An improved numerical method for strong coupling of excitation and contraction models in the heart

Numerical evaluation of cardiac mechanical markers as estimators of the electrical activation time

Recent advances in the development of noninvasive cardiac imaging technologies have made it possible to measure longitudinal and circumferential strains at a high spatial resolution also at intramural level. Local mechanical activation times derived from these strains can be used as noninvasive estimates of electrical activation, in order to determine, eg, the origin of premature ectopic beats during focal arrhythmias or the pathway of reentrant circuits. The aim of this work is to assess the reliability of mechanical activation time markers derived from longitudinal and circumferential strains, denoted by AT_ell and AT_ecc , respectively, by means of three-dimensional cardiac electromechanical simulations. These markers are compared against the electrical activation time (AT_v ), computed from the action potential waveform, and the reference mechanical activation markers derived from the active tension and fiber strain waveforms, denoted by AT_ta and AT_eff , respectively. Our numerical simulations are based on a strongly coupled electromechanical model, including bidomain representation of the cardiac tissue, mechanoelectric (ie, stretch-activated channels) and geometric feedbacks, transversely isotropic strain energy function for the description of passive mechanics and detailed membrane and excitation-contraction coupling models. The results have shown that, during endocardial and epicardial ectopic stimulations, all the mechanical markers considered are highly correlated with AT_v , exhibiting correlation coefficients larger than 0.8. However, during multiple endocardial stimulations, mimicking the ventricular sinus rhythm, the mechanical markers are less correlated with the electrical activation time, because of the more complex resulting excitation sequence. Moreover, the inspection of the endocardial and epicardial isochrones has shown that the AT_ell and AT_ecc mechanical activation sequences reproduce only some qualitative features of the electrical activation sequence, such as the areas of early and late activation, but in some cases, they might yield wrong excitation sources and significantly different isochrones patterns.

Multiphysics computational modelling of the cardiac ventricles

Development of cardiac multiphysics models has progressed significantly over the decades and simulations combining multiple physics interactions have become increasingly common. In this review, we summarise the progress in this field focusing on various approaches of integrating ventricular structures. electrophysiological properties, myocardial mechanics, as well as incorporating blood hemodynamics and the circulatory system. Common coupling approaches are discussed and compared, including the advantages and shortcomings of each. Currently used strategies for patient-specific implementations are highlighted and potential future improvements considered.

Sensitivity analysis of a strongly-coupled human-based electromechanical cardiac model: Effect of mechanical parameters on physiologically relevant biomarkers

The human heart beats as a result of multiscale nonlinear dynamics coupling subcellular to whole organ processes, achieving electrophysiologically-driven mechanical contraction. Computational cardiac modelling and simulation have achieved a great degree of maturity, both in terms of mathematical models of underlying biophysical processes and the development of simulation software. In this study, we present the detailed description of a human-based physiologically-based, and fully-coupled ventricular electromechanical modelling and simulation framework, and a sensitivity analysis focused on its mechanical properties. The biophysical detail of the model, from ionic to whole-organ, is crucial to enable future simulations of disease and drug action. Key novelties include the coupling of state-of-the-art human-based electrophysiology membrane kinetics, excitation-contraction and active contraction models, and the incorporation of a pre-stress model to allow for pre-stressing and pre-loading the ventricles in a dynamical regime. Through high performance computing simulations, we demonstrate that 50% to 200% - 1000% variations in key parameters result in changes in clinically-relevant mechanical biomarkers ranging from diseased to healthy values in clinical studies. Furthermore mechanical biomarkers are primarily affected by only one or two parameters. Specifically, ejection fraction is dominated by the scaling parameter of the active tension model and its scaling parameter in the normal direction ( k ort 2 ); the end systolic pressure is dominated by the pressure at which the ejection phase is triggered ( P ej ) and the compliance of the Windkessel fluid model ( C ); and the longitudinal fractional shortening is dominated by the fibre angle ( ϕ ) and k ort 2 . The wall thickening does not seem to be clearly dominated by any of the considered input parameters. In summary, this study presents in detail the description and implementation of a human-based coupled electromechanical modelling and simulation framework, and a high performance computing study on the sensitivity of mechanical biomarkers to key model parameters. The tools and knowledge generated enable future investigations into disease and drug action on human ventricles.

GEMS: A Fully Integrated PETSc-Based Solver for Coupled Cardiac Electromechanics and Bidomain Simulations

Cardiac contraction is coordinated by a wave of electrical excitation which propagates through the heart. Combined modeling of electrical and mechanical function of the heart provides the most comprehensive description of cardiac function and is one of the latest trends in cardiac research. The effective numerical modeling of cardiac electromechanics remains a challenge, due to the stiffness of the electrical equations and the global coupling in the mechanical problem. Here we present a short review of the inherent assumptions made when deriving the electromechanical equations, including a general representation for deformation-dependent conduction tensors obeying orthotropic symmetry, and then present an implicit-explicit time-stepping approach that is tailored to solving the cardiac mono- or bidomain equations coupled to electromechanics of the cardiac wall. Our approach allows to find numerical solutions of the electromechanics equations using stable and higher order time integration. Our methods are implemented in a monolithic finite element code GEMS (Ghent Electromechanics Solver) using the PETSc library that is inherently parallelized for use on high-performance computing infrastructure. We tested GEMS on standard benchmark computations and discuss further development of our software.

