Estimation of local myocardial stress.

Two formulas are presented for estimating local average circumferential stress in the left ventricle from the cavity pressure and various quantities, available from the angiogram, which characterize the size and shape of the cavity and ventricular wall. The advantages of these formulas are as follows: 1) they are based on thick-wall shell theory; 2) they are intended for application at positions in the ventricular wall other than the base; and 3) they are based on a more general representation of ventricular geometry than a sphere, cylinder, or ellipsoid. Except for one location, both formulas predict average circumferential stresses that agree to within 25% with the corresponding stresses in a finite element model of an aneurysmal ventricle. In addition, at the equator of a thick-wall ellipsoid, the formulas are identical in form to a previously derived formula that has been shown to predict stresses that are in fair to good agreement with measured stresses in the open-chest dog heart.

Effect of chamber eccentricity on equatorial fiber stress during systole.

A survey was conducted of methods of assessing stress in the left ventricle. Although simultaneous measurements have been made of average wall force and thickness in the dog left ventricle (from which stress can be computed), the experimental procedures are not applicable to man. However, two mathematical models based on ellipsoidal representations of ventricular geometry have been shown to predict average circumferential stress reasonably well. Although both of these models are sensitive to errors in average equatorial wall thickness, they appear to be the most reliable models currently available for estimating stress. One of these models was applied to an analysis of midwall equatorial fiber stress and force in the normal conscious dog left ventricle. It was found that variations in chamber eccentricity during systole were much less important in evaluating this stress than variations in the ratio of equatorial wall thickness to semi-minor radius. A formula was derived that relates fiber stress to fiber force. It was found that fiber stress and force decrease more rapidly during ejection than intraventricular pressure, consistent with previous results.

Deformation of the diastolic left ventricle. Nonlinear elastic effects.

A linear incremental finite element model is used to analyze the mechanical behavior of the left ventricle. The ventricle is treated as a heterogeneous, non-linearly elastic, isotropic, thick-walled solid of revolution. A new triaxial constitutive relation for the myocardium is presented which exhibits the observed exponential length-passive tension behavior of left ventricular papillary muscle in the limit of uniaxial tension. This triaxial relation contains three parameters: (a) a "small strain" Young's modulus, (b) a Poisson's ratio, and (c) a parameter which characterizes the nonlinear aspect of the elastic behavior of heart muscle. The inner third and outer two-thirds of the ventricular wall are assumed to have small strain Young's moduli of 30 and 60 g/cm(2), respectively. The Poisson's ratio is assumed to be equal to 0.49 throughout the ventricular wall. In general, the results of this study indicate that while a linearly elastic model for the ventricle may be adequate in terms of predicting pressure-volume relationships, a linear model may have serious limitations with regard to predicting fiber elongation within the ventricular wall. For example, volumes and midwall equatorial circumferential strains predicted by the linear and nonlinear models considered in this study differ by approximately 20 and 90%, respectively, at a transmural pressure of 12 cm H(2)O.

Estimation of regional stress in the left ventricular septum and free wall: an echocardiographic study suggesting a mechanism for asymmetric septal hypertrophy.

Although asymmetric septal hypertrophy is noted in a wide variety of cardiac disorders, its cause remains unclear. One possible mechanism is that the septum is subjected to greater systolic stress because of its flatter (more eccentric) contour. This was investigated noninvasively in nine subjects by estimation of regional myocardial stress from measurements of blood pressure by cuff sphygmomanometry and by echocardiographic examinations of left ventricular shape and dimensions. Analysis of left ventricular cavity shape showed that both the free and septal walls were elliptical, but the septum was more eccentric than the free wall. Using a conceptual model to determine changes in regional systolic stress, the theoretical rate of increase in regional stress relative to pressure (delta S/delta P) was significantly greater in the septum compared to the free wall. Increased hypertrophy of the septum to normalize this increased delta S/delta P may be the cause of asymmetric septal hypertrophy in many disorders associated with elevated left ventricular pressure.

Regional stress in a noncircular cylinder.

Several mathematical formulas are presented for estimating regional average circumferential stress and shear stress in a thick-wall, noncircular cylinder with a plane of symmetry. The formulas require images of exterior and interior chamber silhouettes plus surface pressures. The formulas are primarily intended for application to the left ventricle in the short axis plane near the base (where the meridional radius of curvature is normally much larger than the circumferential radius of curvature) and to blood vessels. The formulas predict stresses in a variety of chambers to within 3% of finite element values determined from a large-scale structural analysis computer program called ANSYS.

Effect of age on passive elastic stiffness of rat heart muscle.

A thick-wall spherical model for the rat left ventricle was used to deduce passive wall stiffness from diastolic pressure-volume data. This was done for rats in three age classes: young (1 mo), adult (17 mo) and old (17 mo). The model was based on finite deformation elasticity theory consistent with the magnitude of observed deformation. A least-squares procedure was used to determine elastic constants in postulated nonlinear stress-stretch relations for the myocardium. It was found that at a given level of stress, wall stiffness for ventricles in the young age class was consistently greater than wall stiffness in the other two classes. In addition, the difference in wall stiffness between rats in the adult and old age classes was found to be approximately 10%.

Effect of shape on pressure-volume relationships of ellipsoidal shells.

Previous theoretical procedures for determining the slope and intercept of the stiffness-stress relationship of the passive myocardium from diastolic pressure-volume (P-V) data assume that the eccentricity of the left ventricle (LV) is invariant. In this study a mathematical model for an ellipsoidal membrane was developed that does not contain that constraint. The model predicts a small (less than 10%) but significant decrease in eccentricity as transmural pressure increases. This result was confirmed by the use of a thick-wall finite element model. The implications of this result are as follows. 1) The slope and intercept of the stiffness-stress relationship of unconstrained ellipsodial shells can be determined by fitting a spherical model to the P-V relationships exhibited by the shells. 2) An ellipsoidal model that assumes that the eccentricity of such shells is invariant for all pressures would predict erroneous intercepts. 3) If the eccentricity of the diastolic LV initially decreases relative to its value at zero transmural pressure, then a thick-wall spherical model may be adequate for determining the slope and intercept of the myocardial stiffness-stress relationship.