On a nonlinear theory for muscle shells: Part II--Application to the beating left ventricle

This paper specializes the nonlinear laminated-muscle-shell theory developed in Part I to cylindrical geometry and computes stresses in arteries and the beating left ventricle. The theory accounts for large strain, material nonlinearity, thick-shell effects, torsion, muscle activation, and residual strain. First, comparison with elasticity solutions for pressurized arteries shows that the accuracy of the shell theory increases as transmural stress gradients and the shell thickness decrease. Residual strain reduces the stress gradients, lowering the error in the predicted peak stress in thick-walled arteries (R/t = 2.8) from about 30 to 10 percent. Second, the canine left ventricle is modeled as a thick-walled laminated cylinder with an internal pressure. Each layer is composed of transversely isotropic muscle with a fiber orientation based on anatomical data. Using a single pseudostrain-energy density function (with time-varying coefficients) for passive and active myocardium, the model predicts strain distributions that agree fairly well with published experimental measurements. The results also show that the peak fiber stress occurs subendocardially near the beginning of ejection and that residual strains significantly alter stress gradients within each lamina, but the magnitude of the peak fiber stress changes by less than 20 percent.     

Left-ventricular shape determines intramyocardial mechanical heterogeneity

Left-ventricular remodeling is considered to be an important mechanism of disease progression leading to mechanical dysfunction of the heart. However, the interaction between the physiological changes in the remodeling process and the associated mechanical dysfunction is still poorly understood. Clinically, it has been observed that the left ventricle often undergoes large shape changes, but the importance of left-ventricular shape as a contributing factor to alterations in mechanical function has not been clearly determined. Therefore, the interaction between left-ventricular shape and systolic mechanical function was examined in a computational finite-element study. Hereto, finite-element models were constructed with varying shapes, ranging from an elongated ellipsoid to a sphere. A realistic transmural gradient in fiber orientation was considered. The passive myocardium was described by an incompressible hyperelastic material law with transverse isotropic symmetry. Activation was governed by the eikonal-diffusion equation. Contraction was incorporated using a Hill model. For each shape, simulations were performed in which passive filling was followed by isovolumic contraction and ejection. It was found that the intramyocardial distributions of fiber stress, strain, and stroke work density were shape dependent. Ejection performance was reduced with increasing sphericity, which was regionally related to a reduction in the active fiber stress development, fiber shortening, and stroke work in the midwall and subepicardial region at the midheight level in the left-ventricular wall. Based on these results, we conclude that a significant interaction exists between left-ventricular shape and regional myofiber mechanics, but the importance for left-ventricular remodeling requires further investigation.     

Influence of left-ventricular shape on passive filling properties and end-diastolic fiber stress and strain

Passive filling is a major determinant for the pump performance of the left ventricle and is determined by the filling pressure and the ventricular compliance. In the quantification of the passive mechanical behaviour of the left ventricle and its compliance, focus has been mainly on fiber orientation and constitutive parameters. Although it has been shown that the left-ventricular shape plays an important role in cardiac (patho-)physiology, the dependency on left-ventricular shape has never been studied in detail. Therefore, we have quantified the influence of left-ventricular shape on the overall compliance and the intramyocardial distribution of passive fiber stress and strain during the passive filling period. Hereto, fiber stress and strain were calculated in a finite element analysis of passive inflation of left ventricles with different shapes, ranging from an elongated ellipsoid to a sphere, but keeping the initial cavity volume constant. For each shape, the wall volume was varied to obtain ventricles with different wall thickness. The passive myocardium was described by an incompressible hyperelastic material law with transverse isotropic symmetry along the muscle fiber directions. A realistic transmural distribution in fiber orientation was assumed. We found that compliance was not altered substantially, but the transmural distribution of both passive fiber stress and strain was highly dependent on regional wall curvature and thickness. A low curvature wall was characterized by a maximum in the transmural fiber stress and strain in the mid-wall region, while a steep subendocardial transmural gradient was present in a high curvature wall. The transmural fiber stress and strain gradients in a low and high curvature wall were, respectively, flattened and steepened by an increase in wall thickness.     