A Multiphysics Biventricular Cardiac Model: Simulations With a Left-Ventricular Assist Device

Computational models have become essential in predicting medical device efficacy prior to clinical studies. To investigate the performance of a left-ventricular assist device (LVAD), a fully-coupled cardiac fluid-electromechanics finite element model was developed, incorporating electrical activation, passive and active myocardial mechanics, as well as blood hemodynamics solved simultaneously in an idealized biventricular geometry. Electrical activation was initiated using a simplified Purkinje network with one-way coupling to the surrounding myocardium. Phenomenological action potential and excitation-contraction equations were adapted to trigger myocardial contraction. Action potential propagation was formulated within a material frame to emulate gap junction-controlled propagation, such that the activation sequence was independent of myocardial deformation. Passive cardiac mechanics were governed by a transverse isotropic hyperelastic constitutive formulation. Blood velocity and pressure were determined by the incompressible Navier-Stokes formulations with a closed-loop Windkessel circuit governing the circulatory load. To investigate heart-LVAD interaction, we reduced the left ventricular (LV) contraction stress to mimic a failing heart, and inserted a LVAD cannula at the LV apex with continuous flow governing the outflow rate. A proportional controller was implemented to determine the pump motor voltage whilst maintaining pump motor speed. Following LVAD insertion, the model revealed a change in the LV pressure-volume loop shape from rectangular to triangular. At higher pump speeds, aortic ejection ceased and the LV decompressed to smaller end diastolic volumes. After multiple cycles, the LV cavity gradually collapsed along with a drop in pump motor current. The model was therefore able to predict ventricular collapse, indicating its utility for future development of control algorithms and pre-clinical testing of LVADs to avoid LV collapse in recipients.

Mechanism for Mechanical Wave Break in the Heart Muscle

Using a reaction-diffusion-mechanics model we identify a mechanism for mechanical wave break in the heart muscle. For a wide range of strengths and durations an external mechanical load causes wave front dissipation leading to formation and breakup of spiral waves. We explain the mechanism, and discuss under which conditions it can cause or abolish cardiac arrhythmias.

Effects of mechanical feedback on the stability of cardiac scroll waves: A bidomain electro-mechanical simulation study

In this work, we investigate the influence of cardiac tissue deformation on re-entrant wave dynamics. We have developed a 3D strongly coupled electro-mechanical Bidomain model posed on an ideal monoventricular geometry, including fiber direction anisotropy and stretch-activated currents (SACs). The cardiac mechanical deformation influences the bioelectrical activity with two main mechanical feedback: (a) the geometric feedback (GEF) due to the presence of the deformation gradient in the diffusion coefficients and in a convective term depending on the deformation rate and (b) the mechano-electric feedback (MEF) due to SACs. Here, we investigate the relative contribution of these two factors with respect to scroll wave stability. We extend the previous works [Keldermann et al., Am. J. Physiol. Heart Circ. Physiol. 299, H134-H143 (2010) and Hu et al., PLoS One 8(4), e60287 (2013)] that were based on the Monodomain model and a simple non-selective linear SAC, while here we consider the full Bidomain model and both selective and non-selective components of SACs. Our simulation results show that the stability of cardiac scroll waves is influenced by MEF, which in case of low reversal potential of non-selective SACs might be responsible for the onset of ventricular fibrillation; GEF increases the scroll wave meandering but does not determine the scroll wave stability.

Personalized Computer Simulation of Diastolic Function in Heart Failure

The search for a parameter representing left ventricular relaxation from non-invasive and invasive diagnostic tools has been extensive, since heart failure (HF) with preserved ejection fraction (HF-pEF) is a global health problem. We explore here the feasibility using patient-specific cardiac computer modeling to capture diastolic parameters in patients suffering from different degrees of systolic HF. Fifty eight patients with idiopathic dilated cardiomyopathy have undergone thorough clinical evaluation, including cardiac magnetic resonance imaging (MRI), heart catheterization, echocardiography, and cardiac biomarker assessment. A previously-introduced framework for creating multi-scale patient-specific cardiac models has been applied on all these patients. Novel parameters, such as global stiffness factor and maximum left ventricular active stress, representing cardiac active and passive tissue properties have been computed for all patients. Invasive pressure measurements from heart catheterization were then used to evaluate ventricular relaxation using the time constant of isovolumic relaxation Tau (τ). Parameters from heart catheterization and the multi-scale model have been evaluated and compared to patient clinical presentation. The model parameter global stiffness factor, representing diastolic passive tissue properties, is correlated significantly across the patient population with τ. This study shows that multi-modal cardiac models can successfully capture diastolic (dys) function, a prerequisite for future clinical trials on HF-pEF.

Effects of Persistent Atrial Fibrillation-Induced Electrical Remodeling on Atrial Electro-Mechanics - Insights from a 3D Model of the Human Atria

Aims

Atrial stunning, a loss of atrial mechanical contraction, can occur following a successful cardioversion. It is hypothesized that persistent atrial fibrillation-induced electrical remodeling (AFER) on atrial electrophysiology may be responsible for such impaired atrial mechanics. This simulation study aimed to investigate the effects of AFER on atrial electro-mechanics.

Methods and Results

A 3D electromechanical model of the human atria was developed to investigate the effects of AFER on atrial electro-mechanics. Simulations were carried out in 3 conditions for 4 states: (i) the control condition, representing the normal tissue (state 1) and the tissue 2–3 months after cardioversion (state 2) when the atrial tissue recovers its electrophysiological properties after completion of reverse electrophysiological remodelling; (ii) AFER-SR condition for AF-remodeled tissue with normal sinus rhythm (SR) (state 3); and (iii) AFER-AF condition for AF-remodeled tissue with re-entrant excitation waves (state 4). Our results indicate that at the cellular level, AFER (states 3 & 4) abbreviated action potentials and reduced the Ca²⁺ content in the sarcoplasmic reticulum, resulting in a reduced amplitude of the intracellular Ca²⁺ transient leading to decreased cell active force and cell shortening as compared to the control condition (states 1 & 2). Consequently at the whole organ level, atrial contraction in AFER-SR condition (state 3) was dramatically reduced. In the AFER-AF condition (state 4) atrial contraction was almost abolished.

Conclusions

This study provides novel insights into understanding atrial electro-mechanics illustrating that AFER impairs atrial contraction due to reduced intracellular Ca²⁺ transients.

