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Departments of 1 Bioengineering and 2 Medicine, University of California, San Diego, La Jolla, California 92093
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ABSTRACT |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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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 |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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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 |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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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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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) |
' 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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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 |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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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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, ', 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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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 .
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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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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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DISCUSSION |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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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 and0.09 (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.
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.
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ACKNOWLEDGEMENTS |
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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.
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FOOTNOTES |
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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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REFERENCES |
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Top
Abstract
Introduction
Methods
Results
Discussion
References
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