A 2D FE model of the heart demonstrates the role of the pericardium in ventricular deformation

During pulmonary artery constriction (PAC), an experimental model of acute right ventricular (RV) pressure overload, the interventricular septum flattens and inverts. Finite element (FE) analysis has shown that the septum is subject to axial compression and bending when so deformed. This study examines the effects of acute PAC on the left ventricular (LV) free wall and the role the pericardium may play in these effects. In eight open-chest anesthetized dogs, LV, RV, aortic, and pericardial pressures were recorded under control conditions and with PAC. Model dimensions were derived from two-dimensional echocardiography minor-axis images of the heart. At control (pericardium closed), FE analysis showed that the septum was concave to the LV; stresses in the LV, RV, and septum were low; and the pericardium was subject to circumferential tension. With PAC, RV end-diastolic pressure exceeded LV pressure and the septum inverted. Compressive stresses developed circumferentially in the septum out to the RV insertion points, forming an arch-like pattern. Sharp bending occurred near the insertion points, accompanied by flattening of the LV free wall. With the pericardium open, the deformations and stresses were different. The RV became much larger, especially with PAC. With PAC, the arch-like circumferential stresses still developed in the septum, but their magnitudes were reduced, compared with the pericardium-closed case. There was no free wall inversion and flattening was less. From these FE results, the pericardium has a significant influence on the structural behavior of the septum and the LV and RV free walls. Furthermore, the deformation of the heart is dependent on whether the pericardium is open or closed.     

A modified Holzapfel-Ogden law for a residually stressed finite strain model of the human left ventricle in diastole

In this work, we introduce a modified Holzapfel-Ogden hyperelastic constitutive model for ventricular myocardium that accounts for residual stresses, and we investigate the effects of residual stresses in diastole using a magnetic resonance imaging–derived model of the human left ventricle (LV). We adopt an invariant-based constitutive modelling approach and treat the left ventricular myocardium as a non-homogeneous, fibre-reinforced, incompressible material. Because in vivo images provide the configuration of the LV in a loaded state even in diastole, an inverse analysis is used to determine the corresponding unloaded reference configuration. The residual stress in this unloaded state is estimated by two different methods. One is based on three-dimensional strain measurements in a local region of the canine LV, and the other uses the opening angle method for a cylindrical tube. We find that including residual stress in the model changes the stress distributions across the myocardium and that whereas both methods yield qualitatively similar changes, there are quantitative differences between the two approaches. Although the effects of residual stresses are relatively small in diastole, the model can be extended to explore the full impact of residual stress on LV mechanical behaviour for the whole cardiac cycle as more experimental data become available. In addition, although not considered here, residual stresses may also play a larger role in models that account for tissue growth and remodelling.     

Finite-element analysis of left ventricular myocardial stresses

A versatile method of finite-element analysis is presented for the determination of the stress distributions in the left ventricular myocardial wall. The instantaneous shapes of the left ventricular myocardial wall, measured at 0,5 mm intervals and at a rate 0f 60 images/sec during a cardiac cycle, are approximated by axisymmetric shells following the approach of Gould et al. and analysed by the method of incremental loadings to account for the changing transmural pressure. The ventricular wall is mathematically divided up into coaxial rings of triagular cross sections so that determination of the stresses at any point within the wall can be achieved by assigning increased number of nodes across the wall thickness in the regions of the left ventricular wall where particular attention is needed. Appropriate boundary conditions are defined at the base of the left ventricle so that it can be treated as a shell with an open end. The computer program, which implements all the stress calculations involved, depends on the dimensions of the left ventricular wall measured from an operator-interactive roengen videometry system. It carries out the sequential formation of the nodes and elements and includes a CALCOMP subroutine to plot the finite-element partitioning of the instantaneous shape. Illustrative results of the end-diastolic stress distributions within the myocardial wall of a metabolically-supported, isolated, working canine left ventricle are given. This technique predicts higher endocardial meridional and hoop wall stresses relative to the stresses in the middle and epicardial region than those obtained with previous models.     

Left Ventricular Stresses in the Intact Human Heart

A set of stress differential equations of equilibrium is presented for a thick prolate spheroid which is the assumed shape for the left ventricle. An analysis for the stresses in the ventricular wall indicates that maximum stresses occur at the inner layers and decrease to a minimum at the epicardial surface, a result that is partially validated by experiment. Simple expressions are available for the evaluation of maximum stresses which occur at the equator and are suitable for small laboratory-oriented digital computers employed in the clinical evaluation of patient status. The surprising result is that Laplace's law yields practical values for mean stresses in thick-walled ventricles.     

