Left-ventricular shape determines intramyocardial mechanical heterogeneity

Left-ventricular remodeling is considered to be an important mechanism of disease progression leading to mechanical dysfunction of the heart. However, the interaction between the physiological changes in the remodeling process and the associated mechanical dysfunction is still poorly understood. Clinically, it has been observed that the left ventricle often undergoes large shape changes, but the importance of left-ventricular shape as a contributing factor to alterations in mechanical function has not been clearly determined. Therefore, the interaction between left-ventricular shape and systolic mechanical function was examined in a computational finite-element study. Hereto, finite-element models were constructed with varying shapes, ranging from an elongated ellipsoid to a sphere. A realistic transmural gradient in fiber orientation was considered. The passive myocardium was described by an incompressible hyperelastic material law with transverse isotropic symmetry. Activation was governed by the eikonal-diffusion equation. Contraction was incorporated using a Hill model. For each shape, simulations were performed in which passive filling was followed by isovolumic contraction and ejection. It was found that the intramyocardial distributions of fiber stress, strain, and stroke work density were shape dependent. Ejection performance was reduced with increasing sphericity, which was regionally related to a reduction in the active fiber stress development, fiber shortening, and stroke work in the midwall and subepicardial region at the midheight level in the left-ventricular wall. Based on these results, we conclude that a significant interaction exists between left-ventricular shape and regional myofiber mechanics, but the importance for left-ventricular remodeling requires further investigation.     

The effect of residual strain on the diastolic function of the left ventricle as predicted by a structural model

The unloaded heart is not stress-free. It is subjected to residual stress and strain. Their extent and influence on the global performance of the left ventricle and on local phenomena in the ventricular wall are studied by model simulation. The analysis focuses on the equatorial region of the ventricle, with an approximate thick-walled cylindrical geometry. The in vivo myocardium is considered to be incompressible, consisting of fibers embedded in a fluid matrix, with transmurally varying anisotropic microstructure in accordance with morphological characteristics.

The results show that residual strain is transmurally distributed with a pattern and magnitude which agree well with measurements. The calculated residual strains are within mean ± one standard deviation of the measured ones. Their magnitude was found to increase with increasing opening angle and with increasing wall thickness. The residual strain was found to have several effects on ventricular function: At volumes higher than the reference one it gives rise to more uniform transmural distributions of stress and intramyocardial pressure; it causes about 50% increase in the ventricular compliance at high volumes and doubles the suction of atrial blood at low volumes, thus facilitating the diastolic filling. In addition, residual strains cause bias of in vivo measured strains from their true values. This may significantly affect physiological interpretation of measured ventricular deformations.

In conclusion, the present structural analysis predicts that residual strain has favorable effect on left-ventricular diastolic performance, and gives rise to more uniform ventricular stress distribution.     

Dependence of local left ventricular wall mechanics on myocardial fiber orientation: a model study.

The dependence of local left ventricular (LV) mechanics on myocardial muscle fiber orientation was investigated using a finite element model. In the model we have considered anisotropy of the active and passive components of myocardial tissue, dependence of active stress on time, strain and strain rate, activation sequence of the LV wall and aortic afterload. Muscle fiber orientation in the LV wall is quantified by the helix fiber angle, defined as the angle between the muscle fiber direction and the local circumferential direction. In a first simulation, a transmural variation of the helix fiber angle from +60 degrees at the endocardium through 0 degrees in the midwall layers to -60 degrees at the epicardium was assumed. In this simulation, at the equatorial level maximum active muscle fiber stress was found to vary from about 110 kPa in the subendocardial layers through about 30 kPa in the midwall layers to about 40 kPa in the subepicardial layers. Next, in a series of simulations, muscle fiber orientation was iteratively adapted until the spatial distribution of active muscle fiber stress was fairly homogeneous. Using a transmural course of the helix fiber angle of +60 degrees at the endocardium, +15 degrees in the midwall layers and -60 degrees at the epicardium, at the equatorial level maximum active muscle fiber stress varied from 52 kPa to 55 kPa, indicating a remarkable reduction of the stress range. Moreover, the change of muscle fiber strain with time was more similar in different parts of the LV wall than in the first simulation. It is concluded that (1) the distribution of active muscle fiber stress and muscle fiber strain across the LV wall is very sensitive to the transmural distribution of the helix fiber angle and (2) a physiological transmural distribution of the helix fiber angle can be found, at which active muscle fiber stress and muscle fiber strain are distributed approximately homogeneously across the LV wall.     

