Predicting defibrillation outcome based on phase of ventricular activity during ICD implantation

Implantable cardioverter-defibrillators (ICDs) are well known medical device for patients who are at a risk of sudden cardiac death caused by ventricular fibrillation (VF). The relationship between VF mechanisms and successful ICD therapy to terminate of VF is still not well understood. The purpose of this work is to evaluate the timing of ICD therapy as a predictor of successful VF termination. Clinical data sets were recorded from the patients who underwent ICD implantation in 6 Canadian centers. Timing of the defibrillation attempt (phase) was analyzed by using the ICD Marker Channel which monitors and displays cardiac events sensed by ICD. Phase, based on the VF period, was divided into 10 equally distributed bins and number of successful defibrillation episodes in each bin was compared. A total of 187 defibrillation attempts were identified from the 65 subjects. 126 of the defibrillation attempts were successful, while 61 failed. The optimal case was observed at a phase value of 1.2pi with 2 successful attempts. The lowest performance rate was found at a phase value of 1.4pi and 1.8pi with 50% (3 and 2 successful attempts, respectively). The probability of success was analyzed by using generalized estimating equations (GEE) approach with an exchangeable correlation structure. The results of the GEE logistic regression model indicate no correlation between successful defibrillation attempts and phase of ventricular activity during VF (p-value = 0.78). From our results, timing of defibrillation shock attempt is not a factor in successful termination of VF for patients undergoing ICD implantation.

Fibrillation complexity as a predictor of successful defibrillation

A major focus of Implantable Cardioverter Defibrillator (ICD) research has been to reduce the defibrillation shock energy to prolong battery life and provide an enhanced quality of life for the patient. We investigated whether the degree of disorganization (complexity) of the electrogram is correlated with defibrillation shock outcome. The study data sets were recorded using the high voltage leads of an ICD during device implantation. A total 57 data segments from 19 patients were analyzed. Beat cycles were identified using a novel wavelet based method. Two algorithms were proposed and implemented to quantify the disorganization of the electrogram signals: Approximate Entropy and Cross Correlation. Entropy Index based on the ApEn method, was able to discriminate successful episodes from failure ones with a specificity of 93% and sensitivity of 100%. Similarity Index based on Cross correlation method, obtained a specificity of 72% and sensitivity of 66%. We conclude that the organization of a VF episode is related to the minimum energy required for successful defibrillation.

Fibrillation complexity as a predictor of successful defibrillation

A major focus of Implantable Cardioverter Defibrillator (ICD) research has been to reduce the defibrillation shock energy to prolong battery life and provide an enhanced quality of life for the patient. We investigated whether the degree of disorganization (complexity) of the electrogram is correlated with defibrillation shock outcome. The study data sets were recorded using the high voltage leads of an ICD during device implantation. A total 57 data segments from 19 patients were analyzed. Beat cycles were identified using a novel wavelet based method. Two algorithms were proposed and implemented to quantify the disorganization of the electrogram signals: Approximate Entropy and Cross Correlation. Entropy Index based on the ApEn method, was able to discriminate successful episodes from failure ones with a specificity of 93% and sensitivity of 100%. Similarity Index based on Cross correlation method, obtained a specificity of 72% and sensitivity of 66%. We conclude that the organization of a VF episode is related to the minimum energy required for successful defibrillation.

Atrial cell action potential parameter fitting using genetic algorithms

Understanding of the considerable variation in action potential (AP) shape throughout the heart is necessary to explain normal and pathological cardiac function. Existing mathematical models reproduce typical APs, but not all measured APs, as fitting the sets of non-linear equations is a tedious process. The study describes the integration of a pre-existing mathematical model of an atrial cell AP with a genetic algorithm to provide an automated tool to generate APs for arbitrary cells by fitting ionic channel conductances. Using the Nygren model as the base, the technique was first verified by starting with random values and fitting the Nygren model to itself with an error of only 0.03%. The Courtemanche model, which has a different morphology from that of the Nygren model, was successfully fitted. The AP duration restitution curve generated by the fit matched that of the target model very well. Finally, experimentally recorded APs were reproduced. To match AP duration restitution behaviour properly, it was necessary simultaneously to fit over several stimulation frequencies. Also, fitting of the upstroke was better if the stimulating current pulse replicated that found in situ as opposed to a rectangular pulse. In conclusion, the modelled parameters were successfully able to reproduce any given atrial AP. This tool can be useful for determining parameters in new AP models, reproducing specific APs, as well as determining the locus of drug action by examining changes in conductance values.

