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.     

A concentric layer model for estimating the energy expenditure of the left ventricle

The energy cost of the left ventricle is quantitatively analyzed on the basis of the following assumptions: (1) The left ventricle is assumed to be an isotropic, homogeneous elastic, thick, spherical shell. (2) The ventricular wall is made up of a finite number of thin concentric shells. (3) The energetics of the left ventricle is in accordance with the second law of thermodynamics. An expression for the work done during ventricular contraction is derived according to the definition of physical work. The energy liberation during isovolumic contraction is formulated parallel to the concepts of heat production in skeletal muscle during isometric contraction. This expression gives the total work done per stroke in terms of mean systolic pressure, end diastolic volume, stroke volume and wall thickness during diastolic phase. Supported by a research fellowship and research grant from the Canadian Heart Foundation.     

Role of tissue structure on ventricular wall mechanics

It is well known that systolic wall thickening in the inner half of the left ventricular (LV) wall is of greater magnitude than predicted by myofiber contraction alone. Previous studies have related the deformation of the LV wall to the orientation of the laminar architecture. Using this method, wall thickening can be interpreted as the sum of contributions due to extension, thickening, and shearing of the laminar sheets. We hypothesized that the thickening mechanics of the ventricular wall are determined by the structural organization of the underlying tissue, and may not be influenced by factors such as loading and activation sequence. To test this hypothesis, we calculated finite strains from biplane cineradiography of transmural markers implanted in apical (n = 22) and basal (n = 12) regions of the canine anterior LV free wall. Strains were referred to three-dimensional laminar microstructural axes measured by histology. The results indicate that sheet angle is of opposite sign in the apical and basal regions, but absolute value differs only in the subepicardium. During systole, shearing and extension of the laminae contribute the most to wall thickening, accounting for >90% (transmural average) at both apex and base. These two types of deformation are also most prominent during diastolic inflation. Increasing afterload has no effect on the pattern of systolic wall thickening, nor does reversing transmural activation sequence. The pattern of wall thickening appears to be a function of the orientation of the laminar sheets, which vary regionally and transmurally. Thus, acute interventions do not appear to alter the contributions of the laminae to wall thickening, providing further evidence that the structural architecture of the ventricular wall is the dominant factor for its regional mechanical function.     

Non-homogeneous analysis of three-dimensional transmural finite deformation in canine ventricular myocardium

A new method has been developed for analyzing transmural distributions of finite deformation in canine ventricular myocardium without the need to assume that the strain in a finite volume of the wall is homogeneous. The three-dimensional nodal geometric parameters of bilinear-cubic or bilinear-quadratic finite elements are fitted by least squares to the measured coordinates of 12-18 radiopaque markers implanted in the left ventricular free wall. For six dog hearts, root-mean-squared errors in the fitted in-plane coordinates ranged from 0.079-0.556 mm in the end-diastolic reference state and 0.142-0.622 mm at end-systole. The corresponding error ranges in the radial coordinate were 0.042-0.264 mm at end-diastole and 0.106-0.279 mm at end-systole. Smoothly continuous transmural profiles of wall strain computed as the element deformed during the cardiac cycle from end-diastole to end-systole showed good agreement with the discrete results of conventional homogeneous analysis. Using the kinematics of a thick-walled incompressible cylinder, overall absolute errors due to the non-homogeneity of myocardial deformation were found to be reduced in the new analysis by 30-35% for typical experimental parameters. Overall relative errors were also reduced (from 23 to 20%). Since measurement errors in the reconstructed marker coordinates were spatially smoothed by the fitting procedure, noise in the computed deformations was also substantially attenuated, and transmural gradients of three-dimensional strain components could be obtained with improved accuracy. Hence physiological factors affected by transmural stress and strain distributions, such as myocardial blood flow, ischemia and hypertrophy, may be better understood.     

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.     

Computational modeling of growth: systemic and pulmonary hypertension in the heart

We introduce a novel constitutive model for growing soft biological tissue and study its performance in two characteristic cases of mechanically induced wall thickening of the heart. We adopt the concept of an incompatible growth configuration introducing the multiplicative decomposition of the deformation gradient into an elastic and a growth part. The key feature of the model is the definition of the evolution equation for the growth tensor which we motivate by pressure-overload-induced sarcomerogenesis. In response to the deposition of sarcomere units on the molecular level, the individual heart muscle cells increase in diameter, and the wall of the heart becomes progressively thicker. We present the underlying constitutive equations and their algorithmic implementation within an implicit nonlinear finite element framework. To demonstrate the features of the proposed approach, we study two classical growth phenomena in the heart: left and right ventricular wall thickening in response to systemic and pulmonary hypertension.     

