Vol. 273, Issue 4, H1968-H1976, October 1997
Three-dimensional residual strain in midanterior canine left
ventricle
Kevin D.
Costa1,
Karen
May-Newman1,
Dyan
Farr2,
Walter G.
O'Dell1,
Andrew D.
McCulloch1, and
Jeffrey H.
Omens2
Departments of 1 Bioengineering
and 2 Medicine, University of
California, San Diego, La Jolla, California 92093
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ABSTRACT |
All previous studies of residual strain in the
ventricular wall have been based on one- or two-dimensional
measurements. Transmural distributions of three-dimensional (3-D)
residual strains were measured by biplane radiography of columns of
lead beads implanted in the midanterior free wall of the canine left
ventricle (LV). 3-D bead coordinates were reconstructed with the
isolated arrested LV in the zero-pressure state and again after local
residual stress had been relieved by excising a transmural block of
tissue. Nonhomogeneous 3-D residual strains were computed by finite
element analysis. Mean ± SD (n = 8) circumferential residual strain indicated that the intact unloaded
myocardium was prestretched at the epicardium (0.07 ± 0.06) and
compressed in the subendocardium (
0.04 ± 0.05). Small but
significant longitudinal shortening and torsional shear residual
strains were also measured. Residual fiber strain was tensile at the
epicardium (0.05 ± 0.06) and compressive in the subendocardium
(0.01 ± 0.04), with residual extension and shortening, respectively, along structural axes parallel and perpendicular to the
laminar myocardial sheets. Relatively small residual shear strains with
respect to the myofiber sheets suggest that prestretching in the plane
of the myocardial laminae may be a primary mechanism of residual stress
in the LV.
zero-stress state; fiber architecture; cleavage planes; cardiac
mechanics; finite element analysis
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INTRODUCTION |
IT IS NOW WELL RECOGNIZED that the resting left
ventricular wall contains residual stress in the absence of luminal or
pericardial pressure or any other external loads (2, 14, 25).
Transmural distributions of the associated residual strains have been
measured in short-axis rings of isolated left ventricle (LV), which
spring open into an arc when residual stress is relieved by a radial cut (14). The stress-free configuration has been characterized by the
"opening angle" of this arc, which is ~45° in the adult rat
(14, 19). The opening angle is species dependent (16), changes during
development of the embryonic chick heart (25), and may be altered by
ventricular growth and hypertrophy (17).
When residual strain is included in mechanical models of LV filling (4,
12, 24), the resulting residual stress is compressive at the
endocardium and tensile at the epicardium, helping to reduce the
endocardial stress concentrations that might otherwise occur during
diastole. At the myocyte level, Rodriguez et al. (19) found that
residual stress causes sarcomeres to be a mean of 0.13 µm shorter at
the endocardium than at the epicardium in the unloaded rat LV. The
implications of this finding for systolic fiber stress can be
appreciated when one considers the steepness of the isometric tension-sarcomere length relation (1, 26). Hence, the presence of
residual stress and strain may significantly influence ventricular mechanical function throughout the cardiac cycle.
To date, measurements of ventricular residual strain have been one
dimensional (19) or two dimensional (14), but myocardial deformations
in the intact heart are three dimensional (8, 27). Therefore, to obtain
a complete description of the stress-free state of the myocardium, we
arrayed radiopaque beads across the anterior wall of the canine LV and
recorded their motion when residual stress was relieved locally. The
measured deformations were used to compute transmural distributions of
three-dimensional residual strain. The components of residual strain
were referred to structural axes constructed from histological
measurements of the three-dimensional myocyte and connective tissue
organization. The analysis showed that there are substantial
three-dimensional components of residual strain not previously
described and that the primary residual stress-bearing structures tend
to align with the local myocardial sheet orientation.
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METHODS |
All animal studies were performed according to the National Institutes
of Health (NIH) "Guide for the Care and Use of Laboratory Animals." All protocols were approved by the Animal Subjects
Committee of the University of California, San Diego, which is
accredited by the American Association for Accreditation of Laboratory
Animal Care.
Experimental protocol.
Eight adult mongrel dogs (21-41 kg) were pretreated with 10 mg
oral nifedipine the day before the study. Each dog was anesthetized to
a surgical plane with pentobarbital sodium (25-30 mg/kg),
intubated, and ventilated with positive pressure. The heart was exposed
by a median sternotomy and bilateral thoracotomy and supported in a
pericardial cradle. Three columns of four to six lead beads (1 mm diam)
were inserted 10 mm apart in a triangular array into the midanterior LV
free wall at approximately two-thirds the distance from base to apex. A
larger (2 mm) surface bead was sutured to the epicardium over each
column. Long-axis reference markers were sutured at the apical dimple
and at the first bifurcation of the left main coronary artery (Fig.