The relative role of patient physiology and device optimisation in cardiac resynchronisation therapy: A computational modelling study

Cardiac resynchronisation therapy (CRT) is an established treatment for heart failure, however the effective selection of patients and optimisation of therapy remain controversial. While extensive research is ongoing, it remains unclear whether improvements in patient selection or therapy planning offers a greater opportunity for the improvement of clinical outcomes. This computational study investigates the impact of both physiological conditions that guide patient selection and the optimisation of pacing lead placement on CRT outcomes. A multi-scale biophysical model of cardiac electromechanics was developed and personalised to patient data in three patients. These models were separated into components representing cardiac anatomy, pacing lead location, myocardial conductivity and stiffness, afterload, active contraction and conduction block for each individual, and recombined to generate a cohort of 648 virtual patients. The effect of these components on the change in total activation time of the ventricles (ΔTAT) and acute haemodynamic response (AHR) was analysed. The pacing site location was found to have the largest effect on ΔTAT and AHR. Secondary effects on ΔTAT and AHR were found for functional conduction block and cardiac anatomy. The simulation results highlight a need for a greater emphasis on therapy optimisation in order to achieve the best outcomes for patients.

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Ventricular conduction block indicates patient response to CRT.

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Placement of CRT pacing leads strongly affects response to therapy.

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Improved treatment planning should be prioritised in order to maximise CRT outcomes.

The cardiac torsion as a sensitive index of heart pathology: A model study

The torsional behaviour of the heart (i.e. the mutual rotation of the cardiac base and apex) was proved to be sensitive to alterations of some cardiovascular parameters, i.e. preload, afterload and contractility. Moreover, pathologies which affect the fibers architecture and cardiac geometry were proved to alter the cardiac torsion pattern. For these reasons, cardiac torsion represents a sensitive index of ventricular performance. The aim of this work is to provide further insight into physiological and pathological alterations of the cardiac torsion by means of computational analyses, combining a structural model of the two ventricles with simple lumped parameter models of both the systemic and the pulmonary circulations. Starting from diagnostic images, a 3D anatomy based geometry of the two ventricles was reconstructed. The myocytes orientation in the ventricles was assigned according to literature data and the myocardium was modelled as an anisotropic hyperelastic material. Both the active and the passive phases of the cardiac cycle were modelled, and different clinical conditions were simulated. The results in terms of alterations of the cardiac torsion in the presence of pathologies are in agreement with experimental literature data. The use of a computational approach allowed the investigation of the stresses and strains in the ventricular wall as well as of the global hemodynamic parameters in the presence of the considered pathologies. Furthermore, the model outcomes highlight how for specific pathological conditions, an altered torsional pattern of the ventricles can be present, encouraging the use of the ventricular torsion in the clinical practice.

Towards Personalized Cardiology: Multi-Scale Modeling of the Failing Heart

BACKGROUND

Despite modern pharmacotherapy and advanced implantable cardiac devices, overall prognosis and quality of life of HF patients remain poor. This is in part due to insufficient patient stratification and lack of individualized therapy planning, resulting in less effective treatments and a significant number of non-responders.

METHODS AND RESULTS

State-of-the-art clinical phenotyping was acquired, including magnetic resonance imaging (MRI) and biomarker assessment. An individualized, multi-scale model of heart function covering cardiac anatomy, electrophysiology, biomechanics and hemodynamics was estimated using a robust framework. The model was computed on n=46 HF patients, showing for the first time that advanced multi-scale models can be fitted consistently on large cohorts. Novel multi-scale parameters derived from the model of all cases were analyzed and compared against clinical parameters, cardiac imaging, lab tests and survival scores to evaluate the explicative power of the model and its potential for better patient stratification. Model validation was pursued by comparing clinical parameters that were not used in the fitting process against model parameters.

CONCLUSION

This paper illustrates how advanced multi-scale models can complement cardiovascular imaging and how they could be applied in patient care. Based on obtained results, it becomes conceivable that, after thorough validation, such heart failure models could be applied for patient management and therapy planning in the future, as we illustrate in one patient of our cohort who received CRT-D implantation.

Control of cardiac alternans in an electromechanical model of cardiac tissue

Electrical alternations in cardiac action potential duration have been shown to be a precursor to arrhythmias and sudden cardiac death. Through the mechanism of excitation-contraction coupling, the presence of electrical alternans induces alternations in the heart muscle contractile activity. Also, contraction of cardiac tissue affects the process of cardiac electric wave propagation through the mechanism of the so-called mechanoelectrical feedback. Electrical excitation and contraction of cardiac tissue can be linked by an electromechanical model such as the Nash-Panfilov model. In this work, we explore the feasibility of suppressing cardiac alternans in the Nash-Panfilov model which is employed for small and large deformations. Several electrical pacing and mechanical perturbation feedback strategies are considered to demonstrate successful suppression of alternans on a one-dimensional cable. This is the first attempt to combine electrophysiologically relevant cardiac models of electrical wave propagation and contractility of cardiac tissue in a synergistic effort to suppress cardiac alternans. Numerical examples are provided to illustrate the feasibility and the effects of the proposed algorithms to suppress cardiac alternans.

Application of the Kalman Filter for Faster Strong Coupling of Cardiovascular Simulations

In this paper, we propose a method for reducing the computational cost of strong coupling for multiscale cardiovascular simulation models. In such a model, individual model modules of myocardial cell, left ventricular structural dynamics, and circulatory hemodynamics are coupled. The strong coupling method enables stable and accurate calculation, but requires iterative calculations which are computationally expensive. The iterative calculations can be reduced, if accurate initial approximations are made available by predictors. The proposed method uses the Kalman filter to estimate accurate predictions by filtering out noise included in past values. The performance of the proposed method was assessed with an application to a previously published multiscale cardiovascular model. The proposed method reduced the number of iterations by 90% and 62% compared with no prediction and Lagrange extrapolation, respectively. Even when the parameters were varied and number of elements of the left ventricular finite-element model increased, the number of iterations required by the proposed method was significantly lower than that without prediction. These results indicate the robustness, scalability, and validity of the proposed method.

