A model of cardiac electrical activity incorporating ionic pumps and concentration changes

A novel biophysically-detailed mathematical model of rabbit Purkinje cell electrophysiology

Purkinje fibres play an important role in cardiac conduction and have been implicated in arrhythmia in presence of diseased states, genetic mutations, or adverse side effects of drugs. For these reasons, the Purkinje assay is commonly used in pre-clinical in vitro drug assessment of arrhythmic risk. Several investigators have pointed out that rabbit Purkinje cells, compared to other species, have a better sensitivity in detecting arrhythmic risk.

Integrative modeling of the cardiac ventricular myocyte

Cardiac electrophysiology is a discipline with a rich 50-year history of experimental research coupled with integrative modeling which has enabled us to achieve a quantitative understanding of the relationships between molecular function and the integrated behavior of the cardiac myocyte in health and disease. In this paper, we review the development of integrative computational models of the cardiac myocyte. We begin with a historical overview of key cardiac cell models that helped shape the field. We then narrow our focus to models of the cardiac ventricular myocyte and describe these models in the context of their subcellular functional systems including dynamic models of voltage-gated ion channels, mitochondrial energy production, ATP-dependent and electrogenic membrane transporters, intracellular Ca dynamics, mechanical contraction, and regulatory signal transduction pathways. We describe key advances and limitations of the models as well as point to new directions for future modeling research. WIREs Syst Biol Med 2011 3 392-413 DOI: 10.1002/wsbm.122

An intuitive safety factor for cardiac propagation

Safety factor is a useful concept for analyzing the propagation of impulses through cardiac tissue, which may have compromised ion channel function or electrical connectivity. Several formulations for its calculation have been proposed and have proved useful in one dimension; however, as we demonstrate, recent attempts to use the same formulation in multiple dimensions have led to questionable conclusions. In this study, we mathematically analyze the latest formulation of safety factor and explain its puzzling behavior. We propose a new formulation that is suitable for any dimension and can be estimated from experimental measurements. Its applicability is verified in two-dimensional simulations.

Competing oscillators in cardiac pacemaking: historical background

Interaction between a membrane oscillator generated by voltage-dependent ion channels and an intracellular calcium signal oscillator was present in the earliest models (1984 to 1985) using representations of the sarcoplasmic reticulum. Oscillatory release of calcium is inherent in the calcium-induced calcium release process. Those historical results fully support the synthesis proposed in the articles in this review series. The oscillator mechanisms do not primarily compete with each; they entrain each other. However, there is some asymmetry: the membrane oscillator can continue indefinitely in the absence of the calcium oscillator. The reverse seems to be true only in pathological conditions. Studies from tissue-level work and on the development of the heart also provide valuable insights into the integrative action of the cardiac pacemaker.

Arrhythmogenesis by single ectopic beats originating in the Purkinje system

Cells in the Purkinje system (PS) are known to be more vulnerable than ventricular myocytes to secondary excitations during the action potential (AP) plateau or repolarization phases, known as early afterdepolarizations (EADs). Since myocytes have a lower intrinsic AP duration than the PS cells to which they are coupled, EADs occurring in distal branches of the PS are more likely to result in propagating ectopic beats. In this study, we use a computer model of the rabbit ventricles and PS to investigate the consequences of EADs occurring at different times and places in the cardiac conduction system. We quantify the role of tissue conductivity and excitability, as well as interaction with sinus excitation, in determining whether an EAD-induced ectopic beat will establish reentrant activity. We demonstrate how a single ectopic beat arising from an EAD in the distal PS can give rise to reentrant arrhythmia; in contrast, EADs in the proximal PS were unable to initiate reentry. Clinical studies have established the PS as a potential substrate for reentry, but the underlying mechanisms of these types of disorder are not well understood, nor are conditions leading to their development clearly defined; this work provides new insights into the role of the PS in such circumstances. Our findings indicate that simulated EADs in the distal PS can induce premature beats, which can lead to the tachycardias involving the conduction system due to interactions with sinus activity or impaired myocardial conduction velocity.

Models of cardiac excitation-contraction coupling in ventricular myocytes

Mathematical and computational modeling of cardiac excitation-contraction coupling has produced considerable insights into how the heart muscle contracts. With the increase in biophysical and physiological data available, the modeling has become more sophisticated with investigations spanning in scale from molecular components to whole cells. These modeling efforts have provided insight into cardiac excitation-contraction coupling that advanced and complemented experimental studies. One goal is to extend these detailed cellular models to model the whole heart. While this has been done with mechanical and electrophysiological models, the complexity and fast time course of calcium dynamics have made inclusion of detailed calcium dynamics in whole heart models impractical. Novel methods such as the probability density approach and moment closure technique which increase computational efficiency might make this tractable.

Stiffness analysis of cardiac electrophysiological models

The electrophysiology in a cardiac cell can be modeled as a system of ordinary differential equations (ODEs). The efficient solution of these systems is important because they must be solved many times as sub-problems of tissue- or organ-level simulations of cardiac electrophysiology. The wide variety of existing cardiac cell models encompasses many different properties, including the complexity of the model and the degree of stiffness. Accordingly, no single numerical method can be expected to be the most efficient for every model. In this article, we study the stiffness properties of a range of cardiac cell models and discuss the implications for their numerical solution. This analysis allows us to select or design numerical methods that are highly effective for a given model and hence outperform commonly used methods.

A computational model of Purkinje fibre single cell electrophysiology: implications for the long QT syndrome

Computer modelling has emerged as a particularly useful tool in understanding the physiology and pathophysiology of cardiac tissues. Models of ventricular, atrial and nodal tissue have evolved and include detailed ion channel kinetics and intercellular Ca(2+) handling. Purkinje fibre cells play a central role in the electrophysiology of the heart and in the genesis of cardiac arrhythmias. In this study, a new computational model has been constructed that incorporates the major membrane currents that have been isolated in recent experiments using Purkinje fibre cells. The model, which integrates mathematical models of human ion channels based on detailed biophysical studies of their kinetic and voltage-dependent properties, recapitulates distinct electrophysiological characteristics unique to Purkinje fibre cells compared to neighbouring ventricular myocytes. These characteristics include automaticity, hyperpolarized voltage range of the action potential plateau potential, and prolonged action potential duration. Simulations of selective ion channel blockade reproduce responses to pharmacological challenges characteristic of isolated Purkinje fibres in vitro, and importantly, the model predicts that Purkinje fibre cells are prone to severe arrhythmogenic activity in patients harbouring long QT syndrome 3 but much less so for other common forms of long QT. This new Purkinje cellular model can be a useful tool to study tissue-specific drug interactions and the effects of disease-related ion channel dysfunction on the cardiac conduction system.

