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.     

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.     

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.     

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.     

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.     

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.     

A modified mathematical model of the anatomy of the cardiac left ventricle

A modification of the mathematical model of the shape and fiber direction field of the left cardiac ventricle is presented. The model was developed based on the idea of nested spiral surfaces. The ventricle is composed of surfaces that model myocardial layers. Each layer is filled with curves corresponding to myocardial fibers. The tangents to these curves form the myofiber direction field. A modified spherical coordinate system is linked with the model left ventricle, where the ventricular boundaries are coordinate surfaces. The model is based on echocardiographic, computed-tomography, or magnetic-resonance-imaging data. For this purpose, four-chamber and two-chamber echocardiography views or sections along the long axis of the left ventricle from these tomographic data in several positions are approximated with a model profile. To construct a 3D model, we then interpolate model parameters by periodic cubic splines and the vector field of the tangents to the model fibers is calculated. For verification of the model, we used diffusion-tensor magneticresonance-imaging data of the human heart.     

Effect of endurance running on cardiac and skeletal muscle in rats

We studied the effect of resistance running on left cardiac ventricle size and rectus femoris muscle fiber composition. Ten male Wistar rats were trained on a treadmill 6 days per week for 12 weeks. Ten rats remained sedentary and served as controls. A higher endurance time (40%) and cardiac hypertrophy in the trained animals were indicators of training efficiency. Morphometric analysis of the left ventricle cross-sectional area, left ventricular wall, and left ventricular cavity were evaluated. The endurance-running group demonstrated a hypertrophy of the ventricular wall (22%) and an increase in the ventricular cavity (25%); (p<0.0001). Semi-quantitative analysis of rectus femoris fiber-type composition and of the oxidative and glycolytic capacity was histochemically performed. Endurance running demonstrated a significant (p<0.01) increase in the relative frequency of Type I (24%), Type IIA (8%) and Type IIX (16%) oxidative fibers, and a decrease in Type IIB (20%) glycolytic fibers. There was a hypertrophy of both oxidative and glycolytic fiber types. The relative cross-sectional area analysis demonstrated an increase in oxidative fibers and a decrease in glycolytic fibers (p<0.0001). Changes were especially evident for Type IIX oxidative-glycolytic fibers. The results of this study indicate that the left ventricle adapts to endurance running by increasing wall thickness and enlargement of the ventricular cavity. Skeletal muscle adapts to training by increasing oxidative fiber Type. This increase may be related to fiber transformation from Type IIB glycolytic to Type IIX oxidative fibers. These results open the possibility for the use of this type of exercise to prevent muscular atrophy associated with age or post-immobilization.     

X-ray diffraction from a left ventricular wall of rat heart

We studied x-ray diffraction from the left ventricular wall of an excised, perfused whole heart of a rat using x rays from the third-generation synchrotron radiation facility, SPring-8. With the beam at right angles to the long axis of the left ventricle, well-oriented, strong equatorial reflections were observed from the epicardium surface. The reflections became vertically split arcs when the beam passed through myocardium deeper in the wall, and rings were observed when the beam passed into the inner myocardium of the wall. These diffraction patterns were explained by employing a layered-spiral model of the arrangement of muscle fibers in the heart. In a quiescent heart with an expanded left ventricle, the muscle fibers at the epicardium surface were found to have a (1,0) lattice spacing smaller than in the rest of the wall. The intensity ratio of the (1,0) and (1,1) equatorial reflections decreased on contraction with a similar time course in all parts of the wall. The results show that it is possible to assign the origin of reflections in a diffraction diagram from a whole heart. This study offers a basis for interpretation of x-ray diffraction from a beating heart under physiologically and pathologically different conditions.     

Dependence of local left ventricular wall mechanics on myocardial fiber orientation: a model study.

