A cylindrical model for studying subendocardial ischaemia in the left ventricle

In this paper a mathematical model of a left ventricle with a cylindrical geometry is presented with the aim of gaining a better understanding of the relationship between subendocardial ischaemia and ST depression. The model is formulated as an infinite cylinder and takes into account the full bidomain nature of cardiac tissue, as well as fibre rotation. A detailed solution method (based on Fourier series, Fourier transforms and a one dimensional finite difference scheme) for the governing equations for electric potential in the tissue and the blood is also presented. The model presented is used to study the effect increasing subendocardial ischaemia has on the epicardial potential distribution as well as the effects of changing the bidomain conductivity values. The epicardial potential distributions obtained with this cylindrical geometry are compared with results obtained using a previously published slab model. Results of the simulations presented show that the morphologies of the epicardial potential distributions are similar between the two geometries, with the main difference being that the cylindrical model predicts slightly higher potentials.     

The effect of boundary conditions on epicardial potential distributions

This study presents a comparison of semi-analytical and numerical solution techniques for solving the passive bidomain equation in simple tissue geometries containing a region of subendocardial ischaemia. When the semi-analytical solution is based on Fourier transforms, recovering the solution from the frequency domain via fast Fourier transforms imposes a periodic boundary condition on the solution of the partial differential equation. On the other hand, the numerical solution uses an insulation boundary condition. When these techniques are applied to calculate the epicardial surface potentials, both yield a three well potential distribution which is identical if fibre rotation within the tissue is ignored. However, when fibre rotation is included, the resulting three-well distribution rotates, but through different angles, depending on the solution method. A quantitative comparison between the semi-analytical and numerical solutiontechniques is presented in terms of the effect fibre rotation has on the rotation of the epicardial potential distribution. It turns out that the Fourier transform approach predicts a larger rotation of the epicardial potential distribution than the numerical solution. The conclusion from this study is that it is not always possible to use analytical or semi-analytical solutions to check the accuracy of numerical solution procedures. For the problem considered here, this checking is only possible when it is assumed that there is no fibre rotation through the tissue.     

The effect of simplifying assumptions in the bidomain model of cardiac tissue: application to ST segment shifts during partial ischaemia

In this study various electrical conductivity approximations used in bidomain models of cardiac tissue are considered. Comparisons are based on epicardial surface potential distributions arising from regions of subendocardial ischaemia situated within the cardiac tissue. Approximations studied are a single conductivity bidomain model, an isotropic bidomain model and equal and reciprocal anisotropy ratios both with and without fibre rotation. It is demonstrated both analytically and numerically that the approximations involving a single conductivity bidomain, an isotropic bidomain or equal anisotropy ratios (ignoring fibre rotation) results in identical epicardial potential distributions for all degrees of subendocardial ischaemia. This result is contrary to experimental observations. It is further shown that by assuming reciprocal anisotropy ratios, epicardial potential distributions vary with the degree of subendocardial ischaemia. However, it is concluded that unequal anisotropy ratios must be used to obtain the true character of experimental observations.     

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.     

Evaluation of extravascular resistance in the unloaded fibrillating left ventricle of the dog

In open-chest anaesthetized dogs the relative magnitude of extravascular resistance in the superficial and deep left ventricular myocardium was estimated from the flow through the isolated vascular segments inserted at different depths of the myocardium. It was confirmed that in the normally working heart extravascular resistance was significantly greater in the deep than in the superficial layer. In the unloaded fibrillating left ventricle no difference in extravascular resistance between these layers could be detected. Since it had been found previously that subendocardial preponderance of ischaemia persists in the unloaded fibrillating left ventricle (Sedek and Micha?owski, submitted for publication), the present observation is a further challenge for the current view that the subendocardium is more vulnerable to ischaemia because extravascular resistance is greater in this layer.     

