiffness of the specimen increases
dramatically.
A few preliminary studies were done with a different perfusate and a different arresting protocol.
Several equatorial slices were taken from hearts that
were arrested after aortic cannulation, with a perfusate that did not contain nifedipine or EGTA. Figure
4 shows the results from three of these experiments
(circles). The first data point (time zero) is taken
within 30 seconds of the radial cutting. It is seen that
the opening angle was increasing during the entire
data acquisition period, presumably due to the onset
of ischemic contracture. Many other experiments
showed similar results. WVhen the heart was arrested
in vivo, with nifedipine and EGTA added to the
perfusate, the opening angle remained constant for
the first few minutes after radial cutting (shown as
triangles in Figure 4). The mean initial opening angle
from 11 of these slices was 45±+100 (1 SD). In all
cases, the opening angle was estimated with the steps
outlined in "Materials and Methods" section.
Distribution of Residual Strain in Equatorial Slices
To simplify the presentation of the results of
surface residual strain measurement, only the princi-
Omens and Fung Residual Strain in Rat Left Ventricle
Q4
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Q3
Cut
Q1
Q2
FIGURE 5. Drawing of slice in the no-load condition showing points along radial (dashed) lines at which strains were
calculated. Triangular regions (triads) used for the strain
calculations are shown for two of the radial lines. The
computed strain is assumed to be at the center of each triad.
The locations of the four 900 quadrants used for averaging are
also shown (QJ-Q4).
pal strain components, in terms of stretch ratios, are
given. We have chosen to plot the stretch ratios
corresponding to the principal strains, as a function
of radius, at different circumferential locations. Along
several radial lines, triads are chosen such that their
centers lie within a given distance from the radial line
(the limit of that distance was set at 7% of the
external radius of the slice). On average, the perpendicular distance from the center of a triad to the
radial line is 3% of the external radius of the slice.
Figure 5 is a drawing of a slice in the no-load state,
showing the positions at which strains were found
along four of the radial lines. The triad regions used
for the strain calculations are shown for two of the
lines. The stretch ratio plots for one equatorial slice
are shown in Figure 6. Each small graph is a radial
distribution of the two principal stretch ratios in the
plane of the slice. Within the equatorial slices, the
principal directions of the residual strain coincide
more or less with the local radial and circumferential
directions. Hence, these stretch ratios will be referred
to as "radial" and "circumferential," although their
directions may differ somewhat from the actual radial
and circumferential directions. As an example, for
the nine stretch ratio distributions shown in Figure 6,
the directions of principal strain differ from the local
estimated circumferential-radial directions by an absolute average of 140 (range 5°-28°).
41
To examine statistical differences in these strain
distributions, we have combined data from five equatorial slices. The transmural stretch ratios have been
divided into three regions, each comprising 33% of
the wall thickness, and are referred to as endocardial,
midwall, and epicardial. In all cases, the wall of the
right ventricle and the left ventricular papillary muscles have been ignored. In the no-load configuration,
each slice has been further divided into four 900
quadrants, with the position of the cut used as a
reference (see Figure 5). Near the center of each
quadrant, a representative stretch ratio distribution
has been found, and the two principal stretch ratios
have been averaged for each of the three transmural
regions. Thus, each slice gives 12 average stretch
ratios in each of the principal directions. Figure 7
shows the means and standard deviations from five
such slices.
The average endocardial circumferential stretch
ratio from all four quadrants was 0.911+0.055 (mean+ 1
SD). A Student's t test shows this to be significantly
less than A=1.00 (p<0.001). Thus a significant compressive endocardial prestrain in the circumferential
direction exists in each of the four quadrants. The
epicardial circumferential stretch ratios were greater
than 1.00 in quadrants 1 and 2 (p=0.006 and 0.012,
respectively), with no significant differences in quadrants 3 and 4. Therefore, a significant difference exists
between endocardial and epicardial circumferential
stretch ratios. The stretch ratio distributions in quadrants 1 and 4 have different patterns. A t test comparing groups from quadrants 1 and 4 shows significant
differences in both the circumferential (p=0.004) and
radial (p=0.011) epicardial stretch ratios, with no
significant differences in the endocardial or midwall
values. Thus, there is evidence for posterior wall
versus anterior wall strain distribution variation.
