Vol. 279, Issue 1, H216-H224, July 2000
Maturation of end-systolic stress-strain relations in chick
embryonic myocardium
Kimimasa
Tobita and
Bradley B.
Keller
Cardiovascular Development Research Program, Department of
Pediatrics, University of Kentucky, Lexington, Kentucky 40536
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ABSTRACT |
The embryonic myocardium increases
functional performance geometrically during cardiac morphogenesis. We
investigated developmental changes in the in vivo end-systolic
stress-strain relations of embryonic chick myocardium in stage 17, 21, and 24 white Leghorn chick embryos (n = 10 for each
stage). End-systolic stress-strain relations were linear in all
developmental stages. End-systolic strain decreased from 0.50 ± 0.02 to 0.31 ± 0.01 (mean ± SE, P < 0.05),
while average end-systolic wall stress was similar at 3.29 ± 0.34 to 4.19 ± 0.43 mmHg (P = 0.14) from stage 17 to
24. Normalized end-systolic myocardial stiffness, a load-independent index of ventricular contractility, increased from 2.98 ± 0.19 to
6.03 ± 0.39 mmHg from stage 17 to 24 (P < 0.05).
Zero-stress midwall volume increased from 0.024 ± 0.002 to
0.124 ± 0.004 µl from stage 17 to 24 (P < 0.05). These results suggest that the embryonic ventricle increases
normalized ventricular "contractility" while maintaining average
end-systolic wall stress over a relatively narrow range during
cardiovascular morphogenesis.
embryonic ventricle; cardiovascular development; end-systolic wall
stress; end-systolic wall strain; contractility
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INTRODUCTION |
THE DEVELOPING
MYOCARDIUM undergoes simultaneous structural and functional
maturation as the avian and mammalian embryonic heart transforms from a
straight tube to a four-chamber heart (11,
21, 23). Embryonic cardiovascular adaptation
to changes in metabolic and hemodynamic demand occurs at the tissue and
cellular levels; however, there are limits to adaptation that result in a normal mature phenotype (11, 13).
Congenital anomalies in cardiovascular structure and function likely
occur because of failures to compensate for altered morphogenesis
(3, 6, 8). Experimental models
in vertebrate embryos result in structural anomalies similar to those
seen in patients (1, 10, 24).
Experimental methods developed for the mature circulation have been
adapted to accurately measure blood pressure, blood flow, and chamber
size and to alter cardiac function or form during development
(4, 11, 18). The developing
myocardium rapidly adjusts growth and morphogenesis in response to
increased or decreased hemodynamic loading conditions (5,
24). Myocardial growth and morphogenesis are controlled in
part by genetic information. However, mechanical factors such as wall
stress and/or strain also likely influence growth and morphogenesis
(5, 15, 26). Several studies in
the mature heart have demonstrated that cardiovascular adaptation
occurs in response to changes in end-diastolic wall stress and/or
strain (7, 19, 20). In the
embryonic heart, Lin and Taber (15) used a growth law that
depends on end-diastolic stress in a model that reproduced experimental
measures of normal growth. These studies suggest that the ventricular
geometry and/or myocardial properties are changed so as to reduce
initial increases in diastolic wall stress or strain. In contrast to
the relationship between end-diastolic stress or strain and ventricular
growth, the relationship between end-systolic stress-strain relations, ventricular contractility, and growth have not been determined for the
developing embryonic heart. Our previous study of stage 24 chick
embryonic ventricle showed that the end-systolic stress-strain relations based on the incremental elastic modulus concept were linear
and that normalized myocardial stiffness reflects ventricular contractility in chick embryo heart (29).
We applied the incremental elastic modulus concept to define
developmental changes in end-systolic stress-strain relations and
ventricular contractility in the chick embryonic ventricle. End-systolic stress-strain relations were linear over a fourfold increase in ventricular mass during development (4). The
embryonic ventricle increases normalized end-systolic myocardial
stiffness, an index of ventricular contractility, while maintaining
average end-systolic wall stress over a relatively narrow range during cardiac morphogenesis.
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MATERIALS AND METHODS |
Embryo preparation and developmental staging.
Vertebrate cardiogenesis follows similar developmental patterns with
varying time lines between species (11). We studied white
Leghorn chicken embryos at Hamburger-Hamilton stage 17 (3 days,
n = 10), stage 21 (3.5 days, n = 10),
and stage 24 (4 days, n = 10) of a 46-stage (21-day)
incubation period (9). At these stages the ventricles
function as a single chamber (23) and ventricular wet
weight increases by about fourfold from stage 17 to 24 (4). Fertile eggs were incubated in a force draft
incubator at 38°C and constant humidity. Each egg was positioned on a
photomacroscope stage under radiant warmers to maintain ambient
temperature between 37 and 38°C. An ~1-cm2 hole was
made in the shell, and the inner shell and extraembryonic membranes
were removed to expose the developing embryo. Embryos that were
dysmorphic or exhibited overt bleeding were excluded from study.
