Site of initial excitation and current threshold as a function of electrode radius in heart muscle

Modeling bipolar stimulation of cardiac tissue

Unipolar stimulation of cardiac tissue is often used in the design of cardiac pacemakers because of the low current required to depolarize the surrounding tissue at rest. However, the advantages of unipolar over bipolar stimulation are not obvious at shorter coupling intervals when the tissue near the pacing electrode is relatively refractory. Therefore, this paper analyzes bipolar stimulation of cardiac tissue. The strength-interval relationship for bipolar stimulation is calculated using the bidomain model and a recently developed parsimonious ionic current model. The strength-interval curves obtained using different electrode separations and arrangements (electrodes placed parallel to the fibers versus perpendicular to the fibers) indicate that bipolar stimulation results in more complex activation patterns compared to unipolar stimulation. An unusually low threshold stimulus current is observed when the electrodes are close to each other (a separation of 1 mm) because of break excitation. Unlike for unipolar stimulation, anode make excitation is not present during bipolar stimulation, and an abrupt switch from anode break to cathode make excitation can cause dramatic changes in threshold with very small changes in the interval. These results could impact the design of implantable pacemakers and defibrillators.

The strength-interval curve in cardiac tissue

The bidomain model describes the electrical properties of cardiac tissue and is often used to simulate the response of the heart to an electric shock. The strength-interval curve summarizes how refractory tissue is excited. This paper analyzes calculations of the strength-interval curve when a stimulus is applied through a unipolar electrode. In particular, the bidomain model is used to clarify why the cathodal and anodal strength-interval curves are different, and what the mechanism of the “dip” in the anodal strength-interval curve is.

MECHANISM OF ANODE BREAK EXCITATION IN THE HEART: THE RELATIVE INFLUENCE OF MEMBRANE AND ELECTROTONIC FACTORS

Two hypotheses for the mechanism of anode break excitation in cardiac tissue are electrotonic interaction between adjacent regions of depolarization and hyperpolarization, and a hyperpolarization-activated membrane current, if. We incorporate membrane kinetics proposed for if into the bidomain model with unequal anisotropy ratios. During unipolar stimulation, we find that:

(1) The mechanisms of cathode make, cathode break, and anode make excitation are insensitive to if.

(2) Both electrotonic interactions and if contribute to anode break excitation. In our simulations, if makes the dominant contribution.

(3) Electrotonic interactions cause the "dip" in the anodal strength-interval curve.

(4) Following anode break excitation, the wave front propagates in the direction perpendicular to the fibers.

(5) if improves the agreement between the measured and calculated strength-interval curves.

We suggest three experiments to determine the mechanism of anode break excitation: measure the site and timing of initial excitation, or use drugs to suppress if.

Subthreshold parameters of cardiac tissue in a bi-layer computer model of heart failure

Current density threshold and liminal area are subthreshold parameters of the cardiac tissue that indicate its susceptibility to external and internal stimulations. Extensive experimental and theoretical research has been conducted to quantify these two parameters in normal conditions for both animal and human models. Here we employed a 2D numerical model of human cardiac tissue to assess these subthreshold parameters under the pathological conditions of heart failure and fibrosis. Stimuli were applied over an area ranging from 0.04 to 1 mm² using various pulse durations. The current density threshold decreased with increasing stimulation area or pulse duration. No significant changes were found in both parameters between control conditions and heart failure in the atrial tissue, while in the ventricular tissue, heart failure resulted in significantly reduced excitability with higher stimulation current magnitudes needed for excitation and larger liminal areas. This results from the specific ionic remodeling in ventricular heart failure that affects both subthreshold active currents such as I(K₁) and connexin 43 conductance. In fibrosis, increased fibroblast to myocyte coupling coefficient had a non-linear influence on current density thresholds, with an initial increase of current magnitude followed by a relaxation phase down to the current magnitude threshold for the control condition with no fibrosis. The results show that subthreshold excitation properties of the myocardium are influenced in a complex, non-linear manner by cardiac pathologies. Such observations may contribute to our understanding of impulse capturing properties, relevant, for example, for the generation of ectopic foci-originated arrhythmias and for the efficient design of cardiac stimulating electrodes.

Effect of electrode surface area on thresholds for AC stimulation and ventricular fibrillation

Unintended, weak AC stimulation (leakage currents) from medical devices can cause blood pressure collapse and ventricular fibrillation (VF), potentially even death. Yet, little is understood about AC cardiac stimulation. The objective of this paper is to establish the relationship between the stimulation and VF thresholds for electrode size and stimulation frequency. Twenty-four retired male breeder guinea pigs were anesthetized with isoflurane, a tracheotomy and thoracotomy were performed, and vitals were monitored using the lead II ECG and an optical plethysmograph. The circular flat ends of eleven stainless steel rods were used as electrodes with areas ranging from 0.1 to 26.79 mm2. In the first study, 60-Hz AC stimuli of 5 s duration were delivered with strengths from 25-3000 microA or until VF was induced. In the second group, the current thresholds at 20, 40, 80, and 160 Hz were determined at electrode areas of 0.2, 2.01, and 16.4 mm2. Reactions were categorized as having no effect, having some effect (EFFECT, typically blood pressure collapse), and inducing VF. On a log-log scale, electrode radii had a piecewise-linear relationship with the current thresholds for EFFECT (p < 0.005) and VF (p < 0.01). The liminal area determined by the piecewise-linear fit was 2.0 and 2.84 mm2 for EFFECT and VF, respectively. Above the liminal area, the threshold increased proportional to r(1.25) and r(0.95) (r = radius of electrode), for EFFECT and VF, respectively. Based on these experimental results, we present a theoretical framework to explain the electrode size-stimulation threshold variation for both low strength AC stimulation and VF initiation.

