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

On the possibility for computing the transmembrane potential in the heart with a one shot method: an inverse problem

We analyze the possibility for using body surface potential maps (BSPMs), a priori information about the voltage distribution in the heart and the bidomain equations to compute the transmembrane potential throughout the myocardium. Our approach is defined in terms of an inverse problem for elliptic partial differential equations (PDEs). More precisely, we formulate it in terms of an output least squares framework in which a goal functional is minimized subject to suitable PDE constraints. The problem is highly unstable and, even under optimal recording conditions, it does not have a unique solution. We propose a methodology for stabilizing and enforcing uniqueness for this inverse problem. Moreover, a fully implicit method for solving the involved minimization problem is presented. In other words, we show how one may solve it in terms of a system consisting of three linear elliptic PDEs, i.e. we derive a so-called one shot method (also commonly referred to as an all-at-once method). Finally, our theoretical findings are illuminated by a series of numerical experiments. These examples indicate that, in the presence of regional ischemia, it might be possible to approximately recover the transmembrane potential during the resting and plateau phases of the heart cycle. This is probably due to the fact that rather accurate a priori information is available during these time intervals. The problem of computing the transmembrane potential at an arbitrary time instance during a heart beat is still an open problem.

Quantitative techniques for steady-state calculation and dynamic integrated modelling of membrane potential and intracellular ion concentrations

The membrane potential (E(m)) is a fundamental cellular parameter that is primarily determined by the transmembrane permeabilities and concentration gradients of various ions. However, ion gradients are themselves profoundly influenced by E(m) due to its influence upon transmembrane ion fluxes and cell volume (V(c)). These interrelationships between E(m), V(c) and intracellular ion concentrations make computational modelling useful or necessary in order to guide experimentation and to achieve an integrated understanding of experimental data, particularly in complex, dynamic, multi-compartment systems such as skeletal and cardiac myocytes. A variety of quantitative techniques exist that may assist such understanding, from classical approaches such as the Goldman-Hodgkin-Katz equation and the Gibbs-Donnan equilibrium, to more recent "current-summing" models as exemplified by cardiac myocyte models including those of DiFrancesco & Noble, Luo & Rudy and Puglisi & Bers, or the "charge-difference" modelling technique of Fraser & Huang so far applied to skeletal muscle. In general, the classical approaches provide useful and important insights into the relationships between E(m), V(c) and intracellular ion concentrations at steady state, providing their core assumptions are fully understood, while the more recent techniques permit the modelling of changing values of E(m), V(c) and intracellular ion concentrations. The present work therefore reviews the various approaches that may be used to calculate E(m), V(c) and intracellular ion concentrations with the aim of establishing the requirements for an integrated model that can both simulate dynamic systems and recapitulate the key findings of classical techniques regarding the cellular steady state. At a time when the number of cellular models is increasing at an unprecedented rate, it is hoped that this article will provide a useful and critical analysis of the mathematical techniques fundamental to each of them.

Modulation of rate by autonomic agonists in SAN cells involves changes in diastolic depolarization and the pacemaker current

Two distinct intracellular mechanisms have been proposed to affect the firing rate of cardiac pacemaker cells: one involves modulation of the I(f) current by the second messenger cAMP, and one relies upon disruption or alteration of SR Ca2+ transients during activity. Although both mechanisms are necessary for proper automaticity and autonomic rate control, the specific contribution of each to pacemaking is still debated. We investigated if the two processes can be separated based on potentially different effects on action potential characteristics during rate modulation. To identify specific I(f)-mediated effects, we used the selective I(f) blocker ivabradine and found that ivabradine (3 microM) slows rate (-16.2%) by selectively reducing (-31.9%) the steepness of early diastolic depolarization (EDD). On the other hand ryanodine (3 microM), used to evaluate the effects of abolishment of SR Ca2+ transients, slowed rate (-31.3%) by depolarizing the take-off potential (TOP, 18.1%) without affecting EDD. We therefore used these two parameters to identify I(f)-based or SR Ca2+ transients-based processes and analyzed the effects on action potential's characteristics of Rp-cAMPs (50 microM), a membrane permeable cAMP analogue directly activating f-channels; we found that Rp-cAMPs accelerates rate by increasing EDD (+42.3%) without modifying TOP. Finally, rate modulation was achieved by muscarinic (ACh 0.01 microM) or beta-adrenergic (Iso 1 microM) stimulation; in both cases, rate changes were associated with modifications of EDD (ACh, -29.3% and Iso, +47.6%) and not of TOP. We conclude that rate-related changes in the EDD induced by autonomic agonists are mediated by I(f) and not by processes involving SR Ca2+ transients.

The vicissitudes of the pacemaker current I Kdd of cardiac purkinje fibers

The mechanisms underlying the pacemaker current in cardiac tissues is not agreed upon. The pacemaker potential in Purkinje fibers has been attributed to the decay of the potassium current I (Kdd). An alternative proposal is that the hyperpolarization-activated current I (f) underlies the pacemaker potential in all cardiac pacemakers. The aim of this review is to retrace the experimental development related to the pacemaker mechanism in Purkinje fibers with reference to findings about the pacemaker mechanism in the SAN as warranted. Experimental data and their interpretation are critically reviewed. Major findings were attributed to K(+) depletion in narrow extracellular spaces which would result in a time dependent decay of the inward rectifier current I (K1). In turn, this decay would be responsible for a "fake" reversal of the pacemaker current. In order to avoid such a postulated depletion, Ba(2+) was used to block the decay of I (K1). In the presence of Ba(2+) the time-dependent current no longer reversed and instead increased with time and more so at potentials as negative as -120 mV. In this regard, the distinct possibility needs to be considered that Ba(2+) had blocked I (Kdd) (and not only I (K1)). That indeed this was the case was demonstrated by studying single Purkinje cells in the absence and in the presence of Ba(2+). In the absence of Ba(2+), I (Kdd) was present in the pacemaker potential range and reversed at E (K). In the presence of Ba(2+), I (Kdd) was blocked and I (f) appeared at potentials negative to the pacemaker range. The pacemaker potential behaves in a manner consistent with the underlying I (Kdd) but not with I (f). The fact that I (f) is activated on hyperpolarization at potential negative to the pacemaker range makes it suitable as a safety factor to prevent the inhibitory action of more negative potentials on pacemaker discharge. It is concluded that the large body of evidence reviewed proves the pacemaker role of I (Kdd) (but not of I (f)) in Purkinje fibers.