Hemodynamic preparation.
We use standard methods for measuring chick embryonic ventricular
pressure and dimensions, as previously described (29). We
used a custom-integrated physiology-morphometry workstation to
simultaneously measure intraventricular pressure and ventricular dimensions. A fluid-filled glass capillary pipette was positioned using
a micromanipulator (Leitz, Wetzlar, Germany) to puncture the developing
right ventricle and measure intraventricular pressure with a servo-null
pressure system (model 900A, World Precision Instruments, Sarasota, FL)
and analog-digital board (AT-MIO 16, National Instruments, Austin, TX).
Video images were acquired using a photomacroscope (model M400, Wild
Leitz, Rockleigh, NJ), videocamera (model 70-series, Dage-MTI, Michigan
City, IN), frame grabber board (LG-3, Scion, Fredrick, MD), and a
custom-programmed eight-bit gray-scale analog-digital image (image size
640 × 480 pixels) acquisition system (LabVIEW, National
Instruments). This custom acquisition system simultaneously captured
video images at 60 Hz and intraventricular pressure at 600 Hz for
4 s. The pressure waveform was decimated from 600 to 60 Hz and
interpolated with the image data. A 50-µm division scribed glass
standard was recorded in the plane of each embryo after imaging for
calibration of image analysis software (LabVIEW, National Instruments).
Ventricular afterload alteration.
Our previous study of acute, near-complete conotruncal occlusion showed
that embryonic ventricular peak systolic pressure and end-diastolic
volume changed simultaneously in response to increased afterload
(14). Arterial tone also changes almost simultaneously in
response to alterations in preload (31). In the present
study, we used gradual conotruncal constriction to increase ventricular
afterload without dramatically changing preload. In this fashion, we
obtained simultaneous ventricular pressure and dimension data from five
to seven cardiac cycles during increasing ventricular afterload by a
gradual conotruncal occlusion in each embryo (29). The
conotruncus was occluded using microforceps mounted on a
micromanipulator. The forceps were closed gradually over 5 s to
narrow the conotruncus until end-diastolic volume visibly increased.
Video image processing.
We used custom-programmed image analysis software (LabVIEW, National
Instruments) to measure ventricular epicardial cross-sectional area.
First, we manually traced the maximum and the minimum epicardial ventricular borders from recorded sequences of each embryo (Fig. 1, left). The number of pixels
and individual pixel values in the area contained between the maximum
and minimum epicardial borders were stored in memory as a region of
interest (ROI; Fig. 1, right, shaded area). The image size
represented by a single pixel ranged from 1.5 to 2.3 × 10
5 mm2 depending on magnification. We
assumed that movement of the embryonic ventricular epicardial border
would be associated with changes in the values of pixels within the
image of the heart. Changes in the ventricular epicardial area from the
minimum area during the cardiac cycle were then identified
automatically by detecting the pixels that changed value in the ROI for
sequential video fields. Total epicardial cross-sectional area in each
video field was then calculated as the sum of the changed area within
the ROI and the minimum epicardial cross-sectional area.
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Fig. 1.
Representative video image of an embryonic ventricle
manually traced to calculate maximum and minimum areas in stage 21 embryo. Ventricular epicardial borders were traced (white lines), and
the information of the number of pixels and each pixel value in the
area between these 2 epicardial borders (gray shaded area) were stored
in memory as a region of interest (ROI). We then determined the number
of pixels that changed in value in the ROI of each sequential video
field during the cardiac cycle. The epicardial cross-sectional area in
each video field was calculated by adding the changed area in ROI to
the minimum epicardial cross-sectional area. CT, conotruncus; V,
ventricle; *, tips of microforceps used for gradual conotruncal
occlusion.
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To determine the reproducibility and accuracy of this semiautomated
method for calculating ventricular epicardial area during the cardiac
cycle, we compared the ventricular cross-sectional epicardial areas
calculated by densitometry with the areas calculated by manual
planimetry, as previously described (14, 29).
Intra- and interobserver error of the area measurement by manual
planimetry is not significant (P > 0.29 and
P > 0.96, respectively). Figure 2 shows representative time-course curves
for epicardial cross-sectional areas calculated by manual planimetry
and densitometry in a representative stage 24 chick embryo. There was
an excellent linear correlation between manual planimetry and
densitometry during the cardiac cycle for each embryo
(r = 0.989; Fig. 3).
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Fig. 2.