Effects of transmural electrical heterogeneities and electrotonic interactions on the dispersion of cardiac repolarization and action potential duration: A simulation study

It has been shown in the literature that myocytes isolated from the ventricular walls at various intramural depths have different action potential durations (APDs). When these myocytes are embedded in the ventricular wall, their inhomogeneous properties affect the sequence of repolarization and the actual distribution of the APDs in the entire wall. In this article, we implement a mathematical model to simulate the combined effect of (a) the non-homogeneous intrinsic membrane properties (in particular the non-homogeneous APDs) and (b) the electrotonic currents that modulate the APDs when the myocytes are embedded in the ventricular myocardium. In particular, we study the effect of (a) and (b) on the excitation and repolarization sequences and on the distribution of APDs in the ventricles. We implement a Monodomain tissue representation that includes orthotropic anisotropy, transmural fiber rotation and homogeneous or heterogeneous transmural intrinsic membrane properties, modeled according to the phase I Luo-Rudy membrane ionic model. Three-dimensional simulations are performed in a cartesian slab with a parallel finite element solver employing structured isoparametric trilinear finite elements in space and a semi-implicit adaptive method in time. Simulations of excitation and repolarization sequences elicited by epicardial or endocardial pacing show that in a homogeneous slab the repolarization pathways approximately follow the activation sequence. Conversely, in the heterogeneous cases considered in this study, we observed two repolarization wavefronts that started from the epi and the endocardial faces respectively and collided in the thickness of the wall and in one case an additional repolarization wave starting from an intramural site. Introducing the heterogeneities along the transmural epi-endocardial direction affected both the repolarization sequence and the APD dispersion, but these effects were clearly discernible only in transmural planes. By contrast, in planes parallel to epi- and endocardium the APD distribution remained remarkably similar to that observed in the homogeneous model. Therefore, the patterns of the repolarization sequence and APD dispersion on the epicardial surface (or any other intramural surface parallel to it) do not reveal the uniform transmural heterogeneity.     

Epicardial suction: a new approach to mechanical testing of the passive ventricular wall

The lack of an appropriate three-dimensional constitutive relation for stress in passive ventricular myocardium currently limits the utility of existing mathematical models for experimental and clinical applications. Previous experiments used to estimate parameters in three-dimensional constitutive relations, such as biaxial testing of excised myocardial sheets or passive inflation of the isolated arrested heart, have not included significant transverse shear deformation or in-plane compression. Therefore, a new approach has been developed in which suction is applied locally to the ventricular epicardium to introduce a complex deformation in the region of interest, with transmural variations in the magnitude and sign of nearly all six strain components. The resulting deformation is measured throughout the region of interest using magnetic resonance tagging. A nonlinear, three-dimensional, finite element model is used to predict these measurements at several suction pressures. Parameters defining the material properties of this model are optimized by comparing the measured and predicted myocardial deformations. We used this technique to estimate material parameters of the intact passive canine left ventricular free wall using an exponential, transversely isotropic constitutive relation. We tested two possible models of the heart wall: first, that it was homogeneous myocardium, and second, that the myocardium was covered with a thin epicardium with different material properties. For both models, in agreement with previous studies, we found that myocardium was nonlinear and anisotropic with greater stiffness in the fiber direction. We obtained closer agreement to previously published strain data from passive filling when the ventricular wall was modeled as having a separate, isotropic epicardium. These results suggest that epicardium may play a significant role in passive ventricular mechanics.     

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.     

A multiscale model for eccentric and concentric cardiac growth through sarcomerogenesis

We present a novel computational model for maladaptive cardiac growth in which kinematic changes of the cardiac chambers are attributed to alterations in cytoskeletal architecture and in cellular morphology. We adopt the concept of finite volume growth characterized through the multiplicative decomposition of the deformation gradient into an elastic part and a growth part. The functional form of its growth tensor is correlated to sarcomerogenesis, the creation and deposition of new sarcomere units. In response to chronic volume-overload, an increased diastolic wall strain leads to the addition of sarcomeres in series, resulting in a relative increase in cardiomyocyte length, associated with eccentric hypertrophy and ventricular dilation. In response to chronic pressure-overload, an increased systolic wall stress leads to the addition of sacromeres in parallel, resulting in a relative increase in myocyte cross sectional area, associated with concentric hypertrophy and ventricular wall thickening. The continuum equations for both forms of maladaptive growth are discretized in space using a nonlinear finite element approach, and discretized in time using the implicit Euler backward scheme. We explore a generic bi-ventricular heart model in response to volume- and pressure-overload to demonstrate how local changes in cellular morphology translate into global alterations in cardiac form and function.     

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.     