Control over action potential, calcium peak and average fluxes in the cyclic quasi-steady-state ion transport system in cardiac myocytes: in silico studies

MCA (metabolic control analysis) was originally developed to deal with steady-state systems. In the present theoretical study, the control analysis is applied to the cyclic quasi-steady-state system of ion transport in cardiac myocytes. It is demonstrated that the metabolic control of particular components (channels, exchangers, pumps) of the system over such quasi-steady-state variables as action potential amplitude, action potential duration, area under the Ca2+ peak and average fluxes through particular channels during one oscillation period can be defined and calculated. It is shown that the control over particular variables in the analysed, periodical system is distributed among many (potentially all) components of the system. Nevertheless, some components seem to exert much more control than other components, and different variables are controlled to the greatest extent by different channels. Finally, it is hypothesized that the Na+ and K+ transport system exerts a significant control over the Ca2+ transport system, but not vice versa.

Drift and breakup of spiral waves in reaction-diffusion-mechanics systems

Rotating spiral waves organize excitation in various biological, physical, and chemical systems. They underpin a variety of important phenomena, such as cardiac arrhythmias, morphogenesis processes, and spatial patterns in chemical reactions. Important insights into spiral wave dynamics have been obtained from theoretical studies of the reaction-diffusion (RD) partial differential equations. However, most of these studies have ignored the fact that spiral wave rotation is often accompanied by substantial deformations of the medium. Here, we show that joint consideration of the RD equations with the equations of continuum mechanics for tissue deformations (RD-mechanics systems), yield important effects on spiral wave dynamics. We show that deformation can induce the breakup of spiral waves into complex spatiotemporal patterns. We also show that mechanics leads to spiral wave drift throughout the medium approaching dynamical attractors, which are determined by the parameters of the model and the size of the medium. We study mechanisms of these effects and discuss their applicability to the theory of cardiac arrhythmias. Overall, we demonstrate the importance of RD-mechanics systems for mathematics applied to life sciences.

Strong Coupling System for the LV Motion Simulation in a Distributed Simulation Environment

A system where model parts can be easily exchanged and modified is of great advantage, especially in a combination of models such as an electrophysiological cell model and a mechanical model to a more complex left ventricular (LV) motion model. The use of a distributed simulation environment is straightforward because each simulation model is calculated by an existing user-friendly simulator. However, the weak coupling calculation usually used in a distributed environment reduces the accuracy of the simulation and results in an unstable simulation of the LV motion. To overcome this problem, we have developed a strong coupling simulation system for the distributed simulation environment. Simulation results for a myocardial tissue and a simple LV shape model are presented to elucidate the effectiveness of our system.

Parametric sensibility study of the sinoatrial node math model

The results of the parametric sensibility study to the sinoatrial node math model was presented. The model was proposed by H. Zhang, A. V. Holden and M.R. Boyett. The sensitivity analysis indicates that the sodium and potassium ionic concentrations need to be controlled in order to maintain the normal behavior of the node. The calcium concentrations changes simulated don't produce significant effects over the operation of the node. One response surface model was developed as a simplification of original model. The diastolic depolarization rate was redefined in order to allow its measure in potential waves for peripheral node cells.

A probability density approach to modeling local control of calcium-induced calcium release in cardiac myocytes

We present a probability density approach to modeling localized Ca2+ influx via L-type Ca2+ channels and Ca2+-induced Ca2+ release mediated by clusters of ryanodine receptors during excitation-contraction coupling in cardiac myocytes. Coupled advection-reaction equations are derived relating the time-dependent probability density of subsarcolemmal subspace and junctional sarcoplasmic reticulum [Ca2+] conditioned on "Ca2+ release unit" state. When these equations are solved numerically using a high-resolution finite difference scheme and the resulting probability densities are coupled to ordinary differential equations for the bulk myoplasmic and sarcoplasmic reticulum [Ca2+], a realistic but minimal model of cardiac excitation-contraction coupling is produced. Modeling Ca2+ release unit activity using this probability density approach avoids the computationally demanding task of resolving spatial aspects of global Ca2+ signaling, while accurately representing heterogeneous local Ca2+ signals in a population of diadic subspaces and junctional sarcoplasmic reticulum depletion domains. The probability density approach is validated for a physiologically realistic number of Ca2+ release units and benchmarked for computational efficiency by comparison to traditional Monte Carlo simulations. In simulated voltage-clamp protocols, both the probability density and Monte Carlo approaches to modeling local control of excitation-contraction coupling produce high-gain Ca2+ release that is graded with changes in membrane potential, a phenomenon not exhibited by so-called "common pool" models. However, a probability density calculation can be significantly faster than the corresponding Monte Carlo simulation, especially when cellular parameters are such that diadic subspace [Ca2+] is in quasistatic equilibrium with junctional sarcoplasmic reticulum [Ca2+] and, consequently, univariate rather than multivariate probability densities may be employed.

From the Hodgkin-Huxley axon to the virtual heart

Experimentally based models of the heart have been developed since 1960, starting with the discovery and modelling of potassium channels. The early models were based on extensions of the Hodgkin-Huxley nerve impulse equations. The first models including calcium balance and signalling were made in the 1980s and have now reached a high degree of physiological detail. During the 1990s these cell models have been incorporated into anatomically detailed tissue and organ models to create the first virtual organ, the Virtual Heart. With over 40 years of interaction between simulation and experiment, the models are now sufficiently refined to begin to be of use in drug development.

A tale of two dogs: analyzing two models of canine ventricular electrophysiology

The extensive development of detailed mathematical models of cardiac myocyte electrophysiology in recent years has led to a proliferation of models, including many that model the same animal species and specific region of the heart and thus would be expected to have similar properties. In this paper we review and compare two recently developed mathematical models of the electrophysiology of canine ventricular myocytes. To clarify their similarities and differences, we also present studies using them in a range of preparations from single cells to two-dimensional tissue. The models are compared with each other and with new and previously published experimental results in terms of a number of their properties, including action potential morphologies; transmembrane currents during normal heart rates and during alternans; alternans onsets, magnitudes, and cessations; and reentry dynamics of spiral waves. Action potential applets and spiral wave movies for the two canine ventricular models are available online as supplemental material. We find a number of differences between the models, including their rate dependence, alternans dynamics, and reentry stability, and a number of differences compared with experiments. Differences between models of the same species and region of the heart are not unique to these canine models. Similar differences can be found in the behavior of two models of human ventricular myocytes and of human atrial myocytes. We provide several possible explanations for the differences observed in models of the same species and region of the heart and discuss the implications for the applicability of models in addressing questions of mechanism in cardiac electrophysiology.

