Larry A. Taber

Stress-Modulated Growth, Residual Stress, and Vascular Heterogeneity

A simple phenomenological model is used to study interrelations between material properties, growth-induced residual stresses, and opening angles in arteries. The artery is assumed to be a thick-walled tube composed of an orthotropic pseudoelastic material. In addition, the normal mature vessel is assumed to have uniform circumferential wall stress, which is achieved here via a mechanical growth law. Residual stresses are computed for three configurations: the unloaded intact artery, the artery after a single transmural cut, and the inner and outer rings of the artery created by combined radial and circumferential cuts. The results show that the magnitudes of the opening angles depend strongly on the heterogeneity of the material properties of the vessel wall and that multiple radial and circumferential cuts may be needed to relieve all residual stress. In addition, comparing computed opening angles with published experimental data for the bovine carotid artery suggests that the material properties change continuously across the vessel wall and that stress, not strain, correlates well with growth in arteries.

Mechanics of ventricular torsion

Recent research suggests that left ventricular torsion is an important indicator of cardiac function. We used two theoretical models to study the mechanics of this phenomenon: a compressible cylinder and an incompressible ellipsoid of revolution. The analyses of both models account for large- strain passive and active material behavior, with a muscle fiber angle that varies linearly from endocardium to epicardium. Relative to the end- diastolic configuration, the predicted torsion exhibits several experimentally observed features, including a peak near end systole, rapid untwisting during isovolumic relaxation, and increased twist near the apex. The magnitude of the twist is sensitive to the fiber architecture, the ventricular geometry, and the compressibility and contractility of the myocardium. In particular, the model predicts that the systolic twist increases with increasing compressibility, contractility, and wall thickness, while it decreases with increasing cavity volume. The peak twist approximately doubles (from about 0.02 to 0.04 rad cm(-1)) with a doubling of myocardial compressibility or with a change in the endocardial/epicardial muscle fiber angles from 90/ -90 degrees to 60/ -60 degrees. The twist is less sensitive to changes in contractility and ventricular geometry. These findings provide a basis for interpreting measurements of ventricular torsion in the clinical setting.

Nonlinear theory of elasticity : applications in biomechanics

Vectors, Dyadics, and Tensors Analysis of Deformation Analysis of Stress Constitutive Relations Biomechanics Applications.

Residual strain in the ventricle of the stage 16-24 chick embryo.

Residual stress and strain, i.e., the stress and strain remaining in a solid when all external loads are removed, may be produced in biological tissues by differential growth. During cardiac development, residual stress and strain may play a role in cardiac morphogenesis by affecting ventricular wall stress. After a transmural radial cut, a passive ventricular cross section opens into a sector, and the size of the opening angle provides a measure of the circumferential residual strain. Residual strains were characterized in this manner for the apical region of the diastolic embryonic chick heart for Hamburger-Hamilton stages 16, 18, 21, and 24 (approximately 2.5, 3.5, 4.0, and 4.5 days, respectively, of a 21-day incubation period). The average opening angle at these stages was 107 +/- 10 degrees, 79 +/- 10 degrees, 73 +/- 11 degrees, and 74 +/- 7 degrees, respectively (n > or = 5 for each stage). These measured angles were correlated with changes in ventricular morphology. Scanning electron micrographs of the apex revealed that the wall of the ventricle is smooth at stage 16. Then at stage 18, myocardial trabeculae develop, forming ridges with primarily a circumferential orientation. By stage 21, the trabeculae develop into a mesh, giving the ventricular wall a spongelike appearance, and the preferred orientation is lost by stage 24. The large decrease in opening angle between stages 16 and 18 corresponded to the onset of trabeculation, which is the greatest change in form during the studied stages. We speculate that residual strain is an important biomechanical factor during cardiac morphogenesis.

The possible role of poroelasticity in the apparent viscoelastic behavior of passive cardiac muscle

This paper investigates the contribution of extracellular fluid flow to the apparent viscoelastic behavior of passive cardiac muscle. The muscle is modeled as an incompressible, isotropic, poroelastic solid saturated by an incompressible viscous fluid. Based on Biots linear and nonlinear consolidation theories, solutions are presented for general time-dependent uniaxial loading of unconfined cylindrical muscle specimens. The nonlinear analysis includes the effects of large strain, material nonlinearity, and strain-dependent permeability. The computed results show that, for axial stretch ratios greater than 1.1, the changing permeability and the loading rate strongly affect the total stress relaxation and the short-time relaxation rate. Comparisons of theoretical and published experimental results show that extracellular fluid flow can account for several observed biomechanical features of passive myocardium, including the insensitivity of stress-strain curves to loading rate and of stress-relaxation curves to the amount of stretch. Theoretical hysteresis loops, however, are too small. Thus, both poroelastic and tissue viscoelastic effects must be considered in studies of passive cardiac muscle.

A nonliner poroelastic model for the trabecular embryonic heart.