The dependence of local left ventricular (LV) mechanics on myocardial muscle fiber orientation was investigated using a finite element model. In the model we have considered anisotropy of the active and passive components of myocardial tissue, dependence of active stress on time, strain and strain rate, activation sequence of the LV wall and aortic afterload. Muscle fiber orientation in the LV wall is quantified by the helix fiber angle, defined as the angle between the muscle fiber direction and the local circumferential direction. In a first simulation, a transmural variation of the helix fiber angle from +60 degrees at the endocardium through 0 degrees in the midwall layers to -60 degrees at the epicardium was assumed. In this simulation, at the equatorial level maximum active muscle fiber stress was found to vary from about 110 kPa in the subendocardial layers through about 30 kPa in the midwall layers to about 40 kPa in the subepicardial layers. Next, in a series of simulations, muscle fiber orientation was iteratively adapted until the spatial distribution of active muscle fiber stress was fairly homogeneous. Using a transmural course of the helix fiber angle of +60 degrees at the endocardium, +15 degrees in the midwall layers and -60 degrees at the epicardium, at the equatorial level maximum active muscle fiber stress varied from 52 kPa to 55 kPa, indicating a remarkable reduction of the stress range. Moreover, the change of muscle fiber strain with time was more similar in different parts of the LV wall than in the first simulation. It is concluded that (1) the distribution of active muscle fiber stress and muscle fiber strain across the LV wall is very sensitive to the transmural distribution of the helix fiber angle and (2) a physiological transmural distribution of the helix fiber angle can be found, at which active muscle fiber stress and muscle fiber strain are distributed approximately homogeneously across the LV wall.     

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.     

A Three-Dimensional Analytical (Rheological) Model of the Human Left Ventricle in Passive-Active States: Nontraumatic Determination of the In Vivo Values of the Rheological Parameters

In this paper a three-dimensional continuum model of a mammalian left ventricle is formulated. The stresses in the model satisfy the conditions of zero stress on the outer (epicardial surface-representing) boundary. The strains of the model are obtained from the actual dynamic geometry measurements (obtained from cineangiocardiography). Since the left ventricular muscle is incompressible, the dilatational strain is zero and hence the (three-dimensional) deviatric stress components are related to the corresponding strain components by Maxwell and Voigt rheological model analogues of one-dimensional systems; the parameters of the model are series and parallel elastic (SE, PE) elements and the contractile element (CE) (representing the sarcomere). The incorporation of the rheological features of the cardiac muscle into the three-dimensional constitutive equations (for the three-dimensional continuum model of the left ventricle) is a feature of this paper. A procedure is presented to determine the parameters of the constitutive equations (i.e., the SE, PE, and the parameters of the force-velocity relation for the CE) for the left ventricle of a subject from data on the dimensions and chamber pressure of the left ventricle. The values of these parameters characterize the rheology of the left ventricular muscle of the subject. In order to demonstrate clinical application of the analyses, in vivo data of the subjects' left ventricular pressure and dimensions are obtained, and the analyses are applied to the data to determine (for each subject) the values and characteristics of the elastic elements and CEs.     

Characterization of the normal cardiac myofiber field in goat measured with MR-diffusion tensor imaging

Cardiac myofiber orientation is a crucial determinant of the distribution of myocardial wall stress. Myofiber orientation is commonly quantified by helix and transverse angles. Accuracy of reported helix angles is limited. Reported transverse angle data are incomplete. We measured cardiac myofiber orientation postmortem in five healthy goat hearts using magnetic resonance-diffusion tensor imaging. A novel local wall-bound coordinate system was derived from the characteristics of the fiber field. The transmural course of the helix angle corresponded to data reported in literature. The mean midwall transverse angle ranged from -12 +/- 4 degrees near the apex to +9.0 +/- 4 degrees near the base of the left ventricle, which is in agreement with the course predicted by Rijcken et al. (18) using a uniform load hypothesis. The divergence of the myofiber field was computed, which is a measure for the extent to which wall stress is transmitted through the myofiber alone. It appeared to be <0.07 mm(-1) throughout the myocardial walls except for the fusion sites between the left and right ventricles and the insertion sites of the papillary muscles.     

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.     