Adaptation of passive rat left ventricle in diastolic dysfunction.

This article deals with providing a theoretical explanation for quantitative changes in the geometry, the opening angle and the deformation parameters of the rat ventricular wall during adaptation of the passive left ventricle in diastolic dysfunction. A large deformation theory is applied to analyse transmural stress and strain distribution in the left ventricular wall considering it to be made of homogeneous, incompressible, transversely isotropic, non-linear elastic material. The basic assumptions made for computing stress distributions are that the average circumferential stress and strain for the adaptive ventricle is equal to the average circumferential stress and strain in the normotensive ventricle, respectively.All the relevant parameters, such as opening angle, twist per unit length, axial extension, internal and external radii and others, in the stress-free, unloaded and loaded states of normotensive, hypertensive and adaptive left ventricle are determined. The circumferential stress and strain distribution through the ventricular wall are also computed. Our analysis predicts that during adaptation, wall thickness and wall mass of the ventricle increase. These results are consistent with experimental findings and are the indications of initiation of congestive heart failure.     

Regional nonuniformity of normal adult human left ventricle

Regional nonuniformity is a feature of both diseased and normal left ventricles (LV). With the use of magnetic resonance (MR) myocardial tagging, we performed three-dimensional strain analysis on 87 healthy adults in local cardiac and fiber coordinate systems (radial, circumferential, longitudinal, and fiber strains) to characterize normal nonuniformities and to test the validity of wall thickening as a parameter of regional function. Regional morphology included wall thickness and radii of curvature measurements. With respect to transmural nonuniformity, subendocardial strains exceeded subepicardial strains. Going from base to apex, wall thickness and circumferential radii of curvature decreased, whereas longitudinal radii of curvature increased. All of the strains increased from LV base to apex, resulting in a higher ejection fraction (EF) at the apex than at the base (70.9 +/- 0.4 vs. 62.4 +/- 0.4%; means +/- SE, P < 0.0001). When we looked around the circumference of the ventricle, the anterior part of the LV was the flattest and thinnest and showed the largest wall thickening (46.6 +/- 1.2%) but the lowest EF (64.7 +/- 0.5%). The posterior LV wall was thicker, more curved, and showed a lower wall thickening (32.8 +/- 1.0%) but a higher EF (71.3 +/- 0.5%). The regional contribution of the LV wall to the ejection of blood is thus highly variable and is not fully characterized by wall thickening alone. Differences in regional LV architecture and probably local stress are possible explanations for this marked functional nonuniformity.     

On a nonlinear theory for muscle shells: Part II--Application to the beating left ventricle

This paper specializes the nonlinear laminated-muscle-shell theory developed in Part I to cylindrical geometry and computes stresses in arteries and the beating left ventricle. The theory accounts for large strain, material nonlinearity, thick-shell effects, torsion, muscle activation, and residual strain. First, comparison with elasticity solutions for pressurized arteries shows that the accuracy of the shell theory increases as transmural stress gradients and the shell thickness decrease. Residual strain reduces the stress gradients, lowering the error in the predicted peak stress in thick-walled arteries (R/t = 2.8) from about 30 to 10 percent. Second, the canine left ventricle is modeled as a thick-walled laminated cylinder with an internal pressure. Each layer is composed of transversely isotropic muscle with a fiber orientation based on anatomical data. Using a single pseudostrain-energy density function (with time-varying coefficients) for passive and active myocardium, the model predicts strain distributions that agree fairly well with published experimental measurements. The results also show that the peak fiber stress occurs subendocardially near the beginning of ejection and that residual strains significantly alter stress gradients within each lamina, but the magnitude of the peak fiber stress changes by less than 20 percent.     