Representative time-course changes in the ventricular
epicardial areas determined by manual planimetry and densitometry in
stage 24 chick embryo. The time-course curves were similar to each
other during the cardiac cycle.
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Fig. 3.
Comparison of manual planimetry and densitometry methods
to determine ventricular epicardial areas. There was significant linear
correlation between manual planimetry and densitometry. SEE, standard
error of the estimate. Dashed line, line of identity
(y = x).
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Ventricular volume was calculated using a simplified ellipsoid of
revolution model (13, 29). The ellipsoid
equation is derived from equations for the cross-sectional area
(A) of an ellipsoid (A = 
DL)
and the volume (V) of an ellipsoid of revolution [V = (4D2L)/3], where D is minor
semiaxis and L is major semiaxis. We assume that the
ventricle is axisymmetric along the major semiaxis and recognize that
this may not be completely true for the embryonic heart. We previously
measured epicardial ventricular major and minor semiaxis dimensions at
maximum and minimum cross-sectional areas in stage 18-24 chick
embryos and found a relatively constant aspect ratio of
L/D = 4/3 throughout the cardiac cycle
(13, 29). We use this fixed-aspect ratio in
converting ellipsoid cross-sectional area to volume.
Ventricular cavity volume (Vc) was calculated as total
volume (Vt) minus ventricular wall volume (Vm).
Ventricular wall volume was determined as the epicardial
cross-sectional area of the ventricle after contracture by topical 2 M
sodium chloride administration (13, 29).
Ventricular wall architecture changes from a smooth endocardial surface
to a porous trabeculated wall. It is difficult to directly measure the
volume changes of a porous wall during the cardiac cycle. The wall
volume obtained from maximum contracture obliterates all space in the
lumen and the pores. Therefore, in calculating ventricular cavity
volume, we assumed that the ventricular wall is a freely deforming
solid body composed of transversely isotropic and incompressible
elastic material during the cardiac cycle. Ventricular internal minor
semiaxis dimension (Di) and wall thickness
(h) were computed by solving the following equations
|
(1)
|
where Li is internal major semiaxis
dimension, and it is assumed that
Li/Di = 4/3.
End-systolic stress-strain relations.
According to the incremental elastic modulus concept (17,
29), we assumed the embryonic ventricle to be a
thick-walled ellipsoidal shell and calculated the end-systolic
stress-strain relation at the equator level, as previously described
for the stage 24 chick embryo (29).
Strain difference (
) is defined as the difference of the
circumferential (
) and radial (r)
strain components at the equator of an ellipsoid (17,
29). Considering the large deformation of the embryonic
ventricle during systole, we used the natural strain definition. The
strain difference is expressed as
|
(2)
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Thus the total strain difference is calculated as
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(3)
|
where Dm, Lm, and
D0,m are minor semiaxis, major semiaxis, and
zero-stress minor semiaxis midwall diameter at the equator, respectively.
Stress difference (
) is defined as the difference of the
circumferential () and radial (r)
stress components (17, 29). These stresses
are averaged over the entire cross section at the equator of an
ellipsoid. Thus the average stress difference () is calculated by
![&sfgr;=&sfgr;<SUB>&thgr;,a</SUB><IT>−&sfgr;</IT><SUB>r,a</SUB><IT>=</IT>(P<IT>D</IT><SUB>m</SUB><IT>/h</IT>)[<IT>1−</IT>(<IT>D</IT><SUP><IT>2</IT></SUP><SUB>m</SUB><IT>/2L</IT><SUP><IT>2</IT></SUP><SUB>m</SUB>)<IT>−</IT>(<IT>3h<SUP>2</SUP>/8D</IT><SUP><IT>2</IT></SUP><SUB>m</SUB>)]](/content/vol279/issue1/fulltext/H216/img008.gif)
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(4)
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where ,a and r,a are the
average circumferential and radial stresses, P is left ventricular
pressure, and h is wall thickness.
Average systolic myocardial stiffness (Eav) is
calculated as
|
(5)
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Equation 5 indicates that the stress-strain
relationship is linear (17, 29).
Normalized end-systolic myocardial stiffness (Eav/G).