Abnormal calcium homeostasis in heart failure with preserved ejection fraction is related to both reduced contractile function and incomplete relaxation: an electromechanically detailed biophysical modeling study

Heart failure with preserved ejection fraction (HFpEF) accounts for about 50% of heart failure cases. It has features of incomplete relaxation and increased stiffness of the left ventricle. Studies from clinical electrophysiology and animal experiments have found that HFpEF is associated with impaired calcium homeostasis, ion channel remodeling and concentric left ventricle hypertrophy (LVH). However, it is still unclear how the abnormal calcium homeostasis, ion channel and structural remodeling affect the electro-mechanical dynamics of the ventricles. In this study we have developed multiscale models of the human left ventricle from single cells to the 3D organ, which take into consideration HFpEF-induced changes in calcium handling, ion channel remodeling and concentric LVH. Our simulation results suggest that at the cellular level, HFpEF reduces the systolic calcium level resulting in a reduced systolic contractile force, but elevates the diastolic calcium level resulting in an abnormal residual diastolic force. In our simulations, these abnormal electro-mechanical features of the ventricular cells became more pronounced with the increase of the heart rate. However, at the 3D organ level, the ejection fraction of the left ventricle was maintained due to the concentric LVH. The simulation results of this study mirror clinically observed features of HFpEF and provide new insights toward the understanding of the cellular bases of impaired cardiac electromechanical functions in heart failure.

The Generalized Hill Model: A Kinematic Approach Towards Active Muscle Contraction

Excitation-contraction coupling is the physiological process of converting an electrical stimulus into a mechanical response. In muscle, the electrical stimulus is an action potential and the mechanical response is active contraction. The classical Hill model characterizes muscle contraction though one contractile element, activated by electrical excitation, and two non-linear springs, one in series and one in parallel. This rheology translates into an additive decomposition of the total stress into a passive and an active part. Here we supplement this additive decomposition of the stress by a multiplicative decomposition of the deformation gradient into a passive and an active part. We generalize the one-dimensional Hill model to the three-dimensional setting and constitutively define the passive stress as a function of the total deformation gradient and the active stress as a function of both the total deformation gradient and its active part. We show that this novel approach combines the features of both the classical stress-based Hill model and the recent active-strain models. While the notion of active stress is rather phenomenological in nature, active strain is micro-structurally motivated, physically measurable, and straightforward to calibrate. We demonstrate that our model is capable of simulating excitation-contraction coupling in cardiac muscle with its characteristic features of wall thickening, apical lift, and ventricular torsion.

Fibroblast proliferation alters cardiac excitation conduction and contraction: a computational study

In this study, the effects of cardiac fibroblast proliferation on cardiac electric excitation conduction and mechanical contraction were investigated using a proposed integrated myocardial-fibroblastic electromechanical model. At the cellular level, models of the human ventricular myocyte and fibroblast were modified to incorporate a model of cardiac mechanical contraction and cooperativity mechanisms. Cellular electromechanical coupling was realized with a calcium buffer. At the tissue level, electrical excitation conduction was coupled to an elastic mechanics model in which the finite difference method (FDM) was used to solve electrical excitation equations, and the finite element method (FEM) was used to solve mechanics equations. The electromechanical properties of the proposed integrated model were investigated in one or two dimensions under normal and ischemic pathological conditions. Fibroblast proliferation slowed wave propagation, induced a conduction block, decreased strains in the fibroblast proliferous tissue, and increased dispersions in depolarization, repolarization, and action potential duration (APD). It also distorted the wave-front, leading to the initiation and maintenance of re-entry, and resulted in a sustained contraction in the proliferous areas. This study demonstrated the important role that fibroblast proliferation plays in modulating cardiac electromechanical behaviour and which should be considered in planning future heart-modeling studies.

Modeling the dispersion in electromechanically coupled myocardium

We present an approach to model the dispersion of fiber and sheet orientations in the myocardium. By utilizing structure parameters, an existing orthotropic and invariant-based constitutive model developed to describe the passive behavior of the myocardium is augmented. Two dispersion parameters are fitted to experimentally observed angular dispersion data of the myocardial tissue. Computations are performed on a unit myocardium tissue cube and on a slice of the left ventricle indicating that the dispersion parameter has an effect on the myocardial deformation and stress development. The use of fiber dispersions relating to a pathological myocardium had a rather big effect. The final example represents an ellipsoidal model of the left ventricle indicating the influence of fiber and sheet dispersions upon contraction over a cardiac cycle. Although only a minor shift in the pressure-volume (PV) loops between the cases with no dispersions and with fiber and sheet dispersions for a healthy myocardium was observed, a remarkably different behavior is obtained with a fiber dispersion relating to a diseased myocardium. In future simulations, this dispersion model for myocardial tissue may advantageously be used together with models of, for example, growth and remodeling of various cardiac diseases.

Computational modeling of chemo-electro-mechanical coupling: a novel implicit monolithic finite element approach

Computational modeling of the human heart allows us to predict how chemical, electrical, and mechanical fields interact throughout a cardiac cycle. Pharmacological treatment of cardiac disease has advanced significantly over the past decades, yet it remains unclear how the local biochemistry of an individual heart cell translates into global cardiac function. Here, we propose a novel, unified strategy to simulate excitable biological systems across three biological scales. To discretize the governing chemical, electrical, and mechanical equations in space, we propose a monolithic finite element scheme. We apply a highly efficient and inherently modular global-local split, in which the deformation and the transmembrane potential are introduced globally as nodal degrees of freedom, whereas the chemical state variables are treated locally as internal variables. To ensure unconditional algorithmic stability, we apply an implicit backward Euler finite difference scheme to discretize the resulting system in time. To increase algorithmic robustness and guarantee optimal quadratic convergence, we suggest an incremental iterative Newton-Raphson scheme. The proposed algorithm allows us to simulate the interaction of chemical, electrical, and mechanical fields during a representative cardiac cycle on a patient-specific geometry, robust and stable, with calculation times on the order of 4 days on a standard desktop computer.

