## 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

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.

## 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-cm^{2} 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} mm^{2} 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.

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). Figure2 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).

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* = π*DL*) and the volume (V) of an ellipsoid of revolution [V = (4π*D ^{2}L*)/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 (V_{c}) was calculated as total volume (V_{t}) minus ventricular wall volume (V_{m}). 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 (*D*
_{i}) and wall thickness (*h*) were computed by solving the following equations
Equation 1
where *L*
_{i} is internal major semiaxis dimension, and it is assumed that*L*
_{i}
*/D*
_{i} = 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 (ε) is defined as the difference of the circumferential (ε_{θ}) and radial (ε_{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
Equation 2
Thus the total strain difference is calculated as
Equation 3where *D*
_{m}, *L*
_{m}, and*D*
_{0,m} are minor semiaxis, major semiaxis, and zero-stress minor semiaxis midwall diameter at the equator, respectively.

Stress difference (ς) is defined as the difference of the circumferential (ς_{θ}) and radial (ς_{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 (ς) is calculated by
Equation 4where ς_{θ,a} and ς_{r,a} are the average circumferential and radial stresses, P is left ventricular pressure, and *h* is wall thickness.

Average systolic myocardial stiffness (*E*
_{av}) is calculated as
Equation 5
*Equation* 5 indicates that the stress-strain relationship is linear (17, 29).

#### Normalized end-systolic myocardial stiffness (E_{av}/G).

Ventricular midwall volume (V_{m}) based on the thick-walled ellipsoidal model is
Equation 6If it is assumed that*D*
_{m}/*L*
_{m} = 3/4, then
Equation 7where *k* = (16/9)π. We can then use V_{m} and V_{0,m} to calculate
Equation 8where V_{0,m} is the zero-stress midwall volume. Thus*Eq.* 5 is expressed as
Equation 9We then converted end-systolic stress-strain relations to end-systolic pressure-volume relations (ESPVR)
Equation 10
where ς_{es}, P_{es}, V_{es}, and*G* are end-systolic stress difference, end-systolic pressure, end-systolic midwall volume, and geometric factor, respectively, and α and β are regression coefficients.

By use of *Eqs. 9
* and *
10
*, ESPVR is expressed as
Equation 11The slope of *Eq. 11
*,*E*
_{av}/*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 ς_{es}-to-ε_{es} ratio reaches a maximum value after the onset of systole. We obtained five to seven different ς_{es}-*D*
_{es} points from gradual conotruncal occlusion in each embryo. We first assumed a value for*D*
_{0,m}, and then the ς_{es} and logarithmic *D*
_{es} points were fitted by linear regression analysis. A new *D*
_{0,m} was obtained by approximate extrapolation to zero stress. This iterative procedure was continued until the value for *D*
_{0,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).

## 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 (Table2). 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; Table3).

#### End-systolic myocardial stiffness.

End-systolic myocardial stiffness (*E*
_{av,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).

Zero-stress midwall volume increased significantly from stage 17 to 24 (*P* < 0.05; Table 3, Fig. 5).

## 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 ). Table4 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.

#### 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 (*E*
_{av}/*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, *E*
_{av}/*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.

## 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 (θ) 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 (*r*
_{0,p}) and wall thickness (*h*
_{0,P}) at the ventricular opening are expressed in the following equations
Equation A1
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).

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.

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 (S_{0,P}), the unloaded physiological state (S_{1}), the loaded state (S_{2}), and the absolute zero-stress state (S_{0,A}). We choose S_{1} as the reference state. S_{0,P} is obtained from S_{1} by cutting the ventricle radially to relieve the passive residual stress, and S_{0,A} represents an additional deformation due to muscle activation under no load. If Λ_{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
Equation A2
where the passive shift ratios α_{i,P}
^{(k)} represent the stretch ratios of S_{1} relative to S_{0,P} and the active shift ratios α_{i,A}
^{(k)} represent the stretch ratios of S_{0},_{P} relative to S_{0},_{A}, and *i* = (1,2, 3) = (φ, θ, *r*) represents the cardiac coordinate system. α_{i,P}
^{(k)} is expressed by the following equations
Equation A3
where θ_{0} is 145°, which is calculated from the opening angle θ = 75° at S_{0,P}, and*R*
^{(k)} and*r*
_{0,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
Equation A4
*Equations 3
* and *
4
* satisfy tissue incompressibility.

We took the average radius of *layer k* at the S_{0,A} from the zero-stress volume of ESPVR in each embryo and assumed α_{i,A}
^{(k)} is constant during the systolic phase. In terms of stretch ratios, the Lagrange strain components are *E _{i,i}
* = (Λ

_{i}

^{2}− 1)/2 relative to S

_{1}.

#### 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
Equation A5where *W*
_{P}
^{(k)} and*W*
_{A}
^{(k)} are the passive and active contributions, respectively (27). For the endocardium and the cardiac jelly,*W*
_{A}
^{(k)} = 0 and, for the myocardium, *W*
_{P}
^{(k)} includes effects due to the extracellular matrix and the passive components of the muscle fibers. *W*
_{A}
^{(k)}includes muscle stiffening during activation by changing the material coefficients based on the time-varying elastance concept, while the coefficients in *W*
_{P}
^{(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
Equation A6
and
Equation A7
where *C* = 1 mmHg is a material constant with unit of stress; *a _{k}
* and

*b*are dimensionless material coefficients and are provided in each layer as

_{k}*a*

_{1}= 0.15,

*a*

_{2}= 0.075,

*a*

_{3}= 0.15,

*b*

_{1}= 0.10,

*b*

_{2}= 0.10, and

*b*

_{3}= 1.2 (27). The dimensionless coefficients

*c*and

_{k}*d*depend on the time

_{k}*t*from the onset of systole. For

*layers 1*and

*2*,

*c*= 0 for all

_{k}*t*, and

*c*

_{3}(myocardial layer) is zero throughout diastole and at

*t*= 0. The slope of the end-systolic stress-strain relations depends on

*c*

_{3}, whereas its shape (isochronal stress-strain curve) depends mainly on

*d*

_{3}.

#### End-systolic wall stress distribution.

Using the strain energy density functions in each embryo, we determined the coefficients *c*
_{3} and*d*
_{3} 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
Equation A8
where *W*, ς_{θ}, ς_{r}, and ς 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 *c*
_{3}and *d*
_{3} 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).

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- Copyright © 2000 the American Physiological Society