Ventricular midwall volume (Vm) based on the thick-walled
ellipsoidal model is
|
(6)
|
If it is assumed that
Dm/Lm = 3/4, then
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(7)
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where k = (16/9). We can then use
Vm and V0,m to calculate
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(8)
|
where V0,m is the zero-stress midwall volume. Thus
Eq. 5 is expressed as
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(9)
|
We then converted end-systolic stress-strain relations to
end-systolic pressure-volume relations (ESPVR)
![=(D<SUB>m</SUB><IT>/h</IT>)[<IT>1−</IT>(<IT>D</IT><SUP><IT>2</IT></SUP><SUB>m</SUB><IT>/2L</IT><SUP><IT>2</IT></SUP><SUB>m</SUB>)<IT>−</IT>(<IT>3h<SUP>2</SUP>/8D</IT><SUP><IT>2</IT></SUP><SUB>m</SUB>)]](/content/vol279/issue1/fulltext/H216/img016.gif)
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(10)
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where es, Pes, Ves, and
G are end-systolic stress difference, end-systolic pressure,
end-systolic midwall volume, and geometric factor, respectively, and
and 
are regression coefficients.
By use of Eqs. 9 and 10, ESPVR is
expressed as
|
(11)
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The slope of Eq. 11,
Eav/G, represents ventricular
contractility (29).
Determination of end-systolic stress-strain points.
By means of modification of time-varying maximum elastance theory
(17, 22), the end-systolic stress-strain
point is defined as a point where the
es-to-es ratio reaches a maximum value after the onset of systole. We obtained five to seven different es-Des points from gradual
conotruncal occlusion in each embryo. We first assumed a value for
D0,m, and then the es and
logarithmic Des points were fitted by linear
regression analysis. A new D0,m was obtained by
approximate extrapolation to zero stress. This iterative procedure was
continued until the value for D0,m converged.
Statistical analysis.
Values are means ± SE. The mean values for each group were
analyzed by single-factor ANOVA. When an assumption of data normality or equal variance was violated, a nonparametric Kruskal-Wallis test was
performed. Individual comparison was performed by a Duncan's multiple
range test. Statistical significance was defined by P < 0.05. Linear regression analysis with the minimum least-square method was performed to analyze the end-systolic stress dimension, stress-strain relations, and geometric factor. To evaluate linearity, we performed an F test in each linear regression analysis.
All calculations were performed using STATISTICA (Statsoft, Tulsa, OK).
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RESULTS |
Hemodynamic data.
Table 1 shows the increase in baseline
heart rate, peak ventricular pressure, and end-diastolic volume that
occurs from stage 17 to 24 (P < 0.05).
End-systolic stress-strain relations.
Figure 4 displays representative
stress-strain relations for stage 17, 21, and 24 chick embryos.
F test indicated that the relations showed no significant
departure from linearity in each group (Table
2). There was no relationship between
end-systolic wall stress and myocardial stiffness, indicating that
end-systolic myocardial stiffness is independent of end-systolic wall
stress during this period of myocardial development (Table 2).
End-systolic strain decreased significantly from stage 17 to 24 (P < 0.05). Average end-systolic wall stress was not
changed significantly (P = 0.14; Table
3).
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Fig. 4.
Ventricular stress-strain loops and end-systolic
stress-strain relations in stage 17, 21, and 24 chick embryos.
End-systolic stress-strain relations were linear in all developmental
stages. The slope, end-systolic myocardial stiffness, increased with
development (P < 0.05). Stress, midwall stress
difference; strain, strain difference.
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Table 3.
End-systolic wall strain, stress, myocardial stiffness, geometric
factor, and zero-stress midwall volume
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End-systolic myocardial stiffness.
End-systolic myocardial stiffness (Eav,max)
increased significantly from stage 17 to 24 (P < 0.05;
Table 3). G was not changed between groups. Normalized
end-systolic myocardial stiffness also increased significantly from
stage 17 to 24 (P < 0.05; Table 3, Fig.
5).
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Fig. 5.
Representative pressure-volume loops and end-systolic
pressure-volume relations based on the incremental elastic modulus
concept. End-systolic pressure-volume relations were curvilinear. The
slope, normalized end-systolic myocardial stiffness, increased
(P < 0.05). Zero-stress midwall volume also increased
in parallel with development (P < 0.05).
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Zero-stress midwall volume increased significantly from stage 17 to 24 (P < 0.05; Table 3, Fig. 5).
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DISCUSSION |
End-systolic stress-strain relations.
Our previous study showed that the end-systolic stress-strain relations
in stage 24 chick embryos were linear and that end-systolic myocardial
stiffness was independent of end-systolic wall stress (29). To calculate the embryonic ventricular wall stress,
we assumed that the embryonic myocardium is a freely deforming body composed of an isotropic, homogeneous, and incompressible
elastic material and then calculated "average midwall stress." In
the present study, we applied the same concept to embryos of different stages, and the same results were observed. However, we must consider the impact of changes in ventricular wall composition during this developmental period on experimental and model results. At stage 17, dorsoventrally aligned trabecular ridges are found in the ventricular
apex, and a significant portion of the myocardial wall is composed of
extracellular matrix, termed cardiac jelly. By stage 21, cardiac jelly
has been resolved and the ventricular wall resembles a coarsely
trabeculated sponge with a thin outer sleeve of compact myocardium. At
stage 24, the myocardial wall has differentiated to contain
asymmetrically oriented coarse and fine trabeculae (23).