1A).
The five surface markers were later used to define a local system of
"cardiac coordinates" aligned with the circumferential
(x1),
longitudinal
(x2), and radial (x3)
axes of the LV (10).
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Fig. 1.
Schematic diagram of method for defining structurally based coordinate
systems used to study 3-dimensional (3-D) residual strain.
A: isolated heart in its intact
unloaded state showing 5 surface markers used to define cardiac
coordinates
(x1,
x2,
x3) aligned with local
circumferential
(x1),
longitudinal
(x2), and
radial (x3)
axes of left ventricle. B: excised
stress-free block of tissue containing transmural bead set. Bold lines
indicate intersection of a hypothetical sheet with each of 3 orthogonal
cardiac coordinate planes. Fiber angle,  , is measured in epicardial
tangent (1-2) plane
(x1 = 0°).
Cleavage plane angles,  ' and ", are measured in
(2-3) and (1-3) transverse planes, respectively
(x3 = 0°).
For all 3 angle measurements, a clockwise rotation relative to 0°
represents a negative angle. C: at
each transmural depth, a sheet angle, , is computed from 3 intersection angles and is used to define a local system of fiber-sheet
coordinates
(xf,
xs,
xn) aligned with
structural axes of myocardial laminae. Fiber axis
xf is obtained by
a rotation about radial
x3 axis through
fiber angle . A subsequent rotation about
xf through sheet
angle yields sheet axis
xs, which is
normal to xf and
lies in sheet plane.
xn is mutually
orthogonal sheet normal axis.
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Each animal was heparinized (3 mg/kg), the great vessels were ligated,
and the heart was arrested by coronary perfusion with a hypothermic
cardioplegic solution containing (in g/l) 60.0 dextran, 9.0 NaCl, 4.48 KCl, 3.0 2,3-butanedione monoxime (BDM), and 0.0002 nifedipine. The
heart was excised and rinsed, and at this point, four of the hearts
were used for a separate isolated heart study that involved passively
inflating the LV several times with and without coronary perfusion (8).
After this study, all testing devices were removed, and the heart was
allowed to float in a bath of room temperature cardioplegic solution.
In this unloaded state, the positions of the lead beads in the LV were
recorded with biplane fluoroscopy on standard 30-Hz VHS videotape using charge-coupled device videocameras (Cohu, 4915-2000). In the four hearts used for the perfusion study, the unloaded configuration was
obtained an average of 114 min after arrest. In the remaining experiments, the unloaded state was measured immediately after isolation of the heart, at an average of 27 min after arrest.
Next, an extra reference marker was attached near one epicardial bead
to facilitate column identification in the radiograms. A block of
tissue spanning the LV wall thickness and containing the entire
transmural bead array, ~2.5 cm square (Fig.
1B), was then cut from the LV free
wall with a scalpel to locally relieve residual stress. The tissue
block was suspended in fluid so that all of the beads were visible in
both views of the biplane X-ray, and video was recorded. At the end of
the experiment, a three-dimensional geometric phantom was recorded and
later used to compute the transformations needed to reconstruct the
three-dimensional coordinates of the beads in the unloaded and
stress-free configurations (8).
Morphological studies.
After radiography, the tissue block was fixed by immersion with 10%
formaldehyde in phosphate buffer for at least 48 h. A transmural sample
of the block was dehydrated, embedded in paraffin, and used to measure
muscle fiber orientation in the circumferential-longitudinal (1-2)
plane. At 8-11 points across the wall, 20-µm-thick sections were
obtained parallel to the epicardium and stained with hematoxylin. By
use of a video camera (Sony, DXC-151) mounted on a light microscope (Nikon, Optiphot-2), low-power (×20) images of the serial tissue sections were acquired onto a microcomputer (Apple, Macintosh Quadra
900) using NIH Image software. At each transmural depth, the mean fiber
angle was obtained from at least five measurements in each image.
Angles were measured relative to one edge of the tissue section, and
transformed so that 0° coincided with the circumferential
(x1) cardiac
axis (Fig. 1B).
Canine ventricular myocardium has recently been shown to have a laminar
organization (6, 20) in which myocytes are grouped by the perimysial
collagen matrix into branching sheets approximately four cells thick.
These myocardial laminae give rise to the cleavage planes that have
long been described in long- and short-axis transmural sections of the
mammalian heart (3, 22). To examine the relationship of this laminar
architecture to residual stress and strain, we used the method of
LeGrice et al. (7) to measure the orientations of myocardial cleavage
planes. Briefly, 1-mm-thick sections were cut from the fixed
stress-free tissue block parallel to the longitudinal-radial (2-3)
and the circumferential-radial (1-3) transverse cardiac coordinate
planes. Each of these slices spanned the entire wall thickness, and the
cleavage planes were visible with low-power reflected light microscopy.
