Fast Simulation of Mechanical Heterogeneity in the Electrically Asynchronous Heart Using the MultiPatch Module

Cardiac electrical asynchrony occurs as a result of cardiac pacing or conduction disorders such as left bundle-branch block (LBBB). Electrically asynchronous activation causes myocardial contraction heterogeneity that can be detrimental for cardiac function. Computational models provide a tool for understanding pathological consequences of dyssynchronous contraction. Simulations of mechanical dyssynchrony within the heart are typically performed using the finite element method, whose computational intensity may present an obstacle to clinical deployment of patient-specific models. We present an alternative based on the CircAdapt lumped-parameter model of the heart and circulatory system, called the MultiPatch module. Cardiac walls are subdivided into an arbitrary number of patches of homogeneous tissue. Tissue properties and activation time can differ between patches. All patches within a wall share a common wall tension and curvature. Consequently, spatial location within the wall is not required to calculate deformation in a patch. We test the hypothesis that activation time is more important than tissue location for determining mechanical deformation in asynchronous hearts. We perform simulations representing an experimental study of myocardial deformation induced by ventricular pacing, and a patient with LBBB and heart failure using endocardial recordings of electrical activation, wall volumes, and end-diastolic volumes. Direct comparison between simulated and experimental strain patterns shows both qualitative and quantitative agreement between model fibre strain and experimental circumferential strain in terms of shortening and rebound stretch during ejection. Local myofibre strain in the patient simulation shows qualitative agreement with circumferential strain patterns observed in the patient using tagged MRI. We conclude that the MultiPatch module produces realistic regional deformation patterns in the asynchronous heart and that activation time is more important than tissue location within a wall for determining myocardial deformation. The CircAdapt model is therefore capable of fast and realistic simulations of dyssynchronous myocardial deformation embedded within the closed-loop cardiovascular system.     

Quantitative Study of the Effect of Tissue Microstructure on Contraction in a Computational Model of Rat Left Ventricle

Tissue microstructure, in particular the alignment of myocytes (fibre direction) and their lateral organisation into sheets, is fundamental to cardiac function. We studied the effect of microstructure on contraction in a computational model of rat left ventricular electromechanics. Different fibre models, globally rule-based or locally optimised to DT-MRI data, were compared, in order to understand whether a subject-specific fibre model would enhance the predictive power of our model with respect to the global ones. We also studied the impact of sheets on ventricular deformation by comparing: (a) a transversely isotropic versus an orthotropic material law and (b) a linear model with a bimodal model of sheet transmural variation. We estimated ejection fraction, wall thickening and base-to-apex shortening and compared them with measures from cine-MRI. We also evaluated Lagrangian strains as local metrics of cardiac deformation. Our results show that the subject-specific fibre model provides little improvement in the metric predictions with respect to global fibre models while material orthotropy allows closer agreement with measures than transverse isotropy. Nonetheless, the impact of sheets in our model is smaller than that of fibres. We conclude that further investigation of the modelling of sheet dynamics is necessary to fully understand the impact of tissue structure on cardiac deformation.     

Pump function of the heart as an optimal control problem

In order to model the pump function of the heart the left ventricle is represented as an elastic thick-walled cylinder contracting symmetrically. The acceleration is included in the mathematical formalism describing the contraction of the myocardium and optimal control theory is used to solve the differential equation of motion of the cylindrical wall in such a way as to minimize a given performance index. Application of the equations to experimental data published in the literature is discussed. The mathematical formalism presents a new way to study the time variation of the volume ejected from the left ventricle. Methods to quantify the pump function of the heart are suggested.     

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.     