Roles of hyperpolarization-activated current If in sinoatrial node pacemaking: insights from bifurcation analysis of mathematical models

To elucidate the roles of hyperpolarization-activated current (I(f)) in sinoatrial node (SAN) pacemaking, we theoretically investigated 1) the effects of I(f) on stability and bifurcation during hyperpolarization of SAN cells; 2) combined effects of I(f) and the sustained inward current (I(st)) or Na(+) channel current (I(Na)) on robustness of pacemaking against hyperpolarization; and 3) whether blocking I(f) abolishes pacemaker activity under certain conditions. Bifurcation analyses were performed for mathematical models of rabbit SAN cells; equilibrium points (EPs), periodic orbits, and their stability were determined as functions of parameters. Unstable steady-state potential region determined with applications of constant bias currents shrunk as I(f) density increased. In the central SAN cell, the critical acetylcholine concentration at which bifurcations, to yield a stable EP and quiescence, occur was increased by smaller I(f), but decreased by larger I(f). In contrast, the critical acetylcholine concentration and conductance of gap junctions between SAN and atrial cells at bifurcations progressively increased with enhancing I(f) in the peripheral SAN cell. These effects of I(f) were significantly attenuated by eliminating I(st) or I(Na), or by accelerating their inactivation. Under hyperpolarized conditions, blocking I(f) abolished SAN pacemaking via bifurcations. These results suggest that 1) I(f) itself cannot destabilize EPs; 2) I(f) improves SAN cell robustness against parasympathetic stimulation via preventing bifurcations in the presence of I(st) or I(Na); 3) I(f) dramatically enhances peripheral cell robustness against electrotonic loads of the atrium in combination with I(Na); and 4) pacemaker activity of hyperpolarized SAN cells could be abolished by blocking I(f).

The role of the funny current in pacemaker activity

Abstract: Pacemaking is a basic physiological process, and the cellular mechanisms involved in this function have always attracted the keen attention of investigators. The "funny" (I(f)) current, originally described in sinoatrial node myocytes as an inward current activated on hyperpolarization to the diastolic range of voltages, has properties suitable for generating repetitive activity and for modulating spontaneous rate. The degree of activation of the funny current determines, at the end of an action potential, the steepness of phase 4 depolarization; hence, the frequency of action potential firing. Because I(f) is controlled by intracellular cAMP and is thus activated and inhibited by beta-adrenergic and muscarinic M2 receptor stimulation, respectively, it represents a basic physiological mechanism mediating autonomic regulation of heart rate. Given the complexity of the cellular processes involved in rhythmic activity, an exact quantification of the extent to which I(f) and other mechanisms contribute to pacemaking is still a debated issue; nonetheless, a wealth of information collected since the current was first described more than 30 years ago clearly agrees to identify I(f) as a major player in both generation of spontaneous activity and rate control. I(f)- dependent pacemaking has recently advanced from a basic, physiologically relevant concept, as originally described, to a practical concept that has several potentially useful clinical applications and can be valuable in therapeutically relevant conditions. Typically, given their exclusive role in pacemaking, f-channels are ideal targets of drugs aiming to pharmacological control of cardiac rate. Molecules able to bind specifically to and block f-channels can thus be used as pharmacological tools for heart rate reduction with little or no adverse cardiovascular side effects. Indeed a selective f-channel inhibitor, ivabradine, is today commercially available as a tool in the treatment of stable chronic angina. Also, several loss-of-function mutations of HCN4 (hyperpolarization-activated, cyclic-nucleotide gated 4), the major constitutive subunit of f-channels in pacemaker cells, are known today to cause rhythm disturbances, such as for example inherited sinus bradycardia. Finally, gene- or cell-based methods for in situ delivery of f-channels to silent or defective cardiac muscle represent novel approaches for the development of biological pacemakers eventually able to replace electronic devices.

Cardiac cell modelling: observations from the heart of the cardiac physiome project

In this manuscript we review the state of cardiac cell modelling in the context of international initiatives such as the IUPS Physiome and Virtual Physiological Human Projects, which aim to integrate computational models across scales and physics. In particular we focus on the relationship between experimental data and model parameterisation across a range of model types and cellular physiological systems. Finally, in the context of parameter identification and model reuse within the Cardiac Physiome, we suggest some future priority areas for this field.

Bistable dynamics of cardiac cell models coupled by dynamic gap junctions linked to cardiac memory

In an earlier study, we suggested that adaptive gap junctions (GJs) might be a basis of cardiac memory, a phenomenon which refers to persistent electrophysiological response of the heart to external pacing. Later, it was also shown that the proposed mechanism of adaptation of GJs is consistent with known electrophysiology of GJs. In the present article, we show that a pair of cardiac cell models coupled by dynamic, voltage-sensitive GJs exhibits bistable behavior under certain conditions. Three kinds of cell pairs are considered: (1) a Noble-Noble cell pair that represents adjacent cells in Purkinje network, (2) a pair of DiFranceso-Noble cells that represents adjacent SA nodal cells, and (3) a model of Noble cell coupled to Luo-Rudy cell model, which represents an interacting pair of a Purkinje fiber and a ventricular myocyte. Bistability is demonstrated in all the three cases. We suggest that this bistability might be an underlying factor behind cardiac memory. Focused analysis of a pair of Noble cell models showed that bistability is obtained only when the properties of GJs "match" with the properties of the pair of cells that is coupled by the GJs. This novel notion of match between GJs and cardiac cell types might give an insight into specialized distributions of various connexin proteins in cardiac tissue.