Images of the tissue sections were captured onto a Macintosh computer,
and cleavage plane angles in each of the two transverse sections were
measured at 1-mm increments from epicardium to endocardium, with 0°
aligned with the radial x3-axis
perpendicular to the epicardial boundary (Fig.
1B).
Following the observations of LeGrice et al. (7), we interpret the
cleavage plane angles as projections of a sheet angle (), which,
together with the fiber angle (), defines the orientation of
myocardial laminae in three dimensions (Fig.
1C). These angles would therefore
satisfy the following trigonometric relationships
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(1a)
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(1b)
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where
' is the cleavage plane angle measured in the (2-3)
plane, " is the cleavage plane angle from the (1-3)
plane, and is the fiber angle from the (1-2) plane. Each of
the two cleavage plane measurements (' and ") was
used to calculate the sheet angle from the appropriate equation
(1a or
1b), with interpolated at the
corresponding relative wall depth from linear least-squares fits to the
measured fiber angles. Equations 1a and 1b indicate that when = 90° (typical near the endocardium), is independent of
', and when = 0° (typical near midwall), equals
' and is independent of ". Therefore, a best
estimate of the transmural distribution of sheet angle was obtained
from a quadratic least-squares fit to the combined data set, with
individual points weighted by the sine or cosine of to reflect the
accuracy of the cleavage plane angle projection based on the local
fiber orientation. A system of local "fiber-sheet coordinates"
(xf, xs,
xn) was then
defined by two consecutive rotations of the cardiac coordinate axes.
First, a rotation about the radial
x3-axis through the interpolated fiber angle yields the fiber axis
xf aligned with
the local muscle fiber orientation in the epicardial-tangent (1-2)
plane. A second rotation about
xf through the
sheet angle yields the sheet axis
xs, which lies
within the sheet plane and is normal to
xf, and the
mutually orthogonal sheet-normal axis
xn, which is
perpendicular to the local sheet plane (Fig. 1C).
Strain analysis.
The two-dimensional pixel coordinates of the marker centroids were
digitized from images acquired with an 8-bit frame grabber (Data
Translation DT2651) on a VAXstation 3200. Images of the stationary bead
set from three consecutive video frames were digitized and averaged in
both the unloaded and stress-free states. The three-dimensional bead
coordinates were computed from the camera transformations found with
the geometric phantom (8). The coordinates of the markers in each
configuration were then transformed into the local cardiac coordinate
system (x1,
x2,
x3) defined in
the intact, unloaded state using the apical and basal reference
markers.
Continuous nonhomogeneous transmural distributions of three-dimensional
finite strain were computed using a modification of the least-squares
finite-element technique described by McCulloch and Omens (9). A
three-dimensional bilinear-quadratic finite element was used to model
the deformation of the bead set. Bead positions in the unloaded
configuration were computed inside a triangular prismatic element (Fig.
2A),
which enclosed the entire bead set and extended transmurally to span
the measured wall thickness in each heart. The deformed element
configuration was then obtained by a least-squares fit to the projected
material coordinates of the beads in the stress-free configuration
(Fig. 2B). Hence, components of the
Lagrangian strain tensor,
Eexp, could be
computed along the transmural centerline of the element from epicardium
to endocardium as previously described (9). Eexp describes
the experimentally observed deformation from the isolated unloaded
heart to the stress-free block of tissue, referred to initial segment
lengths in the unloaded intact state. However, because residual strain
is defined by the inverse deformation from the stress-free state to the
unloaded residually stressed state (14), we computed
eres = Eexp,
which is equal to the residual strain measured with respect to deformed
coordinates (Eulerian definition).
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Fig. 2.
Finite element analysis of 3-D residual strain.
A: bead set in unloaded configuration
inside a high-order finite element, which matches measured wall
thickness. B: updated element
configuration obtained from a least-squares fit to projected bead
coordinates in stress-free state (actual bead coordinates and element
geometries from a representative heart). Lagrangian 3-D finite strains
are computed from A to B and are used to
calculate Eulerian residual strains which describe inverse deformation
from B to A.
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The six components of the symmetric residual strain tensor include
three normal strains, which describe extension or shortening along each
coordinate axis, and three shear strains, which describe changes in the
angle between pairs of coordinate axes that are mutually perpendicular
in the unloaded configuration. The strains were computed at 10%
increments of relative wall depth in each heart (starting at the
epicardium) and averaged. Although strains could be computed throughout
the finite element (i.e., 100% of wall thickness), we did not
extrapolate beyond the subendocardial extent of the transmural bead
array. The strain tensors were also transformed at each depth to
compute components of residual strain with respect to fiber-sheet
coordinates. Relative volume changes in the myocardium contained within
the bead set were found from the determinant of the deformation
gradient tensor,
detFexp, which
represents the local ratio of stress-free to unloaded tissue volume.