Wave intensity analysis of left atrial mechanics and energetics in anesthetized dogs

The left atrium (LA) acts as a booster pump during late diastole, generating the Doppler transmitral A wave and contributing incrementally to left ventricular (LV) filling. However, after volume loading and in certain disease states, LA contraction fills the LV less effectively, and retrograde flow (i.e., the Doppler Ar wave) into the pulmonary veins increases. The purpose of this study was to provide an energetic analysis of LA contraction to clarify the mechanisms responsible for changes in forward and backward flow. Wave intensity analysis was performed at the mitral valve and a pulmonary vein orifice. As operative LV stiffness increased with progressive volume loading, the reflection coefficient (i.e., energy of reflected wave/energy of incident wave) also increased. This reflected wave decelerated the forward movement of blood through the mitral valve and was transmitted through the LA, accelerating retrograde blood flow in the pulmonary veins. Although total LA work increased with volume loading, the forward hydraulic work decreased and backward hydraulic work increased. Thus wave reflection due to increased LV stiffness accounts for the decrease in the A wave and the increase in the Ar wave measured by Doppler.     

Computational modeling of left heart diastolic function: examination of ventricular dysfunction

A computational model that accounts for blood-tissue interaction under physiological flow conditions was developed and applied to a thin-walled model of the left heart. This model consisted of the left ventricle, left atrium, and pulmonary vein flow. The input functions for the model included the pulmonary vein driving pressure and time-dependent relationship for changes in chamber tissue properties during the simulation. The Immersed Boundary Method was used for the interaction of the tissue and blood in response to fluid forces and changes in tissue pathophysiology, and the fluid mass and momentum conservation equations were solved using Patankar's Semi-Implicit Method for Pressure Linked Equations (SIMPLE). This model was used to examine the flow fields in the left heart under abnormal diastolic conditions of delayed ventricular relaxation, delayed ventricular relaxation with increased ventricular stiffness, and delayed ventricular relaxation with an increased atrial contraction. The results obtained from the left heart model were compared to clinically observed diastolic flow conditions, and to the results from simulations of normal diastolic function in this model [1]. Cases involving impairment of diastolic function were modeled with changes to the input functions for fiber relaxation/contraction of the chambers. The three cases of diastolic dysfunction investigated agreed with the changes in diastolic flow fields seen clinically. The effect of delayed relaxation was to decrease the early filling magnitude, and this decrease was larger when the stiffness of the ventricle was increased. Also, increasing the contraction of the atrium during atrial systole resulted in a higher late filling velocity and atrial pressure. The results show that dysfunction can be modeled by changing the relationships for fiber resting-length and/or stiffness. This provides confidence in future modeling of disease, especially changes to chamber properties to examine the effect of local dysfunction on global flow fields.     

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.     

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.     

A Tissue-Level Electromechanical Model of the Left Ventricle: Application to the Analysis of Intraventricular Pressure

The ventricular pressure profile is characteristic of the cardiac contraction progress and is useful to evaluate the cardiac performance. In this contribution, a tissue-level electromechanical model of the left ventricle is proposed, to assist the interpretation of left ventricular pressure waveforms. The left ventricle has been modeled as an ellipsoid composed of twelve mechano-hydraulic sub-systems. The asynchronous contraction of these twelve myocardial segments has been represented in order to reproduce a realistic pressure profiles. To take into account the different energy domains involved, the tissue-level scale and to facilitate the building of a modular model, multiple formalisms have been used: Bond Graph formalism for the mechano-hydraulic aspects and cellular automata for the electrical activation. An experimental protocol has been defined to acquire ventricular pressure signals from three pigs, with different afterload conditions. Evolutionary Algorithms have been used to identify the model parameters in order to minimize the error between experimental and simulated ventricular pressure signals. Simulation results show that the model is able to reproduce experimental ventricular pressure. In addition, electro-mechanical activation times have been determined in the identification process. For example, the maximum electrical activation time is reached, respectively, 96.5, 139.3 and 131.5 ms for the first, second, and third pigs. These preliminary results are encouraging for the application of the model on non-invasive data like ECG, arterial pressure or myocardial strain.     

