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.     

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.     

Origin of axial prestretch and residual stress in arteries

The structural protein elastin endows large arteries with unique biological functionality and mechanical integrity, hence its disorganization, fragmentation, or degradation can have important consequences on the progression and treatment of vascular diseases. There is, therefore, a need in arterial mechanics to move from materially uniform, phenomenological, constitutive relations for the wall to those that account for separate contributions of the primary structural constituents: elastin, fibrillar collagens, smooth muscle, and amorphous matrix. In this paper, we employ a recently proposed constrained mixture model of the arterial wall and show that prestretched elastin contributes significantly to both the retraction of arteries that is observed upon transection and the opening angle that follows the introduction of a radial cut in an unloaded segment. We also show that the transmural distributions of elastin and collagen, compressive stiffness of collagen, and smooth muscle tone play complementary roles. Axial prestresses and residual stresses in arteries contribute to the homeostatic state of stress in vivo as well as adaptations to perturbed loads, disease, or injury. Understanding better the development of and changes in wall stress due to individual extracellular matrix constituents thus promises to provide considerable clinically important insight into arterial health and disease.     

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.     

Differences in distensibility between the anterior and posterior walls of the left ventricle in dogs

Nonuniformity of myocardial systolic and diastolic performance in the normal left ventricle has been recognized by a number of investigators. Lack of homogeneity in diastolic properties might be caused by or related to differences in the distensibility of different regions of the left ventricular (LV) wall. Thus, we compared the end-diastolic transmural pressure-strain relations in both the anterior and posterior LV walls in seven anesthetized dogs during two interventions (pulmonary artery constriction and aortic constriction). Transmural pressure was defined as the difference between LV intracavitary pressure and local pericardial pressure. LV pressure was measured using a micromanometer; pericardial pressures over the LV anterior and posterior walls were measured with balloon transducers. Circumferentially oriented pairs of sonomicrometer crystals were implanted in the midwall of the anterior and posterior walls of the LV to measure segment lengths. Strains were calculated as (L-L0)/L0, where L was the instantaneous segment length and L0 was the segment length when transmural pressure was zero. The pattern of end-diastolic transmural pressure--strain relations was similar in all dogs. The change in strain in the posterior wall was always greater than that in the anterior wall. Opening the pericardium did not affect the difference in distensibility of the anterior and posterior walls. The results suggest that the posterior wall is more compliant than the anterior wall (that is, for a given difference in transmural pressure, the local segment length change of the posterior wall was greater). This seems consistent with other observations, which suggest that the posterior wall might make a greater contribution to diastolic filling.     

The motion of the left ventricle: II

We use the dimensional parameters previously derived (Bull. Math. Biophysics,28, 355–362, 1966), the ventricular pressure expressed as a Fourier series, and several additional assumptions to derive expressions for the mechanical parameters of the ventricle: flow, muscle segment length, surface area, transmural force, wall tension and work. The wall of the ventricle is assumed to consist of three layers of muscle. Each of the mechanical parameters is expressed in terms of Fourier series.     

The theory of the left ventricular aneurysm

A spherical model for the human left ventricle with two different types of aneurysms, circular and rectangular-square, is proposed and meaningful relations are derived between the parameters of the aneurysms and ventricle. Such ventricular parameters as stroke volume, end-diastolic volume, and end-systolic volume are given normal human values to compute values for end-systolic radius and percentage shortening of muscle for various sized circular and rectangular-square aneurysms.     

Effects of myocardial sheetlet sliding on left ventricular function

Left ventricle myocardium has a complex micro-architecture, which was revealed to consist of myocyte bundles arranged in a series of laminar sheetlets. Recent imaging studies demonstrated that these sheetlets re-orientated and likely slided over each other during the deformations between systole and diastole, and that sheetlet dynamics were altered during cardiomyopathy. However, the biomechanical effect of sheetlet sliding is not well-understood, which is the focus here. We conducted finite element simulations of the left ventricle (LV) coupled with a windkessel lumped parameter model to study sheetlet sliding, based on cardiac MRI of a healthy human subject, and modifications to account for hypertrophic and dilated geometric changes during cardiomyopathy remodeling. We modeled sheetlet sliding as a reduced shear stiffness in the sheet-normal direction and observed that (1) the diastolic sheetlet orientations must depart from alignment with the LV wall plane in order for sheetlet sliding to have an effect on cardiac function, that (2) sheetlet sliding modestly aided cardiac function of the healthy and dilated hearts, in terms of ejection fraction, stroke volume, and systolic pressure generation, but its effects were amplified during hypertrophic cardiomyopathy and diminished during dilated cardiomyopathy due to both sheetlet angle configuration and geometry, and that (3) where sheetlet sliding aided cardiac function, it increased tissue stresses, particularly in the myofibre direction. We speculate that sheetlet sliding is a tissue architectural adaptation to allow easier deformations of the LV walls so that LV wall stiffness will not hinder function, and to provide a balance between function and tissue stresses. A limitation here is that sheetlet sliding is modeled as a simple reduction in shear stiffness, without consideration of micro-scale sheetlet mechanics and dynamics.