Statistical analysis.
Data are presented as means ± SD for
n = 8 animals unless otherwise
specified. Statistical analyses were performed using the software
Superanova (version 1.1, Abacus Concepts, Berkeley, CA). Statistical
significance was accepted at P < 0.05.
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RESULTS |
Myocardial fiber and sheet morphology.
The measured fiber and cleavage plane angles (, ',
") are shown in Fig. 3 vs.
relative wall depth for all eight animals. Fiber angle and the
(1-3) cleavage plane angle " both exhibited a gradient
from epicardium to endocardium, whereas the (2-3) angle '
was more uniform, especially in the outer half of the wall. In several
animals, discontinuities in ' and " were observed in
the subendocardium (>80% depth), where papillary muscle and trabeculae caused abrupt changes in the laminar connective tissue structure. A sudden change in subepicardial " was also
observed in one animal. Mean local wall thickness before fixation was
12.8 ± 1.3 mm.
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Fig. 3.
Individual transmural distributions from epicardium (Epi) to
endocardium (Endo) of fiber angle , measured in (1-2) cardiac
coordinate plane, and 2 cleavage plane angles, ' and
", measured in (2-3) and (1-3) coordinate planes,
respectively, from midanterior left ventricular free wall myocardium
fixed in stress-free state. Discontinuities in ' and
" reflect abrupt changes in sheet structure, such as
endocardial trabeculae. ' and " are plotted in
ranges of 0 to 180° and 0 to 180°, respectively.
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Measurements of , ', and " for two animals are
shown in Fig.
4A,
together with the fiber angle regression line. The two distributions of
sheet angle computed from these data with Eqs. 1a and 1b showed
general agreeement through most of the wall (Fig. 4B). To check the validity of the
mathematical model, the fitted distributions of and were
substituted back into Eqs. 1a and 1b, which were then solved for the
corresponding cleavage plane angles. These estimates of ' and
" showed good consistency with the measurements (Fig.
4A, dashed lines), including the
higherorder transmural variations.
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Fig. 4.
Distributions from Epi to Endo of 3 measured angles (A) and
calculated sheet angle (B) for 2 individual animals. A
continuous transmural distribution of fiber angle was obtained from
a linear least-squares fit to data (solid line in
A). Each measurement of 2-3
cleavage plane angle ' was then used to compute a sheet angle
, using an interpolated value of at matching wall depth. This
was repeated using (1-3) angle ", yielding 2 sets of
sheet angle data in B. A continuous
transmural distribution of was then obtained from a weighted
quadratic least-squares fit to combined data set (solid line in
B). To test for self-consistency,
sheet angle equations were solved in reverse for values of '
and " using fitted fiber and sheet angle distributions, and
these results agreed with actual cleavage plane measurements (dashed
lines in A)
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The average fitted fiber and sheet angle distributions ( and ,
respectively) from the eight animals are shown in Fig.
5. Fiber angle varied linearly from
53 ± 20° at the epicardium to 87 ± 25°
at the endocardium, whereas mean values of were 20 ± 23°, 33 ± 14°, and 25 ± 56° at the
epicardium, midwall, and endocardium, respectively. For six of the
eight animals, the reconstructed sheet angle was negative across the
wall, but in two hearts, crossed zero at ~80% wall depth to
yield positive values in the subendocardium, giving rise to the larger
standard deviations in that region. The average root-mean-squared (rms) errors in the least-squares fits were 9 ± 4° for and 23 ± 12° for .
Three-dimensional residual strain.
The average transmural distributions of the six components of
three-dimensional Eulerian residual strain referred to cardiac coordinates are shown in Fig. 6. In all
eight experiments, the bead set spanned at least 70% of the wall
thickness. Mean results deeper in the subendocardium are also reported,
with n = 5 at 80% depth and
n = 4 at 90% depth.
One-factor analysis of covariance (ANCOVA), with the strain component
treated as a nominal factor, revealed a significant interaction of
component and depth on the variation of residual strain
(P < 0.001). Therefore, post hoc linear regression analysis was performed to help characterize the
transmural distribution of each strain component. All residual cardiac
strain components showed significant transmural gradients (P < 0.002). Mean circumferential
strain (eres11) indicated
residual extension at the epicardium with subendocardial shortening
(slope = 0.0014/%depth), whereas radial strain
(eres33) had a slope of
similar magnitude but opposite sign. Longitudinal strain
(eres22) indicated small
residual shortening across most of the wall, and the average transmural
value (50% intercept of linear regression) of 0.03 was
significant (P = 0.0001). Two of the
residual shear strains (eres12 and eres13) had a similar
small transmural gradient (slope = 0.0005/%depth), and
eres12 was consistently
positive, indicating a small net residual torsion of 0.03 (P = 0.0001). However, the mean
transverse residual shear strain in the longitudinal-radial plane
(eres23) was
predominant, with an average transmural value of 0.04
(P = 0.0001).