Effect of ectopic excitation on the ventricular pump function in chicken

The pump function of the heart ventricles was studied in chest-open anaesthetized adult female chickens under sinus rhythm and ectopic excitation of different localization. The intraventricular pressure in the right and left heart ventricles was measured by insertion of catheters through the ventricular free walls. Maximum systolic pressure, end-diastolic pressure, contractility (dP/dtmax) and relaxation (dP/dtmin) of both heart ventricles, and duration of the asynchronous contraction time of the left ventricle were analyzed. It was revealed that reduction of the pump function of the left ventricle tends to be greater under right ventricular ectopic excitation compared with left ventricular one. In comparison with the sinus rhythm, the pump function of the right ventricle was preserved to a greater extent under stimulation of the left ventricular apex and was significantly impaired under right ventricular ectopic excitation. Relaxation of both heart ventricles was more susceptible to ventricular ectopic excitation than contractility, and was more vulnerable in the right ventricle than in the left one. The direction of changes of the pump function of the heart ventricles in chickens under ventricular ectopic excitation was similar to changes of the pump function of mammalian hearts.     

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.     

The dynamics of contraction and walls motion of the calf heart left ventricle

The echocardiographic research of the left ventricular has revealed heterogeneity of thickness of the posterior wall and interventricular septum in three parallel planes in the transverse direction of the left ventricle in calves. The amplitude of systolic motion of the left ventricle posterior wall is larger than that of the interventricular septum at the level of the mitral valve, at the level of the papillary muscles, and at the apical level. The excursion of left ventricular walls in the basal level is twice as large as the mobility of ventricular walls in the apical level. During the contraction of the myocardium, the shortness of the left ventricular transversal diameter is to great extent determined by the degree of contraction of the left ventricular wall rather than of the interventricular septum. The high contractility is revealed in calves.     

PIV measurements of flow in a centrifugal blood pump: time-varying flow

Measurements of the time-varying flow in a centrifugal blood pump operating as a left ventricular assist device (LVAD) are presented. This includes changes in both the pump flow rate as a function of the left ventricle contraction and the interaction of the rotating impeller and fixed exit volute. When operating with a pulsing ventricle, the flow rate through the LVAD varies from 0-11 L/min during each cycle of the heartbeat. Phase-averaged measurements of mean velocity and some turbulence statistics within several regions of the pump, including the inlet, blade passage, exit volute, and diffuser, are reported at 20 phases of the cardiac cycle. The transient flow fields are compared to the constant flow rate condition that was reported previously in order to investigate the transient effects within the pump. It is shown that the quasi-steady assumption is a fair treatment of the time varying flow field in all regions of this representative pump, which greatly simplifies the comprehension and modeling of this flow field. The measurements are further interpreted to identify the effects that the transient nature of the flow field will have on blood damage. Although regions of recirculation and stagnant flow exist at some phases of the cardiac cycle, there is no location where flow is stagnant during the entire heartbeat.     

Modeling flow in collecting lymphatic vessels: one-dimensional flow through a series of contractile elements

The lymphatic system comprises a series of elements, lymphangions, separated by valves and possessed of active, contractile walls to pump interstitial fluid from its collection in the terminal lymphatics back to the main circulation. Despite its importance, there is a dearth of information on the fluid dynamics of the lymphatic system. In this article, we describe linked experimental and computational work aimed at elucidating the biomechanical properties of the individual lymphangions. We measure the static and dynamic mechanical properties of excised bovine collecting lymphatics and develop a one-dimensional computational model of the coupled fluid flow/wall motion. The computational model is able to reproduce the pumping behavior of the real vessel using a simple contraction function producing fast contraction pulses traveling in the retrograde direction to the flow.     

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.     

Models of cardiac electromechanics based on individual hearts imaging data Image-based electromechanical models of the heart

Current multi-scale computational models of ventricular electromechanics describe the full process of cardiac contraction on both the micro- and macro- scales including: the depolarization of cardiac cells, the release of calcium from intracellular stores, tension generation by cardiac myofilaments, and mechanical contraction of the whole heart. Such models are used to reveal basic mechanisms of cardiac contraction as well as the mechanisms of cardiac dysfunction in disease conditions. In this paper, we present a methodology to construct finite element electromechanical models of ventricular contraction with anatomically accurate ventricular geometry based on magnetic resonance and diffusion tensor magnetic resonance imaging of the heart. The electromechanical model couples detailed representations of the cardiac cell membrane, cardiac myofilament dynamics, electrical impulse propagation, ventricular contraction, and circulation to simulate the electrical and mechanical activity of the ventricles. The utility of the model is demonstrated in an example simulation of contraction during sinus rhythm using a model of the normal canine ventricles.     

