Andrew D. McCulloch

Stress-dependent finite growth in soft elastic tissues

Growth and remodeling in tissues may be modulated by mechanical factors such as stress. For example, in cardiac hypertrophy, alterations in wall stress arising from changes in mechanical loading lead to cardiac growth and remodeling. A general continuum formulation for finite volumetric growth in soft elastic tissues is therefore proposed. The shape change of an unloaded tissue during growth is described by a mapping analogous to the deformation gradient tensor. This mapping is decomposed into a transformation of the local zero-stress reference state and an accompanying elastic deformation that ensures the compatibility of the total growth deformation. Residual stress arises from this elastic deformation. Hence, a complete kinematic formulation for growth in general requires a knowledge of the constitutive law for stress in the tissue. Since growth may in turn be affected by stress in the tissue, a general form for the stress-dependent growth law is proposed as a relation between the symmetric growth-rate tensor and the stress tensor. With a thick-walled hollow cylinder of incompressible, isotropic hyperelastic material as an example, the mechanics of left ventricular hypertrophy are investigated. The results show that transmurally uniform pure circumferential growth, which may be similar to eccentric ventricular hypertrophy, changes the state of residual stress in the heart wall. A model of axially loaded bone is used to test a simple stress-dependent growth law in which growth rate depends on the difference between the stress due to loading and a predetermined growth equilibrium stress.

The Cardiac Mechanical Stretch Sensor Machinery Involves a Z Disc Complex that Is Defective in a Subset of Human Dilated Cardiomyopathy

Muscle cells respond to mechanical stretch stimuli by triggering downstream signals for myocyte growth and survival. The molecular components of the muscle stretch sensor are unknown, and their role in muscle disease is unclear. Here, we present biophysical/biochemical studies in muscle LIM protein (MLP) deficient cardiac muscle that support a selective role for this Z disc protein in mechanical stretch sensing. MLP interacts with and colocalizes with telethonin (T-cap), a titin interacting protein. Further, a human MLP mutation (W4R) associated with dilated cardiomyopathy (DCM) results in a marked defect in T-cap interaction/localization. We propose that a Z disc MLP/T-cap complex is a key component of the in vivo cardiomyocyte stretch sensor machinery, and that defects in the complex can lead to human DCM and associated heart failure.

A myocardial lineage derives from Tbx18 epicardial cells

Understanding the origins and roles of cardiac progenitor cells is important for elucidating the pathogenesis of congenital and acquired heart diseases. Moreover, manipulation of cardiac myocyte progenitors has potential for cell-based repair strategies for various myocardial disorders. Here we report the identification in mouse of a previously unknown cardiac myocyte lineage that derives from the proepicardial organ. These progenitor cells, which express the T-box transcription factor Tbx18, migrate onto the outer cardiac surface to form the epicardium, and then make a substantial contribution to myocytes in the ventricular septum and the atrial and ventricular walls. Tbx18-expressing cardiac progenitors also give rise to cardiac fibroblasts and coronary smooth muscle cells. The pluripotency of Tbx18 proepicardial cells provides a theoretical framework for applying these progenitors to effect cardiac repair and regeneration.

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.

Modelling the mechanical properties of cardiac muscle.

A model of passive and active cardiac muscle mechanics is presented, suitable for use in continuum mechanics models of the whole heart. The model is based on an extensive review of experimental data from a variety of preparations (intact trabeculae, skinned fibres and myofibrils) and species (mainly rat and ferret) at temperatures from 20 to 27 degrees C. Experimental tests include isometric tension development, isotonic loading, quick-release/restretch, length step and sinusoidal perturbations. We show that all of these experiments can be interpreted with a four state variable model which includes (i) the passive elasticity of myocardial tissue, (ii) the rapid binding of Ca2+ to troponin C and its slower tension-dependent release, (iii) the kinetics of tropomyosin movement and availability of crossbridge binding sites and the length dependence of this process and (iv) the kinetics of crossbridge tension development under perturbations of myofilament length.

A collocation-Galerkin finite element model of cardiac action potential propagation

A new computational method was developed for modeling the effects of the geometric complexity, nonuniform muscle fiber orientation, and material inhomogeneity of the ventricular wall on cardiac impulse propagation. The method was used to solve a modification to the FitzHugh-Nagumo system of equations. The geometry, local muscle fiber orientation, and material parameters of the domain were defined using linear Lagrange or cubic Hermite finite element interpolation. Spatial variations of time-dependent excitation and recovery variables were approximated using cubic Hermite finite element interpolation, and the governing finite element equations were assembled using the collocation method. To overcome the deficiencies of conventional collocation methods on irregular domains, Galerkin equations for the no-flux boundary conditions were used instead of collocation equations for the boundary degrees-of-freedom. The resulting system was evolved using an adaptive Runge-Kutta method. Converged two-dimensional simulations of normal propagation showed that this method requires less CPU time than a traditional finite difference discretization. The model also reproduced several other physiologic phenomena known to be important in arrhythmogenesis including: Wenckebach periodicity, slowed propagation and unidirectional block due to wavefront curvature, reentry around a fixed obstacle, and spiral wave reentry. In a new result, the authors observed wavespeed variations and block due to nonuniform muscle fiber orientation. The findings suggest that the finite element method is suitable for studying normal and pathological cardiac activation and has significant advantages over existing techniques.<<ETX>>

Substrate Stiffness Affects the Functional Maturation of Neonatal Rat Ventricular Myocytes