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Fig. 6.
Transmural distributions from Epi to Endo of 6 components of 3-D
Eulerian residual strain tensor,
eres, referred to
cardiac coordinates. Circumferential,
eres11, and radial,
eres33, strains showed
opposing transmural gradients, consistent with published 2-D residual
strain studies. Longitudinal strain,
eres22, was small but
significant (P < 0.0001), as were
eres12 and
eres13 shear strains. A
substantial residual transverse shear strain,
eres23, was found in
longitudinal-radial plane. Symbols represent mean values
(n = 8 except as noted), with error
bars indicating ±SD.
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The distribution of
detFexp (not
shown) indicated the change in overall average tissue volume within the
bead set from the unloaded intact state to the stress-free state was
not significant [Ho: mean
(detFexp 1) 
0;
P = 0.50]. However, the
transmural gradient was significant (slope = 0.0011/%depth,
P = 0.0001) and indicated a 6%
decrease in local tissue volume at the epicardium, with a reciprocal
increase in subendocardial volume when residual stress was relieved.
When residual cardiac strains were transformed into fiber-sheet
coordinates (Fig. 7), mean residual strain
in the fiber direction (eresff) indicated ~5%
extension at the epicardium with smaller subendocardial shortening
(slope = 0.0007/%depth, P = 0.002). Sheet-normal strain was negative and relatively uniform across
the wall (eresnn 
0.05), whereas residual strain within the sheet plane showed
uniform extension transverse to the fiber axis with
eresss 0.04 (neither slope
was significant, P > 0.06). All
three shear components
(eresfs,
eresfn, and
eressn) exhibited a small but
significant transmural gradient (P < 0.005). However, none was different from zero on average
(P > 0.06), and except in the inner
and outer 20% of the wall, none of the mean shear strains exceeded
values of ±0.02. Therefore, the principal deformations due to
residual stress may act primarily along sheet structural axes across
most of the LV wall.
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Fig. 7.
Transmural distributions from Epi to Endo of 6 components of 3-D
Eulerian residual strain,
eres, referred to
sheet coordinates. Residual stress gives rise to compression of
adjacent myocardial laminae
(eresnn < 0) and stretching
within sheet plane, particularly transverse to fiber axis
(eresss > 0). With exception
of subepicardial eressn, all 3 shear strains were small. Gradient in fiber strain,
eresff, was consistent with
previously measured changes in sarcomere length due to residual stress
in rat heart. Symbols represent mean values
(n = 8 except as noted),
with error bars indicating ±SD.
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DISCUSSION |
We have presented the first measurements of fully three-dimensional
residual strains in the heart. The findings were consistent with
earlier one- and two-dimensional studies but revealed other significant
components, including substantial negative transverse shear strains.
From measurements of myocyte fiber orientations and cleavage plane
orientations, we defined a local coordinate system based on these
anatomic structures in the tissue. By referral of the strain components
to these microstructural axes, the mechanics of residually stressed
myocardium could be interpreted relative to the underlying tissue
architecture. The relation between the observed residual deformations
and the fibrous, laminar connective tissue organization of the
ventricular wall suggests structural prestressing of myocardial laminae
in the unloaded intact LV myocardium.
Myocardial fiber and sheet morphology.
The three-dimensional nonhomogeneous fibrous architecture of the
myocardium has been well documented (13, 23), and our transmural fiber
angle data from the midanterior LV free wall are consistent with
previous measurements in this region (8, 15). A laminar connective
tissue organization of the myocardium has also long been recognized (3,
22), and quantitative measurements of the structure of these myocardial
sheets have recently become available (6, 7). Our cleavage plane angle
measurements (Fig. 3) were similar to these previous measurements,
although LeGrice et al. (6, 7) reported a steeper transmural gradient in the (2-3) cleavage plane angle ', which may reflect
regional variations between our measurement site and theirs. It is also possible that cleavage plane morphology is different in the stress-free state than in the unloaded (6) or inflated (7) states studied previously.
Three-dimensional residual strain.