Additional Deformation due to an Additional Cut
To answer the question of whether one cut is
enough to relieve all residual stress in a slice of
myocardium, a few experiments were performed in
which a second radial cut was made directly opposite
to the first. This second cut relieves normal and shear
stresses remaining in the specimen along the second
cut surfaces. The second cut induces additional deformations. These additional strains were measured
referenced to the single cut configuration (given in
terms of principal stretch ratios). Figure 8 shows the
stretch ratios at one circumferential location (shown
below plots) after one, and then after a second radial
cut. The stretch ratios due to the second cut are
referred to the configuration after the first radial cut.
The mean stretch ratios along the chosen radius
resulting from the second cut are 0.990+0.007 and
1.019+0.018 (mean+1 SD) for A1 and A2, respectively.
The differences of these numbers from unity are small
enough that they lie within the limits of experimental
accuracy of the method. Hence, the additional strain
due to the second cut can be considered small when
42
Circulation Research Vol 66, No 1, January 1990
o Circumferential
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1.20
1.20
1 8.i1
1.10
1.00
0.90
A Radial
0
0
A
A1.00
0
0.80
0.90
A
0
00
o
0.80
FIGURE 6. Plots ofprincipal stretch ratios of an equatorial slice with a radial cut through the center of the left ventricularfree wall.
The no-load geometry is shown, along with the location of the radial cut. The horizontal axis (radius) of each graph ranges from
endocardium to epicardium, although the measured stretch ratios do not span this entire range. The right ventricular wall is ignored.
The line of zero strain (A= 1. 0) is also shown. The two principal stretch ratios are referred to as circumferential (o) and radial (A)
since the principal angles correspond to these two directions. The stretch ratio values have an estimated uncertainty of 2.8% due
to digitization errors.
compared with that after one cut, and a slice with one
radial cut is considered to be free of stress.
Discussion
When a radial cut is made through an equatorial
slice of the left ventricle, the ring opens up to an arc,
showing the existence of residual stress. Although it
may be possible to measure these residual stresses
directly using some strain gauge devices, an accurate
method to measure the myocardial forces directly is
unknown.719 At the present time, the only way to
accurately assess the residual stresses in the myocardium is to calculate them from analytical or finite
element models once the residual strains have been
measured. Even if the experimental measurements
are precise, the calculation can only be as accurate as
the assumed stress-strain relation.
For diastolic residual strain measurements, ideally
the diastolic mechanical properties of the myocardium should be preserved. Although the mechanical
properties of the potassium-arrested myocardium are
probably different from the diastolic properties of
contracting muscle, our experimental protocol was
designed to measure strain in arrested myocardium
without contracture. Ischemic contracture will alter
the mechanical properties of the tissue and hence
change the measured residual strains. Certain interventions can reduce ischemic contracture,20,21 such as
hyperkalemic arrest and hypothermia.22,23 A calcium
channel blocker may provide increased protection
Omens and Fung Residual Strain in Rat Left Ventricle
Midwall
Endocardial
xi,
43
Epicardial
'\2
0
.c
a)
L-
cn)
1 234
1 234
1 234
1 234
1 234
1 234
Quadrant
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FIGURE 7. Bar graph of principal stretch ratios represented as mean and standard deviation forfive equatorial slices. The slices
have been divided transmurally into three equal regions (Endocardial, Midwall, and Epicardial) and circumferentially divided into
four quadrants (Q1-Q4). The quadrants are shown in Figure 5. Near the center of each quadrant, the transmural distributions of
the two principal stretch ratios were found and the data were averaged for each of the three transmural regions. A1 is referred to as
the "circumferential" principal stretch ratio, and A2 is the "radial. "
against contracture during cardioplegia.24 Another
perfusate additive that may help prevent contracture
is the calcium chelator EGTA. Holubarsch et a125
used a calcium-free Tyrode's solution with 10 mM
EGTA in isolated papillary experiments, and showed
that the active tension was completely abolished
without changing initial resting tensions. These perfusate additives should help the arrested myocardial
tissue retain its diastolic mechanical properties. The
o
Crcumferential
A Radial
o
CircumfarentIl
a Radbl
.g
Radhls
Deformation due
to frst cut
RFu
due to second cut
.*Second cut
FIGURE 8. Principal stretch ratios after one radial cut, and
the addition deformation due to a second radial cut. These
distributions are taken from an area near the first cut (shown
below plots). The stretch ratios after a second cut are referred
to the configuration after the first cut. Within the experimental
accuracy, the second cut does not result in substantial deformations compared with those after the first cut.