     

Variability of the cardiomyocyte ploidy in normal human hearts.

We have performed cytophotometry for DNA in isolated myocytes of the left ventricle from 16 men, aged 19-39 years, who died from various non-cardiac or pulmonary causes. The mean ploidy of myocytes varied from 3.2-3.9 c to 6.6-7.3 c in different layers of the anterior wall of the left ventricle (where c is the haploid DNA content measured by cytophotometry in Feulgen-stained preparations). There was no correlation between the layers. The percentage of binuclear cells varied from 25 to 86% and correlated in every layer with the mean ploidy value of the whole myocyte population. Approximate calculation of total ploidy revealed low values in the ventricles of some individuals, and high values in others. Averaging the values for all the hearts studied obscures this variation. Mean myocyte ploidy in different layers of the anterior wall was similar: in the external layer it was 5.1 +/- 0.3 c, in the middle layer 5.5 +/- 0.3 c and in the inner layer 4.8 +/- 0.4 c. The mean percentage of binuclear myocytes in these three layers was also similar, being 61 +/- 3%, 63 +/- 4% and 54 +/- 5%, respectively. Myocyte ploidy in tissue from the posterior wall of the left ventricle also varied, but was always higher than for the same layer of the anterior wall in the same ventricle. We propose that high or low myocyte ploidy, as well as different proportions of mono- and binucleate cells, can be a factor affecting the course and result of cardiac pathology in the absence of any changes of myocyte genome determined during early ontogenesis and representing a stable characteristic of the individual.     

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.     

Left ventricular wall stress compendium

Left ventricular (LV) wall stress has intrigued scientists and cardiologists since the time of Lame and Laplace in 1800s. The left ventricle is an intriguing organ structure, whose intrinsic design enables it to fill and contract. The development of wall stress is intriguing to cardiologists and biomedical engineers. The role of left ventricle wall stress in cardiac perfusion and pumping as well as in cardiac pathophysiology is a relatively unexplored phenomenon. But even for us to assess this role, we first need accurate determination of in vivo wall stress. However, at this point, 150 years after Lame estimated left ventricle wall stress using the elasticity theory, we are still in the exploratory stage of (i) developing left ventricle models that properly represent left ventricle anatomy and physiology and (ii) obtaining data on left ventricle dynamics. In this paper, we are responding to the need for a comprehensive survey of left ventricle wall stress models, their mechanics, stress computation and results. We have provided herein a compendium of major type of wall stress models: thin-wall models based on the Laplace law, thick-wall shell models, elasticity theory model, thick-wall large deformation models and finite element models. We have compared the mean stress values of these models as well as the variation of stress across the wall. All of the thin-wall and thick-wall shell models are based on idealised ellipsoidal and spherical geometries. However, the elasticity model's shape can vary through the cycle, to simulate the more ellipsoidal shape of the left ventricle in the systolic phase. The finite element models have more representative geometries, but are generally based on animal data, which limits their medical relevance. This paper can enable readers to obtain a comprehensive perspective of left ventricle wall stress models, of how to employ them to determine wall stresses, and be cognizant of the assumptions involved in the use of specific models.     

Left ventricular wall stress compendium

Left ventricular (LV) wall stress has intrigued scientists and cardiologists since the time of Lame and Laplace in 1800s. The left ventricle is an intriguing organ structure, whose intrinsic design enables it to fill and contract. The development of wall stress is intriguing to cardiologists and biomedical engineers. The role of left ventricle wall stress in cardiac perfusion and pumping as well as in cardiac pathophysiology is a relatively unexplored phenomenon. But even for us to assess this role, we first need accurate determination of in vivo wall stress. However, at this point, 150 years after Lame estimated left ventricle wall stress using the elasticity theory, we are still in the exploratory stage of (i) developing left ventricle models that properly represent left ventricle anatomy and physiology and (ii) obtaining data on left ventricle dynamics. In this paper, we are responding to the need for a comprehensive survey of left ventricle wall stress models, their mechanics, stress computation and results. We have provided herein a compendium of major type of wall stress models: thin-wall models based on the Laplace law, thick-wall shell models, elasticity theory model, thick-wall large deformation models and finite element models. We have compared the mean stress values of these models as well as the variation of stress across the wall. All of the thin-wall and thick-wall shell models are based on idealised ellipsoidal and spherical geometries. However, the elasticity model's shape can vary through the cycle, to simulate the more ellipsoidal shape of the left ventricle in the systolic phase. The finite element models have more representative geometries, but are generally based on animal data, which limits their medical relevance. This paper can enable readers to obtain a comprehensive perspective of left ventricle wall stress models, of how to employ them to determine wall stresses, and be cognizant of the assumptions involved in the use of specific models.     