Contribution of myocardium hydraulic skeleton to left ventricular wall interaction and synergy in dogs

The most premature motion change after coronary occlusion is early diastolic thinning of the ischemic left ventricular (LV) wall, with concomitant thickening of the normoperfused wall. We aimed 1). to demonstrate that these early changes are the result of the absence of fluid within the ischemic myocardium (hydraulic skeleton) rather than to cell anoxia and 2). to quantitate the contribution of the lack of hydraulic skeleton to left ventricular asynergy of contraction in seven anesthetized dogs submitted to acute, short-lasting circumflex artery (Cx) occlusion (ischemia) and to perfusion of the Cx with an oxygen-free solution (anoxia). We analyzed the time course of regional work index (WI, area of the LV pressure-wall thickness loop) and regional efficiency (defined as the ratio of WI to the maximum possible work). Interwall asynergy was defined as the difference between the regional efficiency of the anterior and posterior walls. After 9-10 s, posterior wall efficiency decreased 37 +/- 6% with anoxia and 72 +/- 3% with ischemia (P < 0.025), and interwall asynergy was 0 +/- 6% with anoxia and 32 +/- 5% with ischemia (P < 0.05). The contribution of absent hydraulic skeleton to interwall asynergy (calculated as the difference between %asynergy in anoxia and %asynergy in ischemia) was 30 +/- 8% (P < 0.05). In conclusion, the earliest wall motion change observed after acute coronary occlusion, namely ischemic wall thinning concomitant with normoperfused wall thickening during isovolumic relaxation, is the result of the absence of intracoronary fluid. The lack of hydraulic skeleton within the myocardium contributes approximately 30% to interwall asynergy.     

Right and left ventricular wall deformation patterns in normal and left heart hypoplasia chick embryos

The vertebrate embryonic ventricle transforms from a smooth-walled single tube to trabeculated right ventricular (RV) and left ventricular (LV) chambers during cardiovascular morphogenesis. We hypothesized that ventricular contraction patterns change from globally isotropic to chamber-specific anisotropic patterns during normal morphogenesis and that these deformation patterns are influenced by experimentally altered mechanical load produced by chronic left atrial ligation (LAL). We measured epicardial RV and LV wall strains during normal development and left heart hypoplasia produced by LAL in Hamburger-Hamilton stage 21, 24, 27, and 31 chick embryos. Normal RV contracted isotropically until stage 24 and then contracted preferentially in the circumferential direction. Normal LV contracted isotropically at stage 21, preferentially in the longitudinal direction at stages 24 and 27, and then in the circumferential direction at stage 31. LAL altered both RV and LV strain patterns, accelerated the onset of preferential RV circumferential strain patterns, and abolished preferential LV longitudinal strain (P < 0.05 vs. normal). Mature patterns of anisotropic RV and LV deformation develop coincidentally with morphogenesis, and changes in these deformation patterns reflect altered cardiovascular function and/or morphogenesis.     

Tank-treading dynamics of red blood cells in shear flow: On the membrane viscosity rheology

In this article, extensive three-dimensional simulations are conducted for tank-treading (TT) red blood cells (RBCs) in shear flow with different cell viscous properties and flow conditions. Apart from recent numerical studies on TT RBCs, this research considers the uncertainty in cytoplasm viscosity, covers a more complete range of shear flow situations of available experiments, and examines the TT behaviors in more details. Key TT characteristics, including the rotation frequency, deformation index, and inclination angle, are compared with available experimental results of similar shear flow conditions. Fairly good simulation-experiment agreements for these parameters can be obtained by adjusting the membrane viscosity values; however, different rheological relationships between the membrane viscosity and the flow shear rate are noted for these comparisons: shear thinning from the TT frequency, Newtonian from the inclination angle, and shear thickening from the cell deformation. Previous studies claimed a shear-thinning membrane viscosity model based on the TT frequency results; however, such a conclusion seems premature from our results and more carefully designed and better controlled investigations are required for the RBC membrane rheology. In addition, our simulation results reveal complicate RBC TT features and such information could be helpful for a better understanding of in vivo and in vitro RBC dynamics.     

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.