gradual increase in observed opening angle is presumably due to the effects of ischemic contracture.
Our results have shown that arresting the heart with
a hypothermic, hyperkalemic solution with EGTA
and nifedipine delayed the adverse mechanical effects
of contracture, since the opening angle remained
constant for the first few minutes after radial cutting.
We have chosen to present the principal strains
(stretch ratios); hence, normal and shear components
of the strain tensor are not explicitly given. If the
principal angles correspond to the local circumferential and radial directions, the shear strains will be
zero, and the radial and circumferential components
will be equal to the two principal strains. For simplicity, the radial direction is assumed to always pass
through the center of the no-load ventricular slices.
Due to the irregular geometry of the slices, the
estimated local radial and circumferential directions
may not be accurate. Even though our calculated
principal directions were statistically different from
the radial and circumferential directions found directly
from the slice geometry, the example of Figure 6
shows an average absolute difference of about 140. As
an example, if the principal strains are 0.1 and -0.1
(A=1.095 and 0.905), and the principal angle is 140,
the in-plane shear strain is 0.024. Other slices show
similar values for principal directions. The technique
does not account for rigid body motions or rotations.
These quantities do not contribute to the strain
tensor and must be considered separately.
The measured opening angle depends on the location chosen for the center of the ventricle. If the
center is moved 1 mm (on the scale of the slice) in the
anterior-posterior direction, then the opening angle
changes by approximately 20. If the center is moved
1 mm perpendicular to the anterior-posterior direction, the opening angle will change by about 7°. Since
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Circulation Research Vol 66, No 1, January 1990
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the inner diameter of a typical slice is <5 mm, it is
unlikely that the estimated center would differ by
more than 1 mm in two similarly shaped slices.
Variability in principal strains may arise from several
sources. The resolution of the photographs is
extremely high. Errors arise when microsphere positions are digitized. Typically, distances measured
from an enlarged photograph are on the order of 1
cm. The digitization error for this length is ±+0.02 cm.
Using the method described by Kline and
McClintock,26 the uncertainty in the calculated stretch
ratios becomes 2.8%.
The location of the radial cut may affect the
measured residual strains, as well as the opening
angle. The location of the equatorial slice may also
influence the strain distributions. Preliminary results
from slices in different regions (closer to the base or
apex) have shown different residual strain patterns.
Other sources of variability include ischemic contracture in the tissue, frictional interaction between the
specimen and the material on which it rests, and
possible movement of the microspheres, especially
when the radial cut is made.
The residual strains of equatorial slices measured
in this study do not conform to a simple, regular
pattern. Slices from similiar locations in different
hearts do show somewhat predictable strain patterns.
The anterior free ventricular wall (right side of slices)
has a strain distribution close to what might be
expected from pure bending of the ventricular wall: a
transmural gradient of circumferential principal
stretch ratio with endocardial values less than unity
and epicardial values greater that unity. Near the
intraventricular septum, the maximum stretch ratios
are smaller in magnitude. On the posterior ventricular free wall (left side of slices), the strain pattern
does not suggest simple bending of the wall. In all
cases, the endocardial circumferential stretch ratios
are less than unity. This trend is significant for all of
the locations around the circumference. In general,
this points to a compressive residual circumferential
stress for this portion of the left ventricular wall. This
is an important feature when considering stress concentration reduction at the endocardium. The nonaxisymmetric distribution of residual strain may be
due to regional variations in wall thickness and wall
curvature, or possibly caused by mechanical interaction with the papillary muscles.
