The effect of distension of the left ventricle of the heart on the length of the individual myocardial fibers

Based on the ellipsoid model of the left ventricle and the helicoidal course of the left ventricular myocardial fibers, a theory has been developed for calculating the length of the individual myocardial fibers. Numerical solutions of the final equation show that when the left ventricle is distended, the increase in length of the myocardial fibers is not uniform throughout the thickness of the myocardial wall. It was shown that with increasing dimensions of the left ventricle, the distension of the myocardial fibers becomes smaller as one advances from the endocardium to the middle layer of fibers, whereas it increases as one advances from the middle layer to the epicardial layer. The mechanism by which this effect is brought about as well as its physiological implications are discussed.     

Structural finite deformation model of the left ventricle during diastole and systole

A model of left ventricular function is developed based on morphological characteristics of the myocardial tissue. The passive response of the three-dimensional collagen network and the active contribution of the muscle fibers are integrated to yield the overall response of the left ventricle which is considered to be a thick wall cylinder. The deformation field and the distributions of stress and pressure are determined at each point in the cardiac cycle by numerically solving three equations of equilibrium. Simulated results in terms of the ventricular deformation during ejection and isovolumic cycles are shown to be in good qualitative agreement with experimental data. It is shown that the collagen network in the heart has considerable effect on the pressure-volume loops. The particular pattern of spatial orientation of the collagen determines the ventricular recoil properties in early diastole. The material properties (myocardial stiffness and contractility) are shown to affect both the pressure-volume loop and the deformation pattern of the ventricle. The results indicate that microstructural consideration offer a realistic representation of the left ventricle mechanics.     

The preload of individual myocardial fibers

The preload of the indiviuual myocardial fibers of the left ventricle, that is, the stress exerted upon the myocardial fibers at end-diastole, is calculated by means of a set of equations. The development of the equations was based on anatomical data referring to the shape of the left ventricle and the orientation of the myocardial fibers, as well as some assumptions of minor importance. Numerical solution of the equations shows that in general, the preload increases as one advances from the apex to the equator of the left ventricle and then it decreases as one advances toward the base. The preload also changes as one advances from the epicardium to the endocardium in such a way that one can distinguish three zones: one outer, or epicardial, with low preloads, one middle with high preloads and one inner, or endocardial, with low preloads. The physiological significance of the findings as well as the validity of the assumptions on which the theory was based are discussed.     

Adaptation of cardiac structure by mechanical feedback in the environment of the cell: a model study.

In the cardiac left ventricle during systole mechanical load of the myocardial fibers is distributed uniformly. A mechanism is proposed by which control of mechanical load is distributed over many individual control units acting in the environment of the cell. The mechanics of the equatorial region of the left ventricle was modeled by a thick-walled cylinder composed of 6-1500 shells of myocardial fiber material. In each shell a separate control unit was simulated. The direction of the cells was varied so that systolic fiber shortening approached a given optimum of 15%. End-diastolic sarcomere length was maintained at 2.1 microns. Regional early-systolic stretch and global contractility stimulated growth of cellular mass. If systolic shortening was more than normal the passive extracellular matrix stretched. The design of the load-controlling mechanism was derived from biological experiments showing that cellular processes are sensitive to mechanical deformation. After simulating a few hundred adaptation cycles, the macroscopic anatomical arrangement of helical pathways of the myocardial fibers formed automatically. If pump load of the ventricle was changed, wall thickness and cavity volume adapted physiologically. We propose that the cardiac anatomy may be defined and maintained by a multitude of control units for mechanical load, each acting in the cellular environment. Interestingly, feedback through fiber stress is not a compelling condition for such control.     

The myocardium as a composite material: Analysis

As a further attempt to determine the stresses and strains of the individual myocardial fibers, the heart muscle is considered as an orthotropic material. A theory is presented which leads to the expression of the equilibrium conditions for the left ventricle in the form of three simultaneous differential equations. Solution of these equations would give the changes in shape of the left ventricle throughout the cardiac cycle, and, in addition, the stresses and strains of the individual myocardial fibers. It is pointed out, however, that meaningful solutions of the equations cannot be obtained at the present time because of difficulties in experimental determination of certain parameters.     

Inhomogeneous deformation as a source of error in strain measurements derived from implanted markers in the canine left ventricle

This article quantifies the errors inherent in the measurement of myocardial strain in the canine left ventricle when the motion of four radiopaque marker beads is used to determine this strain. These errors are introduced because the strain is strongly inhomogeneous and only an averaged value of this strain can be determined by measuring the displacements of four points with finite separation. In this work, the error in the principal strains has been estimated by modeling the primary deformation components of the left ventricle and comparing the true strains obtained from these models with the strains computed according to the protocol typically used in experimental studies to determine strain from the motion of marker beads. Both a cylindrical and a spherical model of the left ventricle are used. For the cylindrical model, it is found that the traditional tetrahedra used may give errors as high as 20% in the maximum principal strain. A six-marker prism is found to give more consistent results, underestimating the maximum principal strain, which is in the radial direction, by no more than 8% in almost all cases. The spherical model, having double curvature, gives larger errors. In both models, the error in the other two principal strains was usually less than 5%. Furthermore, the principal strain directions were correct to within 6 degrees.     