Therefore, the material properties of the ventricular wall likely
change during these stages.
Using a laminated thick-walled cylindrical shell model that was
composed of three isotropic, pseudoelastic layers representing the
endocardium, the cardiac jelly, and the myocardium, Taber et al.
(27) computed the wall stress distribution of a stage 16 chick embryo. High stress concentrations were shown to occur in the
outer myocardial layer at end systole (27). Yang et al. (30) used a thick-walled, cylindrical shell model composed
of a porous inner layer and a thin compact outer layer of isotropic myocardium to compute the wall stress distribution for a stage 21 chick
embryo. For the trabecular stage 21 myocardium, end-systolic wall
stress decreased monotonically from endocardium to epicardium, and the
stress gradient was nearly uniform across the ventricular wall
(30). To evaluate the impact of nonuniform wall
composition on myocardial wall stress, we recalculated the end-systolic
wall stress distribution at stage 17 according to the method of Taber et al. (see APPENDIX). Table
4 and Fig.
6 show that the average end-systolic
stress difference in the myocardial layer based on the laminated
cylindrical model was similar to the end-systolic midwall stress
difference. These results in the present study suggest that the
incremental elastic modulus concept is useful to assess the
contractility of the developing myocardium.
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Fig. 6.
Distribution of end-systolic wall stress difference
throughout the ventricular wall of a stage 17 chick embryo based on the
pseudostrain-energy density function. Wall stress concentration
occurred at the outer myocardial layer. Stress, end-systolic stress
difference; EN, endocardial layer; CJ, cardiac jelly layer; MC,
myocardial layer; End, endocardium; Epi, epicardium.
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Developmental changes in ventricular contractility.
Average end-systolic myocardial stiffness increased significantly from
stage 17 to 24. Our previous study showed that average end-systolic
myocardial stiffness normalized by the geometric factor
(Eav/G) is an index of ventricular
contractility in the embryonic ventricle (29). The
geometric factor, which represents the effect of the ventricular
dimension-wall thickness relation to wall stress, did not change from
stage 17 to 24, despite significant changes in wall dimensions and
thickness. Therefore, Eav/G also increased in parallel with development in the present study.
Ventricular contractility increased primarily because of decreased
end-systolic strain. Clark et al. (5) showed that
embryonic chick heart alters myocyte division and the number of
myocytes (ventricular cavity volume and mass) without morphological
changes in response to increased afterload. Our results suggest that
the embryonic ventricle does not increase ventricular contractility by
a change of ventricular geometry, but by a change of end-systolic
strain-stiffening relations due to cardiomyocyte, cell-cell, and
cell-matrix maturation. Thus abnormalities in ventricular morphogenesis
could alter ventricular end-systolic myocardial stiffness and
ventricular contractility via altered ventricular geometry and/or
strain-stiffening relations.
Zero-stress midwall volume increased significantly in parallel with
developmental stage. Ventricular dimension and geometry at the
zero-stress point depend on residual stress and strain (26). In the absence of residual stress and strain, the
combined effects of a large deformation and the highly nonlinear
constitutive relations that characterize the behavior of many soft
tissues can create severe stress concentrations, even under normal
loading conditions (2). Previous data on residual strain
in stage 16-24 chick embryonic ventricles showed that residual
strain changes dramatically at the onset of trabeculation, suggesting
that residual strain is sensitive to changes in ventricular structure
(26). For zero stress to correspond to zero strain, the
constitutive relations must be referred to the absolute (passive + active) zero-stress configuration, which depends on the residual
strains and the degree of activation, and not to the passive zero-load configuration (27). Previous study of residual strain has
represented only a passive zero-stress state, and strains were not
changed significantly after the onset of trabeculation
(26). In this study, the zero-stress volume represents the
absolute zero-stress state. Thus changes of zero-stress volume in our
study not only suggest the changes of ventricular wall structure but
also changes of the material properties of the myocardium during morphogenesis.
Assumptions and limitations.
There are several assumptions and limitations in the present study.
First, the embryonic ventricle is assumed to be a thick-walled ellipsoidal shell with a fixed ratio of semiminor and semimajor axis
diameter during systole. Our method of calculation of ventricular cavity volume and wall thickness may not accurately assess absolute volume and wall thickness changes. However, there are no more accurate
methods to assess absolute ventricular volumes and wall thickness in
the embryonic heart. In addition, ventricular geometry changes
dramatically during cardiovascular morphogenesis. After the onset of
ventricular septation at stage 24, the embryonic ventricle
differentiates into right and left ventricles (23). Thus a
simplified geometric model cannot be applied to calculate ventricular
volume and wall thickness after ventricular septation, and more
accurate geometric models will be required to evaluate the
stress-strain relations and ventricular contractility at these later stages.