Effects of the purkinje system and cardiac geometry on biventricular pacing: a model study

Heart failure leads to gross cardiac structural changes. While cardiac resynchronization therapy (CRT) is a recognized treatment for restoring synchronous activation, it is not clear how changes in cardiac shape and size affect the electrical pacing therapy. This study used a human heart computer model which incorporated anatomical structures such as myofiber orientation and a Purkinje system (PS) to study how pacing affected failing hearts. The PS was modeled as a tree structure that reproduced its retrograde activation feature. In addition to a normal geometry, two cardiomyopathies were modeled: dilatation and hypertrophy. A biventricular pacing protocol was tested in the context of atrio-ventricular block. The contribution of the PS was examined by removing it, as well as by increasing endocardial conductivity. Results showed that retrograde conduction into the PS was a determining factor for achieving intraventricular synchrony. Omission of the PS led to an overestimate of the degree of electrical dyssynchrony while assessing CRT. The activation patterns for the three geometries showed local changes in the order of activation of the lateral wall in response to the same pacing strategy. These factors should be carefully considered when determining lead placement and optimizing device parameters in clinical practice.

Modeling cardiac action potential shortening driven by oxidative stress-induced mitochondrial oscillations in guinea pig cardiomyocytes

Ischemia-induced shortening of the cardiac action potential and its heterogeneous recovery upon reperfusion are thought to set the stage for reentrant arrhythmias and sudden cardiac death. We have recently reported that the collapse of mitochondrial membrane potential (DeltaPsi(m)) through a mechanism triggered by reactive oxygen species (ROS), coupled to the opening of sarcolemmal ATP-sensitive potassium (K(ATP)) channels, contributes to electrical dysfunction during ischemia-reperfusion. Here we present a computational model of excitation-contraction coupling linked to mitochondrial bioenergetics that incorporates mitochondrial ROS-induced ROS release with coupling between the mitochondrial energy state and electrical excitability mediated by the sarcolemmal K(ATP) current (I(K,ATP)). Whole-cell model simulations demonstrate that increasing the fraction of oxygen diverted from the respiratory chain to ROS production triggers limit-cycle oscillations of DeltaPsi(m), redox potential, and mitochondrial respiration through the activation of a ROS-sensitive inner membrane anion channel. The periods of transient mitochondrial uncoupling decrease the cytosolic ATP/ADP ratio and activate I(K,ATP), consequently shortening the cellular action potential duration and ultimately suppressing electrical excitability. The model simulates emergent behavior observed in cardiomyocytes subjected to metabolic stress and provides a new tool for examining how alterations in mitochondrial oxidative phosphorylation will impact the electrophysiological, contractile, and Ca(2+) handling properties of the cardiac cell. Moreover, the model is an important step toward building multiscale models that will permit investigation of the role of spatiotemporal heterogeneity of mitochondrial metabolism in the mechanisms of arrhythmogenesis and contractile dysfunction in cardiac muscle.

Uniqueness and stability of action potential models during rest, pacing, and conduction using problem-solving environment

Development and application of physiologically detailed dynamic models of the action potential (AP) and Ca2+ cycling in cardiac cells is a rapidly growing aspect of computational cardiac electrophysiology. Given the large scale of the nonlinear system involved, questions were recently raised regarding reproducibility, numerical stability, and uniqueness of model solutions, as well as ability of the model to simulate AP propagation in multicellular configurations. To address these issues, we reexamined ventricular models of myocyte AP developed in our laboratory with the following results. 1), Recognizing that the model involves a system of differential-algebraic equations, a procedure is developed for estimating consistent initial conditions that insure uniqueness and stability of the solution. 2), Model parameters that can be used to modify these initial conditions according to experimental values are identified. 3), A convergence criterion for steady-state solution is defined based on tracking the incremental contribution of each ion species to the membrane voltage. 4), Singularities in state variable formulations are removed analytically. 5), A biphasic current stimulus is implemented to completely eliminate stimulus artifact during long-term pacing over a broad range of frequencies. 6), Using the AP computed based on 1-5 above, an efficient scheme is developed for computing propagation in multicellular models.

Purkinje-mediated effects in the response of quiescent ventricles to defibrillation shocks

In normal cardiac function, orderly activation of the heart is facilitated by the Purkinje system (PS), a specialized network of fast-conducting fibers that lines the ventricles. Its role during ventricular defibrillation remains unelucidated. Physical characteristics of the PS make it a poor candidate for direct electrical observation using contemporary experimental techniques. This study uses a computer modeling approach to assess contributions by the PS to the response to electrical stimulation. Normal sinus rhythm was simulated and epicardial breakthrough sites were distributed in a manner consistent with experimental results. Defibrillation shocks of several strengths and orientations were applied to quiescent ventricles, with and without PS, and electrical activation was analyzed. All shocks induced local polarizations in PS branches parallel to the field, which led to the rapid spread of excitation through the network. This produced early activations at myocardial sites where tissue was unexcited by the shock and coupled to the PS. Shocks along the apico-basal axis of the heart resulted in a significant abbreviation of activation time when the PS was present; these shocks are of particular interest because the fields generated by internal cardioverter defibrillators tend to have a strong component in the same direction. The extent of PS-induced changes, both temporal and spatial, was constrained by the amount of shock-activated myocardium. Increasing field strength decreased the transmission delay between PS and ventricular tissue at Purkinje-myocardial junctions (PMJs), but this did not have a major effect on the organ-level response. Weaker shocks directly affect a smaller volume of myocardial tissue but easily excite the PS, which makes the PS contribution to far field excitation more substantial than for stronger shocks.

Optimal velocity and safety of discontinuous conduction through the heterogeneous Purkinje-ventricular junction

Slow and discontinuous wave conduction through nonuniform junctions in cardiac tissues is generally considered unsafe and proarrythmogenic. However, the relationships between tissue structure, wave conduction velocity, and safety at such junctions are unknown. We have developed a structurally and electrophysiologically detailed model of the canine Purkinje-ventricular junction (PVJ) and varied its heterogeneity parameters to determine such relationships. We show that neither very fast nor very slow conduction is safe, and there exists an optimal velocity that provides the maximum safety factor for conduction through the junction. The resultant conduction time delay across the PVJ is a natural consequence of the electrophysiological and morphological differences between the Purkinje fiber and ventricular tissue. The delay allows the PVJ to accumulate and pass sufficient charge to excite the adjacent ventricular tissue, but is not long enough for the source-to-load mismatch at the junction to be enhanced over time. The observed relationships between the conduction velocity and safety factor can provide new insights into optimal conditions for wave propagation through nonuniform junctions between various cardiac tissues.