Substrate size as a determinant of fibrillatory activity maintenance in a mathematical model of canine atrium

Tissue size has been considered an important determinant of atrial fibrillation (AF), but recent work has questioned the critical size hypothesis. Here, we use a previously developed mathematical model of the two-dimensional canine atrium with realistic action potential, ionic, and conduction properties to address substrate size effects on the maintenance of fibrillatory activity. Cholinergic AF was simulated at different acetylcholine (ACh) concentrations ([ACh]) and distributions, with substrate area varied 11.1-fold. Automated phase singularity detection was used to facilitate the analysis of arrhythmic activity. The duration of activity induced by a single extrastimulus increased with increasing substrate dimensions. Two general mechanisms underlying activity were observed and were differentially affected by substrate size. For large mean [ACh], single primary rotors anchored in low-[ACh] zones maintained activity and substrate dimensions were not critical. At lower mean [ACh], extensive spiral wave meander prevented the emergence of single stable rotors. Prolonged activity was favored when substrate size permitted a sufficiently large number of simultaneous longer-lasting rotors that extinction of all was unlikely. Thus either single dominant rotor or multiple reentrant spiral generator mechanisms could maintain fibrillatory activity in this model and were differentially dependent on substrate size. These results speak to recent debates about the role in AF of single driver rotors versus multiple reentrant circuit mechanisms by suggesting that either may maintain fibrillatory atrial activity depending on atrial size and electrophysiological properties.

Mechanisms of Atrial Fibrillation Termination by Pure Sodium Channel Blockade in an Ionically-Realistic Mathematical Model

The mechanisms by which Na + -channel blocking antiarrhythmic drugs terminate atrial fibrillation (AF) remain unclear. Classical “leading-circle” theory suggests that Na + -channel blockade should, if anything, promote re-entry. We used an ionically-based mathematical model of vagotonic AF to evaluate the effects of applying pure Na + -current (I Na ) inhibition during sustained arrhythmia. Under control conditions, AF was maintained by 1 or 2 dominant spiral waves, with fibrillatory propagation at critical levels of action potential duration (APD) dispersion. I Na inhibition terminated AF increasingly with increasing block, terminating all AF at 65% block. During 1:1 conduction, I Na inhibition reduced APD (by 13% at 4 Hz and 60% block), conduction velocity (by 37%), and re-entry wavelength (by 24%). During AF, I Na inhibition increased the size of primary rotors and reduced re-entry rate (eg, dominant frequency decreased by 33% at 60% I Na inhibition) while decreasing generation of secondary wavelets by wavebreak. Three mechanisms contributed to I Na block–induced AF termination in the model: (1) enlargement of the center of rotation beyond the capacity of the computational substrate; (2) decreased anchoring to functional obstacles, increasing meander and extinction at boundaries; and (3) reduction in the number of secondary wavelets that could provide new primary rotors. Optical mapping in isolated sheep hearts confirmed that tetrodotoxin dose-dependently terminates AF while producing effects qualitatively like those of I Na inhibition in the mathematical model. We conclude that pure I Na inhibition terminates AF, producing activation changes consistent with previous clinical and experimental observations. These results provide insights into previously enigmatic mechanisms of class I antiarrhythmic drug-induced AF termination. The full text of this article is available online at http://circres.ahajournals.org

Defibrillation Depends on Conductivity Fluctuations and the Degree of Disorganization in Reentry Patterns

Defibrillation depends on conductivity and disorganization.

INTRODUCTION

Cardiac fibrillation is the deterioration of the heart's normally well-organized activity into one or more meandering spiral waves, which subsequently break up into many meandering wave fronts. Delivery of an electric shock (defibrillation) is the only effective way of restoring the normal rhythm. This study focuses on examining whether higher degrees of disorganization requires higher shock strengths to defibrillate and whether microscopic conductivity fluctuations favor shock success.

METHODS AND RESULTS

We developed a three-dimensional computer bidomain model of a block of cardiac tissue with straight fibers immersed in a conductive bath. The membrane behavior was described by the Courtemanche human atrial action potential model incorporating electroporation and an acetylcholine- (ACh) dependent potassium current. Intracellular conductivities were varied stochastically around nominal values with variations of up to 50%. A single rotor reentry was initiated and, by adjusting the spatial ACh variation, the level of organization could be controlled. The single rotor could be stabilized or spiral wave breakup could be provoked leading to fibrillatory-like activity. For each level of organization, multiple shock timings and strengths were applied to compute the probability of shock success as a function of shock strength.

CONCLUSIONS

Our results suggest that the level of the small-scale conductivity fluctuations is a very important factor in defibrillation. A higher variation significantly lowers the required shock strength. Further, we demonstrated that success also heavily depends on the level of organization of the fibrillatory episode. In general, higher levels of disorganization require higher shock strengths to defibrillate.

Simulation of QRST Integral Maps With a Membrane-Based Computer Heart Model Employing Parallel Processing

The simulation of the propagation of electrical activity in a membrane-based realistic-geometry computer model of the ventricles of the human heart, using the governing monodomain reaction-diffusion equation, is described. Each model point is represented by the phase 1 Luo-Rudy membrane model, modified to represent human action potentials. A separate longer duration action potential was used for the M cells found in the ventricular midwall. Cardiac fiber rotation across the ventricular wall was implemented via an analytic equation, resulting in a spatially varying anisotropic conductivity tensor and, consequently, anisotropic propagation. Since the model comprises approximately 12.5 million points, parallel processing on a multiprocessor computer was used to cut down on simulation time. The simulation of normal activation as well as that of ectopic beats is described. The hypothesis that in situ electrotonic coupling in the myocardium can diminish the gradients of action-potential duration across the ventricular wall was also verified in the model simulations. Finally, the sensitivity of QRST integral maps to local alterations in action-potential duration was investigated.