Next, several models can be used to quantify wall stress; however, it
is difficult to compare our results of end-systolic wall stress with
those obtained for the mature left ventricle. We calculated the
end-systolic wall stress at stage 17 chick embryos by two different
methods: average midwall stress by use of the elastic modulus concept
and average wall stress in the outer myocardial layer by use of the
pseudostrain-energy density function. The results were quite similar to
each other. Both results depend on the assumption that the embryonic
wall is an incompressible and homogenous material, at least in the
circumferential direction (27, 29). This
assumption must be viewed with caution until further experimental data
are available.
Finally, the theoretical model of systolic stiffness concept assumes
that stress is a function of strain alone, and viscous and inertial
effects are excluded. However, the embryonic myocardium differs
markedly from the mature heart in ultrastructure with a small volume
fraction of organized extracellular matrix and less anisotropy
(28). Recently, Miller et al. (16) showed that the viscoelastic properties of stage 16 and 18 chick embryonic ventricle significantly differ from those of the mature left ventricle. Thus further study is needed to evaluate the effects of developmental changes in the viscoelasticity and inertial effects on embryonic ventricular end-systolic stress-strain relations and systolic myocardial stiffness.
In conclusion, end-systolic stress-strain relations based on the
incremental elastic modulus concept are linear during rapid ventricular
morphogenesis. End-systolic ventricular wall stresses are maintained
within a relatively narrow range, and the end-systolic myocardial
stiffness, an index of ventricular contractility, increases in parallel
with morphogenesis. Measures of ventricular contractility can now be
incorporated into physiological and numerical models of normal and
experimentally altered developing cardiovascular systems.
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APPENDIX |
Taber et al. (27) used a thick-walled,
pseudoelastic cylindrical shell to determine the wall stress
distribution of a stage 16 chick embryonic ventricle. Using this
method, we calculated the average wall stress difference in the
myocardial layer of the stage 17 chick embryonic ventricle from our
experimental data.
Geometric model.
We assumed the ventricle to be a straight, thick-walled, laminated
cylindrical shell with a constant cross section and pressure-sealed ends that are unconstrained geometrically (27).
Deformation therefore depends only on the radial coordinate. The
midwall radius of the cylinder is chosen as the radius of the
ellipsoidal shell at equator level used in the incremental elastic
modulus concept. Passive embryonic ventricular cross section opens
approximately into a circular sector, relieving the residual stresses.
The average opening angle (
) of 75° was taken from previous data
on stage 16 chick embryonic ventricle (26). We also
assumed that the wall thickness and the length of the middle surface
change little when the ventricle opens by bending (Fig.
7). Midwall radius (R) and
wall thickness (h) at the passive zero state and midwall
radius (r0,p) and wall thickness
(h0,P) at the ventricular opening are expressed
in the following equations
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(A1)
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In all embryos in the study, ventricular pressure was initially
negative at early diastole (diastolic suction) and then became positive
during ventricular filling (12). Therefore, we defined the
passive zero-stress radius (R) as the point at which
ventricular pressure returned from negative to zero during ventricular
filling (Fig. 7).
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Fig. 7.
Schematic diagram of the cross section of embryonic
ventricle in passive zero-stress state (state 0,P), absolute
zero-stress state (state 0,A), unloaded physiological state
(state 1), and loaded state (state 2).
State 0,P is obtained from state 1 by cutting
ventricle radially to relieve the passive residual stress. An average
opening angle () of 75° was taken from previous data on a stage 16 chick embryonic ventricle. State 0,A represents an
additional deformation due to muscle activation under no load.
r0, Midwall radius at state 0,P;
h0, wall thickness at state 0,P;
r0,A, midwall radius at state
0,A; h0,A, wall thickness at state
0,A; R, midwall radius at state 1;
h, wall thickness at state 1; r,
midwall radius at state 2; h2, wall
thickness at state 2.
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The wall of the model consists of three isotropic, incompressible, and
pseudoelastic layers (Fig. 8): the
endocardium (layer 1), the cardiac jelly (layer
2), and the myocardium (layer 3). The endocardium
comprised 10%, the cardiac jelly 70%, and the myocardium 20% of the
wall thickness, and only layer 3 has contractile properties.
Analysis of the model is based on a laminated-shell theory that
includes large displacements and strains, nonlinear constitutive
behavior, thick-shell effects, residual strain, and quasi-static muscle
activation. In this theory, no slippage is allowed between layers, and
muscle viscoelasticity and inertance effects are ignored.