Computational comparison of aortic root stresses in presence of stentless and stented aortic valve bio-prostheses

We provide a computational comparison of the performance of stentless and stented aortic prostheses, in terms of aortic root displacements and internal stresses. To this aim, we consider three real patients; for each of them, we draw the two prostheses configurations, which are characterized by different mechanical properties and we also consider the native configuration. For each of these scenarios, we solve the fluid–structure interaction problem arising between blood and aortic root, through Finite Elements. In particular, the Arbitrary Lagrangian–Eulerian formulation is used for the numerical solution of the fluid-dynamic equations and a hyperelastic material model is adopted to predict the mechanical response of the aortic wall and the two prostheses. The computational results are analyzed in terms of aortic flow, internal wall stresses and aortic wall/prosthesis displacements; a quantitative comparison of the mechanical behavior of the three scenarios is reported. The numerical results highlight a good agreement between stentless and native displacements and internal wall stresses, whereas higher/non-physiological stresses are found for the stented case.     

Heterogeneous transmural proteoglycan distribution provides a mechanism for regulating residual stresses in the aorta

The arterial wall contains a significant amount of charged proteoglycans, which are inhomogeneously distributed, with the greatest concentrations in the intimal and medial layers. The hypothesis of this study is that the transmural distribution of proteoglycans plays a significant role in regulating residual stresses in the arterial wall. This hypothesis was first tested theoretically, using the framework of mixture theory for charged hydrated tissues, and then verified experimentally by measuring the opening angle of rat aorta in NaCl solutions of various ionic strengths. A three-dimensional finite element model of aortic ring, using realistic values of the solid matrix shear modulus and proteoglycan fixed-charge density, yielded opening angles and changes with osmolarity comparable to values reported in the literature. Experimentally, the mean opening angle in isotonic saline (300 mosM) was 15 +/- 17 degrees and changed to 4 +/- 19 degrees and 73 +/- 18 degrees under hypertonic (2,000 mosM) and hypotonic (0 mosM) conditions, respectively (n = 16). In addition, the opening angle in isotonic (300 mosM) sucrose, an uncharged molecule, was 60 +/- 16 degrees (n = 11), suggesting that the charge effect, not cellular swelling, was the major underlying mechanism for these observations. The extent of changes in opening angle under osmotic challenges suggests that transmural heterogeneity of fixed-charge density plays a crucial role in governing the zero-stress configuration of the aorta. A significant implication of this finding is that arterial wall remodeling in response to altered wall stresses may occur via altered deposition of proteoglycans across the wall thickness, providing a novel mechanism for regulating mechanical homeostasis in vascular tissue.     

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.     

Variability of the cardiomyocyte ploidy in normal human hearts

We have performed cytophotometry for DNA in isolated myocytes of the left ventricle from 16 men, aged 19–39 years, who died from various non-cardiac or pulmonary causes. The mean ploidy of myocytes varied from 3.2–3.9 c to 6.6–7.3 c in different layers of the anterior wall of the left ventricle (where c is the haploid DNA content measured by cytophotometry in Feulgenstained preparations). There was no correlation between the layers. The percentage of binuclear cells varied from 25 to 86% and correlated in every layer with the mean ploidy value of the whole myocyte population. Approximate calculation of total ploidy revealed low values in the ventricles of some individuals, and high values in others. Averaging the values for all the hearts studied obscures this variation. Mean myocyte ploidy in different layers of the anterior wall was similar: in the external layer it was 5.1±0.3 c, in the middle layer 5.5±0.3 c and in the inner layer 4.8±0.4 c. The mean percentage of binuclear myocytes in these three layers was also similar, being 61±3%, 63±4% and 54±5%, respectively. Myocyte ploidy in tissue from the posterior wall of the left ventricle also varied, but was always higher than for the same layer of the anterior wall in the same ventricle. We propose that high or low myocyte ploidy, as well as different proportions of mono- and binucleate cells, can be a factor affecting the course and result of cardiac pathology in the absence of any changes of myocyte genome determined during early ontogenesis and representing a stable characteristic of the individual.     