Cardiac near-field morphology during conduction around a microscopic obstacle--a computer simulation study

In a recent paper, we described the behavior of the cardiac electric near-field, E, parallel to the tissue surface during continuous conduction. We found that the tip of E describes a vector-loop during depolarization with the peak field, E, pointing opposite to the direction of propagation, phiI(m). Experimentally recorded loop morphologies of E, however, frequently showed significant deviations from the theoretically predicted behavior. We hypothesized that this variety of morphologies might be caused by conduction obstacles at a microscopic size scale. This study examines the influence of obstacles on the morphology of vector loops of E and whether the peak of distorted loops remains a reliable indicator for the direction of propagation. We used a computer model of a sheet of cardiac tissue with a central conduction obstacle immersed in an unbounded volume conductor. We studied the loop morphologies of E and the differences between the intracellularly determined direction of propagation, phiI(m), and the direction of E, phiE. Distortions of the vector loop were morphologically similar to those observed experimentally. Differences between phiI(m) and phiE were less than 18 degrees at all observation sites. The obstacle led to deformations of the loop morphology, particularly during the initial and terminal phases, and to a lesser degree near the instant of E. We concluded that E is a reliable indicator of phiI(m).

Computational tools for modeling electrical activity in cardiac tissue

Computer models offer many attractive benefits. However, the modeling of cardiac tissue is computationally expensive due to several physical constraints which result in fine spatiotemporal discretization over large spatiotemporal regions. Our laboratory has been actively trying to develop new techniques to make large scale cardiac simulations tractable over the past 15 years. This paper describes the latest modeling software that our group has developed, called Carp (Cardiac arrhythmias research package). It is designed to run in both shared memory and clustered computing environments. Carp aims to be modular and flexible by following a plug-in framework. This allows the latest models and most efficient solvers to be incorporated as well as enabling run-time selection of techniques. Performance results are given for a large-scale simulation which utilized a comprehensive membrane ionic current description.

Mechanisms of termination of atrial fibrillation by Class I antiarrhythmic drugs: evidence from clinical, experimental, and mathematical modeling studies

Sodium channel blocking drugs (Class I antiarrhythmic agents) have been used for the termination of atrial fibrillation (AF) and for sinus rhythm maintenance for almost 100 years. Despite this long history, the mechanisms that underlie their efficacy in AF remain poorly understood. Classic notions about the determinants of cardiac reentry, as embodied in leading circle theory, and of AF, as reflected in the multiple wavelet hypothesis, suggest that cardiac conduction slowing should promote, rather than prevent, AF. This article reviews the evidence (both clinical and experimental) for the efficacy and mechanisms of action of Class I antiarrhythmic agents in AF. Application of mathematical models of AF to the evaluation of Class I mechanisms is discussed, and recent insights into the latter are presented. A better understanding of the ways in which Na+ channel blockers affect AF will be useful, not only for new antiarrhythmic drug development but also for gaining insight into the mechanisms of the arrhythmia.

Development of a computer algorithm for the detection of phase singularities and initial application to analyze simulations of atrial fibrillation

Atrial fibrillation (AF) is a common cardiac arrhythmia, but its mechanisms are incompletely understood. The identification of phase singularities (PSs) has been used to define spiral waves involved in maintaining the arrhythmia, as well as daughter wavelets. In the past, PSs have often been identified manually. Automated PS detection algorithms have been described previously, but when we attempted to apply a previously developed algorithm we experienced problems with false positives that made the results difficult to use directly. We therefore developed a tool for PS identification that uses multiple strategies incorporating both image analysis and mathematical convolution for automated detection with optimized sensitivity and specificity, followed by manual verification. The tool was then applied to analyze PS behavior in simulations of AF maintained in the presence of spatially distributed acetylcholine effects in cell grids of varying size. These analyses indicated that in almost all cases, a single PS lasted throughout the simulation, corresponding to the central-core tip of a single spiral wave that maintained AF. The sustained PS always localized to an area of low acetylcholine concentration. When the grid became very small and no area of low acetylcholine concentration was surrounded by zones of higher concentration, AF could not be sustained. The behavior of PSs and the mechanisms of AF were qualitatively constant over an 11.1-fold range of atrial grid size, suggesting that the classical emphasis on tissue size as a primary determinant of fibrillatory behavior may be overstated. (c) 2002 American Institute of Physics.