Passive material properties of intact ventricular myocardium determined from a cylindrical model

The equatorial region of the canine left ventricle was modeled as a thick-walled cylinder consisting of an incompressible hyperelastic material with homogeneous exponential properties. The anisotropic properties of the passive myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varied linearly across the wall. Simultaneous inflation, extension, and torsion were applied to the cylinder to produce epicardial strains that were measured previously in the potassium-arrested dog heart. Residual stress in the unloaded state was included by considering the stress-free configuration to be a warped cylindrical arc. In the special case of isotropic material properties, torsion and residual stress both significantly reduced the high circumferential stress peaks predicted at the endocardium by previous models. However, a resultant axial force and moment were necessary to cause the observed epicardial deformations. Therefore, the anisotropic material parameters were found that minimized these resultants and allowed the prescribed displacements to occur subject to the known ventricular pressure loads. The global minimum solution of this parameter optimization problem indicated that the stiffness of passive myocardium (defined for a 20 percent equibiaxial extension) would be 2.4 to 6.6 times greater in the fiber direction than in the transverse plane for a broad range of assumed fiber angle distributions and residual stresses. This agrees with the results of biaxial tissue testing. The predicted transmural distributions of fiber stress were relatively flat with slight peaks in the subepicardium, and the fiber strain profiles agreed closely with experimentally observed sarcomere length distributions. The results indicate that torsion, residual stress and material anisotropy associated with the fiber architecture all can act to reduce endocardial stress gradients in the passive left ventricle.     

Diastolic pressure-volume relations and distribution of pressure and fiber extension across the wall of a model left ventricle.

A model for left ventricular diastolic mechanics is formulated that takes into account noneligible wall thickness, incompressibility, finite deformation, nonlinear elastic effects, and the known fiber architecture of the ventricular wall. The model consists of a hollow cylindrical mass of muscle bound between two plates of negligible mass. The wall contains fiber elements that follow a helical course and carry only axial tension. The fiber angle (i.e., helical pitch) is constant along the length of each fiber but varies through the wall in accordance with the known distribution of fiber orientations in the canine left ventricle. To simplify the analysis and reduce the number of degrees of freedom, the anatomic distribution of fiber orientations is divided into a clockwise and counterclockwise system. The reference configuration for the model corresponds to a state in which, by hypothesis, the transmural pressure gradient is zero, the tension is zero for all fibers across the wall, and all fibers are assumed to have a sarcomere length of 1.9 micrometer. This choice of reference configuration is based on the empirical evidence that canine ventricles, fixed in a state of zero transmural pressure gradient and dissected, demonstrate sarcomere lengths between 1.9 and 2.0 micrometer in inner, middle, and outer wall layers, while isolated ventricular muscle bundles are observed to have zero resting tension when the sarcomere length ranges from 1.9 to 2.0 micrometer. An equation representing the global condition for equilibrium is derived and solved numerically. It is found that the model's pressure-volume relation is representative of diastolic filling in vivo over a wide range of filling pressures, and the calculated midwall sarcomere lengths in the model compare favorably with published experimental data. Subendocardial fibers are stretched beyond Lmax even at low filling pressures, i.e., 5 mm Hg, while fibers located between 60-80% of wall thickness extend minimally between 5 and 12 mm Hg. The hydrostatic pressure field within the wall is highly nonlinear. The pressure rises steeply in the subendocardial layers so that the net gain in pressure in the inner third of the wall is 85% of the filling pressure. It is demonstrated that these results are independent of heart size for a family of heart models that are scale models of each other. They are, however, critically dependent on the existence of longitudinally oriented fibers in the endocardial and epicardial regions of heart wall.     

Factors influencing left-ventricular stiffness

The aim of the study was to investigate the relative contributions of geometrical and material factors to overall left-ventricular cavity stiffness. Left-ventricular cavity shapes were reconstructed using a computer and the variation of myocardial elastic modulus was calculated, by the finite element method, through the passive phase of diastole when rising volume coincided with rising pressure. Geometric data were obtained from biplane cineangiography, with micromanometer pressure measurements, for ten patients with left ventricular disease. Dimensional analysis was applied to the initial and derived data from which the influences of myocardial compliance, wall thickness-to-long dimension ratio, and aspect ratio (long-to-short axes) were determined. The ratio between the volume elasticity and the myocardial modulus of elasticity, the normalized stiffness ratio (NSR), is proposed as a useful index of left ventricular mechanical behaviour in diastole. The volume elasticity of the chamber is dependent not only upon the myocardium elastic modulus and the wall thickness ratio, but also on the shape of the chambe. Changes in the thickness/radius ratio of the ventricle have less effect upon its distention than those in the long dimension/radius ratio. The left ventricle becomes more spherical in shpae through diastole and hence becomes stiffer by this geometric mechanism.     