Effects of anisotropy and nonhomogeneity on left ventricular stresses in the intact heart

The qualitative effects of anisotropy and nonhomogeneity are considered in the evaluation of left ventricular stresses in the intact heart. Maximum stresses and their location are significantly dependent on the nonhomogeneity factors and to a lesser degree on anisotropy of the ventricular wall material. If the circumferential elastic modulus is assumed to vary in a parabolic manner through the wall thickness, maximum stresses occur within the endocardial layers, a result in qualitative agreement with experimental studies.     

A geometric model for the myocardium: Biventricular wall stresses in normal and hypertrophied states

Assuming truncated ellipsoidal geometry for the right and left ventricles, a model is developed for the myocardium enabling biventricular mechanical behavior to be studied. Employing pressure-volume data taken from normal dog hearts and from hearts in which the pulmonary artery has been banded over periods of 2–40 weeks, it is shown that: (a) right ventricular wall stresses are higher than left ventricular stresses; (b) right ventricular wall stress increases initially to a maximum after 3–4 weeks followed by a decline to normal and even subnormal levels, attaining a minimum value at 32–33 weeks; (c) left ventricular stresses behave in a similar manner, attaining their maximum and minimum levels after 7–8 weeks and 32–33 weeks respectively. These results suggest that surgical or medical therapy in patients with hypertrophied ventricles might be more appropriate during the period of wall stress reduction.     

Mechanical analysis of the development of left ventricular aneurysms

The left ventricle is modelled as a spherical shell with an infarcted wall segment. The mechanics of the circumstances causing this infarcted segment to develop into an aneurysm is presented. Both the wall stresses and deformations are worked out for aneurysms developing from infarcts of different sizes and percentages of wall damage. The governing equations consist of incompressibility relations, force-equilibrium relations and stress-strain relations. Newton Raphson technique is used to solve these nonlinear simultaneous algebraic equations, for the values of the myocardial stresses in the infarcted segment and the bulge values, in terms of the ventricular geometry and the damage extent (expressed in terms of the damage angle and percentage of wall damage). The results indicate that in general it is innermost layer which is severely stressed and that in the rupture of the ventricle the critical factor involved is the percentage of infarct thickness rather than the angle of damage.     

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.     

In-vivo mitral annuloplasty ring transducer: Implications for implantation and annular downsizing

Mitral annuloplasty has been a keystone to the success of mitral valve repair in functional mitral regurgitation. Understanding the complex interplay between annular-ring stresses and left ventricular function has significant implications for patient-ring selection, repair failure, and patient safety. A step towards assessing these challenges is developing a transducer that can be implanted in the exact method as commercially available rings and can quantify multidirectional ring loading. An annuloplasty ring transducer was developed to measure stresses at eight locations on both the in-plane and out-of-plane surfaces of an annuloplasty ring's titanium core. The transducer was implanted in an ovine subject using 10 sutures at near symmetric locations. At implantation, the ring was observed to undersize the mitral annulus. The flaccid annulus exerted both compressive (−) and tensile stresses (+) on the ring ranging from −3.17 to 5.34 MPa. At baseline hemodynamics, stresses cyclically changed and peaked near mid-systole. Mean changes in cyclic stress from ventricular diastole to mid-systole ranged from −0.61 to 0.46 MPa (in-plane direction) and from −0.49 to 1.13 MPa (out-of-plane direction). Results demonstrate the variability in ring stresses that can be introduced during implantation and the cyclic contraction of the mitral annulus. Ring stresses at implantation were approximately 4 magnitudes larger than the cyclic changes in stress throughout the cardiac cycle. These methods will be extended to ring transducers of differing size and geometry. Upon additional investigation, these data will contribute to improved knowledge of annulus-ring stresses, LV function, and the safer development of mitral repair techniques.     

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.     

Compression of interventricular septum during right ventricular pressure loading

The interventricular septum, which flattens and inverts in conditions such as pulmonary hypertension, is considered by many to be an unstressed membrane, in that its position is assumed to be determined solely by the transseptal pressure gradient. A two-dimensional finite element model was developed to investigate whether compression and bending moments (behavior incompatible with a membrane) exist in the septum during diastole under abnormal loading, i.e., pulmonary artery (PA) constriction. Hemodynamic and echocardiographic data were obtained in six open-chest anesthetized dogs. For both control and PA constriction, the measured left ventricular and right ventricular pressures were applied to a residually stressed mesh. Adjustments were made to the stiffness and end-bending moments until the deformed and loaded residually stressed mesh matched the observed configuration of the septum. During PA constriction, end-bending moments were required to obtain satisfactory matches but not during control. Furthermore, substantial circumferential compressive stresses developed during PA constriction. Such stresses might impede septal blood flow and provoke the unexplained ischemia observed in some conditions characterized by abnormal septal motion.     