Cholinergic atrial fibrillation in a computer model of a two-dimensional sheet of canine atrial cells with realistic ionic properties

Classical concepts of atrial fibrillation (AF) have been rooted in Moe's multiple-wavelet hypothesis and simple cellular-automaton computer model. Recent experimental work has raised questions about the multiple-wavelet mechanism, suggesting a discrete "driver region" underlying AF. We reexplored the theoretical basis for AF with a 2-dimensional computer model of a 5x10-cm sheet of atrial cells with realistic ionic and coupling properties. Vagal actions were formulated based on patch-clamp studies of acetylcholine (ACh) effects. In control, a single extrastimulus resulted in a highly meandering unstable spiral wave. Simulated electrograms showed fibrillatory activity, with a dominant frequency (DF, 6.5 Hz) that correlated with the mean rate. Uniform ACh reduced core meander of the spiral wave by approximately 70% (as measured by the standard deviation of spiral-wave tip position) and accelerated the DF to 17.0 Hz. Simulated vagally induced refractoriness heterogeneity caused wavefront breakup as accelerated reentrant activity in regions of short refractoriness impinged on regions unable to respond in a 1:1 fashion because of longer refractoriness. In 7 simulations spanning the range of conditions giving sustained AF, 5 were maintained by single dominant spiral waves. On average, 3.0+/-1.3 wavelets were present (range, 1 to 7). Most wavelets were short-lived and did not contribute to AF maintenance. In contrast to predictions of the multiple-wavelet hypothesis, but in agreement with recent experimental evidence, our model indicates that AF can result from relatively stable primary spiral-wave generators and is significantly organized. Our results suggest that vagal AF may arise from ACh-induced stabilization of the primary spiral-wave generator and disorganization of the heterogeneous tissue response. The full text of this article is available at http://www.circresaha.org.

Effect of fibre rotation on the initiation of re-entry in cardiac tissue

Transmural rotation of cardiac fibres can have a big influence on the initiation of re-entry in the heart. However, owing to computational demands, this has not been fully explored in a three-dimensional model of cardiac tissue that has a microscopic description of membrane currents, such as the Luo-Rudy model. Using a previously described model that is computationally fast, re-entry in three-dimensional blocks of cardiac tissue is induced by a cross-shock protocol, and the activity is examined. In the study, the effect of the transmural fibre rotation is ascertained by examining differences between a tissue block with no rotation and ones with 1, 2 and 3 degrees of rotation per fibre layer. The direction of the re-entry is significant in establishing whether or not re-entry can be induced, with clockwise re-entry being easier to initiate. Owing to the rotating anisotropy that results in preferential propagation along the fibre axis, the timing of the second stimulus in the cross-shock protocol has to be changed for different rates of fibre rotation. The fibre rotation either increases or decreases the window of opportunity for re-entry, depending on whether the activation front is perpendicular or parallel to the fibre direction. By varying the transmural extent of the S2, it is found that a deeper stimulus has to be applied to the blocks with fibre rotation to create re-entry. Increasing the transmural resistance also tends to reduce the extent of the S2 required to induce re-entry. Results suggest that increasing fibre rotation reduces the susceptibility of the tissue to re-entry, but that more complex spatiotemporal patterns are possible, e.g. stable figure-of-eight re-entries and transient rotors. Three mechanisms of re-entry annihilation are identified: front catchup, filling of the excitable gap and core wander.

Electrophysiological basis of mono-phasic action potential recordings

Monophasic action potentials (MAPs) have been recorded for over a century, however, the exact mechanism responsible for their genesis has yet to be elucidated fully. The goal of the paper is to examine the physical basis of MAP recordings. MAP recordings are simulated by modelling a three-dimensional block of cardiac tissue. The effect of the MAP electrode is modelled by introducing a large, non-specific leakage conductance to the small region under the electrode. From the spread of the electrical activity, the equivalent extracellular current flow can be efficiently determined. These computed current sources are then input into a boundary element model of the tissue to determine the surface potentials. Finally, differences in surface potentials are used to compute waveforms that closely resemble MAP recordings. By varying model parameters, the mechanisms responsible for the MAP are determined, and a theory is put forward that can account for all observations. It is hypothesised that the leakage current causes the formation of a double-layer potential with a strength equal to the difference in transmembrane voltage between the regions under the electrode and those outside the electrode, leading to a recorded potential that mimics the transmembrane voltage outside the electrode region, although offset. Based on experimental MAP recordings, an equivalent leakage channel with a conductance of 0.1 mS cm-2 and a reversal potential of -43 mV is introduced by the electrode.

Computationally efficient model for simulating electrical activity in cardiac tissue with fiber rotation

Transmural rotation of cardiac fibers may have a large influence on the initiation, stabilization, and termination of several life threatening cardiac arrhythmias. However, three-dimensional modeling of reentry in cardiac tissue is computationally demanding, as a tissue on the order of centimeters in size must be used to sustain reentry and several seconds must be simulated. Numerical accuracy requires time steps on the order of microseconds and spatial discretization on the order of microns. Consequently, the resultant numerical systems are extremely large. In this article, a computationally efficient model of a three-dimensional block of cardiac tissue with fiber rotation is presented. Computational speedup is achieved by using a discrete cable model which allowed for system order reduction, and also by using a scheme for tracking the activation wave front which identified regions requiring integration with a small time step. Simulating 1.2 s of activity of the approximately 2 x 10(6) cells constituting a block measuring 2.0 x 4.0 x 0.29 cm was performed in 26 h. Effects of model parameters on performance are discussed. The effect of fiber rotation on the spread of electrical activity after point source stimulation and a cross shock protocol is clearly demonstrated.