Myosin types and fiber types in cardiac muscle. I. Ventricular myocardium

Antisera against bovine atrial myosin were raised in rabbits, purified by affinity chromatography, and absorbed with insolubilized ventricular myosin. Specific anti-bovine atrial myosin (anti-bAm) antibodies reacted selectively with atrial myosin heavy chains, as determined by enzyme immunoassay combined with SDS-gel electrophoresis. In direct and indirect immunofluorescence assay, anti-bAm was found to stain all atrial muscle fibers and a minor proportion of ventricular muscle fibers in the right ventricle of the bovine heart. In contrast, almost all muscle fibers in the left ventricle were unreactive. Purkinje fibers showed variable reactivity. In the rabbit heart, all atrial muscle fibers were stained by anti-bAm, whereas ventricular fibers showed a variable response in both the right and left ventricle, with a tendency for reactive fibers to be more numerous in the right ventricle and in subepicardial regions. Diversification of fiber types with respect to anti-bAm reactivity was found to occur during late stages of postnatal development in the rabbit heart and to be influenced by thyroid hormone. All ventricular muscle fibers became strongly reactive after thyroxine treatment, whereas they became unreactive or poorly reactive after propylthiouracil treatment. These findings are consistent with the existence of different ventricular isomyosins whose relative proportions can vary according to the thyroid state. Variations in ventricular isomyosin composition can account for the changes in myosin Ca2+-activated ATPase activity previously observed in cardiac muscle from hyper- and hypothyroid animals and may be responsible for the changes in the velocity of contraction of ventricular myocardium that occur under these conditions. The differential distribution of ventricular isomyosins in the normal heart suggests that fiber types with different contractile properties may coexist in the ventricular myocardium.     

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.     

Myocardial blood flow regulation relative to left ventricle pressure and volume in anesthetized dogs

The influence of left ventricle pressure and volume changes on coronary blood flow was investigated in eight anesthetized dogs. Coronary artery pressure-flow relationships were determined at two levels of left ventricular pressure and volume. The distribution of blood flow within the myocardium was also determined when these relationships varied. Reducing left ventricle pressures and volumes increased heart rate. Rate-pressure product, diastolic coronary pressure, myocardial O2 consumption, total, subendocardial and subepicardial flow decreased. Hematocrit and blood gas data were unchanged. The pressure-flow relationships were shifted leftward (p = 0.001) but the range of autoregulation was not altered. At low left ventricle pressures and volumes, the lower coronary artery pressure limit was shifted leftward (from 75 to 45 mm Hg (1 mm Hg = 133.3 Pa)), while total, subendocardial, and subepicardial blood flow did not change compared with the control. Below the lower coronary artery pressure limit, subendocardial but not subepicardial flow decreased, resulting in maldistribution of flow across the left ventricular wall. When coronary pressure was reset between control and the lower coronary artery pressure limit, subendocardial flow was restored. These results show that the lower coronary artery pressure limit can be shifted leftward while the distribution of blood flow across the left ventricular wall is preserved.     

Relationship between apexcardiogram,left ventricular pressure and wall stress

Simultaneous measurement of left ventricular dimension and wall thickness by M-mode echocardiography, of left ventricular pressure by a tip-transducer manometer, and of the calibrated apexcardiogram with a pixie beam transducer, were made during acute experiments on anaesthetized dogs. Instantaneo us values for chamber dimensions and wall thickness were obtained throughout the heart cycle by digitizing the echo-mechanocardigrams.From these data myocardial stresses, derived from a thick shell theory (meridional and circumferential components) and from Laplace's law, were computed. Laplace stress is shown to be an adequate expression for average wall stress. Its value was correlated with the calibrated apexcardiogram. The present investigation suggests that to a certain extent, the apexcardiogram not only reflects pressure changes but also dimensional changes of the left ventricle.     

Velocity and turbulence measurements past mitral valve prostheses in a model left ventricle

Thrombogenesis and hemolysis have both been linked to the flow dynamics past heart valve prostheses. To learn more about the particular flow dynamics past mitral valve prostheses in the left ventricle under controlled experimental conditions, an in vitro study was performed. The experimental methods included velocity and turbulent shear stress measurements past caged-ball, tilting disc, bileaflet, and polyurethane trileaflet mitral valves in an acrylic rigid model of the left ventricle using laser Doppler anemometry. The results indicate that all four prosthetic heart valves studied create at least mildly disturbed flow fields. The effect of the left ventricular geometry on the flow development is to produce a stabilizing vortex which engulfs the entire left ventricular cavity, depending on the orientation of the valve. The measured turbulent shear stress magnitudes for all four valves did not exceed the reported value for hemolytic damage. However, the measured turbulent shear stresses were near or exceeded the critical shear stress reported in the literature for platelet lysis, a known precursor to thrombus formation.