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.     

Passive elastic wall stiffness of the left ventricle: A comparison between linear theory and large deformation theory

Assuming a spherical geometry for the left ventricle, passive elastic stiffness-stress relations have been obtained on the basis of linear elasticity theory and large deformation theory. Employing pressure-volume aata taken from rat hearts of various age groups, it is shown that young rat heart muscle (1 month) is stiffer than either adult (7 months) or old rat heart muscle (17 months). Although the qualitative results are similar for both elasticity theories, the large deformation theory gave results in closer agreement with those obtained from papillary muscle studies. These results imply that stiffness of muscleper se can be assessed from left ventricular pressure-volume data.     

Mechanical behaviour of ventricular aneurysms

A mathematical model describes the mechanical behaviour of ventricular aneurysms assuming a spherical geometry for the left ventricle. Employing pressure-volume data obtained from normal dog hearts 1–2 hours after infarction, conditions are obtained on infarct thickness and angle of damage for ventricular rupture to occur. The results indicate that rupture is more likely to occur in the early period following infarction and that the dominant factor is the per cent thickness of the infarct.     

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.     

Ventricular and arterial wall stresses based on large deformation analyses

Assuming a spherical geometry for the left ventricle and a cylindrical geometry for arteries, wall stresses and elastic stiffnesses are evaluated on the basis of a large elastic deformation theory. On the basis of canine pressure-volume data, the numerical results indicate marked gradients of stress in the endocardial layers even for thin-walled vessels, a result not predicted by the classical theory of elasticity. These high gradients of stress are due to the fact that the elastic stiffness of the wall material increases with the stress which reaches maximum levels in the endocardial layers. The high stresses may be responsible for ischemia of the left ventricle and be a triggering mechanism for atherosclerosis.     

等参变换在人体左心室三维有限元机械模型中的应用

基于人体左心室的纤维结构和可以真实反映心电兴奋传播过程的心脏电模型,建立了采用复合材料分析方法的左心室三维等参有限元机械模型,本文主要介绍了等参变换思想、方法在建模分析中的特色应用,如基于纤维结构的等参有限单元离散,考虑非均匀性的材料参数和等参变换、层和数值积分和考虑复杂边界条件及负载特性的逆变换等均是运用等参变换的思路来复杂的生理结构问题,最后给出了基于该方法的模型应用。     

Effect of age on passive elastic stiffness of rat heart muscle.

A thick-wall spherical model for the rat left ventricle was used to deduce passive wall stiffness from diastolic pressure-volume data. This was done for rats in three age classes: young (1 mo), adult (17 mo) and old (17 mo). The model was based on finite deformation elasticity theory consistent with the magnitude of observed deformation. A least-squares procedure was used to determine elastic constants in postulated nonlinear stress-stretch relations for the myocardium. It was found that at a given level of stress, wall stiffness for ventricles in the young age class was consistently greater than wall stiffness in the other two classes. In addition, the difference in wall stiffness between rats in the adult and old age classes was found to be approximately 10%.     

Dynamic morphology of the cardiac ventricle

OBJECTIVE: To describe the outline of the ventricular wall on the basis of physiologic data. STUDY DESIGN: Applying the finite deformation theory, we introduced equilibrium equations to obtain the stress-strain relationship of canine myocardial fibers. This was then applied to drawing the shape of the ventricular wall by the finite element method. RESULTS: The outlines of the ventricle wall corresponded to those seen on an echocardiogram. CONCLUSION: Our method is applicable to the present purpose.     

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.     

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.     

Physiological basis for mechanical time-variance in the heart: special consideration of non-linear function

A relationship between ventricular pressure and volume is developed starting from basic cardiac muscle mechanics. The known and measurable properties of myocardium, such as the Hill law, the periodic excitation-contraction mechanism, and non-linear elasticity of the surrounding elastin and collagen structure, are formulated into a myofibril unit. A cylindrical geometry is chosen to represent the structure of the ventricle, using the myofibril unit as the basic building block. Pressure-volume isochrones computed from this model illustrate non-linear function in the heart which arises from both geometric effects and muscle effects. The above theory and model is linearized to provide a special study case. The behavior that resulted is that of a time-varying elastance, E(t), and, hence, can help in the interpretation of its meaning. It is found that the minimum in E(t) is the consequence of the stiffness of the myocardial fibrous network, adjusted by a geometric factor. In addition, the magnitude of E(t) is governed by myocardial contractility, a geometric factor, and the excitation-contraction mechanism, where time-dependency is imparted by periodic excitation. Since the elastic fibers are the only true elastic elements, the quantity of elastance is determined by controlled volume feedback. A circuit model is provided to illustrate this concept. The non-linear active and passive heart function curves are specified independently. These curves are required to intersect below the resting volume and result in a negative pressure at the intersection. This is found to explain the phenomenon of ventricular suction. In addition, they lead to a time-varying dead volume by virtue of time-dependent isochronal slope. Non-linear function is introduced to the model and is found to explain the variation in curvature of the ventricular isochrones.     

Adaptation of passive rat left ventricle in diastolic dysfunction.