Deformation of the Diastolic Left Ventricle: I. Nonlinear Elastic Effects

A linear incremental finite element model is used to analyze the mechanical behavior of the left ventricle. The ventricle is treated as a heterogeneous, non-linearly elastic, isotropic, thick-walled solid of revolution. A new triaxial constitutive relation for the myocardium is presented which exhibits the observed exponential length-passive tension behavior of left ventricular papillary muscle in the limit of uniaxial tension. This triaxial relation contains three parameters: (a) a “small strain” Young's modulus, (b) a Poisson's ratio, and (c) a parameter which characterizes the nonlinear aspect of the elastic behavior of heart muscle. The inner third and outer two-thirds of the ventricular wall are assumed to have small strain Young's moduli of 30 and 60 g/cm², respectively. The Poisson's ratio is assumed to be equal to 0.49 throughout the ventricular wall. In general, the results of this study indicate that while a linearly elastic model for the ventricle may be adequate in terms of predicting pressure-volume relationships, a linear model may have serious limitations with regard to predicting fiber elongation within the ventricular wall. For example, volumes and midwall equatorial circumferential strains predicted by the linear and nonlinear models considered in this study differ by approximately 20 and 90%, respectively, at a transmural pressure of 12 cm H₂O.     

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.     

Factors influencing left-ventricular stiffness

The aim of the study was to investigate the relative contributions of geometrical and material factors to overall left-ventricular cavity stiffness. Left-ventricular cavity shapes were reconstructed using a computer and the variation of myocardial elastic modulus was calculated, by the finite element method, through the passive phase of diastole when rising volume coincided with rising pressure. Geometric data were obtained from biplane cineangiography, with micromanometer pressure measurements, for ten patients with left ventricular disease. Dimensional analysis was applied to the initial and derived data from which the influences of myocardial compliance, wall thickness-to-long dimension ratio, and aspect ratio (long-to-short axes) were determined. The ratio between the volume elasticity and the myocardial modulus of elasticity, the normalized stiffness ratio (NSR), is proposed as a useful index of left ventricular mechanical behaviour in diastole. The volume elasticity of the chamber is dependent not only upon the myocardium elastic modulus and the wall thickness ratio, but also on the shape of the chambe. Changes in the thickness/radius ratio of the ventricle have less effect upon its distention than those in the long dimension/radius ratio. The left ventricle becomes more spherical in shpae through diastole and hence becomes stiffer by this geometric mechanism.     

血清胱抑素C 水平与糖尿病心室重构关系的临床研究

目的:研究血清胱抑素C水平与糖尿病心室重构的关系。方法:选择2013年10月~2015年10月在我院进行诊治的糖尿病患者90例,检测血清胱抑素C水平,按照糖尿病患者血清胱抑素C水平的中位数,分为正常组(胱抑素C水平1.65mg/L)和升高组(胱抑素C水平1.65mg/L)。行超声心动图检测左室舒张末内径、左房内径、左室舒张末容积、室间隔厚度和左室后壁厚度,并计算出左室质量指数。对两组的这些指标进行比较,并分析血清胱抑素C与糖尿病心室重构的相关性。结果:与正常组相比,升高组的胱抑素C、左室舒张末内径、左房内径、室间隔厚度、左室后壁厚度、左室质量指数和脑钠肽水平均明显增高(P0.05);经过相关性分析,血清胱抑素C水平与左室舒张末内径、左房内径、室间隔厚度、左室后壁厚度、左室质量指数和脑钠肽均呈正相关(P0.05);Logistic回归分析显示左室舒张末内径、左房内径、室间隔厚度、左室后壁厚度、左室质量指数和脑钠肽等是胱抑素C水平升高的危险因素。结论:血清胱抑素C水平与糖尿病患者的心功能和心室重构具有明显相关性,可作为衡量糖尿病患者心室重构程度的一项参考指标。     

The echocardiographic study of the rabbit heart left ventricle morpho-functional parameters

Morphometric and functional parameters of the heart left ventricle in rabbits during systole and diastole were investigated by the method of echocardiography. Morphometric parameters were studied on three levels: the mitral valve, the papillary muscles and the apical level. The internal dimension of the left ventricle uniformly decreases in three parallel planes during systole, its maximal reduction being observed on the apical level. During the contraction phase, the posterior wall thickness of the left ventricular and the interventricular septum thickness increases on the basal level to a greater extent than on the apical one. During systole, the interventricular septum movement is greater than the left ventricular posterior wall motion. During the heart cycle, the form of the left ventricular cavity changes from an ellipsoid in diastole to elliptic paraboloid in systole.     