Excitation-contraction coupling between human atrial myocytes with fibroblasts and stretch activated channel current: a simulation study

Myocytes have been regarded as the main objectives in most cardiac modeling studies and attracted a lot of attention. Connective tissue cells, such as fibroblasts (Fbs), also play crucial role in cardiac function. This study proposed an integrated myocyte-I_sac-Fb electromechanical model to investigate the effect of Fbs and stretch activated ion channel current (I_sac) on cardiac electrical excitation conduction and mechanical contraction. At the cellular level, an active Fb model was coupled with a human atrial myocyte electrophysiological model (including I_sac) and a mechanical model. At the tissue level, electrical excitation conduction was coupled with an elastic mechanical model, in which finite difference method (FDM) was used to solve the electrical excitation equations, while finite element method (FEM) was used for the mechanics equations. The simulation results showed that Fbs and I_sac coupling caused diverse effects on action potential morphology during repolarization, depolarized the resting membrane potential of the human atrial myocyte, slowed down wave propagation, and decreased strains in fibrotic tissue. This preliminary simulation study indicates that Fbs and I_sac have important implications for modulating cardiac electromechanical behavior and should be considered in future cardiac modeling studies.

In silico investigation of the short QT syndrome, using human ventricle models incorporating electromechanical coupling

INTRODUCTION

Genetic forms of the Short QT Syndrome (SQTS) arise due to cardiac ion channel mutations leading to accelerated ventricular repolarization, arrhythmias and sudden cardiac death. Results from experimental and simulation studies suggest that changes to refractoriness and tissue vulnerability produce a substrate favorable to re-entry. Potential electromechanical consequences of the SQTS are less well-understood. The aim of this study was to utilize electromechanically coupled human ventricle models to explore electromechanical consequences of the SQTS.

METHODS AND RESULTS

The Rice et al. mechanical model was coupled to the ten Tusscher et al. ventricular cell model. Previously validated K(+) channel formulations for SQT variants 1 and 3 were incorporated. Functional effects of the SQTS mutations on [Ca(2+)] i transients, sarcomere length shortening and contractile force at the single cell level were evaluated with and without the consideration of stretch-activated channel current (I sac). Without I sac, at a stimulation frequency of 1Hz, the SQTS mutations produced dramatic reductions in the amplitude of [Ca(2+)] i transients, sarcomere length shortening and contractile force. When I sac was incorporated, there was a considerable attenuation of the effects of SQTS-associated action potential shortening on Ca(2+) transients, sarcomere shortening and contractile force. Single cell models were then incorporated into 3D human ventricular tissue models. The timing of maximum deformation was delayed in the SQTS setting compared to control.

CONCLUSION

The incorporation of I sac appears to be an important consideration in modeling functional effects of SQT 1 and 3 mutations on cardiac electro-mechanical coupling. Whilst there is little evidence of profoundly impaired cardiac contractile function in SQTS patients, our 3D simulations correlate qualitatively with reported evidence for dissociation between ventricular repolarization and the end of mechanical systole.

Beta-adrenergic stimulation maintains cardiac function in Serca2 knockout mice

Previous studies on Serca2 knockout (KO) mice showed that cardiac function is sustained in vivo for several weeks after knockout, whereas SERCA protein levels decrease and calcium dynamics are significantly impaired. In this study, we reconcile observed cellular and organ level contractile function using a cardiac multiscale model. We identified and quantified the changes in cellular function that are both consistent with observations and able to compensate for the decrease in SERCA. Calcium transients were used as input for multiscale computational simulations to predict whole-organ response. Although this response matched experimental pressure-volume (PV) measurements in healthy mice, the reduced magnitude calcium transients observed in KO cells were insufficient to trigger ventricular ejection. To replicate the effects of elevated catecholamine levels observed in vivo, cells were treated with isoproterenol. Incorporation of the resulting measured β-adrenergically stimulated calcium transients into the model resulted in a close match with experimental PV loops. Changes in myofilament properties, when considered in isolation, were not able to increase tension development to levels consistent with measurements, further confirming the necessity of a high β-adrenergic state. Modeling additionally indicated that increased venous return observed in the KO mice helps maintain a high ejection fraction via the Frank-Starling effect. Our study shows that increased β-adrenergic stimulation is a potentially highly significant compensatory mechanism by which cardiac function is maintained in Serca2 KO mice, producing the increases in both systolic and diastolic calcium, consistent with the observed contractile function observed in experimental PV measurements.

A discrete electromechanical model for human cardiac tissue: effects of stretch-activated currents and stretch conditions on restitution properties and spiral wave dynamics

We introduce an electromechanical model for human cardiac tissue which couples a biophysical model of cardiac excitation (Tusscher, Noble, Noble, Panfilov, 2006) and tension development (adjusted Niederer, Hunter, Smith, 2006 model) with a discrete elastic mass-lattice model. The equations for the excitation processes are solved with a finite difference approach, and the equations of the mass-lattice model are solved using Verlet integration. This allows the coupled problem to be solved with high numerical resolution. Passive mechanical properties of the mass-lattice model are described by a generalized Hooke's law for finite deformations (Seth material). Active mechanical contraction is initiated by changes of the intracellular calcium concentration, which is a variable of the electrical model. Mechanical deformation feeds back on the electrophysiology via stretch-activated ion channels whose conductivity is controlled by the local stretch of the medium. We apply the model to study how stretch-activated currents affect the action potential shape, restitution properties, and dynamics of spiral waves, under constant stretch, and dynamic stretch caused by active mechanical contraction. We find that stretch conditions substantially affect these properties via stretch-activated currents. In constantly stretched medium, we observe a substantial decrease in conduction velocity, and an increase of action potential duration; whereas, with dynamic stretch, action potential duration is increased only slightly, and the conduction velocity restitution curve becomes biphasic. Moreover, in constantly stretched medium, we find an increase of the core size and period of a spiral wave, but no change in rotation dynamics; in contrast, in the dynamically stretching medium, we observe spiral drift. Our results may be important to understand how altered stretch conditions affect the heart's functioning.

Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations

Mathematical models of cardiac electro-mechanics typically consist of three tightly coupled parts: systems of ordinary differential equations describing electro-chemical reactions and cross-bridge dynamics in the muscle cells, a system of partial differential equations modelling the propagation of the electrical activation through the tissue and a nonlinear elasticity problem describing the mechanical deformations of the heart muscle. The complexity of the mathematical model motivates numerical methods based on operator splitting, but simple explicit splitting schemes have been shown to give severe stability problems for realistic models of cardiac electro-mechanical coupling. The stability may be improved by adopting semi-implicit schemes, but these give rise to challenges in updating and linearising the active tension. In this paper we present an operator splitting framework for strongly coupled electro-mechanical simulations and discuss alternative strategies for updating and linearising the active stress component. Numerical experiments demonstrate considerable performance increases from an update method based on a generalised Rush-Larsen scheme and a consistent linearisation of active stress based on the first elasticity tensor.

At the heart of computational modelling

The link between experimental data and biophysically based mathematical models is key to computational simulation meeting its potential to provide physiological insight. However, despite the importance of this link, scrutiny and analysis of the processes by which models are parameterised from data are currently lacking. While this situation is common to many areas of physiological modelling, to provide a concrete context, we use examples drawn from detailed models of cardiac electro-mechanics. Using this biophysically detailed cohort of models we highlight the specific issues of model parameterization and propose this process can be separated into three stages: observation, fitting and validation. Finally, future research challenges and directions in this area are discussed.

Genotype-Phenotype Map Characteristics of an In silico Heart Cell

Understanding the causal chain from genotypic to phenotypic variation is a tremendous challenge with huge implications for personalized medicine. Here we argue that linking computational physiology to genetic concepts, methodology, and data provides a new framework for this endeavor. We exemplify this causally cohesive genotype–phenotype (cGP) modeling approach using a detailed mathematical model of a heart cell. In silico genetic variation is mapped to parametric variation, which propagates through the physiological model to generate multivariate phenotypes for the action potential and calcium transient under regular pacing, and ion currents under voltage clamping. The resulting genotype-to-phenotype map is characterized using standard quantitative genetic methods and novel applications of high-dimensional data analysis. These analyses reveal many well-known genetic phenomena like intralocus dominance, interlocus epistasis, and varying degrees of phenotypic correlation. In particular, we observe penetrance features such as the masking/release of genetic variation, so that without any change in the regulatory anatomy of the model, traits may appear monogenic, oligogenic, or polygenic depending on which genotypic variation is actually present in the data. The results suggest that a cGP modeling approach may pave the way for a computational physiological genomics capable of generating biological insight about the genotype–phenotype relation in ways that statistical-genetic approaches cannot.

Multi-scale computational models of familial hypertrophic cardiomyopathy: genotype to phenotype

Familial hypertrophic cardiomyopathy (FHC) is an inherited disorder affecting roughly one in 500 people. Its hallmark is abnormal thickening of the ventricular wall, leading to serious complications that include heart failure and sudden cardiac death. Treatment is complicated by variation in the severity, symptoms and risks for sudden death within the patient population. Nearly all of the genetic lesions associated with FHC occur in genes encoding sarcomeric proteins, indicating that defects in cardiac muscle contraction underlie the condition. Detailed biophysical data are increasingly available for computational analyses that could be used to predict heart phenotypes based on genotype. These models must integrate the dynamic processes occurring in cardiac cells with properties of myocardial tissue, heart geometry and haemodynamic load in order to predict strain and stress in the ventricular walls and overall pump function. Recent advances have increased the biophysical detail in these models at the myofilament level, which will allow properties of FHC-linked mutant proteins to be accurately represented in simulations of whole heart function. The short-term impact of these models will be detailed descriptions of contractile dysfunction and altered myocardial strain patterns at the earliest stages of the disease-predictions that could be validated in genetically modified animals. Long term, these multi-scale models have the potential to improve clinical management of FHC through genotype-based risk stratification and personalized therapy.

Electromechanical models of the ventricles

Computational modeling has traditionally played an important role in dissecting the mechanisms for cardiac dysfunction. Ventricular electromechanical models, likely the most sophisticated virtual organs to date, integrate detailed information across the spatial scales of cardiac electrophysiology and mechanics and are capable of capturing the emergent behavior and the interaction between electrical activation and mechanical contraction of the heart. The goal of this review is to provide an overview of the latest advancements in multiscale electromechanical modeling of the ventricles. We first detail the general framework of multiscale ventricular electromechanical modeling and describe the state of the art in computational techniques and experimental validation approaches. The powerful utility of ventricular electromechanical models in providing a better understanding of cardiac function is then demonstrated by reviewing the latest insights obtained by these models, focusing primarily on the mechanisms by which mechanoelectric coupling contributes to ventricular arrythmogenesis, the relationship between electrical activation and mechanical contraction in the normal heart, and the mechanisms of mechanical dyssynchrony and resynchronization in the failing heart. Computational modeling of cardiac electromechanics will continue to complement basic science research and clinical cardiology and holds promise to become an important clinical tool aiding the diagnosis and treatment of cardiac disease.

Efficient computational methods for strongly coupled cardiac electromechanics

Strongly coupled cardiac electromechanical models can further our understanding of the relative importance of feedback mechanisms in the heart, but computational challenges currently remain a major obstacle, which limit their widespread use. To address this issue, we present a set of efficient computational methods including an efficient adaptive cell model integration scheme and a solution method for the monodomain equations that maintains high conduction velocity for time steps greater than 0.1 ms. We also present a novel method for increasing the efficiency of simulating electromechanical coupling, which shows a significant reduction in computational cost of the mechanical component on a personalized left ventricular geometry with an active contraction cell model reparametrized for human cells.

A study on prediction methods for a cardiovascular strong-coupling simulation

We investigated numerical methods for predictors in a multiscale cardiovascular simulation model. The proposed method predicts initial approximations for the iterative convergence calculations of the strong coupling method using the smoothing spline to remove errors from values of past timesteps and using the linear and second-order extrapolation. The new coupling algorithm was used for coupling a left ventricular finite element model to a myocardial excitation-contraction model. We performed experiments with different values for the smoothing parameter λ and with linear and second-order extrapolations. λ = 1 with the linear extrapolation gave the best results. It reduced computation time by 91% compared to the strong coupling method. With the use of the smoothing spline, distance between the initial approximation and converged solution reduced by 62%, while the average number of iterations reduced by 32%. The smoothing spline can be used to improve the accuracy of predictors and reduce the number of iterations needed for the computation of the convergence procedure.