Hodgkin–Huxley type ion channel characterization: An improved method of voltage clamp experiment parameter estimation

The Hodgkin-Huxley formalism for quantitative characterization of ionic channels is widely used in cellular electrophysiological models. Model parameters for these individual channels are determined from voltage clamp experiments and usually involve the assumption that inactivation process occurs on a time scale which is infinitely slow compared to the activation process. This work shows that such an assumption may lead to appreciable errors under certain physiological conditions and proposes a new numerical approach to interpret voltage clamp experiment results. In simulated experimental protocols the new method was shown to exhibit superior accuracy compared to the traditional least squares fitting methods. With noiseless input data the error in gating variables and time constants was less than 1%, whereas the traditional methods generated upwards of 10% error and predicted incorrect gating kinetics. A sensitivity analysis showed that the new method could tolerate up to approximately 15% perturbation in the input data without unstably amplifying error in the solution. This method could also assist in designing more efficient experimental protocols, since all channel parameters (gating variables, time constants and maximum conductance) could be determined from a single voltage step.

The emergence of a general theory of the initiation and strength of the heartbeat

Sarcoplasmic reticulum (SR) Ca(2+) cycling, that is, the Ca(2+) clock, entrained by externally delivered action potentials has been a major focus in ventricular myocyte research for the past 5 decades. In contrast, the focus of pacemaker cell research has largely been limited to membrane-delimited pacemaker mechanisms (membrane clock) driven by ion channels, as the immediate cause for excitation. Recent robust experimental evidence, based on confocal cell imaging, and supported by numerical modeling suggests a novel concept: the normal rhythmic heart beat is governed by the tight integration of both intracellular Ca(2+) and membrane clocks. In pacemaker cells the intracellular Ca(2+) clock is manifested by spontaneous, rhythmic submembrane local Ca(2+) releases from SR, which are tightly controlled by a high degree of basal and reserve PKA-dependent protein phosphorylation. The Ca(2+) releases rhythmically activate Na(+)/Ca(2+) exchange inward currents that ignite action potentials, whose shape and ion fluxes are tuned by the membrane clock which, in turn, sustains operation of the intracellular Ca(2+) clock. The idea that spontaneous SR Ca(2+) releases initiate and regulate normal automaticity provides the key that reunites pacemaker and ventricular cell research, thus evolving a general theory of the initiation and strength of the heartbeat.

The Role of the Hyperpolarization-Activated Inward Current$I_rm f$in Arrhythmogenesis: A Computer Model Study

Atrial fibrillation is the most common cardiac arrhythmia. Structural cardiac defects such as fibrosis and gap junction remodeling lead to a reduced cellular electrical coupling and are known to promote atrial fibrillation. It has been observed that the expression of the hyperpolarization-activated current If is increased under pathological conditions. Recent experimental data indicate a possible contribution of If to arrhythmogenesis. In this paper, the role of If in action potential propagation in normal and in pathological tissue is investigated by means of computer simulations. The effect of diffuse fibrosis and gap junction remodeling is simulated by reducing cellular coupling nonuniformly. As expected, the conduction velocity decreases when cellular coupling is reduced. In the presence of If the conduction velocity increases both in normal and in pathological tissue. In our simulations, ectopic activity is present in regions with high expression of If and is facilitated by cellular uncoupling. We conclude that an increased If may facilitate propagation of the action potential. Hence, If may prevent conduction slowing and block. Overexpression of If may lead to ectopic activity, especially when cellular coupling is reduced under pathological conditions.

Computational biology in the study of cardiac ion channels and cell electrophysiology

The cardiac cell is a complex biological system where various processes interact to generate electrical excitation (the action potential, AP) and contraction. During AP generation, membrane ion channels interact nonlinearly with dynamically changing ionic concentrations and varying transmembrane voltage, and are subject to regulatory processes. In recent years, a large body of knowledge has accumulated on the molecular structure of cardiac ion channels, their function, and their modification by genetic mutations that are associated with cardiac arrhythmias and sudden death. However, ion channels are typically studied in isolation (in expression systems or isolated membrane patches), away from the physiological environment of the cell where they interact to generate the AP. A major challenge remains the integration of ion-channel properties into the functioning, complex and highly interactive cell system, with the objective to relate molecular-level processes and their modification by disease to whole-cell function and clinical phenotype. In this article we describe how computational biology can be used to achieve such integration. We explain how mathematical (Markov) models of ion-channel kinetics are incorporated into integrated models of cardiac cells to compute the AP. We provide examples of mathematical (computer) simulations of physiological and pathological phenomena, including AP adaptation to changes in heart rate, genetic mutations in SCN5A and HERG genes that are associated with fatal cardiac arrhythmias, and effects of the CaMKII regulatory pathway and beta-adrenergic cascade on the cell electrophysiological function.

Late sodium current in the pathophysiology of cardiovascular disease: consequences of sodium-calcium overload

Late sodium current in cardiac cells is very small compared with the fast component, but as it flows throughout the action potential it may make a substantial contribution to sodium loading during each cardiac cycle. Late sodium current may contribute to triggering arrhythmia in two ways: by causing repolarisation failure (early after depolarisations); and by triggering late after depolarisations attributable to calcium oscillations in sodium-calcium overload conditions. Reduction of late sodium current would therefore be expected to have therapeutic benefits, particularly in disease states such as ischaemia in which sodium-calcium overload is a major feature.