An anatomic basis for fetal right ventricular dominance and arterial pressure sensitivity

Fetal right ventricular dominance of flow and arterial pressure sensitivity were recently recognized but controversial findings. We investigated ventricular volumes, weights and dimensions in order to understand if there were anatomic differences between the ventricles which might explain these differential functional findings in the fetal sheep. Forty-four near term lambs and their hearts were weighed. Right and left ventricular free wall weights were not different. Volumes were measured by generating in vitro pressure-volume relations and by casting the two ventricles after fixation at equal, physiologic pressures. Right ventricular volume was greater than left ventricular volume by both techniques. Ventricular interaction and a restraining effect of the pericardium were present. Measurements of the fixed ventricles and their casts revealed the following: left ventricular wall thickness was slightly greater than right ventricular wall thickness; lateral ventricular diameters were not different but anteroposterior ventricular diameters were much greater in the right than left ventricle. Because of these findings, the right ventricular circumferential radii of curvature were greater than for the left ventricle as was the radius to wall thickness ratio. Greater right ventricular volume and radius to wall thickness ratio may be important factors in right ventricular flow dominance and greater sensitivity to arterial pressure.     

Multiphysics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method

To relate the subcellular molecular events to organ level physiology in heart, we have developed a three-dimensional finite-element-based simulation program incorporating the cellular mechanisms of excitation-contraction coupling and its propagation, and simulated the fluid-structure interaction involved in the contraction and relaxation of the human left ventricle. The FitzHugh-Nagumo model and four-state model representing the cross-bridge kinetics were adopted for cellular model. Both ventricular wall and blood in the cavity were modeled by finite element mesh. An arbitrary Lagrangian Eulerian finite element method with automatic mesh updating has been formulated for large domain changes, and a strong coupling strategy has been taken. Using electrical analog of pulmonary circulation and left atrium as a preload and the windkessel model as an afterload, dynamics of ventricular filling as well as ejection was simulated. We successfully reproduced the biphasic filling flow consisting of early rapid filling and atrial contraction similar to that reported in clinical observation. Furthermore, fluid-structure analysis enabled us to analyze the wave propagation velocity of filling flow. This simulator can be a powerful tool for establishing a link between molecular abnormality and the clinical disorder at the macroscopic level.     

A framework for biomechanics simulations using four-chamber cardiac models

Computational cardiac models have been extensively used to study different cardiac biomechanics; specifically, finite-element analysis has been one of the tools used to study the internal stresses and strains in the cardiac wall during the cardiac cycle. Cubic-Hermite finite element meshes have been used for simulating cardiac biomechanics due to their convergence characteristics and their ability to capture smooth geometries compactly–fewer elements are needed to build the cardiac geometry–compared to linear tetrahedral meshes. Such meshes have previously been used only with simple ventricular geometries with non-physiological boundary conditions due to challenges associated with creating cubic-Hermite meshes of the complex heart geometry. However, it is critical to accurately capture the different geometric characteristics of the heart and apply physiologically equivalent boundary conditions to replicate the in vivo heart motion. In this work, we created a four-chamber cardiac model utilizing cubic-Hermite elements and simulated a full cardiac cycle by coupling the 3D finite element model with a lumped circulation model. The myocardial fiber-orientations were interpolated within the mesh using the Log-Euclidean method to overcome the singularity associated with interpolation of orthogonal matrices. Physiologically equivalent rigid body constraints were applied to the nodes along the valve plane and the accuracy of the resulting simulations were validated using open source clinical data. We then simulated a complete cardiac cycle of a healthy heart and a heart with acute myocardial infarction. We compared the pumping functionality of the heart for both cases by calculating the ventricular work. We observed a 20% reduction in acute work done by the heart immediately after myocardial infarction. The myocardial wall displacements obtained from the four-chamber model are comparable to actual patient data, without requiring complicated non-physiological boundary conditions usually required in truncated ventricular heart models.     

Ventricular mechanics in diastole: material parameter sensitivity

Models of ventricular mechanics have been developed over the last 20 years to include finite deformation theory, anisotropic and inhomogeneous material properties and an accurate representation of ventricular geometry. As computer performance and the computational efficiency of the models improve, clinical application of these heart mechanics models is becoming feasible. One such application is to estimate myocardial material properties by adjusting the constitutive parameters to match wall deformation from MRI or ultrasound measurements, together with a measurement (or estimate) of ventricular pressure. Pigs are now the principal large animal model for these studies and in this paper we present the development of a new three-dimensional finite element model of the heart based on measurements of the geometry and the fibre and sheet orientations of pig hearts. The end-diastolic deformation of the model is computed using the "pole-zero" constitutive law which we have previously used to model the mechanics of passive myocardial tissue specimens. The sensitivities of end-diastolic fibre-sheet material strains and heart shape to changes in the material parameters are computed for the parameters of the pole-zero law in order to assess the utility of the models for inverse material property determination.