In vivo porcine left atrial wall stress: Computational model

Most computational models of the heart have so far concentrated on the study of the left ventricle, mainly using simplified geometries. The same approach cannot be adopted to model the left atrium, whose irregular shape does not allow morphological simplifications. In addition, the deformation of the left atrium during the cardiac cycle strongly depends on the interaction with its surrounding structures. We present a procedure to generate a comprehensive computational model of the left atrium, including physiological loads (blood pressure), boundary conditions (pericardium, pulmonary veins and mitral valve annulus movement) and mechanical properties based on planar biaxial experiments. The model was able to accurately reproduce the in vivo dynamics of the left atrium during the passive portion of the cardiac cycle. A shift in time between the peak pressure and the maximum displacement of the mitral valve annulus allows the appendage to inflate and bend towards the ventricle before the pulling effect associated with the ventricle contraction takes place. The ventricular systole creates room for further expansion of the appendage, which gets in close contact with the pericardium. The temporal evolution of the volume in the atrial cavity as predicted by the finite element simulation matches the volume changes obtained from CT scans. The stress field computed at each time point shows remarkable spatial heterogeneity. In particular, high stress concentration occurs along the appendage rim and in the region surrounding the pulmonary veins.     

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

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

Modelling skeletal muscle fibre orientation arrangement

Skeletal muscle tissues have complex geometries. In addition, the complex fibre orientation arrangement makes it quite difficult to create an accurate finite element muscle model. There are many possible ways to specify the complex fibre orientations in a finite element model, for example defining a local element coordinate system. In this paper, an alternative method using ABAQUS, which is combination of the finite element method and the non-uniform rational B-spline solid representation, is proposed to calculate the initial fibre orientations. The initial direction of each muscle fibre is specified as the tangent direction of the NURBS curve which the fibre lies on, and the directions of the deformed fibres are calculated from the initial fibre directions, the deformation gradients and the fibre stretch ratios. Several examples are presented to demonstrate the ability of the proposed method. Results show that the proposed method is able to characterise both the muscle complex fibre orientation arrangement and its complex mechanical response.     

Modelling skeletal muscle fibre orientation arrangement

Skeletal muscle tissues have complex geometries. In addition, the complex fibre orientation arrangement makes it quite difficult to create an accurate finite element muscle model. There are many possible ways to specify the complex fibre orientations in a finite element model, for example defining a local element coordinate system. In this paper, an alternative method using ABAQUS, which is combination of the finite element method and the non-uniform rational B-spline solid representation, is proposed to calculate the initial fibre orientations. The initial direction of each muscle fibre is specified as the tangent direction of the NURBS curve which the fibre lies on, and the directions of the deformed fibres are calculated from the initial fibre directions, the deformation gradients and the fibre stretch ratios. Several examples are presented to demonstrate the ability of the proposed method. Results show that the proposed method is able to characterise both the muscle complex fibre orientation arrangement and its complex mechanical response.     

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。     

Ischemia-induced changes in sarcolemmal (Na+, K+)-ATPase, K+-pNPPase, sialic acid, and phospholipid in the dog and effects of the nisoldipine and chlorpromazine treatment

This work was undertaken to study functional and structural changes of the cardiac sarcolemmal membrane which was isolated from the ischemic lesion in the dog. The sarcolemmal fraction was prepared, by adopting the method devised by Reeves and Sutko , from the right ventricle and the subendocardial and subepicardial layers of the left ventricle. Ischemic lesion was produced by occlusion of a branch of the left anterior descending coronary artery for a period of 1.5 hr in the thoracotomized dog, followed by release of the occlusion for 3 hr. Nisoldipine, 5 micrograms/kg, was given twice intravenously, and chlorpromazine was infused at a rate of 10 micrograms/kg X min, in addition to the administration of twice bolus doses of 400 micrograms/kg each. Nisoldipine significantly decreased the incidence of premature ventricular contractions and microvascular hemorrhage. Sarcolemmal purity was monitored by using enzyme and chemical markers; the results indicated that the membrane preparation was tenfold purified over the homogenate. Although the activities of ouabain-sensitive (Na+, K+)-ATPase and ouabain-sensitive K+-p-nitrophenylphosphatase ( pNPPase ) of the sarcolemmal preparation isolated from the subendocardial layer were similar to those from the subepicardial layer in the nonischemic left ventricle, a significant decrease in these activities was observed only when the sarcolemmal fraction isolated from the subendocardial layer of ischemic area was compared with that from the subendocardial layer of nonischemic area. In contrast, the sialic acid content of the sarcolemma from the ischemic subendocardial layer was significantly increased compared to that of the nonischemic subendocardial layer. No such changes occurred in sarcolemma prepared from the ischemic subepicardial layer. The total phospholipid content as well as phosphatidylcholine and -ethanolamine contents of the sarcolemmal membrane prepared from the subendocardial layer of ischemic area were significantly decreased compared to nonischemic area. Nisoldipine prevented the ischemia-induced alterations in sarcolemmal (Na+, K+)-ATPase, pNPPase , sialic acid and phospholipids of the subendocardial layer. Chlorpromazine showed a less consistent effect than did Nisoldipine under our experimental conditions. Our study thus demonstrates that the lipid component and function of cardiac sarcolemmal membrane are altered in the early ischemic lesion and that these alterations are nonuniform in distribution and are alleviated by some pharmacological intervention.     