Residual strains describe the deformation from the stress-free state to
the unloaded state of a body (from B
to A in Fig. 2). We have chosen to
present Eulerian residual strains,
eres, because the
deformed (intact, unloaded) state of the myocardium represents a
well-defined reproducible reference configuration in which the cardiac
coordinates are easily related to the LV geometry. However, Lagrangian
residual strains,
Eres, referred to
the undeformed (stress-free) configuration are equally valid and are
more consistent with previous studies. So, for comparison, we also
computed Eres for
each heart (21), which was straightforward once the deformation gradient tensor was extracted from the finite element analysis. In both
the cardiac and sheet coordinate systems, the Lagrangian normal strain
profiles were shifted upward (indicating greater lengthening and
smaller shortening) compared with the corresponding Eulerian strain
profiles, whereas the trend was inconsistent for the shear strains. The
difference between the mean Lagrangian and Eulerian strain profiles was
typically ~0.02 or less for the normal strains and <0.01 strain
units for the shear components. These small differences were always
within 1 SD of the mean Eulerian strains and hence would not affect our
conclusions.
The distributions of circumferential
(eres11) and radial
(eres33) residual
strains agree with previous two-dimensional measurements from the study by Omens and Fung (14), in which equatorial rings of isolated rat heart
sprang open into an arc after being radially transected to relieve
residual stress. On the basis of their mean principal stretch ratios in
the anterior portion of the rat LV, Eulerian residual strains in the
subepicardial and subendocardial one-thirds of the heart wall would be
0.07 and 0.09 (circumferential) and 0.09 and 0.12 (radial), respectively. We found similarly opposing transmural
gradients of eres11 and
eres33 (Fig. 6),
characteristic of the opening angle experiments. The somewhat smaller
magnitudes of residual strain in this study suggest a smaller opening
angle in the dog than in the rat, consistent with measurements from our
laboratory (unpublished data, 1994) of an opening angle of 25 ± 4° (mean ± SD, n = 3) for an
equatorial section of the potassium-arrested dog heart, compared with
45 ± 10° in the rat (14). Residual stress was also found to
cause small but significant longitudinal shortening,
eres22, which has been
neglected in previous two-dimensional studies. In addition, the
three-dimensional measurements revealed a small positive torsional
residual shear strain, eres12. This is consistent with measurements in the normal mouse LV (16), in
which an opening angle of 25 ± 9° was accompanied by an
out-of-plane warp angle of 3.4 ± 1.2° so that the anterior side
of the LV moved toward the apex when residual stress was relieved. The
three-dimensional measurements also revealed a substantial negative
longitudinal-radial transverse residual shear strain,
eres23, which had not
previously been reported.
Residual strain referred to fiber-sheet coordinates indicated
subepicardial stretching of muscle fibers in the presence of residual
stress, with subendocardial fiber shortening. This is consistent with
the study of Rodriguez et al. (19), who reported sarcomere extension in
the subepicardium and sarcomere shortening in the subendocardium due to
residual stress in an equatorial ring of rat LV, with no measurable
change in midwall sarcomere lengths compared with the stress-free
state. Using the stress-free sarcomere length distribution reported in
that study combined with our mean values of
eresff, we calculated a
transmural gradient of sarcomere length in the unloaded canine heart of
0.13 µm/normalized wall thickness, which compares with the
value of 0.114 ± 0.054 µm reported for the rat (19).
Considering the smaller opening angle in the dog than in the rat, one
might have expected a smaller gradient in sarcomere length as well. However, Rodriguez et al. (19) commented that they may have underestimated the actual sarcomere length gradient in the intact unloaded rat heart because the effects of longitudinal residual stress
were neglected, which presumably would have the greatest influence in
the epicardial and endocardial regions where muscle fiber orientations
have a substantial longitudinal component. Our mean values of
eres22 indicated that, relative to the stress-free state, the intact unloaded myocardium was
stretched longitudinally at the epicardium and compressed in the
subendocardium, supporting the hypothesis of a steeper transmural sarcomere length gradient in the intact residually stressed
LV than would be measured in a partially stress-relieved equatorial
ring of myocardium.