Endogenous driving and synchronization in cardiac and uterine virtual tissues: bifurcations and local coupling

Cardiac and uterine muscle cells and tissue can be either autorhythmic or excitable. These behaviours exchange stability at bifurcations produced by changes in parameters, which if spatially localized can produce an ectopic pacemaking focus. The effects of these parameters on cell dynamics have been identified and quantified using continuation algorithms and by numerical solutions of virtual cells. The ability of a compact pacemaker to drive the surrounding excitable tissues depends on both the size of the pacemaker and the strength of electrotonic coupling between cells within, between, and outside the pacemaking region. We investigate an ectopic pacemaker surrounded by normal excitable tissue. Cell-cell coupling is simulated by the diffusion coefficient for voltage. For uniformly coupled tissues, the behaviour of the hybrid tissue can take one of the three forms: (i) the surrounding tissue electrotonically suppresses the pacemaker; (ii) depressed rate oscillatory activity in the pacemaker but no propagation; and (iii) pacemaker driving propagations into the excitable region. However, real tissues are heterogeneous with spatial changes in cell-cell coupling. In the gravid uterus during early pregnancy, cells are weakly coupled, with the cell-cell coupling increasing during late pregnancy, allowing synchronous contractions during labour. These effects are investigated for a caricature uterine tissue by allowing both excitability and diffusion coefficient to vary stochastically with space, and for cardiac tissues by spatial gradients in the diffusion coefficient.

Cardiac electrophysiology and tissue structure: bridging the scale gap with a joint measurement and modelling paradigm

Significant tissue structures exist in cardiac ventricular tissue that are of supracellular dimension. It is hypothesized that these tissue structures contribute to the discontinuous spread of electrical activation, may contribute to arrhymogenesis and also provide a substrate for effective cardioversion. However, the influences of these mesoscale tissue structures in intact ventricular tissue are difficult to understand solely on the basis of experimental measurement. Current measurement technology is able to record at both the macroscale tissue level and the microscale cellular or subcellular level, but to date it has not been possible to obtain large volume, direct measurements at the mesoscales. To bridge this scale gap in experimental measurements, we use tissue-specific structure and mathematical modelling. Our models have enabled us to consider key hypotheses regarding discontinuous activation. We also consider the future developments of our intact tissue experimental programme.

Sodium/calcium exchanger: influence of metabolic regulation on ion carrier interactions

The Na(+)/Ca(2+) exchanger's family of membrane transporters is widely distributed in cells and tissues of the animal kingdom and constitutes one of the most important mechanisms for extruding Ca(2+) from the cell. Two basic properties characterize them. 1) Their activity is not predicted by thermodynamic parameters of classical electrogenic countertransporters (dependence on ionic gradients and membrane potential), but is markedly regulated by transported (Na(+) and Ca(2+)) and nontransported ionic species (protons and other monovalent cations). These modulations take place at specific sites in the exchanger protein located at extra-, intra-, and transmembrane protein domains. 2) Exchange activity is also regulated by the metabolic state of the cell. The mammalian and invertebrate preparations share MgATP in that role; the squid has an additional compound, phosphoarginine. This review emphasizes the interrelationships between ionic and metabolic modulations of Na(+)/Ca(2+) exchange, focusing mainly in two preparations where most of the studies have been carried out: the mammalian heart and the squid giant axon. A surprising fact that emerges when comparing the MgATP-related pathways in these two systems is that although they are different (phosphatidylinositol bisphosphate in the cardiac and a soluble cytosolic regulatory protein in the squid), their final target effects are essentially similar: Na(+)-Ca(2+)-H(+) interactions with the exchanger. A model integrating both ionic and metabolic interactions in the regulation of the exchanger is discussed in detail as well as its relevance in cellular Ca(i)(2+) homeostasis.

A mathematical model of ion movements in grey matter during a stroke

The development of cytotoxic oedema during a stroke consists in cell swelling and shrinking of the extracellular space. This phenomenon is triggered by ion movements through voltage-gated channels, exchangers and pumps. During ischaemia, sodium, calcium and chloride enter the neurons whereas potassium and glutamate are expelled out of the cells. A mathematical model is proposed to represent the long-term dynamics of membrane potentials, cell volumes and ionic concentrations in intracellular and extracellular spaces during a stroke and to study the influence of each ionic current on cell swelling. The model relies on electrophysiological mechanisms and takes into account the behaviour of two types of cells: neurons and also astrocytes known to play a key role in the excitotoxic process in grey matter. The results obtained when a severe or a moderate ischaemia is simulated are consistent with those observed in the in vitro and in vivo experiments. As this model appears to be robust, it is used to perform illustrative simulations aimed at studying the effect of some channel blockers on cell swelling. This approach may help to explore new therapeutic strategies in order to reduce stroke damage.

Simulating patterns of excitation, repolarization and action potential duration with cardiac Bidomain and Monodomain models

Parallel numerical simulations of excitation and recovery in three-dimensional myocardial domains are presented. The simulations are based on the anisotropic Bidomain and Monodomain models, including intramural fiber rotation and orthotropic or axisymmetric anisotropy of the intra- and extra-cellular conductivity tensors. The Bidomain model consist of a system of two reaction-diffusion equations, while the Monodomain model consists of one reaction-diffusion equation. Both models are coupled with the phase I Luo-Rudy membrane model describing the ionic currents. Simulations of excitation and repolarization sequences on myocardial slabs of different sizes show how the distribution of the action potential durations (APD) is influenced by both the anisotropic electrical conduction and the fiber rotation. This influence occurs in spite of the homogeneous intrinsic properties of the cell membrane. The APD dispersion patterns are closely correlated to the anisotropic curvature of the excitation wavefront.

Contributions of Purkinje-myocardial coupling to suppression and facilitation of early afterdepolarization-induced triggered activity

Electrical loading by ventricular myocardium modulates conduction system repolarization near Purkinje-ventricular junctions (PVJs). We investigated how that loading suppresses and facilitates early afterdepolarizations (EADs) under conditions where there is a high degree of functional coupling between tissue types, which is consistent with the anatomic arrangement at the peripheral conduction system-myocardial interface. Experiments were completed in eight rabbit right ventricular (RV) free wall preparations. Free-running Purkinje strands were locally superfused, and action potentials were recorded from strands. RV free walls were bathed in normal solution. Surface electrograms were recorded near strand insertions into downstream free wall myocardium. Detailed histology was performed to assemble a computer model with interspersed Purkinje and ventricular myocytes weakly coupled throughout the region. Delays from Purkinje upstrokes to downstream peripheral conduction system and myocardial activation were comparable between experiments and simulations, supporting model node-to-node electrical coupling, i.e., the functional coupling. Purkinje action potential duration (APD) prolongation with localized isoproterenol in experiments and calcium current enhancement in simulations failed to establish EADs. With myocardial APD prolongation by delayed rectifier potassium current inhibition or L-type calcium current enhancement accompanying Purkinje APD prolongation in simulations, however, EAD-induced triggered activity developed. Collectively, our findings suggest competing contributions of the myocardial sink when there is a high degree of functional coupling between tissue types, with the transition from suppression to facilitation of EAD-induced triggered activity depending critically upon myocardial APD prolongation.