Modeling left ventricular dynamics with characteristic deformation modes

A computationally efficient method is described for simulating the dynamics of the left ventricle (LV) in three dimensions. LV motion is represented as a combination of a limited number of deformation modes, chosen to represent observed cardiac motions while conserving volume in the LV wall. The contribution of each mode to wall motion is determined by a corresponding time-dependent deformation variable. The principle of virtual work is applied to these deformation variables, yielding a system of ordinary differential equations for LV dynamics, including effects of muscle fiber orientations, active and passive stresses, and surface tractions. Passive stress is governed by a transversely isotropic elastic model. Active stress acts in the fiber direction and incorporates length–tension and force–velocity properties of cardiac muscle. Preload and afterload are represented by lumped vascular models. The variational equations and their numerical solutions are verified by comparison to analytic solutions of the strong form equations. Deformation modes are constructed using Fourier series with an arbitrary number of terms. Greater numbers of deformation modes increase deformable model resolution but at increased computational cost. Simulations of normal LV motion throughout the cardiac cycle are presented using models with 8, 23, or 46 deformation modes. Aggregate quantities that describe LV function vary little as the number of deformation modes is increased. Spatial distributions of stress and strain change as more deformation modes are included, but overall patterns are conserved. This approach yields three-dimensional simulations of the cardiac cycle on a clinically relevant time-scale.

     

Flow-induced wall shear stress in abdominal aortic aneurysms: Part I--steady flow hemodynamics

Numerical predictions of blood flow patterns and hemodynamic stresses in Abdominal Aortic Aneurysms (AAAs) are performed in a two-aneurysm, axisymmetric, rigid wall model using the spectral element method. Homogeneous, Newtonian blood flow is simulated under steady conditions for the range of Reynolds numbers 10 < or =Re < or =2265. Flow hemodynamics are quantified by calculating the distributions of wall pressure (p(w)), wall shear stress (tau(w)), Wall Shear Stress Gradient (WSSG). A correlation between maximum values of hemodynamic stresses and Reynolds number is established, and the spatial distribution of WSSG is considered as a hemodynamic force that may cause damage to the arterial wall at an intermediate stage of AAA growth. The temporal distribution of hemodynamic stresses in pulsatile flow and their physical implications in AAA rupture are discussed in Part II of this paper.     

Upregulation of angiotensin AT1a receptors mRNA in the heart and renal medulla after myocardial infarction in rats.

The myocardial infarct causes prolonged activation of the renin-angiotensin system and profoundly influences cardiac performance and renal excretory capabilities. The aim of the present study was to determine whether the myocardial infarct is also associated with an altered expression of AT1a receptors (AT1aR) mRNA in the heart and the kidney. To this end male Sprague-Dawley rats were subjected either to the left coronary artery ligation or to the sham surgery. Four weeks after the surgery the animals were sacrificed. In 11 infarcted and 10 sham-operated rats expression of AT1aR mRNA in the walls of the left and right ventricle of the heart, and in the renal cortex and renal medulla was determined by semiquantitative PCR method. In another group of 10 infarcted and 14 sham-operated rats the diameter of cardiomyocytes in the left and right cardiac ventricle was determined. The size of the infarct in the rats used for mRNA determination and for morphometric measurements was equal to 29.4 +/- 1.8% and to 31.0 +/- 1.2 % of the left ventricular wall, respectively. Expression of AT1aR mRNA was significantly greater in the left (P< 0.01) and right ventricle (P<0.03) of the heart in the infarcted than in the sham operated rats. AT1aR mRNA expression was also significantly greater (P<0.02) in the renal medulla of the infarcted rats than in the renal medulla of the sham operated rats whereas no significant difference was found in the renal cortex. The myocardial infarct was associated with a significant increase of diameter of cardiomyocytes of the left ventricle of the heart (P< 0.0001), however there was no significant correlation between changes in AT1aR mRNA expression and diameter of cardiomyocytes. The results provide evidence that the myocardial infarct results in significant and prolonged upregulation of AT1a receptors mRNA expression in the heart and in the medullary region of the kidney.     

Distribution of stresses in the left ventricular wall of the intact heart

The left ventricle is modelled as a prolate spheroid of viscoelastic material with an aim to demonstrate the qualitative effects of anistropy and nonhomogeneity in the calculation of intact ventricular wall stresses. The pericardial pressure is accounted for in the analysis and an attempt is made to examine to what extent this influences the ventricular stresses. Numerical results are also obtained by computing the analytical expressions derived through the analysis.