Relation between left ventricular cavity pressure and volume and systolic fiber stress and strain in the wall.

Pumping power as delivered by the heart is generated by the cells in the myocardial wall. In the present model study global left-ventricular pump function as expressed in terms of cavity pressure and volume is related to local wall tissue function as expressed in terms of myocardial fiber stress and strain. On the basis of earlier studies in our laboratory, it may be concluded that in the normal left ventricle muscle fiber stress and strain are homogeneously distributed. So, fiber stress and strain may be approximated by single values, being valid for the whole wall. When assuming rotational symmetry and homogeneity of mechanical load in the wall, the dimensionless ratio of muscle fiber stress (sigma f) to left-ventricular pressure (Plv) appears to depend mainly on the dimensionless ratio of cavity volume (Vlv) to wall volume (Vw) and is quite independent of other geometric parameters. A good (+/- 10%) and simple approximation of this relation is sigma f/Plv = 1 + 3 Vlv/Vw. Natural fiber strain is defined by ef = In (lf/lf,ref), where lf,ref indicates fiber length (lf) in a reference situation. Using the principle of conservation of energy for a change in ef, it holds delta ef = (1/3)delta In (1 + 3Vlv/Vw).     

Fiber stress profiles in the left ventricle of the heart during diastole: Effects of distension and hypertrophy

Pressure-volume and volume-dimensions relationships, obtained from excised dog left ventricles were used for calculating the stresses acting along the longitudinal axis of the individual myocardial fibers. The calculations were based on a set of empirical and theoretical equations. The pressure-volume relationship as well as the volume-dimensions relationships for the excised left ventricle were expressed in the form of empirical equations; the fiber orientation was written as a function of the fiber location within the left ventricular wall; finally, the fiber stress was determined by means of theoretically derived formulas. Simultaneous solutions for the fibers of a meridian cut through the left ventricular myocardial shell were obtained by means of a digital computer and presented in the form of diagrams. The results showed that at low degrees of distension of the left ventricle there are two zones of higher stresses at the equatorial area, one near the epicardium and one near the endocardium. As the distension proceeds under the effect of progressively increasing intraventricular pressure, these two zones become less well defined, whereas a new zone of higher stresses appears near the apex. At high degrees of distension, the ventricle assumes a more spherical shape and the equatorial zones of higher stresses are replaced by zones of lower stresses. Increase in the myocardial mass results in appearance of the equatorial lower stress zones at lower degrees of distension.     

An anatomic basis for fetal right ventricular dominance and arterial pressure sensitivity

Fetal right ventricular dominance of flow and arterial pressure sensitivity were recently recognized but controversial findings. We investigated ventricular volumes, weights and dimensions in order to understand if there were anatomic differences between the ventricles which might explain these differential functional findings in the fetal sheep. Forty-four near term lambs and their hearts were weighed. Right and left ventricular free wall weights were not different. Volumes were measured by generating in vitro pressure-volume relations and by casting the two ventricles after fixation at equal, physiologic pressures. Right ventricular volume was greater than left ventricular volume by both techniques. Ventricular interaction and a restraining effect of the pericardium were present. Measurements of the fixed ventricles and their casts revealed the following: left ventricular wall thickness was slightly greater than right ventricular wall thickness; lateral ventricular diameters were not different but anteroposterior ventricular diameters were much greater in the right than left ventricle. Because of these findings, the right ventricular circumferential radii of curvature were greater than for the left ventricle as was the radius to wall thickness ratio. Greater right ventricular volume and radius to wall thickness ratio may be important factors in right ventricular flow dominance and greater sensitivity to arterial pressure.     

Transmural distribution of glucose metabolizing enzymes across the left and the right ventricle heart walls in three different mammalian species

The transmural distribution of five glucose metabolizing enzymes (hexokinase; glucose-6-phosphate dehydrogenase; phosphofructokinase; aldolase; and lactate dehydrogenase) were explored in the left and in the right ventricle wall of rat, ox and pig hearts. The levels of most of these enzyme activities were different in the different animal species and (within the same species) in the two ventricles. Most of these enzyme activities were found to be non-uniformly distributed across the left (but not across the right) ventricle wall. Differences in the transmural distribution of enzyme activities were detected among the three examined mammalian species.     