A computationally efficient method for determining the size and location of myocardial ischemia

The purpose of this paper is to introduce a new method for solving the inverse problem of locating ischemic regions in the heart. The electrical activity in the human heart is modeled by the bidomain equations, which can be expanded to compute the potentials on the body surface. The associated inverse problem is to use ECG recordings to gain information about ischemias. We propose an algorithm for doing this, combining the level set method with a simpler minimization problem. Instead of trying to determine the shape, as in the level set method, we simply make the approximation that the ischemia is spherical. The effects of ischemia on the electrical attributes of heart tissue are examined. The new method is tested with computer simulations on synthetic body surface potential maps (BSPMs) in a realistic geometry, using realistic values for the parameters. It is shown to be, in some respects, superior to the level set approach and may be a step toward a practical algorithm useful in medical diagnostics.

Arrhythmogenic mechanisms of the Purkinje system during electric shocks: a modeling study

BACKGROUND

The function of the Purkinje system (PS) is to ensure fast and uniform activation of the heart. Although this vital role during sinus rhythm is well understood, this is not the case when shocks are applied to the heart, especially in the case of failed defibrillation. The PS activates differently from the myocardium, has different electrophysiological properties, and provides alternate propagation pathways; thus, there are many ways in which it can contribute to postshock behavior.

OBJECTIVE

The purpose of this study was to elucidate the role of the PS in the initiation and maintenance of postshock arrhythmias.

METHODS

A computer model of the ventricles including the PS was subjected to different reentry induction protocols.

RESULTS

The PS facilitated reentry induction at relatively weaker shocks. Disconnecting the PS from the ventricles during the postshock interval revealed that the PS helps stabilize early-stage reentry by providing focal breakthroughs. During later stages, the PS contributed to reentry by leading to higher frequency rotors. The PS also promoted wave front splitting during reentry due to electrotonic coupling, which prolongs action potential durations at PS-myocyte junctions. The presence of a PS results in the anchoring of reentrant activations that propagate through the pathways provided by the PS.

CONCLUSIONS

The PS is proarrhythmic in that it provides pathways that prolong activity, and it plays a supplementary role in maintaining the later stages of reentry (>800 ms).

Hyperpolarization-activated cation channels: from genes to function

Hyperpolarization-activated cyclic nucleotide-gated (HCN) channels comprise a small subfamily of proteins within the superfamily of pore-loop cation channels. In mammals, the HCN channel family comprises four members (HCN1-4) that are expressed in heart and nervous system. The current produced by HCN channels has been known as I(h) (or I(f) or I(q)). I(h) has also been designated as pacemaker current, because it plays a key role in controlling rhythmic activity of cardiac pacemaker cells and spontaneously firing neurons. Extensive studies over the last decade have provided convincing evidence that I(h) is also involved in a number of basic physiological processes that are not directly associated with rhythmicity. Examples for these non-pacemaking functions of I(h) are the determination of the resting membrane potential, dendritic integration, synaptic transmission, and learning. In this review we summarize recent insights into the structure, function, and cellular regulation of HCN channels. We also discuss in detail the different aspects of HCN channel physiology in the heart and nervous system. To this end, evidence on the role of individual HCN channel types arising from the analysis of HCN knockout mouse models is discussed. Finally, we provide an overview of the impact of HCN channels on the pathogenesis of several diseases and discuss recent attempts to establish HCN channels as drug targets.

A model of action potentials and fast Ca2+ dynamics in pancreatic beta-cells

We examined the ionic mechanisms mediating depolarization-induced spike activity in pancreatic beta-cells. We formulated a Hodgkin-Huxley-type ionic model for the action potential (AP) in these cells based on voltage- and current-clamp results together with measurements of Ca(2+) dynamics in wild-type and Kv2.1 null mouse islets. The model contains an L-type Ca(2+) current, a "rapid" delayed-rectifier K(+) current, a small slowly-activated K(+) current, a Ca(2+)-activated K(+) current, an ATP-sensitive K(+) current, a plasma membrane calcium-pump current and a Na(+) background current. This model, coupled with an equation describing intracellular Ca(2+) homeostasis, replicates beta-cell AP and Ca(2+) changes during one glucose-induced spontaneous spike, the effects of blocking K(+) currents with different inhibitors, and specific complex spike in mouse islets lacking Kv2.1 channels. The currents with voltage-independent gating variables can also be responsible for burst behavior. Original features of this model include new equations for L-type Ca(2+) current, assessment of the role of rapid delayed-rectifier K(+) current, and Ca(2+)-activated K(+) currents, demonstrating the important roles of the Ca(2+)-pump and background currents in the APs and bursts. This model provides acceptable fits to voltage-clamp, AP, and Ca(2+) concentration data based on in silico analysis.

From in vivo plasma composition to in vitro cardiac electrophysiology and in silico virtual heart: the extracellular calcium enigma