This article deals with providing a theoretical explanation for quantitative changes in the geometry, the opening angle and the deformation parameters of the rat ventricular wall during adaptation of the passive left ventricle in diastolic dysfunction. A large deformation theory is applied to analyse transmural stress and strain distribution in the left ventricular wall considering it to be made of homogeneous, incompressible, transversely isotropic, non-linear elastic material. The basic assumptions made for computing stress distributions are that the average circumferential stress and strain for the adaptive ventricle is equal to the average circumferential stress and strain in the normotensive ventricle, respectively.All the relevant parameters, such as opening angle, twist per unit length, axial extension, internal and external radii and others, in the stress-free, unloaded and loaded states of normotensive, hypertensive and adaptive left ventricle are determined. The circumferential stress and strain distribution through the ventricular wall are also computed. Our analysis predicts that during adaptation, wall thickness and wall mass of the ventricle increase. These results are consistent with experimental findings and are the indications of initiation of congestive heart failure.     

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.     

A model for one-dimensional morphoelasticity and its application to fibroblast-populated collagen lattices

The mechanical behaviour of solid biological tissues has long been described using models based on classical continuum mechanics. However, the classical continuum theories of elasticity and viscoelasticity cannot easily capture the continual remodelling and associated structural changes in biological tissues. Furthermore, models drawn from plasticity theory are difficult to apply and interpret in this context, where there is no equivalent of a yield stress or flow rule. In this work, we describe a novel one-dimensional mathematical model of tissue remodelling based on the multiplicative decomposition of the deformation gradient. We express the mechanical effects of remodelling as an evolution equation for the effective strain, a measure of the difference between the current state and a hypothetical mechanically relaxed state of the tissue. This morphoelastic model combines the simplicity and interpretability of classical viscoelastic models with the versatility of plasticity theory. A novel feature of our model is that while most models describe growth as a continuous quantity, here we begin with discrete cells and develop a continuum representation of lattice remodelling based on an appropriate limit of the behaviour of discrete cells. To demonstrate the utility of our approach, we use this framework to capture qualitative aspects of the continual remodelling observed in fibroblast-populated collagen lattices, in particular its contraction and its subsequent sudden re-expansion when remodelling is interrupted.     

The time-varying elastic properties of the left ventricular muscle

We combine two techniques in order to discuss the time-varying elastic properties of the left ventricular muscle. An analytic model for the shape and forces in the left ventricle is combined with the Fourier series representations of certain of the ventricular dimensions and pressure to derive expressions for the stress and strain in the left ventricle. The strain is thus a function of the elastic material properties, which are then expressed as functions of time by using Fourier series. The only data needed for a numerical study using these techniques are closed-chest determinations of the ventricular dimensions and the ventricular pressure.     

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.     

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.     

Towards a biomechanics-based technique for assessing myocardial contractility: an inverse problem approach

This work presents the initial development and implementation of a novel 3D biomechanics-based approach to measure the mechanical activity of myocardial tissue, as a potential non-invasive tool to assess myocardial function. This technique quantifies the myocardial contraction forces developed within the ventricular myofibers in response to electro-physiological stimuli. We provide a 3D finite element formulation of a contraction force reconstruction algorithm, along with its implementation using magnetic resonance (MR) data. Our algorithm is based on an inverse problem solution governed by the fundamental continuum mechanics principle of conservation of linear momentum, under a first-order approximation of elastic and isotropic material conditions. We implemented our technique using a subject-specific ventricle model obtained by extracting the left ventricular anatomical features from a set of high-resolution cardiac MR images acquired throughout the cardiac cycle using prospective electrocardiographic gating. Cardiac motion information was extracted by non-rigid registration of the mid-diastole reference image to the remaining images of a 4D dataset. Using our technique, we reconstructed dynamic maps that show the contraction force distribution superimposed onto the deformed ventricle model at each acquired frame in the cardiac cycle. Our next objective will consist of validating this technique by showing the correlation between the presence of low contraction force patterns and poor myocardial functionality.     

Calcium and magnesium levels in isolated mitochondria from human cardiac biopsies

Summary A non-enzymatic method is presented for isolating mitochondria from small-sized human cardiac samples, including ventricular needle biopsies of 15–25 mg of wet weight. Electron microscopy demonstrates that these fractions are rich in structurally well preserved mitochondria. Calcium and magnesium levels of fractions are determined by atomic absorption flame spectroscopy. Comparative analyses are made in similar fractions of the mouse ventricle. Calcium concentrations of mitochondria isolated in the presence of ruthenium red do not differ significantly between the human auricle and ventricle, averaging 61 nmol Ca/mg protein and 68 nmol Ca/mg protein, respectively. Mitochondrial calcium level is lower in the mouse ventricular fractions, averaging 7 nmol Ca/mg protein. Mitochondrial magnesium amounts to slightly less than 60% of the calcium levels in the human heart, while it exceeds the calcium level by more than 100 per cent in the mouse heart. There is no significant difference of mitochondrial calcium between normal auricles, and, auricles of patients with increased right atrial mean pressure and/or volume overload.This work was supported by grants from The Norwegian Council on Cardiovascular Disease and from The Norwegian Research Council for Science and the Humanities