A two-phase finite element model of the diastolic left ventricle

A porous medium finite element model of the passive left ventricle is presented. The model is axisymmetric and allows for finite deformation, including torsion about the axis of symmetry. An anisotropic quasi-linear viscoelastic constitutive relation is implemented in the model. The model accounts for changing fibre orientation across the myocardial wall. During passive filling, the apex rotates in a clockwise direction relative to the base for an observer looking from apex to base. Within an intraventricular pressure range of 0-3 kPa the rotation angle of all nodes remained below 0.1 rad. Diastolic viscoelasticity of myocardial tissue is shown to reduce transmural differences of preload-induced sarcomere stretch and to generate residual stresses in an unloaded ventricular wall, consistent with the observation of opening angles seen when the heart is slit open. It is shown that the ventricular model stiffens following an increase of the intracoronary blood volume. At a given left ventricular volume, left ventricular pressure increases from 1.5 to 2.0 kPa when raising the intracoronary blood volume from 9 to 14 ml (100 g)-1 left ventricle.     

Stress Distribution in the Canine Left Ventricle during Diastole and Systole

A model is proposed for stress analysis of the left ventricular wall (LV wall) based on the realistic assumption that the myocardium is essentially composed of fiber elements which carry only axial tension and vary in orientation through the wall. Stress analysis based on such a model requires an extensive study of muscle fiber orientation and curvature through the myocardium. Accordingly, the principal curvatures were studied at a local site near the equator in ten dog hearts rapidly fixed in situ at end diastole and end systole; the fiber orientation for these hearts had already been established in a previous study. The principal radii of curvature were (a) measured by fitting templates to the endocardial and epicardial wall surfaces in the circumferential and longitudinal directions and (b) computed from measured lengths of semiaxes of ellipsoids of revolution representing the LV wall (“ellipsoid” data). The wall was regarded as a tethered set of nested shells, each having a unique fiber orientation. Results indicate the following. (a) Fiber curvature, k, is maximum at midwall at end systole; this peak shifts towards endocardium at end diastole. (b) The pressure or radial stress through the wall decreases more rapidly near the endocardium than near the epicardium at end diastole and at end systole when a constant tension is assumed for each fiber through the wall. (c) At end diastole the curve for the circumferential stress vs. wall thickness is convex with a maximum at midwall. In the longitudinal direction the stress distribution curve is concave with a minimum at midwall. Similar distributions are obtained at end systole when a constant tension is assumed for each fiber through the wall. (d) The curvature and stress distributions obtained by direct measurements at a selected local site agree well with those computed from “ellipsoid” data.     

Inhomogeneity in the response to mechanical stimulation: cardiac muscle function and gene expression

Mechanical stimulation has important consequences for myocardial function. However, this stimulation and the response to it, is not uniform. The right ventricle is thinner walled and operates at lower pressure than the left ventricle. Within the ventricles, differences in the orientation of myocardial fibres exist. These differences produce inhomogeneity in the stress and strain between and across the ventricles. Possibly as a result of these variations in mechanical stimulation, there are well characterised inhomogeneities in gene expression and protein function within the ventricular myocardium, for example in the transient outward K+ current and its associated Kv channels. Perhaps not surprisingly, it is becoming apparent that gradients of expression and function exist for proteins that are intimately involved in the response to mechanical stimulation in the heart, for example in the left ventricle of the rat there is a transmural gradient in mRNA and current density of the mechanosensitive two-pore domain K+ channel TREK-1 (ENDO>EPI). In healthy hearts it is assumed that these gradients are important for normal function and therefore that their disruption in diseased myocardium is involved in the dysfunction that occurs.     

Optimizing ventricular fibers: uniform strain or stress,but not ATP consumption,leads to high efficiency

The aim of this study was to investigate the influence of fiber orientation in the left ventricular (LV) wall on the ejection fraction, efficiency, and heterogeneity of the distributions of developed fiber stress, strain and ATP consumption. A finite element model of LV mechanics was used with active properties of the cardiac muscle described by the Huxley-type cross-bridge model. The computed variances of sarcomere length (SL(var)), developed stress (DS(var)), and ATP consumption (ATP(var)) have several minima at different transmural courses of helix fiber angle. We identified only one region in the used design space with high ejection fraction, high efficiency of the LV and relatively small SL(var), DS(var), and ATP(var). This region corresponds to the physiological distribution of the helix fiber angle in the LV wall. Transmural fiber angle can be predicted by minimizing SL(var) and DS(var), but not ATP(var). If ATP(var) was minimized, then the transverse fiber angle was considerably underestimated. The results suggest that ATP consumption distribution is not regulating the fiber orientation in the heart.     