Imaging the heart: computer 3-dimensional anatomic models of the heart

Since the 1960s, models of the action potential in various cardiac cell types have been developed, and since the 1990s, 3-dimensional anatomic (or geometric) models of various cardiac structures have been developed. We are approaching the time when, for one species, we should have a complete set of action potential and anatomic models for the various cardiac tissues and then we will have realized the aim of constructing a "virtual heart" with accurate anatomy and electrophysiology. However, already the two types of model are beginning to be used in tandem to reconstruct the activation sequence of the heart both during sinus rhythm and arrhythmias.

Calmodulin antagonist W7 directly inhibits f-type current in rabbit sino-atrial cells

As reported for cyclic nucleotide-gated channels in sensory neurons, we investigated the action of Ca2+-calmodulin and calmodulin antagonist (W7) on the apparent affinity of pacemaker (I(f)) channels for cAMP. In this study, we used the patch-clamp technique in inside-out macro-patch configuration in rabbit sino-atrial cells. Intracellular calmodulin perfusion had no effect on f-channel activity and did not change the cAMP-induced I(f) activation shift. Nevertheless, W7 decreased maximal conductance and induced a voltage shift of the current activation curve towards negative potentials. W7 did not modify the positive shift caused by cAMP, and cAMP did not prevent the effects of W7. Contrary to the cyclic nucleotide-gated channel, the f-channel is not directly modulated by Ca2+-calmodulin. The data suggest that W7 alters the voltage-dependent properties of I(f)independent of cAMP binding. This agent opens the pathway for a new family of bradycardic drugs.

Physiology and pharmacology of the cardiac pacemaker (“funny”) current

First described over a quarter of a century ago, the cardiac pacemaker "funny" (I(f)) current has been extensively characterized since, and its role in cardiac pacemaking has been thoroughly demonstrated. A similar current, termed I(h), was later described in different types of neurons, where it has a variety of functions and contributes to the control of cell excitability and plasticity. I(f) is an inward current activated by both voltage hyperpolarization and intracellular cAMP. In the heart, as well as generating spontaneous activity, f-channels mediate autonomic-dependent modulation of heart rate: beta-adrenergic stimulation accelerates, and vagal stimulation slows, cardiac rate by increasing and decreasing, respectively, the intracellular cAMP concentration and, consequently, the f-channel degree of activation. Four isoforms of hyperpolarization-activated, cyclic nucleotide-gated (HCN) channels have been cloned more recently and shown to be the molecular correlates of native f-channels in the heart and h-channels in the brain. Individual HCN isoforms have kinetic and modulatory properties which differ quantitatively. A comparison of their biophysical properties with those of native pacemaker channels provides insight into the molecular basis of the pacemaker current properties and, together with immunolabelling and other detection techniques, gives information on the pattern of HCN isoform distribution in different tissues. Because of their relevance to cardiac pacemaker activity, f-channels are a natural target of drugs aimed at the pharmacological control of heart rate. Several agents developed for their ability to selectively reduce heart rate act by a specific inhibition of f-channel function; these substances have a potential for the treatment of diseases such as angina and heart failure. In the near future, devices based on the delivery of f-channels in situ, or of a cellular source of f-channels (biological pacemakers), will likely be developed for use in therapies for diseases of heart rhythm with the aim of replacing electronic pacemakers.

Modelling of calcium dynamics in brain energy metabolism and Alzheimer's disease

Functional imaging techniques play a major role in the study of brain activation by monitoring the changes in blood flow and energy metabolism. In order to interpret functional neuroimaging data better, the existing mathematical models describing the links that may exist between electrical activity, energy metabolism and hemodynamics in literature are thoroughly analyzed for their advantages and disadvantages in terms of their prediction of available experimental data. Then, these models are combined within a single model that includes membrane ionic currents, glycolysis, mitochondrial activity, exchanges through the blood-brain barrier, as well as brain hemodynamics. Particular attention is paid to the transport and storage of calcium ions in neurons since calcium is not only an important molecule for signalling in neurons, but it is also essential for memory storage. Multiple efforts have underlined the importance of calcium dependent cellular processes in the biochemical characterization of Alzheimer's disease (AD), suggesting that abnormalities in calcium homeostasis might be involved in the pathophysiology of the disease. The ultimate goal of this study is to investigate the hypotheses about the physiological or biochemical changes in health and disease and to correlate them to measurable physiological parameters obtained from functional neuroimaging data as in the time course of blood oxygenation level dependent (BOLD) signal. When calcium dynamics are included in the model, both BOLD signal and metabolite concentration profiles are shown to exhibit temporal behaviour consistent with the experimental data found in literature. In the case of Alzheimer's disease, the effect of halved cerebral blood flow increase results in a negative BOLD signal implying suppressed neural activity.

Modelling of calcium handling in airway myocytes

Airway myocytes are the primary effectors of airway reactivity which modulates airway resistance and hence ventilation. Stimulation of airway myocytes results in an increase in the cytosolic Ca(2+) concentration ([Ca(2+)](i)) and the subsequent activation of the contractile apparatus. Many contractile agonists, including acetylcholine, induce [Ca(2+)](i) increase via Ca(2+) release from the sarcoplasmic reticulum through InsP(3) receptors. Several models have been developed to explain the characteristics of InsP(3)-induced [Ca(2+)](i) responses, in particular Ca(2+) oscillations. The article reviews the modelling of the major structures implicated in intracellular Ca(2+) handling, i.e., InsP(3) receptors, SERCAs, mitochondria and Ca(2+)-binding cytosolic proteins. We developed theoretical models specifically dedicated to the airway myocyte which include the major mechanisms responsible for intracellular Ca(2+) handling identified in these cells. These biocomputations pointed out the importance of the relative proportion of InsP(3) receptor isoforms and the respective role of the different mechanisms responsible for cytosolic Ca(2+) clearance in the pattern of [Ca(2+)](i) variations. We have developed a theoretical model of membrane conductances that predicts the variations in membrane potential and extracellular Ca(2+) influx. Stimulation of this model by simulated increase in [Ca(2+)](i) predicts membrane depolarisation, but not great enough to trigger a significant opening of voltage-dependant Ca(2+) channels. This may explain why airway contraction induced by cholinergic stimulation does not greatly depend on extracellular calcium. The development of such models of airway myocytes is important for the understanding of the cellular mechanisms of airway reactivity and their possible modulation by pharmacological agents.