Residual strain is defined with respect to the
zero-stress state. The tissue is completely free of
stress only if an arbitrary cut does not cause a
deformation. In our myocardial slices, the stretch
ratios resulting from a second radial cut are small
compared with those obtained after the first cut,
particularly in regions further removed from the
second cut. This is expected from the well-known St.
Venant principle27 because the resultant normal and
shear forces and bending moment must vanish over
the second cut. The St. Venant principle states that
in an elastic body, the influence of any system of
loads with zero resultant forces and moments will
vanish as the distance from the loads increases
indefinitely. In all probability, however, local residual
stress still exists in the slice even after two radial cuts
have been made. The only way to relieve all of these
local residual forces may be with innumerable "microscopic" cuts. At the present time, this is not experimentally feasible. Using a continuum approach, these
local stress "concentrations" are considered to exist
on a much smaller scale than the dimensions of the
slice. From this point of view, we may assume that the
stress-free state is closely represented by an equatorial slice with a single radial cut.
The opening of the radial cut may be influenced by
other factors besides the residual stress such as
localized tissue damage near the cut surfaces, interstitial fluid movement and changes in interstitial
pressure, and ischemic contracture. Even if our preparations are free from such effects, the components
of the ventricular wall that "create" the residual
stress have not been identified. Likely candidates
include the myocytes and/or the collagen network.
Studying residual stress and strain in the hypertrophic heart, wherein the collagen and muscle cells
may change in size, shape, or rheological properties,
may help resolve the question as to which components are important in the generation of the residual
forces. Residual stress has many possible implications to the physiology of the left ventricle. Direct
implications may include altering the diastolic stress
distribution in the ventricle in such a way as to lower
the endocardial tensile stress components, changes in
hypertrophic adaptation, and alteration of myocardial blood flow patterns. Indirectly, residual stress
can affect dynamic features of the heart such as force
of contraction, stroke volume, and tissue oxygen
consumption by changing the end-diastolic stress
distribution, that is, the preload on each muscle fiber.
It is evident that residual stress may be an important
factor in many facets of cardiac dynamics.
The present description of residual strain in the
left ventricle is far from complete. Three-dimensional
residual strain fields must be found for the entire left
ventricle. Although the present method cannot measure strains perpendicular to the plane of the slice, in
all probability residual stresses and strains exist in
this direction and should be quantified before a
three-dimensional model for residual stress is formulated. Even though an accurate model for residual
stress in the left ventricle has not been developed, the
zero-stress/zero-strain reference state should be carefully considered in analyses of ventricular wall stress
and ventricular mechanics. A valid model predicting
left ventricular residual stress should include the
large deformations of a nonlinear elastic material
anisotropic with respect to a fiber axis changing
continuously through the three-dimensional geometry. Many cardiologists, using relatively simple linear
models for stress, often wrongly assume that wall
stress is zero when the ventricular pressure is zero.
In conclusion, two-dimensional residual strain distributions have been found for equatorial slices of
Omens and Fung Residual Strain in Rat Left Ventricle
potassium-arrested rat hearts. Residual stresses are
relieved in a thin slice with a single radial cut
opposite to the right ventricle. A second radial cut
resulted in small deformations, indicating that a thin
slice with a single radial cut is essentially stress free.
The slices consistently opened up to an arc with an
opening angle of about 45°. The residual principal
strain distributions corresponding to this opening
were calculated and showed significant differences on
the posterior and anterior walls. In all cases, the
endocardial circumferential principal strains were
negative, which suggests a compressive residual circumferential stress component. A residual stress
distribution of this nature may reduce tensile endocardial stress concentrations predicted by mathematical
models of ventricular wall stress using nonlinear
constitutive material properties.
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
Acknowledgment
We wish to acknowledge Dr. Andrew D. McCulloch for his helpful discussions and review of this
manuscript.
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KEY WORDS * residual stress * initial stress * initial strain
zero-stress state
.
Residual strain in rat left ventricle.
J H Omens and Y C Fung
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Circ Res. 1990;66:37-45
doi: 10.1161/01.RES.66.1.37
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