Characterization of the normal cardiac myofiber field in goat measured with MR-diffusion tensor imaging

Cardiac myofiber orientation is a crucial determinant of the distribution of myocardial wall stress. Myofiber orientation is commonly quantified by helix and transverse angles. Accuracy of reported helix angles is limited. Reported transverse angle data are incomplete. We measured cardiac myofiber orientation postmortem in five healthy goat hearts using magnetic resonance-diffusion tensor imaging. A novel local wall-bound coordinate system was derived from the characteristics of the fiber field. The transmural course of the helix angle corresponded to data reported in literature. The mean midwall transverse angle ranged from -12 +/- 4 degrees near the apex to +9.0 +/- 4 degrees near the base of the left ventricle, which is in agreement with the course predicted by Rijcken et al. (18) using a uniform load hypothesis. The divergence of the myofiber field was computed, which is a measure for the extent to which wall stress is transmitted through the myofiber alone. It appeared to be <0.07 mm(-1) throughout the myocardial walls except for the fusion sites between the left and right ventricles and the insertion sites of the papillary muscles.     

Asynchrony of ventricular activation affects magnitude and timing of fiber stretch in late-activated regions of the canine heart

Abnormal electrical activation of the left ventricle results in mechanical dyssynchrony, which is in part characterized by early stretch of late-activated myofibers. To describe the pattern of deformation during "prestretch" and gain insight into its causes and sequelae, we implanted midwall and transmural arrays of radiopaque markers into the left ventricular anterolateral wall of open-chest, isoflurane-anesthetized, adult mongrel dogs. Biplane cineradiography (125 Hz) was used to determine the time course of two- and three-dimensional strains while pacing from a remote, posterior wall site. Strain maps were generated as a function of time. Electrical activation was assessed with bipolar electrodes. Posterior wall pacing generated prestretch at the measurement site, which peaked 44 ms after local electrical activation. Overall magnitudes and transmural gradients of strain were reduced when compared with passive inflation. Fiber stretch was larger at aortic valve opening compared with end diastole (P < 0.05). Fiber stretch at aortic valve opening was weakly but significantly correlated with local activation time (r(2) = 0.319, P < 0.001). With a short atrioventricular delay, fiber lengths were not significantly different at the time of aortic valve opening during ventricular pacing compared with atrial pacing. However, ejection strain did significantly increase (P < 0.05). We conclude that the majority of fiber stretch occurs after local electrical activation and mitral valve closure and is different from passive inflation. The increased shortening of these regions appears to be because of a reduced afterload rather than an effect of length-dependent activation in this preparation.     

Epicardial suction: a new approach to mechanical testing of the passive ventricular wall

The lack of an appropriate three-dimensional constitutive relation for stress in passive ventricular myocardium currently limits the utility of existing mathematical models for experimental and clinical applications. Previous experiments used to estimate parameters in three-dimensional constitutive relations, such as biaxial testing of excised myocardial sheets or passive inflation of the isolated arrested heart, have not included significant transverse shear deformation or in-plane compression. Therefore, a new approach has been developed in which suction is applied locally to the ventricular epicardium to introduce a complex deformation in the region of interest, with transmural variations in the magnitude and sign of nearly all six strain components. The resulting deformation is measured throughout the region of interest using magnetic resonance tagging. A nonlinear, three-dimensional, finite element model is used to predict these measurements at several suction pressures. Parameters defining the material properties of this model are optimized by comparing the measured and predicted myocardial deformations. We used this technique to estimate material parameters of the intact passive canine left ventricular free wall using an exponential, transversely isotropic constitutive relation. We tested two possible models of the heart wall: first, that it was homogeneous myocardium, and second, that the myocardium was covered with a thin epicardium with different material properties. For both models, in agreement with previous studies, we found that myocardium was nonlinear and anisotropic with greater stiffness in the fiber direction. We obtained closer agreement to previously published strain data from passive filling when the ventricular wall was modeled as having a separate, isotropic epicardium. These results suggest that epicardium may play a significant role in passive ventricular mechanics.