Residual strains in cardiac coordinates indicated substantial shearing
deformations in transverse planes of the LV wall, the same planes in
which we measured the orientation of cleavage planes separating
adjacent layers or "sheets" of myocardium. In light of the
observations of LeGrice et al. (7), these cleavage planes were
interpreted as two-dimensional projections of a fully three-dimensional laminar connective tissue organization of the myocardium. Individual myocardial sheets are about four cells thick and are bounded by a dense
perimysial connective tissue matrix, which provides tight coupling
within the sheet, with adjacent sheets connected by a loose network of
collagen fibers (5, 6). It has been postulated that such an
organization facilitates rearrangement of cell bundles, providing a
mechanism for the large changes in ventricular wall thickness that
occur during the cardiac cycle (3, 22). Such sliding or slippage of
myocardial sheets would be consistent with macroscopic transverse
shearing deformations, as recently demonstrated in the subendocardium
during systole (7). However, when residual strains were referred to a
coordinate system based on the local three-dimensional structural axes
of the myocardial laminae, small sheet shear strains through most of
the LV wall implied little relative motion of adjacent layers of
myocardium due to residual stress. Instead, the residual strains
indicated compression normal to the sheets
(eresnn < 0), which may
reduce the gaps or clefts between adjacent myocardial laminae or may reflect thinning of the sheets themselves. Reciprocal extension along
the sheet axis (eresss > 0)
indicated muscle fibers may be transversely stretched or separated
within the sheet plane in the presence of residual stress. Therefore, the measured transverse shear strains in cardiac coordinates arose from
shortening and extension along oblique structural axes oriented parallel and perpendicular to the myocardial laminae. Because of its
organization relative to the myocardial sheets, the perimysial extracellular connective tissue matrix may be an important residual stress-bearing structure in the myocardium. Because the myocytes themselves are by definition aligned with the myocardial sheets, intracellular load-bearing components of the cytoskeleton may also play
a role; further research is needed to determine the contributions of
these individual structures to residual stress in the ventricular wall.
The residual strains measured in this study are smaller than typical
strains during ventricular filling (15) and ejection (27). This,
together with the fact that the stiffness of resting myocardium is
lowest around the zero-stress state due to the material nonlinearity,
suggests that associated residual stresses may be substantially smaller
than ventricular wall stresses under physiological loading conditions.
Nevertheless, the effects of residual stress and strain on
ventricular mechanical function can be significant. Several model
studies suggest residual compression may help to decrease
subendocardial stresses during LV filling (4, 12, 24). Furthermore,
because it affects sarcomere length (19), residual stress may influence
systolic mechanics due to the strong dependence of peak active tension
on sarcomere length in vitro (1, 26). For example, a stress-free
sarcomere length of 1.84 µm (19) would decrease by 0.03 µm in the
residually stressed subendocardium according to our measurements of
residual fiber strain. From the data from ter Keurs et al. (26) in
isolated rat trabeculae at an extracellular calcium concentration of
0.5 mM, this change would correspond to a decrease in peak isometric tension of ~5 mN/mm2,
representing 5-10% of the expected peak tension for end-systolic sarcomere lengths in the range of 2.2-1.9 µm (18).
Effects of edema due to perfusion.
The first four of the eight experiments were conducted after a separate
isolated heart perfusion study (8) in which a mean 14% change in
tissue volume was reported due to edema. To assess whether there
was a statistically significant difference in residual strain
between the perfused and nonperfused groups, one-factor ANCOVA was
performed for each strain component, using the perfusion state as a
nominal factor. Differences in transmural gradients between the two
groups were quantified by post hoc linear regression, and
Scheffé's S procedure was used for post hoc comparison of means.
In the fiber-sheet coordinate system, four of the six residual strain
components (eresff,
eresnn, eresfn, and
eressn) differed significantly
between the perfused and unperfused groups
(P 
0.05). Of these, perfusion only
altered the transmural gradient of
eresfn
(P = 0.005). The mean transmural gradient of eresfn in the four
unperfused hearts was ~40% lower than that shown for the entire
group in Fig. 7. Moreover, the mean values of the residual shears,
eresfn and
eressn, were even smaller in
the unperfused group. This provides more support to our
conclusion that residual stress gives rise to negligible shear
strains in sheet coordinates, implicating sheet axes as principal axes
of residual strain. The average value of
eresnn was 0.03 for the nonperfused hearts compared with 0.05 for the combined group. The average value of eresff
was 0.03 in the nonperfused hearts compared with 0.00 for the
combined data. Thus, although the effects of edema due to perfusion had
some significant effects on the measured residual strains, the
conclusions based on the combined data were supported or strengthened
by the results from the four hearts that were not perfused.
We also found a significant effect of perfusion on the mean value of
detFexp, which
changed slightly from 1.00 for the nonperfused hearts to 0.97 for the
perfused hearts (P = 0.008),
indicating a small net loss of volume from the edematous tissue when it
was cut out of the heart to relieve residual stress. However, there was
no effect of perfusion on the transmural gradient of
detFexp
(P = 0.24). In both groups, there was
a small but significant gradient of
detFexp, which
suggests that compressive endocardial residual stress and tensile
epicardial residual stress may shift the balance of volume in the wall.
If this volume is attributed to the relatively mobile intracoronary
fluid, this finding may explain why the endocardium has a higher
apparent capacitance than the epicardium (8).
Sources of error.