Cardiac structure and electrical activation: models and measurement

1. Our group has developed finite element models of ventricular anatomy that incorporate detailed structural information. These have been used to study normal electrical activation and re-entrant arrhythmia. 2. A model based on the actual three-dimensional microstructure of a transmural left ventricular (LV) segment predicts that cleavage planes between muscle layers may give rise to non-uniform, anisotropic electrical propagation and also provide a substrate for bulk resetting of the myocardium during defibrillation. 3. The model predictions are consistent with the results of preliminary experiments in which a novel fibre-optic probe is used to record transmembrane potentials at multiple intramural sites in the intact heart. Extracellular potentials are recorded at adjacent LV sites in these studies. 4. We conclude that structural discontinuities in ventricular myocardium may play a role in the initiation of re-entrant arrhythmia and discuss future studies that address this hypothesis.

Multi-scale modelling and the IUPS physiome project

We review the development of models of cellular and tissue function and in particular address issues of multi-scale modelling, including the transition from stochastic models to continuum models and the incorporation of cell and tissue structure. The heart is used as an example of linking models at the molecular level to cell, tissue and organ level function.

Using models of the myocyte for functional interpretation of cardiac proteomic data

There has been significant progress towards the development of highly integrative computational models of the cardiac myocyte over the past decade. Models now incorporate descriptions of voltage-gated ionic currents and membrane transporters, mechanisms of calcium-induced calcium release and intracellular calcium cycling, mitochondrial ATP production and its coupling to energy-requiring membrane transport processes and mechanisms of force generation. There is an extensive literature documenting both the reconstructive and predictive abilities of these models and there is no question that an interplay between quantitative modelling and experimental investigation has become a central component of modern cardiovascular research. As data regarding the cardiovascular proteome in both health and disease emerge, integrative models of the myocyte are becoming useful tools for interpreting the functional significance of changes in protein expression and post-translational modifications (PTMs). Data of particular importance include information on: (a) changes of expressed protein level, (b) changes of protein PTMs, (c) protein localization, and (d) protein-protein interactions, as it is often possible to incorporate and interpret the functional significance of such findings using computational models. We provide two examples of how models may be used in this fashion. In the first example, we show how information on altered expression of the sarcoplasmic reticulum Ca2+-ATPase, when interpreted through the use of a computational model, has provided key insights into fundamental mechanisms regulating cardiac action potential duration. In the second example, we show how information on the effects of phosphorylation of L-type Ca2+ channels, when interpreted through the use of a model, provides insights on how this post-translational modification alters the properties of excitation-contraction coupling and risk for arrhythmia.

New insights into the molecular and cellular workings of the cardiac Na+/Ca2+exchanger

The cardiac Na+/Ca2+exchanger (NCX1) is almost certainly the major Ca2+extrusion mechanism in cardiac myocytes, although the driving force for Ca2+extrusion is quite small. To explain multiple recent results, it is useful to think of the exchanger as a slow Ca2+buffer that can reverse its function multiple times during the excitation-contraction cycle (ECC). An article by the group of John Reeves brings new insights to this function by analyzing the role of regulatory domains of NCX1 that mediate its activation by a rise of cytoplasmic Ca2+. It was demonstrated that the gating reactions are operative just in the physiological range of Ca2+changes, a few fold above resting Ca2+level, and that they prevent the exchanger from damping out the influence of mechanisms that transiently increase Ca2+levels. Furthermore, exchangers with deleted regulatory domains are shown to reduce resting Ca2+to lower levels than achieved by wild-type exchangers. A study by the group of Kenneth Philipson demonstrated that the NCX1 regulatory domain can bind and respond to Ca2+changes on the time scale of the ECC in rat myocytes. At the same time, studies of transgenic mice and NCX1 knockout mice generated by the Philipson group revealed that large changes of NCX1 activity have rather modest effects on ECC. Simple simulations predict these results very well: murine cardiac ECC is very sensitive to small changes of the Na+gradient, very sensitive to changes of the sarcoplasmic reticulum Ca2+pump activity, and very insensitive to changes of NCX1 activity. It is speculated that the NCX1 gating reactions not only regulate coupled 3Na+:1Ca2+exchange but also control the exchanger’s Na+leak function that generates background Na+influx and depolarizing current in cardiac myocytes.

Computational physiology and the Physiome Project

Bioengineering analyses of physiological systems use the computational solution of physical conservation laws on anatomically detailed geometric models to understand the physiological function of intact organs in terms of the properties and behaviour of the cells and tissues within the organ. By linking behaviour in a quantitative, mathematically defined sense across multiple scales of biological organization--from proteins to cells, tissues, organs and organ systems--these methods have the potential to link patient-specific knowledge at the two ends of these spatial scales. A genetic profile linked to cardiac ion channel mutations, for example, can be interpreted in relation to body surface ECG measurements via a mathematical model of the heart and torso, which includes the spatial distribution of cardiac ion channels throughout the myocardium and the individual kinetics for each of the approximately 50 types of ion channel, exchanger or pump known to be present in the heart. Similarly, linking molecular defects such as mutations of chloride ion channels in lung epithelial cells to the integrated function of the intact lung requires models that include the detailed anatomy of the lungs, the physics of air flow, blood flow and gas exchange, together with the large deformation mechanics of breathing. Organizing this large body of knowledge into a coherent framework for modelling requires the development of ontologies, markup languages for encoding models, and web-accessible distributed databases. In this article we review the state of the field at all the relevant levels, and the tools that are being developed to tackle such complexity. Integrative physiology is central to the interpretation of genomic and proteomic data, and is becoming a highly quantitative, computer-intensive discipline.