Spatiotemporal evolution of ventricular fibrillation

Sudden cardiac death is the leading cause of death in the industrialized world, with the majority of such tragedies being due to ventricular fibrillation. Ventricular fibrillation is a frenzied and irregular disturbance of the heart rhythm that quickly renders the heart incapable of sustaining life. Rotors, electrophysiological structures that emit rotating spiral waves, occur in several systems that all share with the heart the functional properties of excitability and refractoriness. These re-entrant waves, seen in numerical solutions of simplified models of cardiac tissue, may occur during ventricular tachycardias. It has been difficult to detect such forms of re-entry in fibrillating mammalian ventricles. Here we show that, in isolated perfused dog hearts, high spatial and temporal resolution mapping of optical transmembrane potentials can easily detect transiently erupting rotors during the early phase of ventricular fibrillation. This activity is characterized by a relatively high spatiotemporal cross-correlation. During this early fibrillatory interval, frequent wavefront collisions and wavebreak generation are also dominant features. Interestingly, this spatiotemporal pattern undergoes an evolution to a less highly spatially correlated mechanism that lacks the epicardial manifestations of rotors despite continued myocardial perfusion.

A method for visualization of ventricular fibrillation: Design of a cooled fiberoptically coupled image intensified CCD data acquisition system incorporating wavelet shrinkage based adaptive filtering

The measurement of cardiac transmembrane potential changes with voltage sensitive dyes is in increasing use. Detection of these very small fluorescent alterations using large multiplexed arrays, such as charge coupled device (CCD) cameras at high sampling rates, has proven challenging and usually requires significant averaging to improve the signal-to-noise ratio. To minimize the damage of living tissue stained with voltage sensitive dyes, excitation photon exposure must be limited, with the inevitable consequence of diminishing the fluorescence that is generated. State-of-the-art high frame rate CCD cameras have read noise levels in the 5-10 e(-) rms range, which is at least two orders of magnitude above that required to detect voltage sensitive dye alterations at individual pixels corresponding to 1 mm(2) heart regions illuminated with levels of 100 mW/cm(2) at frame rates approaching 1000 frames/sec. Image intensification is thus required prior to photon quantification. We report here the development of such a data acquisition system using commercially available hardware. Additionally, in the past ten years, a mathematical theory of multiresolution has been developed, and new building blocks called wavelets, allow a signal to be observed at different resolutions. Wavelet analysis also makes possible a new method of extricating signals from noise. We have incorporated spatially adaptive filters based on wavelet denoising of individual pixels to significantly reduce the multiple noise sources present in the acquired data. (c) 1998 American Institute of Physics.

Interactions between adjacent fibers in a cardiac muscle bundle

A strand of cardiac muscle was modeled as a small bundle of individual fibers surrounded by a large volume conductor. The bundle is a uniform assembly of small identical cylindrical fibers, arranged as a series of concentric layers, and its behavior is examined in the presence (coupled bundle) or absence (uncoupled bundle) of transverse resistive coupling between adjacent fibers. Individual fibers are continuous cables of excitable membrane, with circumferential segmentation into 12 equal patches to make the membrane potential changes dependent upon the local interstitial potential. The minimum spacing (d) between adjacent fibers is used to modify the interstitial microstructural organization and the intracellular volume fraction (fi). When d is small enough (d < 0.01 micron), fi remains unchanged at its maximum of about 90%, the interstitial potential is large, the transverse interstitial resistance is high, and the proximity effect arising from the close juxtaposition of adjacent fibers is important. A surface fiber of the uncoupled bundle exhibits little sensitivity to changes in the interstitial microstructure, owing to the dominant influence of the external volume conductor, whereas the central fiber shows a large decrease in velocity, substantial waveshape modifications, and a large increase in interstitial potential as d is reduced. In the coupled bundle, all fibers adopt the same velocity during uniform propagation, owing to the strong transverse resistive coupling; when d is reduced in the range of d < 0.01 micron, the velocity and interstitial potential changes are less pronounced than in the uncoupled bundle. When d is large enough (d > 0.01 micron), the bundle behavior (coupled and uncoupled) approaches that obtained with a bidomain formulation.

Propagation on a central fiber surrounded by inactive fibers in a multifibered bundle model

We studied uniform propagation on a central active fiber surrounded by inactive fibers in a multifibered bundle model lying in a large volume conductor. The behavior of a fully active bundle is considered in a companion paper. The bundle is formed by concentric layers of small cylindrical fibers (radius 5 microns), with a uniform minimum distance (d) between any two adjacent fibers, to yield a bundle radius of about 72 microns. Individual fibers are identical continuous cables of excitable membrane based on a modified Beeler-Reuter model. The intracellular volume fraction (fi) increases to a maximum of about 90% as d is reduced and remains unchanged for d < 0.01 micron. In the range of d < 0.01 micron, the central fiber is effectively shielded from external effects by the first concentric layer of inactive fibers, and a large capacitive load current flows across the surrounding inactive membranes. In addition, the fiber proximity produces a circumferentially nonuniform current density (proximity effect) that is equivalent to an increased average longitudinal interstitial resistance. The conduction velocity is reduced as d becomes smaller in the range of d < 0.1 micron, the interstitial potential becomes larger, and both the maximum rate of rise and time constant of the foot of the upstroke are increased. On the other hand, for d > 0.1 micron, there are negligible changes in the shape of the upstroke, and the behavior of the central fiber is close to that of a uniform cable in a restricted volume conductor. For d larger than about 1.2 microns, the active fiber environment is close to an unbounded isotropic volume conductor.