In spite of its potential impact on simulation results, the problem of setting the appropriate Ca(2+) concentration ([Ca(2+)](o)) in computational cardiac models has not yet been properly considered. Usually [Ca(2+)](o) values are derived from in vitro electrophysiology. Unfortunately, [Ca(2+)](o) in the experiments is set significantly far (1.8 or 2 mM) from the physiological [Ca(2+)] in blood (1.0-1.3 mM). We analysed the inconsistency of [Ca(2+)](o) among in vivo, in vitro and in silico studies and the dependence of cardiac action potential (AP) duration (APD) on [Ca(2+)](o). Laboratory measurements confirmed the difference between standard extracellular solutions and normal blood [Ca(2+)]. Experimental data on human atrial cardiomyocytes confirmed literature data, demonstrating an inverse relationship between APD and [Ca(2+)](o). Sensitivity analysis of APD on [Ca(2+)](o) for five of the most used cardiac cell models was performed. Most of the models responded with AP prolongation to increases in [Ca(2+)](o), i.e. opposite to the AP shortening observed in vitro and in vivo. Modifications to the Ten Tusscher-Panfilov model were implemented to demonstrate that qualitative consistency among in vivo, in vitro and in silico studies can be achieved. The Courtemanche atrial model was used to test the effect of changing [Ca(2+)](o) on quantitative predictions about the effect of K(+) current blockade. The present analysis suggests that (i) [Ca(2+)](o) in cardiac AP models should be changed from 1.8 to 2 mM to approximately 1.15 mM in order to reproduce in vivo conditions, (ii) the sensitivity to [Ca(2+)](o) of ventricular AP models should be improved in order to simulate real conditions, (iii) modifications to the formulation of Ca(2+)-dependent I(CaL) inactivation can make models more suitable to analyse AP when [Ca(2+)](o) is set to lower physiological values, and (iv) it could be misleading to use non-physiological high [Ca(2+)](o) when the quantitative analysis of in vivo pathophysiological mechanisms is the ultimate aim of simulation.

Mathematical models of the electrical action potential of Purkinje fibre cells

Early development of ionic models for cardiac myocytes, from the pioneering modification of the Hodgkin-Huxley giant squid axon model by Noble to the iconic DiFrancesco-Noble model integrating voltage-gated ionic currents, ion pumps and exchangers, Ca(2+) sequestration and Ca(2+)-induced Ca(2+) release, provided a general description for a mammalian Purkinje fibre (PF) and the framework for modern cardiac models. In the past two decades, development has focused on tissue-specific models with an emphasis on the sino-atrial (SA) node, atria and ventricles, while the PFs have largely been neglected. However, achieving the ultimate goal of creating a virtual human heart will require detailed models of all distinctive regions of the cardiac conduction system, including the PFs, which play an important role in conducting cardiac excitation and ensuring the synchronized timing and sequencing of ventricular contraction. In this paper, we present details of our newly developed model for the human PF cell including validation against experimental data. Ionic mechanisms underlying the heterogeneity between the PF and ventricular action potentials in humans and other species are analysed. The newly developed PF cell model adds a new member to the family of human cardiac cell models developed previously for the SA node, atrial and ventricular cells, which can be incorporated into an anatomical model of the human heart with details of its electrophysiological heterogeneity and anatomical complexity.

Modelling sarcoplasmic reticulum calcium ATPase and its regulation in cardiac myocytes

When developing large-scale mathematical models of physiology, some reduction in complexity is necessarily required to maintain computational efficiency. A prime example of such an intricate cell is the cardiac myocyte. For the predictive power of the cardiomyocyte models, it is vital to accurately describe the calcium transport mechanisms, since they essentially link the electrical activation to contractility. The removal of calcium from the cytoplasm takes place mainly by the Na(+)/Ca(2+) exchanger, and the sarcoplasmic reticulum Ca(2+) ATPase (SERCA). In the present study, we review the properties of SERCA, its frequency-dependent and beta-adrenergic regulation, and the approaches of mathematical modelling that have been used to investigate its function. Furthermore, we present novel theoretical considerations that might prove useful for the elucidation of the role of SERCA in cardiac function, achieving a reduction in model complexity, but at the same time retaining the central aspects of its function. Our results indicate that to faithfully predict the physiological properties of SERCA, we should take into account the calcium-buffering effect and reversible function of the pump. This 'uncomplicated' modelling approach could be useful to other similar transport mechanisms as well.

Simulation of cardiac electrophysiology on next-generation high-performance computers

Models of cardiac electrophysiology consist of a system of partial differential equations (PDEs) coupled with a system of ordinary differential equations representing cell membrane dynamics. Current software to solve such models does not provide the required computational speed for practical applications. One reason for this is that little use is made of recent developments in adaptive numerical algorithms for solving systems of PDEs. Studies have suggested that a speedup of up to two orders of magnitude is possible by using adaptive methods. The challenge lies in the efficient implementation of adaptive algorithms on massively parallel computers. The finite-element (FE) method is often used in heart simulators as it can encapsulate the complex geometry and small-scale details of the human heart. An alternative is the spectral element (SE) method, a high-order technique that provides the flexibility and accuracy of FE, but with a reduced number of degrees of freedom. The feasibility of implementing a parallel SE algorithm based on fully unstructured all-hexahedra meshes is discussed. A major computational task is solution of the large algebraic system resulting from FE or SE discretization. Choice of linear solver and preconditioner has a substantial effect on efficiency. A fully parallel implementation based on dynamic partitioning that accounts for load balance, communication and data movement costs is required. Each of these methods must be implemented on next-generation supercomputers in order to realize the necessary speedup. The problems that this may cause, and some of the techniques that are beginning to be developed to overcome these issues, are described.

Role of Purkinje system in cardiac arrhythmias

The mechanisms responsible for postshock behavior of the heart are poorly understood. Below threshold shocks may induce arrhythmias that are difficult to contain. Cardiac vulnerability to shocks and defibrillation efficacy are largely determined by the postshock activity during the occurrence of a brief electrically quiescent period, known as the isoelectric window (IW) and activations following the IW period. This paper presents a detailed computer simulation study that underlines the role of the Purkinje system (PS) in postshock arrhythmogenesis. Reentry was induced in an anatomically realistic rabbit heart model using three different shocking protocols. Regions of vulnerability were determined for each protocol with and without PS. The role of PS during reentry was studied by isolating the PS from myocardium at various instances. The earliest post-shock activations were observed originating from the PS which initiated the reentry. The PS was shown to facilitate the reentry induction at weaker shocks. The PS also helped to stabilize the reentry in the early stages but did not play any significant role in the later stages. This research provides valuable insights into the postshock arrhythmogenesis and maintenance, and extends the discussion on the occurrence of IW as observed during clinical and experimental studies.

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.