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.     

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.     

A Tissue-Level Electromechanical Model of the Left Ventricle: Application to the Analysis of Intraventricular Pressure

The ventricular pressure profile is characteristic of the cardiac contraction progress and is useful to evaluate the cardiac performance. In this contribution, a tissue-level electromechanical model of the left ventricle is proposed, to assist the interpretation of left ventricular pressure waveforms. The left ventricle has been modeled as an ellipsoid composed of twelve mechano-hydraulic sub-systems. The asynchronous contraction of these twelve myocardial segments has been represented in order to reproduce a realistic pressure profiles. To take into account the different energy domains involved, the tissue-level scale and to facilitate the building of a modular model, multiple formalisms have been used: Bond Graph formalism for the mechano-hydraulic aspects and cellular automata for the electrical activation. An experimental protocol has been defined to acquire ventricular pressure signals from three pigs, with different afterload conditions. Evolutionary Algorithms have been used to identify the model parameters in order to minimize the error between experimental and simulated ventricular pressure signals. Simulation results show that the model is able to reproduce experimental ventricular pressure. In addition, electro-mechanical activation times have been determined in the identification process. For example, the maximum electrical activation time is reached, respectively, 96.5, 139.3 and 131.5 ms for the first, second, and third pigs. These preliminary results are encouraging for the application of the model on non-invasive data like ECG, arterial pressure or myocardial strain.     

Deformation of the diastolic left ventricle—II. Nonlinear geometric effects

A finite element model for the rat left ventricle has been developed which is based on finite deformation elasticity theory: i.e. the model is not limited by assumptions relating to the magnitudes of extensions, shears and angles of rotation which are inherent in the classical theory of elasticity. This model represents the ventricle as a heterogeneous, nonlinearly elastic, isotropic thick-wall solid of revolution. For the representation of myocardial elasticity used in this study, the model predicts overall ventricular stiffnesses at physiological pressures which are 20–30 per cent lower than those obtained with a model based on the classical theory. However, extentions predicted by the two theories differ by as much as 100 per cent in certain portions of the ventricular wall.     

Assessment of caspase 3 activity in rabbit myocardial tissue during experimental hemodynamic overload of the left ventricle of the heart

It is well known that chronic overload of the cardiac left ventricle is accompanied by an increase in the cardiomyocyte apoptosis rate. However, direction and extent of changes in programmed cell death under an acute overload of the left ventricle still requires detailed investigation (as its pathogenesis significantly differs from chronic overload). Caspase-3 activity has been investigated in left ventricle myocardium of rabbits on days 1, 3, and 5 after modeling of left ventricle hemodynamic overload caused by experimental stenosis of the ascending aorta. Control group included intact animals. It was found that caspase-3 activity significantly increased in both ventricles on day 1; it increased more than twofold above control values on day 3 and decreased up to nearly control values on day 5. Based on these data it was concluded that the acute hemodynamic overload of the left ventricle may be a cause of increased apoptosis in the myocardial tissue of both cardiac ventricles during first days of the pathological process.     

Discrimination of Recent Ischemic Myocardial Changes in WHHLMI Rabbits from the Findings of Postmortem Degeneration.

To distinguish recent ischemic myocardial changes in myocardial infarction-prone Watanabe heritable hyperlipidemic (WHHLMI) rabbits from general postmortem myocardial degeneration, we examined hearts of WHHLMI rabbits after sudden death and postmortem hearts of non-atherogenic rabbits. Hearts of 8 WHHLMI rabbits were excised within 30 min of sudden death and hearts of 27 non-atherosclerotic rabbits were excised at designated periods after sacrifice. A large number of myocardial cells from WHHLMI rabbits exhibited features characteristic of ischemia (intercellular gap, intracellular edema, eosinophilia, disappearance of myocardial cells, indistinct nuclei, wavy myocardial fibers) simultaneously at regions close to proximal occluded coronary arteries. Although postmortem hearts of non-atherosclerotic rabbits exhibited similar characteristics, several features characteristic of autolyzed myocytes were also randomly observed in the left ventricle wall. Each feature was detected independently in myocardial cells or regions of the ventricle wall. In conclusion, we found several unique characteristics associated with myocardial infarction which enable discrimination between recent ischemic myocardial changes and myocardial degeneration following death.     

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.     