Prediction of fibre architecture and adaptation in diseased carotid bifurcations

Many studies have used patient-specific finite element models to estimate the stress environment in atherosclerotic plaques, attempting to correlate the magnitude of stress to plaque vulnerability. In complex geometries, few studies have incorporated the anisotropic material response of arterial tissue. This paper presents a fibre remodelling algorithm to predict the fibre architecture, and thus anisotropic material response in four patient-specific models of the carotid bifurcation. The change in fibre architecture during disease progression and its affect on the stress environment in the plaque were predicted. The mean fibre directions were assumed to lie at an angle between the two positive principal strain directions. The angle and the degree of dispersion were assumed to depend on the ratio of principal strain values. Results were compared with experimental observations and other numerical studies. In non-branching regions of each model, the typical double helix arterial fibre pattern was predicted while at the bifurcation and in regions of plaque burden, more complex fibre architectures were found. The predicted change in fibre architecture in the arterial tissue during plaque progression was found to alter the stress environment in the plaque. This suggests that the specimen-specific anisotropic response of the tissue should be taken into account to accurately predict stresses in the plaque. Since determination of the fibre architecture in vivo is a difficult task, the system presented here provides a useful method of estimating the fibre architecture in complex arterial geometries.     

Biphasic finite element modeling of hydrated soft tissue contact using an augmented Lagrangian method

A study of biphasic soft tissues contact is fundamental to understanding the biomechanical behavior of human diarthrodial joints. To date, biphasic-biphasic contact has been developed for idealized geometries and not been accessible for more general geometries. In this paper a finite element formulation is developed for contact of biphasic tissues. The augmented Lagrangian method is used to enforce the continuity of contact traction and fluid pressure across the contact interface, and the resulting method is implemented in the commercial software COMSOL Multiphysics. The accuracy of the implementation is verified using 2D axisymmetric problems, including indentation with a flat-ended indenter, indentation with spherical-ended indenter, and contact of glenohumeral cartilage layers. The biphasic finite element contact formulation and its implementation are shown to be robust and able to handle physiologically relevant problems.     

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.     

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.     

Subendocardial and subepicardial pressure-flow relations in the rat heart in diastolic and systolic arrest

Ischemic heart disease is more apparent in the subendocardial than in subepicardial layers. We investigated coronary pressure-flow relations in layers of the isolated rat left ventricle, using 15 microm microspheres during diastolic and systolic arrest in the vasodilated coronary circulation. A special cannula allowed for selective determination of left main stem pressure-flow relations. Arterio-venous shunt flow was derived from microspheres in the venous effluent. We quantitatively investigated the pressure-flow relations in diastolic arrest (n=8), systolic arrest at normal contractility (n=8) and low contractility (n=6). In all three groups normal and large ventricular volume was studied. In diastolic arrest, at a perfusion pressure of 90 mmHg, subendocardial flow is larger than subepicardial flow, i.e., the endo/epi ratio is approximately 1.2. In systolic arrest the endo/epi ratio is approximately 0.3, and subendocardial flow and subepicardial flow are approximately 12% and approximately 55% of their values during diastolic arrest. The endo/epi ratio in diastolic arrest decreases with increasing perfusion pressure, while in systole the ratio increases. The slope of the pressure-flow relations, i.e., inverse of resistance, changes by a factor of approximately 5.3 in the subendocardium and by a factor approximately 2.2 in the subepicardium from diastole to systole. Lowering contractility affects subendocardial flow more than subepicardial flow, but both contractility and ventricular volume changes have only a limited effect on both subendocardial and subepicardial flow. The resistance (inverse of slope) of the total left main stem pressure-flow relation changes by a factor of approximately 3.4 from diastolic to systolic arrest. The zero-flow pressure increases from diastole to systole. Thus, coronary perfusion flow in diastolic arrest is larger than systolic arrest, with the largest difference in the subendocardium, as a result of layer dependent increases in vascular resistance and intercept pressure. Shunt flow is larger in diastolic than in systolic arrest, and increases with perfusion pressure. We conclude that changes in contractility and ventricular volume have a smaller effect on pressure-flow relations than diastolic-systolic differences. A synthesis of models accounting for the effect of cardiac contraction on perfusion is suggested.     