Efficient fully implicit time integration methods for modeling cardiac dynamics

Implicit methods are well known to have greater stability than explicit methods for stiff systems, but they often are not used in practice due to perceived computational complexity. This paper applies the backward Euler (BE) method and a second-order one-step two-stage composite backward differentiation formula (C-BDF2) for the monodomain equations arising from mathematically modeling the electrical activity of the heart. The C-BDF2 scheme is an L-stable implicit time integration method and easily implementable. It uses the simplest forward Euler and BE methods as fundamental building blocks. The nonlinear system resulting from application of the BE method for the monodomain equations is solved for the first time by a nonlinear elimination method, which eliminates local and nonsymmetric components by using a Jacobian-free Newton solver, called Newton--Krylov solver. Unlike other fully implicit methods proposed for the monodomain equations in the literature, the Jacobian of the global system after the nonlinear elimination has much smaller size, is symmetric and possibly positive definite, which can be solved efficiently by standard optimal solvers. Numerical results are presented demonstrating that the C-BDF2 scheme can yield accurate results with less CPU times than explicit methods for both a single patch and spatially extended domains.

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.

Genesis and regulation of the heart automaticity

The heart automaticity is a fundamental physiological function in higher organisms. The spontaneous activity is initiated by specialized populations of cardiac cells generating periodical electrical oscillations. The exact cascade of steps initiating the pacemaker cycle in automatic cells has not yet been entirely elucidated. Nevertheless, ion channels and intracellular Ca(2+) signaling are necessary for the proper setting of the pacemaker mechanism. Here, we review the current knowledge on the cellular mechanisms underlying the generation and regulation of cardiac automaticity. We discuss evidence on the functional role of different families of ion channels in cardiac pacemaking and review recent results obtained on genetically engineered mouse strains displaying dysfunction in heart automaticity. Beside ion channels, intracellular Ca(2+) release has been indicated as an important mechanism for promoting automaticity at rest as well as for acceleration of the heart rate under sympathetic nerve input. The potential links between the activity of ion channels and Ca(2+) release will be discussed with the aim to propose an integrated framework of the mechanism of automaticity.

Computational models of the heart and their use in assessing the actions of drugs

Models of cardiac cells are sufficiently well developed to answer questions concerning the actions of drugs on repolarization and the initiation of arrhythmias. These models can be used to characterize drug-receptor action profiles that would be expected to avoid arrhythmia and so help to identify drugs that may be safer. Several examples of such action profiles are presented here, including a recently-developed blocker of persistent sodium current, ranolazine. The models have also been incorporated into tissue and organ models that enable arrhythmia to be modelled also at these levels. Work at these levels can reproduce both re-entrant arrhythmia and fibrillation.

Minimal model for human ventricular action potentials in tissue

Modeling the dynamics of wave propagation in human ventricular tissue and studying wave stability require models that reproduce realistic characteristics in tissue. We present a minimal ventricular (MV) human model that is designed to reproduce important tissue-level characteristics of epicardial, endocardial and midmyocardial cells, including action potential (AP) amplitudes and morphologies, upstroke velocities, steady-state action potential duration (APD) and conduction velocity (CV) restitution curves, minimum APD, and minimum diastolic interval. The model is then compared with three previously published human ventricular cell models, the Priebe and Beuckelmann (PB), the Ten Tusscher-Noble-Noble-Panfilov (TNNP), and the Iyer-Mazhari-Winslow (IMW). For the first time, the stability of reentrant waves for all four models is analyzed, and quantitative comparisons are made among the models in single cells and in tissue. The PB, TNNP, and IMW models exhibit quantitative differences in APD and CV rate adaptation, as well as completely different reentrant wave dynamics of quasi-breakup, stability, and breakup, respectively. All the models have dominant frequencies comparable to clinical values except for the IMW model, which has a large range of frequencies extending beyond the clinical range for both ventricular tachycardia (VT) and ventricular fibrillation (VF). The TNNP and IMW models possess a large degree of short-term memory and we show for the first time the existence of memory in CV restitution. The MV model also can be fitted to reproduce the dynamics of other models and is computationally more efficient: the times required to simulate the MV, TNNP, PB and IMW models follow the ratio 1:31:50:8084.

CellExcite: an efficient simulation environment for excitable cells

Background

Brain, heart and skeletal muscle share similar properties of excitable tissue, featuring both discrete behavior (all-or-nothing response to electrical activation) and continuous behavior (recovery to rest follows a temporal path, determined by multiple competing ion flows). Classical mathematical models of excitable cells involve complex systems of nonlinear differential equations. Such models not only impair formal analysis but also impose high computational demands on simulations, especially in large-scale 2-D and 3-D cell networks. In this paper, we show that by choosing Hybrid Automata as the modeling formalism, it is possible to construct a more abstract model of excitable cells that preserves the properties of interest while reducing the computational effort, thereby admitting the possibility of formal analysis and efficient simulation.

Results

We have developed CellExcite, a sophisticated simulation environment for excitable-cell networks. CellExcite allows the user to sketch a tissue of excitable cells, plan the stimuli to be applied during simulation, and customize the diffusion model. CellExcite adopts Hybrid Automata (HA) as the computational model in order to efficiently capture both discrete and continuous excitable-cell behavior.

Conclusions

The CellExcite simulation framework for multicellular HA arrays exhibits significantly improved computational efficiency in large-scale simulations, thus opening the possibility for formal analysis based on HA theory. A demo of CellExcite is available at .

Behaviour of the purkinje system during defibrillation-strength shocks

During normal sinus rhythm, orderly activation of the heart is facilitated by a specialized network of fibres lining the ventricles called the Purkinje system (PS). Characteristic features of the PS encourage coordinated depolarization of spatially disparate endocardial sites. Although the basic role of the PS is well understood, many questions regarding its behaviour, especially during the process of defibrillation, remain unanswered. Purkinje fibres react differently during large electrical shocks than the myocardium on which they run because they are oriented in different directions than the endocardial fibres, they possess distinct electrophysiology, and they are part of a system that is one-dimensional in nature. Because of the small size of Purkinje fibres and their positioning on the endocardium, in vivo observation of PS-related phenomena remains problematic. Therefore, computer modelling offers a unique opportunity to investigate the role of the PS during defibrillation. In this paper, the effects of defibrillation-strength shocks on a finite element model of the ventricles coupled to a distinct PS are ascertained. Results indicate that the presence of the PS has a profound impact on the course of activation in the ventricles. During shocks, depolarizations are elicited at bends and bifurcations in the PS. Subsequently, this activity spreads throughout the PS in all directions, creating numerous regions of myocardial depolarization and accelerating the excitation of the whole structure. These excitations are explained by the cable-like nature of Purkinje fibres, which exposes them to vastly different electrical field effects than bulk myocardium due to abrupt conductivity tensor changes.