Akinetic myocardial infarcts must contain contracting myocytes: finite-element model study

Infarcted segments of myocardium demonstrate functional impairment ranging in severity from hypokinesis to dyskinesis. We sought to better define the contributions of passive material properties (stiffness) and active properties (contracting myocytes) to infarct thickening. Using a finite-element (FE) model, we tested the hypothesis that infarcted myocardium must contain contracting myocytes to be akinetic and not dyskinetic. A three-dimensional FE mesh of the left ventricle was developed with echocardiographs from a reperfused ovine anteroapical infarct. The nonlinear stress-strain relationship for the diastolic myocardium was anisotropic with respect to the local muscle fiber direction, and an elastance model for active fiber stress was incorporated. The diastolic stiffness (C) and systolic material property (isometric tension at longest sarcomere length and peak intracellular calcium concentration, T(max)) of the uninfarcted remote myocardium were assumed to be normal (C = 0.876 kPa, T(max) = 135.7 kPa). Diastolic and systolic properties of the infarct necessary to produce akinesis, defined as an average radial strain between -0.01 and 0.01, were determined by assigning a range of diastolic stiffnesses and scaling infarct T(max) to represent the percentage of contracting myocytes between 0% and 100%. As C was increased to 11 times normal (C = 10 kPa) the percentage of T(max) necessary for akinesis increased from 20% to 50%. Without contracting myocytes, C = 250 kPa was necessary to achieve akinesis. If infarct stiffness is <285 times normal, contracting myocytes are required to prevent dyskinetic infarct wall motion.     

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.     

Anisotropic hydraulic permeability under finite deformation

The structural organization of biological tissues and cells often produces anisotropic transport properties. These tissues may also undergo large deformations under normal function, potentially inducing further anisotropy. A general framework for formulating constitutive relations for anisotropic transport properties under finite deformation is lacking in the literature. This study presents an approach based on representation theorems for symmetric tensor-valued functions and provides conditions to enforce positive semidefiniteness of the permeability or diffusivity tensor. Formulations are presented, which describe materials that are orthotropic, transversely isotropic, or isotropic in the reference state, and where large strains induce greater anisotropy. Strain-induced anisotropy of the permeability of a solid-fluid mixture is illustrated for finite torsion of a cylinder subjected to axial permeation. It is shown that, in general, torsion can produce a helical flow pattern, rather than the rectilinear pattern observed when adopting a more specialized, unconditionally isotropic spatial permeability tensor commonly used in biomechanics. The general formulation presented in this study can produce both affine and nonaffine reorientations of the preferred directions of material symmetry with strain, depending on the choice of material functions. This study addresses a need in the biomechanics literature by providing guidelines and formulations for anisotropic strain-dependent transport properties in porous-deformable media undergoing large deformations.     

Estimates of regional work in the canine left ventricle

Assessment of the magnitude of regional myocardial work requires knowledge of regional fiber stress and fiber shortening. The theoretical development and experimental validation of a method is presented which used values of estimated active and passive fiber stress according to a fluid-fiber model, and measured fiber strain values. This enables the construction of regional stress-strain diagrams, a regional analog of the pressure-volume area model by Suga and co-investigators, which can be linked to regional oxygen consumption. In the left ventricle, either normally or asynchronously activated, the method yields reliable data on strain and active and passive fiber stress. The relation between estimated regional work and myocardial oxygen demand is in quantitative agreement with previously reported relations between global oxygen demand and measured pressure-volume area. During coronary artery occlusion, however, these values were less reliable, which might be due to inaqdequate knowledge of the (passive) material properties of the myocardium.     

A mixed finite element formulation for a non-linear,transversely isotropic material model for the cardiac tissue

In this paper we present a mixed finite element method for modeling the passive properties of the myocardium. The passive properties are described by a non-linear, transversely isotropic, hyperelastic material model, and the myocardium is assumed to be almost incompressible. Single-field, pure displacement-based formulations are known to cause numerical difficulties when applied to incompressible or slightly compressible material cases. This paper presents an alternative approach in the form of a mixed formulation, where a separately interpolated pressure field is introduced as a primary unknown in addition to the displacement field. Moreover, a constraint term is included in the formulation to enforce (almost) incompressibility. Numerical results presented in the paper demonstrate the difficulties related to employing a pure displacement-based method, applying a set of physically relevant material parameter values for the cardiac tissue. The same problems are not experienced for the proposed mixed method. We show that the mixed formulation provides reasonable numerical results for compressible as well as nearly incompressible cases, also in situations of large fiber stretches. There is good agreement between the numerical results and the underlying analytical models.     

A mixed finite element formulation for a non-linear, transversely isotropic material model for the cardiac tissue

In this paper we present a mixed finite element method for modeling the passive properties of the myocardium. The passive properties are described by a non-linear, transversely isotropic, hyperelastic material model, and the myocardium is assumed to be almost incompressible. Single-field, pure displacement-based formulations are known to cause numerical difficulties when applied to incompressible or slightly compressible material cases. This paper presents an alternative approach in the form of a mixed formulation, where a separately interpolated pressure field is introduced as a primary unknown in addition to the displacement field. Moreover, a constraint term is included in the formulation to enforce (almost) incompressibility. Numerical results presented in the paper demonstrate the difficulties related to employing a pure displacement-based method, applying a set of physically relevant material parameter values for the cardiac tissue. The same problems are not experienced for the proposed mixed method. We show that the mixed formulation provides reasonable numerical results for compressible as well as nearly incompressible cases, also in situations of large fiber stretches. There is good agreement between the numerical results and the underlying analytical models.     