Rodent Working Heart Model for the Study of Myocardial Performance and Oxygen Consumption

Isolated working heart models have been used to understand the effects of loading conditions, heart rate and medications on myocardial performance in ways that cannot be accomplished in vivo. For example, inotropic medications commonly also affect preload and afterload, precluding load-independent assessments of their myocardial effects in vivo. Additionally, this model allows for sampling of coronary sinus effluent without contamination from systemic venous return, permitting assessment of myocardial oxygen consumption. Further, the advent of miniaturized pressure-volume catheters has allowed for the precise quantification of markers of both systolic and diastolic performance. We describe a model in which the left ventricle can be studied while performing both volume and pressure work under controlled conditions. In this technique, the heart and lungs of a Sprague-Dawley rat (weight 300-500 g) are removed en bloc under general anesthesia. The aorta is dissected free and cannulated for retrograde perfusion with oxygenated Krebs buffer. The pulmonary arteries and veins are ligated and the lungs removed from the preparation. The left atrium is then incised and cannulated using a separate venous cannula, attached to a preload block. Once this is determined to be leak-free, the left heart is loaded and retrograde perfusion stopped, creating the working heart model. The pulmonary artery is incised and cannulated for collection of coronary effluent and determination of myocardial oxygen consumption. A pressure-volume catheter is placed into the left ventricle either retrograde or through apical puncture. If desired, atrial pacing wires can be placed for more precise control of heart rate. This model allows for precise control of preload (using a left atrial pressure block), afterload (using an afterload block), heart rate (using pacing wires) and oxygen tension (using oxygen mixtures within the perfusate).     

Hypoxia-induced downregulation of beta-adrenergic receptors in rat heart.

To test the desensitization hypothesis of cardiac beta-adrenergic receptors (beta-AR) in chronic hypoxia, the effect of 1, 3, 7, 15, and 21 days of exposure to hypobaric hypoxia (380 Torr) was evaluated in Wistar rats. Exposure to hypoxia for 1-15 days did not induce any change in right and left ventricular beta-AR density (Bmax) determined with [125I]iodocyanopindolol or in antagonist affinity. After 21 days, Bmax decreased by 24% in the left ventricle. In contrast, no change in beta-AR was shown in the right hypertrophied ventricle. Agonist affinity in the left ventricle was not altered, as shown by the analysis of displacement curves of isoproterenol (normoxia 185 +/- 26 nM, hypoxia 170 +/- 11 nM). Moreover, there was no significant decrease in adenylate cyclase activity (pmol.mg-1.min-1) in the left ventricle. In the right ventricle, a 21-day exposure to hypoxia led to a decrease in basal and maximal activity when stimulated by isoproterenol. A decrease in tissue norepinephrine content was observed after 7 days of hypoxia. In conclusion, these data support the beta-AR downregulation hypothesis as one of the mechanisms of myocardial adaptation to high altitude occurring after 2-3 wk of exposure to hypoxia. The regulation pathways of beta-AR may differ between left nonhypertrophied and right hypertrophied ventricles. No evidence of profound abnormality of signal transduction was shown.     

Transmural heterogeneity of diffusion anisotropy in the sheep myocardium characterized by MR diffusion tensor imaging

The orientation of MRI-measured diffusion tensor in the myocardium has been directly correlated to the tissue fiber direction and widely characterized. However, the scalar anisotropy indexes have mostly been assumed to be uniform throughout the myocardial wall. The present study examines the fractional anisotropy (FA) as a function of transmural depth and circumferential and longitudinal locations in the normal sheep cardiac left ventricle. Results indicate that FA remains relatively constant from the epicardium to the midwall and then decreases (25.7%) steadily toward the endocardium. The decrease of FA corresponds to 7.9% and 12.9% increases in the secondary and tertiary diffusion tensor diffusivities, respectively. The transmural location of the FA transition coincides with the location where myocardial fibers run exactly circumferentially. There is also a significant difference in the midwall-endocardium FA slope between the septum and the posterior or lateral left ventricular free wall. These findings are consistent with the cellular microstructure from histological studies of the myocardium and suggest a role for MR diffusion tensor imaging in characterization of not only fiber orientation but, also, other tissue parameters, such as the extracellular volume fraction.     

A novel porous mechanical framework for modelling the interaction between coronary perfusion and myocardial mechanics

The strong coupling between the flow in coronary vessels and the mechanical deformation of the myocardial tissue is a central feature of cardiac physiology and must therefore be accounted for by models of coronary perfusion. Currently available geometrically explicit vascular models fail to capture this interaction satisfactorily, are numerically intractable for whole organ simulations, and are difficult to parameterise in human contexts. To address these issues, in this study, a finite element formulation of an incompressible, poroelastic model of myocardial perfusion is presented. Using high-resolution ex vivo imaging data of the coronary tree, the permeability tensors of the porous medium were mapped onto a mesh of the corresponding left ventricular geometry. The resultant tensor field characterises not only the distinct perfusion regions that are observed in experimental data, but also the wide range of vascular length scales present in the coronary tree, through a multi-compartment porous model. Finite deformation mechanics are solved using a macroscopic constitutive law that defines the coupling between the fluid and solid phases of the porous medium. Results are presented for the perfusion of the left ventricle under passive inflation that show wall-stiffening associated with perfusion, and that show the significance of a non-hierarchical multi-compartment model within a particular perfusion territory.