A mathematical treatment of integrated Ca dynamics within the ventricular myocyte

We have developed a detailed mathematical model for Ca2+ handling and ionic currents in the rabbit ventricular myocyte. The objective was to develop a model that: 1), accurately reflects Ca-dependent Ca release; 2), uses realistic parameters, particularly those that concern Ca transport from the cytosol; 3), comes to steady state; 4), simulates basic excitation-contraction coupling phenomena; and 5), runs on a normal desktop computer. The model includes the following novel features: 1), the addition of a subsarcolemmal compartment to the other two commonly formulated cytosolic compartments (junctional and bulk) because ion channels in the membrane sense ion concentrations that differ from bulk; 2), the use of realistic cytosolic Ca buffering parameters; 3), a reversible sarcoplasmic reticulum (SR) Ca pump; 4), a scheme for Na-Ca exchange transport that is [Na]i dependent and allosterically regulated by [Ca]i; and 5), a practical model of SR Ca release including both inactivation/adaptation and SR Ca load dependence. The data describe normal electrical activity and Ca handling characteristics of the cardiac myocyte and the SR Ca load dependence of these processes. The model includes a realistic balance of Ca removal mechanisms (e.g., SR Ca pump versus Na-Ca exchange), and the phenomena of rest decay and frequency-dependent inotropy. A particular emphasis is placed upon reproducing the nonlinear dependence of gain and fractional SR Ca release upon SR Ca load. We conclude that this model is more robust than many previously existing models and reproduces many experimental results using parameters based largely on experimental measurements in myocytes.

Modeling the Heart

Models of the heart have been developed since 1960, starting with the discovery and modeling of potassium channels. The first models of calcium balance were made in the 1980s and have now reached a high degree of physiological detail. During the 1990s, these cell models were incorporated into anatomically detailed tissue and organ models.

Simulation of ATP metabolism in cardiac excitation-contraction coupling

To obtain insights into the mechanisms underlying the membrane excitation and contraction of cardiac myocytes, we developed a computer model of excitation-contraction coupling (Kyoto model: Jpn. J. Physiol. 53 (2003) 105). This model was further expanded by incorporating pivotal reactions of ATP metabolism; the model of mitochondrial oxidative phosphorylation by Korzeniewski and Zoladz (Biophys. Chem. 92 (2001) 17). The ATP-dependence of contraction, and creatine kinase and adenylate kinase were also incorporated. After minor modifications, the steady-state condition was well established for all the variables, including the membrane potential, contraction, and the ion and metabolite concentrations in sarcoplasmic reticulum, mitochondria and cytoplasm. Concentrations of major metabolites were close to the experimental data. Responses of the new model to anoxia were similar to experimental results of the P-31 NMR study in whole heart. This model serves as a prototype for developing a more comprehensive model of excitation-contraction-metabolism coupling.

Simulation of five intracellular Ca2+-regulation mechanisms in response to voltage-clamp pulses

We report the simulation of five C2+ regulation mechanisms in response to voltage-clamp pulses applied at the membrane. We obtain the currents generated and also record the C2+ concentration changes in the whole cell during and after the pulse. Our simulator includes models for the L-type voltage-dependent C2+ channels, the C2+ pump, the Na+/C2+ exchanger and a fast buffering system, along with diffusion assuming a spheric cell shape.

Computer model of action potential of mouse ventricular myocytes

We have developed a mathematical model of the mouse ventricular myocyte action potential (AP) from voltage-clamp data of the underlying currents and Ca2+ transients. Wherever possible, we used Markov models to represent the molecular structure and function of ion channels. The model includes detailed intracellular Ca2+ dynamics, with simulations of localized events such as sarcoplasmic Ca2+ release into a small intracellular volume bounded by the sarcolemma and sarcoplasmic reticulum. Transporter-mediated Ca2+ fluxes from the bulk cytosol are closely matched to the experimentally reported values and predict stimulation rate-dependent changes in Ca2+ transients. Our model reproduces the properties of cardiac myocytes from two different regions of the heart: the apex and the septum. The septum has a relatively prolonged AP, which reflects a relatively small contribution from the rapid transient outward K+ current in the septum. The attribution of putative molecular bases for several of the component currents enables our mouse model to be used to simulate the behavior of genetically modified transgenic mice.

Basic mechanisms of cardiac impulse propagation and associated arrhythmias

Propagation of excitation in the heart involves action potential (AP) generation by cardiac cells and its propagation in the multicellular tissue. AP conduction is the outcome of complex interactions between cellular electrical activity, electrical cell-to-cell communication, and the cardiac tissue structure. As shown in this review, strong interactions occur among these determinants of electrical impulse propagation. A special form of conduction that underlies many cardiac arrhythmias involves circulating excitation. In this situation, the curvature of the propagating excitation wavefront and the interaction of the wavefront with the repolarization tail of the preceding wave are additional important determinants of impulse propagation. This review attempts to synthesize results from computer simulations and experimental preparations to define mechanisms and biophysical principles that govern normal and abnormal conduction in the heart.

Transverse propagation of action potentials between parallel chains of cardiac muscle and smooth muscle cells in PSpice simulations

BACKGROUND

We previously examined transverse propagation of action potentials between 2 and 3 parallel chain of cardiac muscle cells (CMC) simulated using the PSpice program. The present study was done to examine transverse propagation between 5 parallel chains in an expanded model of CMC and smooth muscle cells (SMC).

METHODS

Excitation was transmitted from cell to cell along a strand of 5 cells not connected by low-resistance tunnels (gap-junction connexons). The entire surface membrane of each cell fired nearly simultaneously, and nearly all the propagation time was spent at the cell junctions, the junctional delay time being about 0.3-0.5 ms (CMC) or 0.8-1.6 ms (SMC). A negative cleft potential (Vjc) develops in the narrow junctional clefts, whose magnitude depends on the radial cleft resistance (Rjc), which depolarizes the postjunctional membrane (post-JM) to threshold. Propagation velocity (theta) increased with amplitude of Vjc. Therefore, one mechanism for the transfer of excitation from one cell to the next is by the electric field (EF) that is generated in the junctional cleft when the pre-JM fires. In the present study, 5 parallel stands of 5 cells each (5 x 5 model) were used.