Constitutive stress--strain relations for the myocardium in diastole

The importance of stress-strain myocardial constitutive relations is that they provide a criterion for behavior in vivo. Our purpose was to develop constitutive equations which are valid in diastole. The myocardium was assumed to be composed of a nonlinear viscoelastic, inhomogeneous, anisotropic (transversely isotropic) and incompressible material operating under adiabatic and isothermal conditions. The expressions contain five moduli. Two are fixed by the restriction of incompressibility, one is estimated, the remaining two refer to directions along and perpendicular to a fiber. Both possess a bimodal variation with intermodal switching occurring in late rapid filling and diastasis. They are functions of time and material constants. These constants can be observed. A dynamic test is suggested. Constitutive statements complete a set of equations sufficient for the solution of a class of boundary value problems. One type is formulated. They also permit the determination of stress from measured strain. Examples are given.     

Isotropy and anisotropy of the arterial wall

The passive biomechanical response of intact cylindrical rat carotid arteries is studied in vitro and compared with the mechanical response of rubber tubes. Using true stress and natural strain in the definition of the incremental modulus of elasticity, the tissue wall properties are analyzed over wide ranges of simultaneous circumferential and longitudinal deformations. The type of loading chosen is 'physiological' i.e. symmetric: the cylindrical segments are subjected to internal pressure and axial prestretch without torsion or shear. Several aspects pertaining to the choice of parameters characterizing the material are discussed and the analysis pertaining to the deformational behavior of a hypothetical compliant tube with Hookean wall material is presented. The experimental results show that while rubber response can be adequately represented as linearly elastic and isotropic, the overall response of vascular tissue is highly non-linear and anisotropic. However, for states of deformation that occur in vivo, the elasticity of arteries is quite similar to that of rubber tubes and as such the arterial wall may be viewed as incrementally isotropic for the range of deformations that occur in vivo.     

Stress Distribution in the Canine Left Ventricle during Diastole and Systole

A model is proposed for stress analysis of the left ventricular wall (LV wall) based on the realistic assumption that the myocardium is essentially composed of fiber elements which carry only axial tension and vary in orientation through the wall. Stress analysis based on such a model requires an extensive study of muscle fiber orientation and curvature through the myocardium. Accordingly, the principal curvatures were studied at a local site near the equator in ten dog hearts rapidly fixed in situ at end diastole and end systole; the fiber orientation for these hearts had already been established in a previous study. The principal radii of curvature were (a) measured by fitting templates to the endocardial and epicardial wall surfaces in the circumferential and longitudinal directions and (b) computed from measured lengths of semiaxes of ellipsoids of revolution representing the LV wall (“ellipsoid” data). The wall was regarded as a tethered set of nested shells, each having a unique fiber orientation. Results indicate the following. (a) Fiber curvature, k, is maximum at midwall at end systole; this peak shifts towards endocardium at end diastole. (b) The pressure or radial stress through the wall decreases more rapidly near the endocardium than near the epicardium at end diastole and at end systole when a constant tension is assumed for each fiber through the wall. (c) At end diastole the curve for the circumferential stress vs. wall thickness is convex with a maximum at midwall. In the longitudinal direction the stress distribution curve is concave with a minimum at midwall. Similar distributions are obtained at end systole when a constant tension is assumed for each fiber through the wall. (d) The curvature and stress distributions obtained by direct measurements at a selected local site agree well with those computed from “ellipsoid” data.     

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.     

Active and passive stresses in the myocardium

A mathematical approach that can be used to calculate the passive stress in the ventricular wall is presented. The active fiber stress (force/unit area) generated by the muscular fibers in the ventricular wall is expressed by means of body force (force/unit volume of the myocardium). It is shown that the total intramyocardial passive stress induced in the passive medium of the myocardium can be expressed as the sum of a passive stress induced by the left ventricular pressure and a passive stress induced by the active fiber stress. Applications to experimental data published in the literature are given. New results are presented that show the relation among those two components of the intramyocardial passive stress. New relations between the intramyocardial passive stress, the slope (elastance) of the pressure-volume relation, and the residual volume are also derived. The results obtained give a better understanding of some aspects of the mechanics of cardiac contraction and can provide a more detailed interpretation of clinical conditions.     