Models of cardiac tissue electrophysiology: progress, challenges and open questions

Models of cardiac tissue electrophysiology are an important component of the Cardiac Physiome Project, which is an international effort to build biophysically based multi-scale mathematical models of the heart. Models of tissue electrophysiology can provide a bridge between electrophysiological cell models at smaller scales, and tissue mechanics, metabolism and blood flow at larger scales. This paper is a critical review of cardiac tissue electrophysiology models, focussing on the micro-structure of cardiac tissue, generic behaviours of action potential propagation, different models of cardiac tissue electrophysiology, the choice of parameter values and tissue geometry, emergent properties in tissue models, numerical techniques and computational issues. We propose a tentative list of information that could be included in published descriptions of tissue electrophysiology models, and used to support interpretation and evaluation of simulation results. We conclude with a discussion of challenges and open questions.

Electromechanical wavebreak in a model of the human left ventricle

In the present report, we introduce an integrative three-dimensional electromechanical model of the left ventricle of the human heart. Electrical activity is represented by the ionic TP06 model for human cardiac cells, and mechanical activity is represented by the Niederer-Hunter-Smith active contractile tension model and the exponential Guccione passive elasticity model. These models were embedded into an anatomic model of the left ventricle that contains a detailed description of cardiac geometry and the fiber orientation field. We demonstrated that fiber shortening and wall thickening during normal excitation were qualitatively similar to experimental recordings. We used this model to study the effect of mechanoelectrical feedback via stretch-activated channels on the stability of reentrant wave excitation. We found that mechanoelectrical feedback can induce the deterioration of an otherwise stable spiral wave into turbulent wave patterns similar to that of ventricular fibrillation. We identified the mechanisms of this transition and studied the three-dimensional organization of this mechanically induced ventricular fibrillation.

The role of biofluid mechanics in the assessment of clinical and pathological observations: sixth International Bio-Fluid Mechanics Symposium and Workshop, March 28-30, 2008 Pasadena, California

Biofluid mechanics is increasingly applied in support of diagnosis and decision-making for treatment of clinical pathologies. Exploring the relationship between blood flow phenomena and pathophysiological observations is enhanced by continuing advances in the imaging modalities, measurement techniques, and capabilities of computational models. When combined with underlying physiological models, a powerful set of tools becomes available to address unmet clinical needs, predominantly in the direction of enhanced diagnosis, as well as assessment and prediction of treatment outcomes. This position paper presents an overview of current approaches and future developments along this theme that were discussed at the 5th International Biofluid Symposium and Workshop held at the California Institute of Technology in 2008. The introduction of novel mechanical biomarkers in device design and optimization, and applications in the characterization of more specific and focal conditions such as aneurysms, are at the center of attention. Further advances in integrative modeling, incorporating multiscale and multiphysics techniques are also discussed.

Current progress in patient-specific modeling

We present a survey of recent advancements in the emerging field of patient-specific modeling (PSM). Researchers in this field are currently simulating a wide variety of tissue and organ dynamics to address challenges in various clinical domains. The majority of this research employs three-dimensional, image-based modeling techniques. Recent PSM publications mostly represent feasibility or preliminary validation studies on modeling technologies, and these systems will require further clinical validation and usability testing before they can become a standard of care. We anticipate that with further testing and research, PSM-derived technologies will eventually become valuable, versatile clinical tools.

Coupling multi-physics models to cardiac mechanics

We outline and review the mathematical framework for representing mechanical deformation and contraction of the cardiac ventricles, and how this behaviour integrates with other processes crucial for understanding and modelling heart function. Building on general conservation principles of space, mass and momentum, we introduce an arbitrary Eulerian-Lagrangian framework governing the behaviour of both fluid and solid components. Exploiting the natural alignment of cardiac mechanical properties with the tissue microstructure, finite deformation measures and myocardial constitutive relations are referred to embedded structural axes. Coupling approaches for solving this large deformation mechanics framework with three dimensional fluid flow, coronary hemodynamics and electrical activation are described. We also discuss the potential of cardiac mechanics modelling for clinical applications.

Coupling contraction, excitation, ventricular and coronary blood flow across scale and physics in the heart

In this study, we review the development and application of multi-physics and multi-scale coupling in the construction of whole-heart physiological models. Through an examination of recent computational modelling developments, we analyse the significance of coupling mechanisms for the increased understanding of cardiac function in the areas of excitation-contraction, coronary blood flow and ventricular fluid mechanical coupling. Within these physiological domains, we demonstrate and discuss the importance of model parametrization, imaging-based model anatomy and computational implementation.

CHASTE: incorporating a novel multi-scale spatial and temporal algorithm into a large-scale open source library

Recent work has described the software engineering and computational infrastructure that has been set up as part of the Cancer, Heart and Soft Tissue Environment (CHASTE) project. CHASTE is an open source software package that currently has heart and cancer modelling functionality. This software has been written using a programming paradigm imported from the commercial sector and has resulted in a code that has been subject to a far more rigorous testing procedure than that is usual in this field. In this paper, we explain how new functionality may be incorporated into CHASTE. Whiteley has developed a numerical algorithm for solving the bidomain equations that uses the multi-scale (MS) nature of the physiology modelled to enhance computational efficiency. Using a simple geometry in two dimensions and a purpose-built code, this algorithm was reported to give an increase in computational efficiency of more than two orders of magnitude. In this paper, we begin by reviewing numerical methods currently in use for solving the bidomain equations, explaining how these methods may be developed to use the MS algorithm discussed above. We then demonstrate the use of this algorithm within the CHASTE framework for solving the monodomain and bidomain equations in a three-dimensional realistic heart geometry. Finally, we discuss how CHASTE may be developed to include new physiological functionality--such as modelling a beating heart and fluid flow in the heart--and how new algorithms aimed at increasing the efficiency of the code may be incorporated.