Fibre Orientation in Human Fetal Heart and Ventricular Mechanics : A Small Perturbation Analysis

The study of the topological organisation of myocardial cells is a basic requirement for understanding the mechanical design of the normal and pathological heart. Anatomical observations show that cardiac muscle tissue has a highly specialized architecture. We have made new quantitative measurements of fibre orientation through the heart wall by means of polarized light analysis on some thick sections of human fetal heart embedded in a resin and polymerized. A small perturbation method to find an equilibrium solution in a cylindrical left ventricular (LV) geometry with fibres running on toroidal shells of revolution is used to investigate the mechanical behaviour of three human fetal hearts (FH) of 14, 20 and 33 weeks of gestational age. The results of fibre strains and stresses presented for end-systolic state show significant differences when compared to results of the cylindrical geometry with regular helicoidal fibres running on cylindrical surfaces. Moreover, the toroidal shells of revolution explain shear stresses and strains in the transverse plane which also exist in the adult heart.     

A 2D finite element model of the interventricular septum under normal and abnormal loading

The interventricular septum is the structure that separates the left and right ventricles of the heart. Under normal loading conditions, it is concave to the left ventricle, but under abnormal loading the septum flattens and occasionally inverts. In the past, the septum has frequently been modelled as integral to the left ventricle with the effects of pressure from the right ventricle being ignored. Under abnormal loading, the septum has been described as behaving equivalent to a "flapping sail". There has been no consideration of structural behaviour under these conditions. A 2-D plane stress FE model of the septum was used to investigate the difference in structural behaviour of the septum during diastole between normal and abnormal loading. The biaxial stress patterns that develop are distinctively disparate. Under normal loading, the septum behaves much like a thick-walled cylinder subject to internal and external pressure, with the resulting stresses being circumferential tension and radial compression, both varying with radius. These stresses are very low throughout most of diastole. However, under abnormal loading, the septum behaves in an arch-like fashion, with high compressive stresses almost circumferential in direction, combined with radial compression. We conclude that right ventricular pressures cause bending effects in the wall of the heart, and that under abnormal loading, the compressive stresses that develop in the septum may lead to an understanding of certain, previously unexplained, pathological conditions.     

Experimental measurement of dynamic fluid shear stress on the ventricular surface of the aortic valve leaflet

Aortic valve (AV) calcification is a highly prevalent disease with serious impact on mortality and morbidity. The exact causes and mechanisms of AV calcification are unclear, although previous studies suggest that mechanical forces play a role. It has been clinically demonstrated that calcification preferentially occurs on the aortic surface of the AV. This is hypothesized to be due to differences in the mechanical environments on the two sides of the valve. It is thus necessary to characterize fluid shear forces acting on both sides of the leaflet to test this hypothesis. The current study is one of two studies characterizing dynamic shear stress on both sides of the AV leaflets. In the current study, shear stresses on the ventricular surface of the AV leaflets were measured experimentally on two prosthetic AV models with transparent leaflets in an in vitro pulsatile flow loop using two-component Laser Doppler Velocimetry (LDV). Experimental measurements were utilized to validate a theoretical model of AV ventricular surface shear stress based on the Womersley profile in a straight tube, with corrections for the opening angle of the valve leaflets. This theoretical model was applied to in vivo data based on MRI-derived volumetric flow rates and valve dimension obtained from the literature. Experimental results showed that ventricular surface shear stress was dominated by the streamwise component. The systolic shear stress waveform resembled a half-sinusoid during systole and peaks at 64–71 dyn/cm², and reversed in direction at the end of systole for 15–25?ms, and reached a significant negative magnitude of 40–51 dyn/cm². Shear stresses from the theoretical model applied to in vivo data showed that shear stresses peaked at 77–92 dyn/cm² and reversed in direction for substantial period of time (108–110?ms) during late systole with peak negative shear stress of 35–38 dyn/cm².     

A 2D Finite Element Model of the Interventricular Septum Under Normal and Abnormal Loading

Abstract

The interventricular septum is the structure that separates the left and right ventricles of the heart. Under normal loading conditions, it is concave to the left ventricle, but under abnormal loading the septum flattens and occasionally inverts. In the past, the septum has frequently been modelled as integral to the left ventricle with the effects of pressure from the right ventricle being ignored. Under abnormal loading, the septum has been described as behaving equivalent to a “flapping sail”. There has been no consideration of structural behaviour under these conditions. A 2-D plane stress FE model of the septum was used to investigate the difference in structural behaviour of the septum during diastole between normal and abnormal loading. The biaxial stress patterns that develop are distinctively disparate. Under normal loading, the septum behaves much like a thick-walled cylinder subject to internal and external pressure, with the resulting stresses being circumferential tension and radial compression, both varying with radius. These stresses are very low throughout most of diastole. However, under abnormal loading, the septum behaves in an arch-like fashion, with high compressive stresses almost circumferential in direction, combined with radial compression. We conclude that right ventricular pressures cause bending effects in the wall of the heart, and that under abnormal loading, the compressive stresses that develop in the septum may lead to an understanding of certain, previously unexplained, pathological conditions.     