A modified Holzapfel-Ogden law for a residually stressed finite strain model of the human left ventricle in diastole

In this work, we introduce a modified Holzapfel-Ogden hyperelastic constitutive model for ventricular myocardium that accounts for residual stresses, and we investigate the effects of residual stresses in diastole using a magnetic resonance imaging–derived model of the human left ventricle (LV). We adopt an invariant-based constitutive modelling approach and treat the left ventricular myocardium as a non-homogeneous, fibre-reinforced, incompressible material. Because in vivo images provide the configuration of the LV in a loaded state even in diastole, an inverse analysis is used to determine the corresponding unloaded reference configuration. The residual stress in this unloaded state is estimated by two different methods. One is based on three-dimensional strain measurements in a local region of the canine LV, and the other uses the opening angle method for a cylindrical tube. We find that including residual stress in the model changes the stress distributions across the myocardium and that whereas both methods yield qualitatively similar changes, there are quantitative differences between the two approaches. Although the effects of residual stresses are relatively small in diastole, the model can be extended to explore the full impact of residual stress on LV mechanical behaviour for the whole cardiac cycle as more experimental data become available. In addition, although not considered here, residual stresses may also play a larger role in models that account for tissue growth and remodelling.     

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.     

On the sensitivity of wall stresses in diseased arteries to variable material properties

Accurate estimates of stress in an atherosclerotic lesion require knowledge of the material properties of its components (e.g., normal wall, fibrous plaque, calcified regions, lipid pools) that can only be approximated. This leads to considerable uncertainty in these computational predictions. A study was conducted to test the sensitivity of predicted levels of stress and strain to the parameter values of plaque used in finite element analysis. Results show that the stresses within the arterial wall, fibrous plaque, calcified plaque, and lipid pool have low sensitivities for variation in the elastic modulus. Even a +/- 50% variation in elastic modulus leads to less than a 10% change in stress at the site of rupture. Sensitivity to variations in elastic modulus is comparable between isotropic nonlinear, isotropic nonlinear with residual strains, and transversely isotropic linear models. Therefore, stress analysis may be used with confidence that uncertainty in the material properties generates relatively small errors in the prediction of wall stresses. Either isotropic nonlinear or anisotropic linear models provide useful estimates, however the predictions in regions of stress concentration (e.g., the site of rupture) are somewhat more sensitive to the specific model used, increasing by up to 30% from the isotropic nonlinear to orthotropic model in the present example. Changes resulting from the introduction of residual stresses are much smaller.     

The effect of residual strain on the diastolic function of the left ventricle as predicted by a structural model

The unloaded heart is not stress-free. It is subjected to residual stress and strain. Their extent and influence on the global performance of the left ventricle and on local phenomena in the ventricular wall are studied by model simulation. The analysis focuses on the equatorial region of the ventricle, with an approximate thick-walled cylindrical geometry. The in vivo myocardium is considered to be incompressible, consisting of fibers embedded in a fluid matrix, with transmurally varying anisotropic microstructure in accordance with morphological characteristics.

The results show that residual strain is transmurally distributed with a pattern and magnitude which agree well with measurements. The calculated residual strains are within mean ± one standard deviation of the measured ones. Their magnitude was found to increase with increasing opening angle and with increasing wall thickness. The residual strain was found to have several effects on ventricular function: At volumes higher than the reference one it gives rise to more uniform transmural distributions of stress and intramyocardial pressure; it causes about 50% increase in the ventricular compliance at high volumes and doubles the suction of atrial blood at low volumes, thus facilitating the diastolic filling. In addition, residual strains cause bias of in vivo measured strains from their true values. This may significantly affect physiological interpretation of measured ventricular deformations.

In conclusion, the present structural analysis predicts that residual strain has favorable effect on left-ventricular diastolic performance, and gives rise to more uniform ventricular stress distribution.     

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.     

Validation of three-dimensional conduction models using experimental mapping: are we getting closer?

The anisotropic material properties, irregular geometry, and specialized conduction system of the heart all affect the three-dimensional (3D) spread of electrical activation. A limited number of research groups have tried accounting for these features in 3D conduction models to investigate more thoroughly their observations of cardiac electrical activity in 3D experimental preparations. The full potential of these large scale conduction models, however, has not been realized because of a lack of quantitative validation with experiment. Such validation is critical in order to use the models to predict the electrical response of the myocardium to drugs or electrical stimulation. In this paper, a quantitative, experimental validation of paced activation in a 3D conduction model of a 3 cm × 3 cm × 1 cm section of the ventricular wall is presented. Epicardial and intramural pacing stimuli were applied in the center of a 528 channel electrode plaque sutured to the left ventricle in dogs. Unipolar electrograms were recorded at 2 kHz during and after pacing. Fiber directions within the tissue below the electrodes were estimated histologically and from pace-mapping. Simulated epicardial electrograms were computed for surface paced beats using our 3D bidomain model of the mapped tissue volume incorporating the measured fiber directions. Extracellular potentials and isochronal maps resulting from paced activations in both model and experiment were directly compared. Preliminary results demonstrate that our 3D model reproduces qualitatively such key features of the experimental data as electrogram morphologies and epicardial conduction velocities. Though quantitative agreement between model and experiment was only moderate, the validation approach described herein is an essential first step in assessing the predictive capability of present day conduction models.