The role of the Frank-Starling law in the transduction of cellular work to whole organ pump function: a computational modeling analysis

We have developed a multi-scale biophysical electromechanics model of the rat left ventricle at room temperature. This model has been applied to investigate the relative roles of cellular scale length dependent regulators of tension generation on the transduction of work from the cell to whole organ pump function. Specifically, the role of the length dependent Ca(2+) sensitivity of tension (Ca(50)), filament overlap tension dependence, velocity dependence of tension, and tension dependent binding of Ca(2+) to Troponin C on metrics of efficient transduction of work and stress and strain homogeneity were predicted by performing simulations in the absence of each of these feedback mechanisms. The length dependent Ca(50) and the filament overlap, which make up the Frank-Starling Law, were found to be the two dominant regulators of the efficient transduction of work. Analyzing the fiber velocity field in the absence of the Frank-Starling mechanisms showed that the decreased efficiency in the transduction of work in the absence of filament overlap effects was caused by increased post systolic shortening, whereas the decreased efficiency in the absence of length dependent Ca(50) was caused by an inversion in the regional distribution of strain.

Towards predictive modelling of the electrophysiology of the heart

The simulation of cardiac electrical function is an example of a successful integrative multiscale modelling approach that is directly relevant to human disease. Today we stand at the threshold of a new era, in which anatomically detailed, tomographically reconstructed models are being developed that integrate from the ion channel to the electromechanical interactions in the intact heart. Such models hold high promise for interpretation of clinical and physiological measurements, for improving the basic understanding of the mechanisms of dysfunction in disease, such as arrhythmias, myocardial ischaemia and heart failure, and for the development and performance optimization of medical devices. The goal of this article is to present an overview of current state-of-art advances towards predictive computational modelling of the heart as developed recently by the authors of this article. We first outline the methodology for constructing electrophysiological models of the heart. We then provide three examples that demonstrate the use of these models, focusing specifically on the mechanisms for arrhythmogenesis and defibrillation in the heart. These include: (1) uncovering the role of ventricular structure in defibrillation; (2) examining the contribution of Purkinje fibres to the failure of the shock; and (3) using magnetic resonance imaging reconstructed heart models to investigate the re-entrant circuits formed in the presence of an infarct scar.

A numerical method for cardiac mechanoelectric simulations

Much effort has been devoted to developing numerical techniques for solving the equations that describe cardiac electrophysiology, namely the monodomain equations and bidomain equations. Only a limited selection of publications, however, address the development of numerical techniques for mechanoelectric simulations where cardiac electrophysiology is coupled with deformation of cardiac tissue. One problem commonly encountered in mechanoelectric simulations is instability of the coupled numerical scheme. In this study, we develop a stable numerical scheme for mechanoelectric simulations. A number of convergence tests are carried out using this stable technique for simulations where deformations are of the magnitude typically observed in a beating heart. These convergence tests demonstrate that accurate computation of tissue deformation requires a nodal spacing of around 1 mm in the mesh used to calculate tissue deformation. This is a much finer computational grid than has previously been acknowledged, and has implications for the computational efficiency of the resulting numerical scheme.

Effect of transmurally heterogeneous myocyte excitation-contraction coupling on canine left ventricular electromechanics

The excitation-contraction coupling properties of cardiac myocytes isolated from different regions of the mammalian left ventricular wall have been shown to vary considerably, with uncertain effects on ventricular function. We embedded a cell-level excitation-contraction coupling model with region-dependent parameters within a simple finite element model of left ventricular geometry to study effects of electromechanical heterogeneity on local myocardial mechanics and global haemodynamics. This model was compared with one in which heterogeneous myocyte parameters were assigned randomly throughout the mesh while preserving the total amount of each cell subtype. The two models displayed nearly identical transmural patterns of fibre and cross-fibre strains at end-systole, but showed clear differences in fibre strains at earlier points during systole. Haemodynamic function, including peak left ventricular pressure, maximal rate of left ventricular pressure development and stroke volume, were essentially identical in the two models. These results suggest that in the intact ventricle heterogeneously distributed myocyte subtypes primarily impact local deformation of the myocardium, and that these effects are greatest during early systole.

Modelling and measuring electromechanical coupling in the rat heart

Tension-dependent binding of Ca(2+) to troponin C in the cardiac myocyte has been shown to play an important role in the regulation of Ca(2+) and the activation of tension development. The significance of this regulatory mechanism is quantified experimentally by the quantity of Ca(2+) released following a rapid change in the muscle length. Using a computational, coupled, electromechanics cell model, we have confirmed that the tension dependence of Ca(2+) binding to troponin C, rather than cross-bridge kinetics or the rate of Ca(2+) uptake by the sarcoplasmic reticulum, determines the quantity of Ca(2+) released following a length step. This cell model has been successfully applied in a continuum model of the papillary muscle to analyse experimental data, suggesting the tension-dependent binding of Ca(2+) to troponin C as the likely pathway through which the effects of localized impaired tension generation alter the Ca(2+) transient. These experimental results are qualitatively reproduced using a three-dimensional coupled electromechanics model. Furthermore, the model predicts that changes in the Ca(2+) transient in the viable myocardium surrounding the impaired region are amplified in the absence of tension-dependent binding of Ca(2+) to troponin C.

Cardiac troponin T mutations promote life-threatening arrhythmias

Mutations in contractile proteins in heart muscle can cause anatomical changes that result in cardiac arrhythmias and sudden cardiac death. However, a conundrum has existed because mutations in one such contractile protein, a so-called Ca2+ sensor troponin T (TnT), can promote ventricular rhythm disturbances even in the absence of hypertrophy or fibrosis. Thus, these mutations must enhance abnormal electrophysiological events via alternative means. In this issue of the JCI, Baudenbacher et al. report a novel mechanism to explain this puzzle (see the related article beginning on page 3893). They show that a selected TnT mutation in the adult mouse heart can markedly increase the sensitivity of cardiac muscle myofilaments to Ca2+ and enhance the susceptibility to arrhythmia, even in the absence of anatomical deformities. As these same mutations can cause some forms of arrhythmias in humans, these findings are of both basic and translational significance.