An Analysis and Comparison of Convergence and Uniqueness of Time-Independent Bone Adaptation Models

A recently presented solution method for the bidomain model (Johnston et al. 2006), which involves the application of direct current for studying electrical potential in a slab of cardiac tissue, is extended here to allow the use of an applied alternating current. The advantage of using AC current, in a four-electrode method for determining cardiac conductivities, is that instead of using ‘close’ and ‘wide’ electrode spacings to make potential measurements, increasing the frequency of the AC current redirects a fraction of the current from the extracellular space into the intracellular space.

The model is based on the work of Le Guyader et al. (2001), but is able to include the effects of the fibre rotation between the epicardium and the endocardium on the potentials. Also, rather than using a full numerical technique, the solution method uses Fourier series and a simple one dimensional finite difference scheme, which has the advantage of allowing the potentials to be calculated only at points, such as the measuring electrodes, where they are required.

The new alternating current model, which includes intracellular capacitance, is used with a particular four-electrode configuration, to show that the potential measured is affected by changes in fibre rotation. This is significant because it indicates that it is necessary to include fibre rotation in models, which are to be used in conjunction with measuring arrays that are more complex than those involving simply surface probes or a single vertical probe.     

A solution method for the determination of cardiac potential distributions with an alternating current source

A recently presented solution method for the bidomain model (Johnston et al. 2006), which involves the application of direct current for studying electrical potential in a slab of cardiac tissue, is extended here to allow the use of an applied alternating current. The advantage of using AC current, in a four-electrode method for determining cardiac conductivities, is that instead of using 'close' and 'wide' electrode spacings to make potential measurements, increasing the frequency of the AC current redirects a fraction of the current from the extracellular space into the intracellular space. The model is based on the work of Le Guyader et al. (2001), but is able to include the effects of the fibre rotation between the epicardium and the endocardium on the potentials. Also, rather than using a full numerical technique, the solution method uses Fourier series and a simple one dimensional finite difference scheme, which has the advantage of allowing the potentials to be calculated only at points, such as the measuring electrodes, where they are required. The new alternating current model, which includes intracellular capacitance, is used with a particular four-electrode configuration, to show that the potential measured is affected by changes in fibre rotation. This is significant because it indicates that it is necessary to include fibre rotation in models, which are to be used in conjunction with measuring arrays that are more complex than those involving simply surface probes or a single vertical probe.     

A new approach to the determination of cardiac potential distributions: application to the analysis of electrode configurations

This paper presents a mathematical model and new solution technique for studying the electric potential in a slab of cardiac tissue. The model is based on the bidomain representation of cardiac tissue and also allows for the effects of fibre rotation between the epicardium and the endocardium. A detailed solution method, based on Fourier Series and a simple one-dimensional finite difference scheme, for the governing equations for electric potential in the tissue and the blood, is also presented. This method has the advantage that the potential can be calculated only at points where it is required, such as the measuring electrodes. The model is then used to study various electrode configurations which have been proposed to determine cardiac tissue conductivity parameters. Three electrode configurations are analysed in terms of electrode spacing, placement position and the effect of including fibre rotation: the usual surface four-electrode configuration; a single vertical analogue of this and a two probe configuration, which has the current electrodes on one probe and the measuring electrodes on the other, a fixed distance away. It is found that including fibre rotation has no effect on the potentials measured in the first two cases; however, in the two probe case, non-zero fibre rotation causes a significant drop in the voltage measured. This leads to the conclusion that it is necessary to include the effects of fibre rotation in any model which involves the use of multiple plunge electrodes.