A reduced differential model of the electrical activity of cardiac Purkinje fibres

A reduced order differential model of cardiac Purkinje fibres action potential, with only eight state variables, is presented. Its structure, derived from basic physical principles can be used for the main other cardiac cell types, a useful property for some model-based signal or image processing applications. The electrical activity of cardiac Purkinje fibres is reconstructed using particular values of the parameters. This model of the membrane excitation mechanism and intracellular calcium dynamics describes the principal ionic current underlying autorhythmicity; calcium uptake and release from the sarcoplasmic reticulum; effects of the binding of calcium on myoplasmic proteins which affect the Nernst potential of calcium, and then the membrane potential. The model allows realistic modelling of cardiac Purkinje fibres action potential, total ionic current, CICR dependence on intracellular calcium concentrations. Simulations illustrate the role of the inward sodium current as the dominant mechanism underlying pacemaker depolarization during spontaneous activity.

The funny current: cellular basis for the control of heart rate

The 'funny' (pacemaker, I(f)) current, first described almost 30 years ago in sinoatrial node (SAN) myocytes, is a mixed sodium/potassium inward current, activated on hyperpolarisation in the diastolic range of voltages. 'Funny' (f) channels are activated by intracellular cyclic adenosine monophosphate (cAMP) concentrations according to a mechanism mediating regulation of heart rate by the autonomic nervous system, as well as by voltage hyperpolarisation. Structural subunits of native f-channels are the hyperpolarisation-activated cyclic nucleotide-gated (HCN) channels; of the four HCN isoforms known, HCN4 is the most highly expressed in SAN tissue. The I(f) current is a natural target in the search for drugs aimed specifically at affecting heart rate, given its function in pacemaking. Increased heart rate has a negative influence on clinical outcome in patients with cardiovascular disease, and indeed is also an established risk factor for cardiovascular and all-cause mortality in the general population. Clearly, therefore, independent reduction of heart rate, through inhibition of the I(f) current, appears to be a suitable therapeutic option for patients with ischaemic heart disease.beta-Adrenoceptor antagonists (beta-blockers) reduce intracellular cAMP levels, and a substantial part of their negative chronotropic effect is therefore attributable to a reduction of the I(f) current. However, neither beta-blockers nor Ca(2+) channel antagonists, both of which have traditionally been used to reduce myocardial ischaemia, are 'pure' heart rate-lowering drugs. These agents may, in fact, have adverse cardiovascular and noncardiovascular effects.Conversely, the novel heart rate-reducing agent ivabradine is a specific blocker of f-channels, hence a selective inhibitor of the pacemaker I(f) current in the SAN. Ivabradine slows heart rate by reducing the I(f) current-regulated steepness of the diastolic depolarisation in SAN myocytes, thereby increasing diastolic duration, without altering action potential duration or causing negative inotropy. As such, ivabradine is particularly useful in patients with chronic stable angina pectoris. Further clinical studies are ongoing to evaluate the efficacy of ivabradine in patients with coronary heart disease, left ventricular dysfunction and heart failure. This short article reviews the current state of knowledge of the properties of the 'funny' current in relation to exploitation of the I(f) function in pacemaking generation and modulation for the pharmacological control of heart rate.

Mathematical modeling of mechanically modulated rhythm disturbances in homogeneous and heterogeneous myocardium with attenuated activity of na+ -k+ pump

A mathematical model of the cardiomyocyte electromechanical function is used to study contribution of mechanical factors to rhythm disturbances in the case of the cardiomyocyte calcium overload. Particular attention is paid to the overload caused by diminished activity of the sodium-potassium pump. It is shown in the framework of the model, where mechano-calcium feedback is accounted for that myocardium mechanics may significantly enhance arrhythmogenicity of the calcium overload. Specifically, a role of cross-bridge attachment/detachment processes, a role of mechanical conditions of myocardium contractions (length, load), and a role of myocardium viscosity in the case of simulated calcium overload have been revealed. Underlying mechanisms are analyzed. Several approaches are designed in the model and compared to each other for recovery of the valid myocardium electrical and mechanical performance in the case of the partially suppressed sodium-potassium pump.

Computationally efficient strategy for modeling the effect of ion current modifiers

Electrophysiological studies often seek to relate changes in ion current properties caused by a chemical modifier to changes in cellular properties. Therefore, quantifying concentration-dependent effects of modifiers on ion currents is a topic of importance. In this paper, we sought a mathematical method for using ion current data to predict the effect of several theoretical ion current modifiers on cellular and tissue properties that is computationally efficient without compromising predictive power. We focused on the K+ current I(K,r) as an example case due to its link to long QT syndrome and arrhythmias, but these methods should be generally applicable to other electrophysiological studies. We compared predictions using a Markov model with mass action binding of the modifiers to specific conformational states of the channel to predictions generated by two simplified models. We investigated scaling I(K,r) conductance, and found that although this method produced predictions that agreed qualitatively with the more complicated model, it did not generate quantitatively consistent predictions for all modifiers tested. Our simulations showed that a more computationally efficient Hodgkin-Huxley model that incorporates the effect of modifiers through functional changes in the current produced quantitatively consistent predictions of concentration-dependent changes in cell and tissue properties for all modifiers tested.

An overview of cardiac systems biology

The cardiac system has been a major target for intensive studies in the multi-scale modeling field for many years. Reproduction of the action potential and the ionic currents of single cardiomyocytes, as well as the construction of a whole organ model is well established. Still, there are major hurdles to overcome in creating a realistic and predictive functional cardiac model due to the lack of a profound understanding of the complex molecular interactions and their outcomes controlling both normal and pathological cardiophysiology. The recent advent of systems biology offers the conceptual and practical frameworks to tackle such biological complexities. This review provides an overview of major themes in the developing field of cardiac systems biology, summarizing some of the high-throughput experiments and strategies used to integrate the datasets, and various types of computational approaches used for developing useful quantitative models capable of predicting complex biological behavior.