LeGrice et al. (7) have discussed the limitations of measuring
projections of the three-dimensional sheet structure in three separate
two-dimensional planes. Accurate reconstruction of the sheet
orientation requires structural homogeneity throughout the region of
the measurements. Agreement between sheet angles computed from
' and " indicated that this assumption was
reasonable, and self-consistency between the measured and calculated
cleavage plane angles (Fig. 4A)
further supports this notion. Some specific discrepancies between
the two sets of sheet angle data were anticipated on the basis of
the mathematical model of the cleavage planes. Therefore, reliability
of the projected cleavage plane angles was incorporated into the
analysis. In two animals, a systematic difference of ~40° existed
between the sheet angles calculated from ' and "
across the wall, indicating relatively rapid changes in sheet structure
within the local measurement volume. In these cases, the quadratic fit
fell between the two distributions of and was assumed to be a
reasonable estimate of the average sheet angle distribution in the
region of the bead set.
Several measures were taken to minimize experimental artifacts that
might influence the measured residual strains. To reduce the effects of
gravity on the experimental preparation, the intact hearts and tissue
blocks were suspended in a bath of fluid. To delay the onset of
ischemic contracture, the animals were pretreated with oral nifedipine,
and the heart was arrested in vivo, and the coronary circulation was
briefly perfused with a hypothermic, hyperkalemic cardioplegic solution
containing 2,3-BDM, which protects the isolated tissue from cutting
injury (11). Thus the use of this preparation should eliminate any
residual physiological cross-bridge formation and preserve passive
material properties. Although progressive contracture would cause the
calculated residual strain to change with time, differences between the
reported strains and those measured during the subsequent 2-3 min
were within the accuracy of our experimental methods (9). We thus infer
that ischemic contracture did not adversely affect the results of this
study.
Accuracy of the residual strains requires that the excised tissue block
represents the true zero-stress state of the tissue. Previous
two-dimensional residual strain studies in ventricular cross sections
have shown no significant change in the zero-stress state subsequent to
the first radial cut (14). By comparison, in the present study, the
zero-stress state was effectively approximated as the state following
two closely spaced radial cuts in the ventricular cross section. Thus
it may not be the same as the stress-free state used in previous
studies, but it should be a better approximation to it.
Finally, accuracy of the computed residual strains obtained by
nonhomogeneous finite element analysis was improved by projecting the
actual three-dimensional marker coordinates in the unloaded reference
state directly into the initial finite element, thus eliminating errors
due to the initial approximation of the reference configuration
required by our previous strain analysis method (9). Additionally, rms
errors in the least-squares fit to the deformed (stress free) bead set
coordinates were reduced, ranging from 0.05 to 0.24 mm for the eight
hearts, compared with errors as large as 0.34 mm obtained using the
previous method. Despite these improvements, differences in strain
between the two techniques were typically <0.02, which is similar to
the estimated variation due to the measurement accuracy (9).
In conclusion, three-dimensional residual strains in passive
midanterior canine LV myocardium were consistent with earlier one- and
two-dimensional studies. However, the presence of substantial transverse shear strains with small torsional and longitudinal strains
reveals a more complex distribution of residual stress and strain than
previously identified. When referred to a coordinate system based on
the underlying fibrous and laminar architecture of the myocardium, the
residual strains suggest epicardial fiber tension and endocardial fiber
compression in the unloaded intact LV, although myocytes in much of the
midwall may experience relatively little residual stress along the
muscle fiber axis. The residual strains also indicated both tension and
compression normal to the local fiber axis, with residual stresses
being borne mainly by structures lying parallel and perpendicular to
the laminar bundles of myofibers. For example, the perimysial
extracellular connective tissue matrix may be a primary residual
stress-bearing structure in passive myocardium. Therefore, pathological
conditions, such as fibrosis or matrix degradation during stunning, may
alter residual stresses in the heart and hence may provide useful
models for future studies of the structural basis of residual stress and strain.
| |
ACKNOWLEDGEMENTS |
We thank Dr. James Covell for the use of his laboratory and Rish
Pavelec and Monica Adams for technical assistance. We are indebted to
Dr. Ian LeGrice for sharing his experience and insight with regard to
the laminar structure of the myocardium. We also thank Jim Wilson for
helping to develop the strain analysis method.
| |
FOOTNOTES |
This research was supported by National Heart, Lung, and Blood
Institute Grants HL-41603 (A. D. McCulloch) and HL-32583 (J. W. Covell). K. D. Costa was supported by National Heart, Lung, and Blood
Institute Predoctoral Training Grant HL-07089 (S. Chien). This support
is gratefully acknowledged.
Address for reprint requests: A. D. McCulloch, Dept. of Bioengineering,
University of Califonia, San Diego, 9500 Gilman Dr., La Jolla, CA
92093-0412.
Received 30 December 1996; accepted in final form 27 May 1997.
| |
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Abstract
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