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Fig. 8.
Schematic cross section of the laminated cylindrical
shell model for a stage 17 embryonic heart. The wall of the model
consists of 3 isotropic, incompressible, and pseudoelastic layers.
Endocardium comprised 10%, cardiac jelly 70%, and myocardium 20% of
the wall thickness.
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The muscle-shell theory accounts for two primary time-dependent
phenomena during activation: 1) stiffening of the muscle
tissue and 2) changes in the zero-stress muscle length
through activation-strain parameters or active shift ratios
(27). To consider muscle stiffening, we assumed that the
form of the pseudostrain-energy density function of the myocardium
remains the same throughout the cardiac cycle, but the material
coefficients of the active part change continuously during activation.
Second, we also assumed that the zero-stress muscle length-strain
parameters are constant during muscle activation. These two assumptions
are based on the time-varying elastance theory (22). Shift
ratios characterize the difference between the passive and absolute
(passive + active) zero-stress state. In this regard, it is
convenient to define four global states for the ventricle (Fig. 7): the
passive zero-stress state (S0,P), the unloaded
physiological state (S1), the loaded state
(S2), and the absolute zero-stress state
(S0,A). We choose S1 as the reference state.
S0,P is obtained from S1 by cutting the
ventricle radially to relieve the passive residual stress, and
S0,A represents an additional deformation due to muscle
activation under no load. If 
i are stretch
ratios relative to the reference state, then the stretch ratios of
layer k relative to the passive and active
zero-stress state are, respectively
|
(A2)
|
where the passive shift ratios
i,P(k) represent the stretch
ratios of S1 relative to S0,P and the active
shift ratios i,A(k) represent
the stretch ratios of S0,P relative to
S0,A, and i = (1,
2, 3) = (
, , r)
represents the cardiac coordinate system.
i,P(k) is expressed by the
following equations
|
(A3)
|
where 0 is 145°, which is calculated from the
opening angle = 75° at S0,P, and
R(k) and
r0,P(k) are the measured average
radii for layer k.
Next, we assumed that muscle activation is isotropic in a direction
parallel to the epicardium
![&agr;<SUP>(<IT>k</IT>)</SUP><SUB>1,A</SUB><IT>=&agr;</IT><SUP>(<IT>k</IT>)</SUP><SUB><IT>2,</IT>A</SUB><IT>=</IT>[<IT>&agr;</IT><SUP>(<IT>k</IT>)</SUP><SUB><IT>3,</IT>A</SUB>]<SUP><IT>−0.5</IT></SUP><IT>≡&agr;</IT>](/content/vol279/issue1/fulltext/H216/img026.gif)
|
(A4)
|
Equations 3 and 4 satisfy tissue incompressibility.
We took the average radius of layer k at the
S0,A from the zero-stress volume of ESPVR in each embryo
and assumed i,A(k) is constant
during the systolic phase. In terms of stretch ratios, the Lagrange
strain components are Ei,i = (i2 
1)/2 relative to S1.
Pseudostrain energy density functions.
Each of the cardiac tissues is treated as incompressible, isotropic,
and pseudoelastic with material properties characterized by a
pseudostrain-energy density function. We assumed that this function,
per unit volume of material in the reference state, has the form
![<IT>W</IT><SUP>(<IT>k</IT>)</SUP><IT>=</IT><IT>W</IT><SUP>(<IT>k</IT>)</SUP><SUB>P</SUB>[<IT>&Lgr;</IT><SUP>(<IT>k</IT>)</SUP><SUB><IT>i,</IT>P</SUB>]<IT>+</IT><IT>W</IT><SUP>(<IT>k</IT>)</SUP><SUB>A</SUB>[<IT>&Lgr;</IT><SUP>(<IT>k</IT>)</SUP><SUB><IT>i,</IT>A</SUB>]](/content/vol279/issue1/fulltext/H216/img027.gif)
|
(A5)
|
where WP(k) and
WA(k) are the passive and
active contributions, respectively (27). For the
endocardium and the cardiac jelly, WA(k) = 0 and, for the
myocardium, WP(k) includes
effects due to the extracellular matrix and the passive components of
the muscle fibers. WA(k)
includes muscle stiffening during activation by changing the material
coefficients based on the time-varying elastance concept, while the
coefficients in WP(k) are
kept constant.