A theoretical model is presented for the primitive right ventricle of the stage 21 chick embryo. At this stage of development, the wall of the heart is trabecular with direct intramyocardial blood flow. The model is a pressurized fluid-filled cylinder composed of a porous inner layer of isotropic myocardium and a relatively thin compact outer layer of transversely isotropic myocardium. The analysis is based on nonlinear poroelasticity theory, modified to include residual strain and muscle activation. Correlating theoretical and experimental pressure-volume loops and epicardial strains gives first-approximation constitutive relations for stage 21 embryonic myocardium. The results from the model suggest three primary conclusions: (1) Some muscle fibers likely are aligned in the compact layer, with a fiber angle approximately + 10 deg from the circumferential direction. (2) Blood is drawn into the wall of the ventricle during diastolic filling and isovolumic contraction and is squeezed out of the wall during systolic ejection, giving a primitive intramyocardial circulation before the coronary arteries form. As the heart rate increases, the transmural blood-flow velocity increases, but the volume of blood exchanged with the lumen per beat decreases. (3) Residual strain affects transmural stress distributions, producing nearly uniform stresses in the porous layer, where the peak end-systolic stress occurs. These results improve our understanding of the relation between form and function in the developing heart and provide directions for biological experiments to study cardiac morphogenesis.

Cardiac mechanics in the stage-16 chick embryo

A theoretical model is presented for the tubular heart of the stage-16 chick embryo (2.3 days of a 21-day incubation period). The model is a thick-walled, pseudoelastic cylindrical shell composed of three isotropic layers: the endocardium, the cardiac jelly, and the myocardium. The analysis is based on a shell theory that accounts for large deformation, material nonlinearity, residual strain, and muscle activation, with material properties inferred from available experimental data. We also measured epicardial strains from recorded motions of microspheres on the primitive right ventricles of stage-16 white Leghorn chick embryos. Relative to end diastole, peak axial and circumferential Lagrange strains occurred near end systole and had similar values. The magnitudes of these strains varied along the longitudinal axis of the heart (-0.16 +/- 0.08), being larger near the ends of the primitive right ventricle and smaller near midventricle. The in-plane shear strain was less than 0.05. Comparison of theoretical and experimental strains during the cardiac cycle shows generally good agreement. In addition, the model gives strong stress concentrations in the myocardial layer at end systole.

Epicardial strains in embryonic chick ventricle at stages 16 through 24.

Embryonic cardiac development depends, in part, on the local biomechanical environment. Tracking the motions of microspheres attached to the embryonic chick ventricle, we computed two-dimensional epicardial strains at Hamburger-Hamilton stages 16, 18, 21, and 24 (2.5, 3.5, 4.0, and 4.5 days, respectively, of a 21-day incubation period). First, in a cross-sectional study, strains were measured in separate embryos at each stage (n > or = 19 per stage). Then, in a longitudinal study, strains were measured serially on the same heart, with the eggs resealed and reincubated between successive stages (n > or = 4 per stage). Although the heart undergoes major changes in mass, morphology, and loading during the studied stages, both studies showed that peak circumferential and longitudinal strains relative to end diastole were similar in magnitude (0.13 to 0.16) and did not change significantly across the stage range. The peak principal strains also showed no significant changes, with magnitudes of approximately 0.11 and 0.18. The shear strains were small, and their signs varied from one heart to another. These results suggest that wall strain is maintained within a relatively narrow range during primary cardiac morphogenesis.

Mechanical effects of looping in the embryonic chick heart

During early embryonic development, the heart bends into a curved tube in a vital morphogenetic process called looping. Since looping involves poorly understood biomechanical forces that are difficult to measure, this paper presents a theoretical model for the tubular chick heart, whose development is similar to that of the human heart. Representing the basic morphology of the looped ventricle, the model is a thick-walled, isotropic, pressurized curved tube composed of three layers representing the myocardium, cardiac jelly, and endocardium. The model is analyzed with nonlinear elasticity theory, modified to include residual strain and muscle activation, and material properties are determined by correlating theoretical and experimental pressure-volume relations. The results show that longitudinal curvature significantly influences the biomechanical behavior of the embryonic heart. As the curvature increases, the compliance of the tube increases, especially at end systole. Stress concentrations, which develop in the endocardium during diastole and in the myocardium during systole, also increase with the curvature. The largest wall stress during the cardiac cycle occurs near the beginning of systolic ejection in the myocardial layer at the inner curvature of the tube. Relative to end diastole, the model predicts epicardial strains that are nearly equal in the circumferential and meridional directions, in agreement with experimental measurements. These results provide insight into the interrelation between biomechanical forces and morphogenesis during cardiac looping.

On approximate large strain relations for a shell of revolution

Abstract A series of geometric and constitutive relations is studied for large axisymmetric strain of elastic shells of revolution. The kinematic assumption employs a modified Kirchhoff hypothesis which accounts for thickness changes but neglects transverse shear deformation. Calculations are presented for cylindrical and spherical shells composed of incompressible materials with two types of strain energy density function: Mooney-Rivlin (rubber) and exponential (biological tissue). Comparison of results for large bending at a clamped edge demonstrates the accuracy and limitations of various approximations for stress and strain. The computations indicate that the stress resultants are quite sensitive to the details of the asymmetric motion of points relative to the reference surface.