RESULTS

With electrical stimulation of the first cell of the first strand (cell A1), propagation rapidly spread down that chain and then jumped to the second strand (B chain), followed by jumping to the third, fourth, and fifth strands (C, D, E chains). The rapidity by which the parallel chains became activated depended on the longitudinal resistance of the narrow extracellular cleft between the parallel strands (Rol2); the higher the Rol2 resistance, the faster the theta. The transverse resistance of the cleft (Ror2) had almost no effect. Increasing Rjc decreases the total propagation time (TPT) over the 25-cell network. When the first cell of the third strand (cell C1) was stimulated, propagation spread down the C chain and jumped to the other two strands (B and D) nearly simultaneously.

CONCLUSIONS

Transverse propagation of excitation occurred at multiple points along the chain as longitudinal propagation was occurring, causing the APs in the contiguous chains to become bunched up. Transverse propagation was more erratic and labile in SMC compared to CMC. Transverse transmission of excitation did not require low-resistance connections between the chains, but instead depended on the value of Rol2. The tighter the packing of the chains facilitated transverse propagation.

A simulation analysis on mechanisms of damped oscillation in retinal rod photoreceptor cells

The different actions of two I(h) channel blockers, zatebradine (UL-FS 49) and ZD7288, on rod photoresponses were analysed by computer simulation using a newly revised ionic current model of the rod photoreceptor, based on Hodgkin-Huxley equations. The model, adjusted to fit the experimental results of amphibian rods, shows that both of the blockers enhance the light-induced membrane hyperpolarization. Our model can also predict a mechanism of a damped oscillation arising during the recovery phase appeared only in the presence of zatebradine which, unlike ZD7288, reduces both I(h) and I(Kv). We suggest that the oscillation can appear due to the alternative activation of voltage-dependent Ca(2+) current (I(Ca)) and calcium-dependent current (I(K(Ca)) and I(Cl(Ca))) when I(Kv) is blocked, with I(K(Ca)) having a stronger effect than I(Cl(Ca)).

Modeling Total Heart Function

Computational models of the electrical and mechanical function of the heart are reviewed. These models attempt to explain the integrated function of the heart in terms of ventricular anatomy, the structure and material properties of myocardial tissue, the membrane ion channels, and calcium handling and myofilament mechanics of cardiac myocytes. The models have established the computational framework for linking the structure and function of cardiac cells and tissue to the integrated behavior of the intact heart, but many more aspects of physiological function, including metabolic and signal transduction pathways, need to be included before significant progress can be made in understanding many disease processes.

Genome informatics: current status and future prospects

This article reviews recent advances in genomics and informatics relevant to cardiovascular research. In particular, we review the status of (1) whole genome sequencing efforts in human, mouse, rat, zebrafish, and dog; (2) the development of data mining and analysis tools; (3) the launching of the National Heart, Lung, and Blood Institute Programs for Genomics Applications and Proteomics Initiative; (4) efforts to characterize the cardiac transcriptome and proteome; and (5) the current status of computational modeling of the cardiac myocyte. In each instance, we provide links to relevant sources of information on the World Wide Web and critical appraisals of the promises and the challenges of an expanding and diverse information landscape.

A numerical scheme for modeling wavefront propagation on a monolayer of arbitrary geometry

The majority of models of wavefront propagation in cardiac tissue have assumed relatively simple geometries. Extensions to complicated three-dimensional (3-D) representations are computationally challenging due to issues related both to problem size and to the correct implementation of flux conservation. In this paper, we present a generalized finite difference scheme (GDFS) to simulate the reaction-diffusion system on a 3-D monolayer of arbitrary shape. GDFS is a vertex-centered variant of the finite-volume method that ensures local flux conservation. Owing to an effectively lower dimensionality, the overall computation time is reduced compared to full 3-D models at the same spatial resolution. We present the theoretical background to compute both the wavefront conduction and local electrograms using a matrix formulation. The same matrix is used for both these quantities. We then give some results of simulation for simple monolayers and complex monolayers resembling a human atria.

Role of individual ionic current systems in ventricular cells hypothesized by a model study

Individual ion channels or exchangers are described with a common set of equations for both the sinoatrial node pacemaker and ventricular cells. New experimental data are included, such as the new kinetics of the inward rectifier K+ channel, delayed rectifier K+ channel, and sustained inward current. The gating model of Shirokov et al. (J Gen Physiol 102: 1005-1030, 1993) is used for both the fast Na+ and L-type Ca2+ channels. When combined with a contraction model (Negroni and Lascano: J Mol Cell Cardiol 28: 915-929, 1996), the experimental staircase phenomenon of contraction is reconstructed. The modulation of the action potential by varying the external Ca2+ and K+ concentrations is well simulated. The conductance of I(CaL) dominates membrane conductance during the action potential so that an artificial increase of I(to), I(Kr), I(Ks), or I(KATP) magnifies I(CaL) amplitude. Repolarizing current is provided sequentially by I(Ks), I(Kr), and I(K1). Depression of ATP production results in the shortening of action potential through the activation of I(KATP). The ratio of Ca2+ released from SR over Ca2+ entering via I(CaL) (Ca2+ gain = approximately 15) in excitation-contraction coupling well agrees with the experimental data. The model serves as a predictive tool in generating testable hypotheses.

A mathematical study of volume shifts and ionic concentration changes during ischemia and hypoxia

The response of tissue to ischemia (cessation of blood flow and deprivation of oxygen) includes swelling of the intracellular space, shrinkage of the extracellular space, and an increase in the extracellular potassium concentration. The responses of cardiac and brain tissue to ischemia are qualitatively different in that cardiac tissue shows a rise in extracellular potassium over several minutes from about 5 to 10-12 mM followed by a plateau, while brain tissue shows a similar initial rise followed by a very rapid increase in extracellular potassium to levels of 50-80 mM. During hypoxia the flow of blood (or perfusate) is maintained and, while there is a substantial efflux of potassium from cells, there is little accumulation of potassium in the interstitium. A mathematical model is proposed and studied to try to elucidate the mechanism(s) underlying the increase in extracellular potassium, and the different time courses seen in neural and cardiac tissue. The model involves a Hodgkin-Huxley-type description of transmembrane ion currents, allows for ion concentrations as well as volumes to change for both the intracellular and extracellular space, and includes coupling of damaged tissue to nearby healthy tissue. The model produces a response to ischemia much like that seen in neural tissue, and the mechanism underlying this response in the model is determined. The same mechanism is not present in cardiac ion models, and this may explain the qualitative difference in response shown in cardiac tissue.