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.

SUMMARY. To examine transmural finite deformation in the wall of the canine left ventricle, closely spaced columns of lead beads were implanted at a single site on the left ventricular free wall. The three-dimensional coordinates of these myocardial markers were obtained with highspeed biplane cineradiography. Any four noncoplanar markers forming small tetrahedral volumes («*0.1 cc) were used to calculate finite normal and shear strains with respect to a cardiac coordinate system at end diastole. Due to the symmetry of the finite strain tensor, the algebraic eigenvalue problem could be solved to compute principal strains and the directions of the principal axes of deformation with respect to the reference coordinates. An examination of the principal strains in a number of tetrahedra in five animals indicates that deformation increases with depth beneath the epicardium. For example, the transmural variation of principal shortening strain averages -0.014 ± 0.009 per 10% increment in thickness from epicardium to endocardium. Furthermore, shortening and thickening strains at midwall and deeper are too large (0.10 to 0.40) to be described accurately by infinitesimal theory. These strains are often accompanied by substantial in-plane and transverse shears which are not predicted by typical membrane or shell theories, indicating that these theories must be applied with caution when computing indices of regional ventricular performance. The directions of the principal axes of shortening vary substantially less than the fiber direction varies across the wall (20°-40° compared with 100°-140° for fiber direction), supporting the concept that there are substantial interactions between neighboring fibers in the left ventricular wall. (Circ Res 57: 152-163, 1985)

To determine the relation between local myofiber anatomy and local deformation in the wall of the left ventricle, both three-dimensional transmural deformation and myofiber orientation were examined in the anterior free wall of seven canine left ventricles. Deformation was measured by imaging columns of implanted radiopaque markers with high-speed, biplane cineradiography (16 mm, 120 frames/sec). Hearts were fixed at end diastole and sectioned parallel to the local epicardial tangent plane to determine the transmural distribution of fiber directions at the site of strain measurement. The principal direction of deformation associated with the greatest shortening was compared with the local fiber direction in the outer (21 ± 8 % of the wall thickness from the epicardium) and inner (65 ±9%) halves of the wall. Although the fiber direction varied substantially with depth from the epicardium, the principal direction did not. In the outer half of the wall, fiber direction averaged-8 ± 24°, while the principal direction averaged-33 ±24° from circumferential (counterclockwise angles are positive). In the inner half, fiber direction averaged 69 ±10°, while the principal direction averaged-22 ±21°. Therefore, while fiber and principal directions were not substantially different hi the outer half, the greatest shortening occurred orthogonally to the fiber direction hi the inner half. Normal and shear strains measured in a cardiac coordinate system (circumferential, longitudinal, and radial coordinates) were rotated (transformed) to "fiber" coordinates hi both halves of the wall. In the outer half, normal strains observed in the fiber (-0.09 ±0.04) and cross-fiber (-0. 0 4 ±0.04) directions were not significantly different (paired t test, p<0.05). In the inner half, more than twice as much strain occurred in the cross-fiber (-0.17 ± 0.03) than in the fiber direction (-0.06 ± 0.06). Moreover, the only shear strain that remained substantial after transformation was transverse shear in the plane of the fiber and radial coordinates. These results suggest that both reorientation and cross-sectional shape changes of myofibers or the interstitium may contribute to the large wall thickenings observed during contraction, particularly in the inner hah* of the ventricular wall. (Circulation Research 1988;63:550-562) T he way in which myofibers lying at different depths in the heart wall and having different orientations interact during contraction is not known. It is well known, however, that there is an extensive collagenous network surrounding the myocyte and collagen struts between myocytes. 12 Moreover, the direction of the myofibers, which varies continuously with depth spanning more than 100° across the anterior free wall of the canine left ventricle, 34 is neither altered by large changes in ventricular mass or shape 3 nor affected greatly by

Models of contracting ventricular myocardium were used to study the effects of different assumptions concerning active tension development on the distributions of stress and strain in the equatorial region of the intact left ventricle during systole. Three models of cardiac muscle contraction were incorporated in a cylindrical model for passive left ventricular mechanics developed previously [Guccione et al. ASME Journal of Biomechanical Engineering, Vol. 113, pp. 42-55 (1991)]. Systolic sarcomere length and fiber stresses predicted by a general "deactivation" model of cardiac contraction [Guccione and McCulloch, ASME Journal of Biomechanical Engineering, Vol. 115, pp. 72-81 (1993)] were compared with those computed using two less complex models of active fiber stress: In a time-varying "elastance" model, isometric tension development was computed from a function of peak intracellular calcium concentration, time after contraction onset and sarcomere length; a "Hill" model was formulated by scaling this isometric tension using the force-velocity relation derived from the deactivation model. For the same calcium ion concentration, the sarcomeres in the deactivation model shortened approximately 0.1 microns less throughout the wall at end-systole than those in the other models. Thus, muscle fibers in the intact ventricle are subjected to rapid length changes that cause deactivation during the ejection phase of a normal cardiac cycle. The deactivation model predicted rather uniform transmural profiles of fiber stress and cross-fiber stress distributions that were almost identical to those of the radial component. These three components were indistinguishable from the principal stresses. Transmural strain distributions predicted at end-systole by the deactivation model agreed closely with experimental measurements from the anterior free wall of the canine left ventricle.

We developed a device that applies homogeneous equibiaxial strains of 0-10% to a cell culture substrate and quantitatively verified transmission of substrate deformation to cultured cardiac cells. Clamped elastic membranes in both single-well and multiwell versions of the device are uniformly stretched by indentation with a plastic ring, resulting in strain that is directly proportional to the pitch-to-radius ratio. Two-dimensional deformations were measured by tracking fluorescent microspheres attached to the substrate and to cultured adult rat cardiac fibroblasts. For nominal stretches up to 18%, strains along circumferential and radial axes were equal in magnitude and homogeneously distributed with negligible shear. For 5% stretch, circumferential and radial strains in the substrate were 0.046 +/- 0.005 and 0.048 +/- 0.004 [not significant (NS)], respectively, and shear strain was 0.001 +/- 0.003 (NS). Calibration of both single-well and multiwell versions permits strain selection by device rotation. The reproducible application and quantification of homogeneous equibiaxial strain in cultured cells provides a quantitative approach for correlating mechanical stimuli to cellular transduction mechanisms.