Cardiac cells mature in the first postnatal week, concurrent with altered extracellular mechanical properties. To investigate the effects of extracellular stiffness on cardiomyocyte maturation, we plated neonatal rat ventricular myocytes for 7 days on collagen-coated polyacrylamide gels with varying elastic moduli. Cells on 10 kPa substrates developed aligned sarcomeres, whereas cells on stiffer substrates had unaligned sarcomeres and stress fibers, which are not observed in vivo. We found that cells generated greater mechanical force on gels with stiffness similar to the native myocardium, 10 kPa, than on stiffer or softer substrates. Cardiomyocytes on 10 kPa gels also had the largest calcium transients, sarcoplasmic calcium stores, and sarcoplasmic/endoplasmic reticular calcium ATPase2a expression, but no difference in contractile protein. We hypothesized that inhibition of stress fiber formation might allow myocyte maturation on stiffer substrates. Treatment of maturing cardiomyocytes with hydroxyfasudil, an inhibitor of RhoA kinase and stress fiber-formation, resulted in enhanced force generation on the stiffest gels. We conclude that extracellular stiffness near that of native myocardium significantly enhances neonatal rat ventricular myocytes maturation. Deviations from ideal stiffness result in lower expression of sarcoplasmic/endoplasmic reticular calcium ATPase, less stored calcium, smaller calcium transients, and lower force. On very stiff substrates, this adaptation seems to involve RhoA kinase.

Finite element stress analysis of left ventricular mechanics in the beating dog heart

A three-dimensional finite element model was used to explore whether or not transmural distributions of end-diastolic and end-systolic fiber stress are uniform from the apex to the base of the canine left ventricular wall. An elastance model for active fiber stress was incorporated in an axisymmetric model that accurately represented the geometry and fiber angle distribution of the anterior free wall. The nonlinear constitutive equation for the resting myocardium was transversely isotropic with respect to the local fiber axis. Transmural distributions of end-diastolic fiber stress became increasingly nonuniform from midventricle toward the apex or the base. At a typical diastolic left ventricular pressure (1 kPa), the differences between largest and smallest fiber stresses were only 0.5 kPa near midventricle, compared with 4.6 kPa at the apex, and 3.3 kPa at the base. Transmural fiber stress differences at end-systole (14 kPa) were relatively small in regions from the base to the midventricle (13-22 kPa), but were larger between midventricle and the apex (30-43 kPa). All six three-dimensional end-diastolic strain components were within or very close to one standard deviation of published measurements through the midanterior left ventricular free wall of the passive canine heart [Omens et al., Am. J. Physiol. 261, H918-H928 (1991)]. End-systolic in-plane normal and shear strains also agreed closely with published experimental measurements in the beating dog heart [Waldman et al., Circ. Res. 63, 550-562 (1988)]. The results indicate that, unlike in the midventricle region that has been studied most fully, there may be significant regional nonhomogeneity of fiber stress in the normal left ventricle associated with regional variations in shape and fiber angle.

Three-dimensional analysis of regional cardiac function: a model of rabbit ventricular anatomy

The three-dimensional geometry and anisotropic properties of the heart give rise to nonhomogeneous distributions of stress, strain, electrical activation and repolarization. In this article we review the ventricular geometry and myofiber architecture of the heart, and the experimental and modeling studies of three-dimensional cardiac mechanics and electrophysiology. The development of a three-dimensional finite element model of the rabbit ventricular geometry and fiber architecture is described in detail. Finally, we review the experimental results, from the level of the cell to the intact organ, which motivate the development of coupled three-dimensional models of cardiac electromechanics and mechanoelectric feedback.

Laminar fiber architecture and three-dimensional systolic mechanics in canine ventricular myocardium

Previous studies suggest that the laminar architecture of left ventricular myocardium may be critical for normal ventricular mechanics. However, systolic three-dimensional deformation of the laminae has never been measured. Therefore, end-systolic finite strains relative to end diastole, from biplane radiography of transmural markers near the apex and base of the anesthetized open-chest canine anterior left ventricular free wall (n = 6), were referred to three-dimensional laminar microstructural axes reconstructed from histology. Whereas fiber shortening was uniform [-0.07 +/- 0.04 (SD)], radial wall thickening increased from base (0. 10 +/- 0.09) to apex (0.14 +/- 0.13). Extension of the laminae transverse to the muscle fibers also increased from base (0.08 +/- 0. 07) to apex (0.11 +/- 0.08), and interlaminar shear changed sign [0. 05 +/- 0.07 (base) and -0.07 +/- 0.09 (apex)], reflecting variations in laminar architecture. Nevertheless, the apex and base were similar in that at each site laminar extension and shear contributed approximately 60 and 40%, respectively, of mean transmural thickening. Kinematic considerations suggest that these dual wall-thickening mechanisms may have distinct ultrastructural origins.Previous studies suggest that the laminar architecture of left ventricular myocardium may be critical for normal ventricular mechanics. However, systolic three-dimensional deformation of the laminae has never been measured. Therefore, end-systolic finite strains relative to end diastole, from biplane radiography of transmural markers near the apex and base of the anesthetized open-chest canine anterior left ventricular free wall ( n = 6), were referred to three-dimensional laminar microstructural axes reconstructed from histology. Whereas fiber shortening was uniform [-0.07 ± 0.04 (SD)], radial wall thickening increased from base (0.10 ± 0.09) to apex (0.14 ± 0.13). Extension of the laminae transverse to the muscle fibers also increased from base (0.08 ± 0.07) to apex (0.11 ± 0.08), and interlaminar shear changed sign [0.05 ± 0.07 (base) and -0.07 ± 0.09 (apex)], reflecting variations in laminar architecture. Nevertheless, the apex and base were similar in that at each site laminar extension and shear contributed ∼60 and 40%, respectively, of mean transmural thickening. Kinematic considerations suggest that these dual wall-thickening mechanisms may have distinct ultrastructural origins.