Calculation of transmembrane current from extracellular potential recordings: a model study

INTRODUCTION

A mathematical/computer model of cardiac tissue was used to study the estimation of transmembrane current (EIm) from extracellular potential recordings.

METHODS AND RESULTS

The simulated EIm of transmembrane current was compared with the simulated transmembrane current (Im), and both simulated values were compared with experimentally derived EIm obtained during sinus rhythm and ventricular fibrillation in dogs. We found that although EIm measurements slightly overestimate the duration of the Im waveform, they provide a reasonable approximation of Im during normal conduction and during decremental conduction and conduction block.

CONCLUSIONS

There is a very clear linear correlation between the time spent at or below 25% of the peak inward transmembrane current (Im25), its corresponding estimate (EIm25), the peak inward Im and EIm, and the peak ionic current, providing some evidence that EIm25 may be a suitable in vivo measure of peak ionic current.

Simulation of two-dimensional anisotropic cardiac reentry: effects of the wavelength on the reentry characteristics

A two-dimensional sheet model was used to study the dynamics of reentry around a zone of functional block. The sheet is a set of parallel, continuous, and uniform cables, transversely interconnected by a brick-wall arrangement of fixed resistors. In accord with experimental observations on cardiac tissue, longitudinal propagation is continuous, whereas transverse propagation exhibits discontinuous features. The width and length of the sheet are 1.5 and 5 cm, respectively, and the anisotropy ratio is fixed at approximately 4:1. The membrane model is a modified Beeler-Reuter formulation incorporating faster sodium current dynamics. We fixed the basic wavelength and action potential duration of the propagating impulse by dividing the time constants of the secondary inward current by an integer K. Reentry was initiated by a standard cross-shock protocol, and the rotating activity appeared as curling patterns around the point of junction (the q-point) of the activation (A) and recovery (R) fronts. The curling R front always precedes the A front and is separated from it by the excitable gap. In addition, the R front is occasionally shifted abruptly through a merging with a slow-moving triggered secondary recovery front that is dissociated from the A front and q-point. Sustained irregular reentry associated with substantial excitable gap variations was simulated with short wavelengths (K = 8 and K = 4). Unsustained reentry was obtained with a longer wavelength (K = 2), leading to a breakup of the q-point locus and the triggering of new activation fronts.

A model study of extracellular stimulation of cardiac cells

Point source extracellular stimulation of a myocyte model was used to study the efficacy of excitation of cardiac cells, taking into account the shape of the pulse stimulus and its time of application in the cardiac cycle. The myocyte was modeled as a small cylinder of membrane (10 microns in diameter and 100 microns in length) capped at both ends and placed in an unbounded volume conductor. A Beeler-Reuter model modified for the Na+ dynamics served to simulate the membrane ionic current. The stimulus source was located on the cylinder axis, close to the myocyte (50 microns) in order to generate a nonlinear extracellular field (phi e). The low membrane impedance associated with the high frequency component of the make and break of the rectangular current pulse leads to a current flow across the membrane and an abrupt change in intracellular potential (phi i). Because the intracellular space is very small, phi i is nearly uniform over the length of the myocyte and the membrane potential (V = phi i-phi e) is governed by the applied field phi e. There is then a longitudinal gradient of membrane polarization which is the inverse of the gradient of extracellular potential. With an anodal (positive) pulse, for instance, the proximal portion of the myocyte is hyperpolarized and the distal portion is depolarized. Based on this principle and considering the voltage-dependent activation/inactivation dynamics of the membrane, it is shown that a cathodal (negative) pulse is the most efficacious stimulus at diastolic potentials, an anodal current is preferable during the plateau phase of the action potential, and a biphasic pulse is optimal during the relative refractory phase. Thus a biphasic pulse would constitute the best choice for maximum efficacy at all phases of the action potential.

On the interpretation of voltage-clamp data using the Hodgkin-Huxley model

This paper describes a method to extract membrane model parameters from experimental voltage-clamp records. The underlying theory is based on two premises: (1) the membrane dynamics can be described by a Hodgkin-Huxley (HH) model, and (2) the most reliable data provided by voltage clamp experiments are peak current (Ip) measurements. First, the steady-state characteristics of activation (x infinity) and inactivation (z infinity) must be estimated, and it is shown that Ip data provided by standard voltage-clamp stimulation protocols are sufficient for this purpose for the case of well-separated activation (tau x) and inactivation (tau z) time constants, tau x << tau z. Next, we propose a test (R test) to establish the suitability of the HH model to represent the data. When the HH model is applicable (successful R test), the procedure yields the degree of the gating variables, a range of maximum membrane conductance (g) values, and a tau x/tau z ratio that relates x infinity and z infinity to the Ip data. When additional information is available, such as the time of occurrence of Ip or an estimate of tau z from the late portion of the ionic current response, one can narrow down the value of g and estimate all the HH parameters and functions. Otherwise, when the R test is not successful, one can conclude that x infinity and z infinity have been incorrectly estimated because tau x and tau z are not sufficiently separated or that the HH model is not applicable to the data.