Three-dimensional diastolic blood flow in the left ventricle

Three-dimensional blood flow in a human left ventricle is studied via a computational analysis with magnetic resonance imaging of the cardiac motion. Formation, growth and decay of vortices during the myocardial dilation are analyzed with flow patterns on various diametric planes. They are dominated by momentum transfer during flow acceleration and deceleration through the mitral orifice. The posterior and anterior vortices form an asymmetric annular vortex at the mitral orifice, providing a smooth transition for the rapid inflow to the ventricle. The development of core vortex accommodates momentum for deceleration and for acceleration at end diastolic atrial contraction. The rate of energy dissipation and that of work done by viscous stresses are small; they are approximately balanced with each other. The kinetic energy flux and the rate of work done by pressure delivered to blood from ventricular dilation is well balanced by the total energy influx at the mitral orifice and the rate change of kinetic energy in the ventricle.     

Complex distributions of residual stress and strain in the mouse left ventricle: experimental and theoretical models

 Most soft biological tissues, including ventricular myocardium, are not stress free when all external loads are removed. Residual stress has implications for mechanical performance of the heart, and may be an indicator of patterns of regional growth and remodeling. Cross-sectional rings of arrested ventricles opened up when a radial cut was made (initial mean opening angles were 64 ± 17°), but further circumferential cuts revealed the presence of additional residual stresses in the tissue with further opening of the rings. In normal mouse hearts, the inner half of a short-axis ring opened more than the outer half, and this change was dependent on apex–base location. At the apex the inner section vs. outer section opening angles were 226 ± 47° vs. 89 ± 28°, while at the base the same two angles were 160 ± 30° vs. 123 ± 35°. A simple theoretical cylindrical shell model with incompressible hyperelastic material properties was used to model the experimental deformations based on the cutting experiments. The model predicts different residual stress fields depending on the nature of the opening after the circumferential cut (which is done after the conventional radial cut). The observed opening angles were consistent with steep stress gradients near the endocardium compared with those predicted if the first cut was assumed to relieve all residual stresses. These results imply a more complex distribution of residual stress and strain in ventricular myocardium than previously thought. Received: 23 May 2002 / Accepted: 30 September 2002 We would like to acknowledge the surgical skills and data analysis of Zuangjie Li. This work was supported in part by National Heart, Lung, and Blood Institute Grants HL-43026 and HL-64321.     

Lactate dehydrogenase changes following several cardiac hypertrophic stresses.

The effects of four different cardiac hypertrophic stresses on cardiac lactate dehydrogenase (LDH) isozyme composition and activity were examined. Altitude-induced right ventricular hypertrophy was accompanied by an increase of 10% in the M subunit of LDH in right ventricle, left ventricle, and atria. Left ventricular hypertrophy induced by aortic stenosis also produced isozyme changes in the ventricles, but of only half the magnitude. Biventricular hypertrophy, induced by running or swimming, was accompanied by 4-5% increases in M LDH in the ventricles only. We conclude that changes in LDH activity are directly related to changes in the M subunit in all three portions of the heart. No changes in H subunit were noted under any of the stresses. It appears that the magnitude of changes in cardiac LDH isozyme composition are only marginally related to extent of hypertrophy.     

Theoretical models in mechanics of the left ventricle

Different rheological concepts and theoretical studies have been recently presented using models of myocardial mechanics. Complex analysis of the mechanical behavior of the left ventricular wall have been developed in order to estimate the local stresses and deformations that occur during the heart cycle as well as the ventricular stroke volume and pressure. Theoretical models have taken into account non-linear and viscoelastic passive properties of the myocardium tissue, when subjected to large deformations, through given strain energy functions or stress-strain relations. Different prolate spheroid geometries have been considered for such thick shell cardiac structure. During the active state of the contraction, the rheological behavior of the fibers has been described using different muscle models and relationships between fiber tension and strain, and activation degree. A forthcoming approach for bridging the gap between the knowledge of the muscle fiber microrheological properties and the study of the mechanical behavior of the entire ventricle, consists in including anisotropic and inhomogeneous effects through fiber direction field.     

Thermal stresses in frozen organs

An analysis was performed to determine the thermal stresses in the solid region of an organ frozen so that a constant cooling rate is imposed on its outer surface. The analysis shows that at the instant the water freezes at a certain location in the organ, compressive azimuthal stresses develop in the region close to the change of phase front. The compressive asimuthal stresses decrease and become tensile at that location as the change of phase front propagates further. The radial stresses are always compressive. It is hypothesized that those stresses might induce mechanical damage to the cellular components of the organ. The analysis shows that the magnitude of these stresses is a function of the material properties and the product of the outer surface cooling rate and the square of the outer surface radius.