Multi-scale computational modelling in biology and physiology

Recent advances in biotechnology and the availability of ever more powerful computers have led to the formulation of increasingly complex models at all levels of biology. One of the main aims of systems biology is to couple these together to produce integrated models across multiple spatial scales and physical processes. In this review, we formulate a definition of multi-scale in terms of levels of biological organisation and describe the types of model that are found at each level. Key issues that arise in trying to formulate and solve multi-scale and multi-physics models are considered and examples of how these issues have been addressed are given for two of the more mature fields in computational biology: the molecular dynamics of ion channels and cardiac modelling. As even more complex models are developed over the coming few years, it will be necessary to develop new methods to model them (in particular in coupling across the interface between stochastic and deterministic processes) and new techniques will be required to compute their solutions efficiently on massively parallel computers. We outline how we envisage these developments occurring.

A comparison of non-standard solvers for ODEs describing cellular reactions in the heart

Mathematical models for the electrical activity in cardiac cells are normally formulated as systems of ordinary differential equations (ODEs). The equations are nonlinear and describe processes occurring on a wide range of time scales. Under normal accuracy requirements, this makes the systems stiff and therefore challenging to solve numerically. As standard implicit solvers are difficult to implement, explicit solvers such as the forward Euler method are commonly used, despite their poor efficiency. Non-standard formulations of the forward Euler method, derived from the analytical solution of linear ODEs, can give significantly improved performance while maintaining simplicity of implementation. In this paper we study the performance of three non-standard methods on two different cell models with comparable complexity but very different stiffness characteristics.

Construction of a computer model to investigate sawtooth effects in the Purkinje system

The sawtooth effect refers to how one end of a cardiac cell is depolarized, while the opposite end is hyperpolarized, upon exposure to an exogenous electric field. Although hypothesized, it has not been observed in tissue. The Purkinje system is a one-dimensional (1-D) cable-like system residing on the endocardial surface of the heart and is the most obvious candidate for the manifestation of this phenomenon. This paper describes a computer modeling study of the effect of electric fields on the Purkinje system. Starting with a three-dimensional geometrically realistic, finite element, ventricular description, a Purkinje system is constructed which adheres to general physiological principles. Electrical activity in the Purkinje is described by use of 1-D cubic Hermite finite elements. Such a formulation allows for accurate modeling of loading effects at the Purkinje-myocyte junctions, and for preserving the discrete nature of the system. The response of a strand of Purkinje cells to defibrillation strength shocks is computed under several conditions. Also, the response of the isolated Purkinje network is illustrated. Results indicate that the geometry of the Purkinje system is the greatest determinant for far field excitation of the system. Given parameters within the plausible physiological range, the 1-D nature of the Purkinje system may lead to sawtooth potentials which are large enough to affect excitation. Thus, the Purkinje system is capable of affecting the defibrillation process, and warrants further experimentation to elucidate its role.

The influence of activation time on contraction force of myocardial tissue: a simulation study

The efficiency of heart pump function greatly depends on synchronized contraction of myocardial muscle. In this work, contraction simulation of an excitable ventricular tissue cable was constructed to study the influence of excitation patterns on tissue contraction. The tissue cable is composed of elements which contract when excited by an external stimulus. In each calculation step, contraction force of each element is determined by a ventricular cell model. The mechanical deformation is then solved by finite element method and states of cells are updated accordingly. Several factors such as the starting position of the stimulation signal and the conduction velocity of gap-junctions affect contraction behavior. Simulation results show that the activation time, i.e. the time period the stimulation signal needs to spread over the tissue, is a dominant parameter for determining tissue contraction force. Contraction force of myocardial tissue increases monotonically with a decrease in activation time. This result suggests that minimization of activation time might be important for achieving effective tissue contraction.

Modelling of the ventricular conduction system

The His-Purkinje conduction system initiates the normal excitation of the ventricles and is a major component of the specialized conduction system of the heart. Abnormalities and propagation blocks in the Purkinje system result in abnormal excitation of the heart. Experimental findings suggest that the Purkinje network plays an important role in ventricular tachycardia and fibrillation, which is the major cause of sudden cardiac death. Nowadays an important area in the study of cardiac arrhythmias is anatomically accurate modelling. The majority of current anatomical models have not included a description of the Purkinje network. As a consequence, these models cannot be used to study the important role of the Purkinje system in arrhythmia initiation and maintenance. In this article we provide an overview of previous work on modelling of the Purkinje system and report on the development of a His-Purkinje system for our human ventricular model. We use the model to simulate the normal activation pattern as well as abnormal activation patterns resulting from bundle branch block and bundle branch reentry.

Computer modelling of the sinoatrial node

Over the past decades patch-clamp experiments have provided us with detailed information on the different types of ion channels that are present in the cardiac cell membrane. Sophisticated cardiac cell models based on these data can help us understand how the different types of ion channels act together to produce the cardiac action potential. In the field of biological pacemaker engineering, such models provide important instruments for the assessment of the functional implications of changes in density of specific ion channels aimed at producing stable pacemaker activity. In this review, an overview is given of the progress made in cardiac cell modelling, with particular emphasis on the development of sinoatrial (SA) nodal cell models. Also, attention is given to the increasing number of publicly available tools for non-experts in computer modelling to run cardiac cell models.

A guide to modelling cardiac electrical activity in anatomically detailed ventricles

One of the most recent trends in cardiac electrophysiology is the development of integrative anatomically accurate models of the heart, which include description of cardiac activity from sub-cellular and cellular level to the level of the whole organ. In order to construct this type of model, a researcher needs to collect a wide range of information from books and journal articles on various aspects of biology, physiology, electrophysiology, numerical mathematics and computer programming. The aim of this methodological article is to survey recent developments in integrative modelling of electrical activity in the ventricles of the heart, and to provide a practical guide to the resources and tools that are available for work in this exciting and challenging area.