A model study of electric field interactions between cardiac myocytes

The transmission of excitation via electric field coupling was studied in a model comprising two myocytes abutted end-to-end and placed in an unbounded volume conductor. Each myocyte was modeled as a small cylinder of membrane (10 microns in diameter and 100 microns in length) capped at both ends. A Beeler-Reuter model modified for the Na+ current dynamics served to simulate the membrane ionic current. There was no resistive coupling between the myocytes and the intercellular junction consisted of closely apposed pre- and post-junctional membranes, separated by a uniform cleft distance. The membrane current crossing the prejunctional membrane during the action potential upstroke tends to flow out of the cleft, but it is partly prevented from doing so by the shunt resistance constituted by the cleft volume conductor. The prejunctional upstroke gives rise to a pulse of positive potential within the cleft which induces a small capacitive current across the post-junctional membrane to yield a small positive change in the intracellular potential in the post-junctional cell. The net result is an hyperpolarization of the post-junctional cleft membrane and a slight depolarization of the rest of the cell membrane since the extracellular potential outside of the cell is zero. The magnitude of this depolarization is quite small for a flat junctional membrane and it can be increased by membrane folding and interdigitation, so as to increase the junctional membrane area by a factor of 10 or more. Even then the post-junctional depolarization does not reach threshold when the extracellular potential around the post-junctional cell is effectively zero. Threshold depolarization occurs in the presence of a large decrease of post-junctional load, by increasing the junctional membrane capacitance and/or decreasing the volume of the post-junctional cell. Assuming that the normal resistive coupling between two cardiac myocytes is 1-4 M omega, our model study indicates that electric field coupling would then be about two orders of magnitude smaller. However, substantial enhancement of the efficacy of electric field transmission was observed in the case of cells with substantial junctional membrane folding.

Propagation of activation in cardiac muscle

The hypothesis of local circuit current flow underlying propagation of activation in cardiac muscle has been extensively documented by one-dimensional and two-dimensional simulation studies. The assumptions of spatially uniform membrane capacitance and membrane ionic properties yield simulation results that are in good agreement with experimental observations in healthy cardiac muscle, thereby indicating that differences in propagation velocity and action potential upstroke between longitudinal and transverse directions can be explained solely on the basis of anisotropic intercellular coupling. Two-dimensional model studies of anisotropic propagation have also stressed the more efficient charging of the membrane capacitance and higher safety factor of propagation in the transverse direction. These conditions favor the occurrence of longitudinal unidirectional block and the initiation of reentry via transverse propagation. The authors simulated rotating waves initiated by properly phased transverse and longitudinal plane waves in a two-dimensional sheet model. Sustained propagation requires a minimum anisotropy ratio, corresponding to a velocity ratio of about 4:1. It was found, for uniform anisotropy, that the central focus wandered slightly. A higher anisotropy ratio favors a more stable rotating pattern and a more restricted movement of the central focus.

Structural complexity effects on transverse propagation in a two-dimensional model of myocardium

A thin sheet of cardiac tissue was modeled as a set of resistively coupled excitable cables with membrane dynamics described by the modified Beeler Reuter model. Transverse connections have a resistance Rn and are regularly distributed with a spacing delta on any given cable, to provide alternating input and output junctions. Flat wave longitudinal propagation corresponds to propagation along a single continuous cable since all units of the network are functionally isolated due to the absence of transverse current flow. Events on a given cable during flat transverse propagation include electrotonic spread of potential from input to output junctions, action potential initiation at input junctions, and collision at output junctions. The propagating two-dimensional transverse wavefront is an undulating transmembrane potential surface with highs at the input junctions and lows at the output junctions. The action potential upstroke is also modulated in a periodic manner with minimum and maximum Vmax at the input and output junctions respectively. Thus, the network is capable of a diversity of dynamic behavior spatially distributed in relation to the specific pattern of transverse connections chosen. Overall, the behavior of the network model is in good agreement with available structural and electrophysiological data on myocardium. In addition, this network topology allows to handle more easily parameters governing propagation and to avoid very large matrices which are costly in computational effort and overall computer time.