According to the method of Taber et al. (27), we used the
passive and active pseudostrain energy density functions in the following equations
|
(A6)
|
and
![<IT>W</IT><SUP>(<IT>k</IT>)</SUP><SUB>A</SUB><IT>/C=</IT>[<IT>c<SUB>k</SUB></IT>(<IT>t</IT>)<IT>/d<SUB>k</SUB></IT>(<IT>t</IT>)][I<SUP>(<IT>k</IT>)</SUP><SUB>A</SUB><IT>−3</IT>]<SUP><IT>d<SUB>k</SUB></IT>(<IT>t</IT>)</SUP>](/content/vol279/issue1/fulltext/H216/img031.gif)
|
(A7)
|
where C = 1 mmHg is a material constant with
unit of stress; ak and bk
are dimensionless material coefficients and are provided in each layer
as a1 = 0.15, a2 = 0.075, a3 = 0.15, b1 = 0.10, b2 = 0.10, and
b3 = 1.2 (27). The
dimensionless coefficients ck and
dk depend on the time t from the
onset of systole. For layers 1 and 2,
ck = 0 for all t, and
c3 (myocardial layer) is zero throughout
diastole and at t = 0. The slope of the end-systolic stress-strain relations depends on c3, whereas
its shape (isochronal stress-strain curve) depends mainly on
d3.
End-systolic wall stress distribution.
Using the strain energy density functions in each embryo, we determined
the coefficients c3 and
d3 from the midwall stress difference. If the
ventricular wall is a freely deforming solid body composed of
transversely isotropic and incompressible elastic material during the
cardiac cycle, the average midwall stress difference calculated by the
pseudostrain energy density functions must be same as that of the
elastic modulus concept. Previous study showed that, by comparison with
the elastic modulus concept, the wall stress distribution in the
arterial wall calculated by the pseudostrain energy density functions
was accurate (25). If it is assumed that the ventricular
wall is a freely deforming solid body, which is the same assumption in
the elastic modulus concept, midwall stress difference is expressed as
|
(A8)
|
where W, , r, and
are absolute pseudostrain-energy density (unit of mmHg stress),
midwall circumferential stress, midwall radial stress, and midwall
stress difference, respectively.
From Eq. 8 and the midwall stress difference calculated by
the elastic modulus concept, we determined c3
and d3 at the end-systolic points in each
embryo. The wall stress distribution was computed in each layer (Fig.
6). The average wall stress difference at the myocardial layer was
similar to the average midwall stress difference of the elastic modulus
concept (Table 4).
| |
FOOTNOTES |
Address for reprint requests and other correspondence: K. Tobita, Dept. of Pediatrics, University of Kentucky, 800 Rose
St., Rm. MN472, Lexington, KY 40536-0298 (E-mail:
ktobi0{at}pop.uky.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Received 12 August 1999; accepted in final form 12 January 2000.
| |
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Am J Physiol Heart Circ Physiol 279(1):H216-H224
0363-6135/00 $5.00
Copyright © 2000 the American Physiological Society
TOP
ABSTRACT
INTRODUCTION
![<IT>W</IT><SUP>(<IT>k</IT>)</SUP><SUB>P</SUB><IT>/C=</IT>(<IT>a<SUB>k</SUB>/b<SUB>k</SUB></IT>)[<IT>e</IT><SUP><IT>b<SUB>k</SUB></IT>(I<IT>−3</IT>)</SUP><IT>−1</IT>]](/content/vol279/issue1/fulltext/H216/img028.gif)
![I<SUP>(<IT>k</IT>)</SUP><SUB>P</SUB><IT>=</IT>[<IT>&Lgr;</IT><SUP>(<IT>k</IT>)</SUP><SUB><IT>1,</IT>P</SUB>]<SUP><IT>2</IT></SUP><IT>+</IT>[<IT>&Lgr;</IT><SUP>(<IT>k</IT>)</SUP><SUB><IT>2,</IT>P</SUB>]<SUP><IT>2</IT></SUP><IT>+</IT>[<IT>&Lgr;</IT><SUP>(<IT>k</IT>)</SUP><SUB><IT>3,</IT>P</SUB>]<SUP><IT>2</IT></SUP>](/content/vol279/issue1/fulltext/H216/img030.gif)
![I<SUP>(<IT>k</IT>)</SUP><SUB>A</SUB><IT>=</IT>[<IT>&Lgr;</IT><SUP>(<IT>k</IT>)</SUP><SUB><IT>1,</IT>A</SUB>]<SUP><IT>2</IT></SUP><IT>+</IT>[<IT>&Lgr;</IT><SUP>(<IT>k</IT>)</SUP><SUB><IT>2,</IT>A</SUB>]<SUP><IT>2</IT></SUP><IT>+</IT>[<IT>&Lgr;</IT><SUP>(<IT>k</IT>)</SUP><SUB><IT>3,</IT>A</SUB>]<SUP><IT>2</IT></SUP>](/content/vol279/issue1/fulltext/H216/img032.gif)