Directional characteristics of action potential propagation in cardiac muscle. A model study

Propagation of an elliptic excitation wave front was studied in a two-dimensional model of a thin sheet of cardiac muscle. The sheet model of 2.5 x 10 mm consisted of a set of 100 parallel cables coupled through a regular array of identical transverse resistors. The membrane dynamics was represented by a modified Beeler-Reuter model. We defined the charging factor (CF) to represent by a single number the proportion of input current used to charge the membrane locally below threshold and showed that CF is inversely correlated with the time constant of the foot of the action potential (tau foot) during propagation on a cable. A safety factor of propagation (SF) was also defined for the upstroke of the action potential, with SF directly correlated with the maximum rate of depolarization (Vmax) and, for cablelike propagation, with propagation velocity. Propagation along the principal longitudinal axis of the elliptic wave front is cablelike but, in comparison with a flat wave front, transverse current flow provides a drag effect that somewhat reduces the propagation velocity, Vmax, SF, and CF. With a longitudinal-to-transverse velocity ratio of 3:1 or more, the wave front propagating along the principal transverse axis is essentially flat and is characterized by multiple collisions between successive pairs of input junctions on a given cable; Vmax, SF, and CF are larger than for longitudinal propagation, but CF is no longer correlated with tau foot. There are transient increases in propagation velocity and Vmax with distance from the stimulation site along both principal axes until stablized values are achieved, and a similar transient decrease in tau foot. Away from the principal axes, the action potential characteristics change progressively along the elliptic wave front.(ABSTRACT TRUNCATED AT 250 WORDS)

Computer model of excitation and recovery in the anisotropic myocardium. I. Rectangular and cubic arrays of excitable elements

A computer model of propagated excitation and recovery in anisotropic cardiac tissue is presented that consists of a large number of excitable elements whose subthreshold interactions are governed by the anisotropic bidomain theory but whose suprathreshold behavior (action potential) is largely preassigned. The model's performance was first tested in a two-dimensional configuration with uniform anisotropy; this method allowed comparison of simulated isochrones of excitation and extracellular electrograms with the results of experimental in vitro studies of cardiac tissue. Next the model was used to study propagated excitation in a three-dimensional region representing the anisotropic properties of the ventricular wall, with attention to the effects produced by variable fiber direction from "endocardium" to "epicardium."

Computer model of excitation and recovery in the anisotropic myocardium. II. Excitation in the simplified left ventricle

A computer model of propagated excitation and recovery in anisotropic cardiac tissue was described in the first report of this series. The model consists of a large number of excitable elements whose subthreshold interactions are governed by the anisotropic bidomain theory but whose suprathreshold behavior (action potential) is largely preassigned. As described in the first report, the model's performance was tested in rectangular and cubic arrays of excitable elements. This second report deals with three-dimensional simulations in a simplified left ventricle with anisotropy; specifically, the activation process in the "normal" ventricle is described (exemplified by the activation sequences started from various endocardial, intramural, and epicardial sites). To further substantiate our model's validity, we compare simulated epicardial and body-surface potential distributions with experimental findings in isolated canine hearts and with clinical evidence provided by electrocardiographic body-surface mapping.

Computer model of excitation and recovery in the anisotropic myocardium. III. Arrhythmogenic conditions in the simplified left ventricle

A computer model of propagated excitation and recovery in anisotropic cardiac tissue has been described in the first two reports of this series. The model consists of a large number of excitable elements whose subthreshold interactions are governed by the anisotropic bidomain theory but whose suprathreshold behavior (action potential) is largely preassigned. As described in the previous two reports, the model's performance was tested in rectangular and cubic arrays of excitable elements and in the "normal" three-dimensional simplified left ventricle with anisotropy. The present report deals with arrhythmogenic conditions in the simplified left ventricle with anisotropy and ventricular-gradient properties; specifically, we studied activation and recovery in the presence of an ischemic region and under various stimulation protocols. The aim of these simulations was to elucidate the role of reentry in the genesis of ventricular tachycardia. Our simulations produced reentrant activation as a result of appropriate endocardial stimulation.

A new cable model formulation based on Green's theorem

We describe an alternative formulation of the cable equation to model excitation in a cylinder of cardiac fiber. The formulation uses Green's theorem to develop equations for the extracellular and intracellular potential on either side of the excitable membrane, the dynamics of which are described by a Hodgkin-Huxley type model, without assuming that the radial current is zero. These equations are discretized to yield a system of linear equations which are solved at each instant in time. We found no qualitative differences between this approach and the standard cable model for parameters within accepted physiological limits. When the cable diameter is of the same order as the length constant the new formulation takes into account the intracellular potential change in the radial direction and gives an accurate expression of the conduction velocity.

The effect of torso geometry on magnetocardiographic isofield maps

Using a computer model of a realistically shaped human torso with lungs and intraventricular blood masses, we have assessed how torso geometry and composition affect the extracorporal magnetic field produced by a current dipole in the centre of the ventricular mass. The magnetic induction vector B arising from the dipole has been calculated at points of a precordial measuring grid and the influence of boundaries has been assessed qualitatively, by comparing contour maps of the B component normal to the torso's frontal plane. We found that the maps reflected relatively faithfully the underlying dipolar source for the homogeneous torso and even for the torso with lungs. However, the intraventricular blood masses caused a noticeable rotation of the maps' extrema. Both lungs and blood masses tended to swing the distribution towards the distribution that would have been caused by a dipole oriented along the anatomical axis of the heart.