Transcranial direct current stimulation: a computer-based human model study

OBJECTIVES

Interest in transcranial direct current stimulation (tDCS) in clinical practice has been growing, however, the knowledge about its efficacy and mechanisms of action remains limited. This paper presents a realistic magnetic resonance imaging (MRI)-derived finite element model of currents applied to the human brain during tDCS.

EXPERIMENTAL DESIGN

Current density distributions were analyzed in a healthy human head model with varied electrode montages. For each configuration, we calculated the cortical current density distributions. Analogous studies were completed for three pathological models of cortical infarcts.

PRINCIPAL OBSERVATIONS

The current density magnitude maxima injected in the cortex by 1 mA tDCS ranged from 0.77 to 2.00 mA/cm(2). The pathological models revealed that cortical strokes, relative to the non-pathological solutions, can elevate current density maxima and alter their location.

CONCLUSIONS

These results may guide optimized tDCS for application in normal subjects and patients with focal brain lesions.

Basic assessment of paced activation sequence mapping: implications for practical use

Some experiences support the use of atrial paced activation sequence mapping, but there is no systematic study assessing its spatial resolution, reproducibility, and influence of pacing parameters. The aim of this study was to evaluate these issues by using a 24-pole catheter positioned at the atrial aspect of the tricuspid and mitral annuli in 15 patients. Bipolar pacing was performed at two sites (right and left atria), 2 cycle lengths (300 and 500 ms) and two outputs (twice and tenfold the late diastolic threshold voltage for 2-ms pulses). The elapsed time between the atrial activation at the two dipoles adjacent to the pacing dipole (activation time [AT]) was measured during each pacing sequence. Changes in cycle length did not modify the AT. The increase in voltage slightly modified the AT (maximum -2 ms at the RA; 95% CI -3 to -1 ms) due to a greater shortening of the conduction time to the dipole located next to the anode. The 95% limits of the intraobserver and interobserver agreements in the AT measurement were -2 to 3 ms and -3 to 3 ms, respectively. The spatial resolution was studied in ten patients by measuring the AT during pacing from each dipole of a 20-pole catheter with a 1-3-1 mm interelectrode distance. The mean AT change was 10 +/- 4 ms per 6 mm of pacing site displacement (95% CI 8-11 ms, range 2.5-20 ms). In conclusion, paced atrial activation sequence analysis is reproducible, accurate, and relatively independent of pacing parameters.

Field Stimulation of 2-D Sheets of Excitable Tissue

This paper develops equations for the transmembrane potentials (Vm) that occur in two-dimensional (2-D) sheets of tissue in response to field stimulation from an electrode near but not on the surface of the tissue. Comparison of results with those for one dimension shows that an additional term is present in the 2-D equations that influences the evolution of Vm in the interval between the end of the stimulus and the active propagation that may follow. The results provide an analytical framework for understanding Vm in response to field stimulation in two dimensions, both during the tissue's critical linear phase and thereafter.

Artifacts, assumptions, and ambiguity: Pitfalls in comparing experimental results to numerical simulations when studying electrical stimulation of the heart

Insidious experimental artifacts and invalid theoretical assumptions complicate the comparison of numerical predictions and observed data. Such difficulties are particularly troublesome when studying electrical stimulation of the heart. During unipolar stimulation of cardiac tissue, the artifacts include nonlinearity of membrane dyes, optical signals blocked by the stimulating electrode, averaging of optical signals with depth, lateral averaging of optical signals, limitations of the current source, and the use of excitation-contraction uncouplers. The assumptions involve electroporation, membrane models, electrode size, the perfusing bath, incorrect model parameters, the applicability of a continuum model, and tissue damage. Comparisons of theory and experiment during far-field stimulation are limited by many of these same factors, plus artifacts from plunge and epicardial recording electrodes and assumptions about the fiber angle at an insulating boundary. These pitfalls must be overcome in order to understand quantitatively how the heart responds to an electrical stimulus. (c) 2002 American Institute of Physics.

How electrode size affects the electric potential distribution in cardiac tissue

We investigate the effect of electrode size on the transmembrane potential distribution in the heart during electrical stimulation. The bidomain model is used to calculate the transmembrane potential in a three-dimensional slab of cardiac tissue. Depolarization is strongest under the electrode edge. Regions of depolarization are adjacent to regions of hyperpolarization. The average ratio of peak depolarization to peak hyperpolarization is a function of electrode radius, but over a broad range is close to three.

Electrical stimulation of cardiac tissue by a bipolar electrode in a conductive bath

A three-dimensional (3-D) computer simulation of the electrical stimulation of passive cardiac tissue from a bipolar electrode placed within a conductive bath is presented. Through the bidomain model, the syncytial and anisotropic properties of cardiac tissue are taken into account; tissues with equal anisotropy and no transverse coupling are also considered. The membrane is represented by a capacitor and passive resistor in parallel. Located within an isotropic bath, the bipolar electrode is oriented either perpendicular or parallel to the tissue surface. For anisotropic tissue with a small cathode-tissue separation, the tissue surface is highly depolarized under the cathode with the depolarization persisting a considerable distance from the electrode in the transverse fiber direction. Adjacent to this region in the longitudinal direction, areas of hyperpolarization exist. At large distances from the cathode, the tissue surface is hyperpolarized in all directions when the electrode axis is perpendicular to the tissue. In the parallel case, surface depolarization creates buried regions of hyperpolarization. For the perpendicular configuration, the ratio of the steady-state maximum depolarization to steady-state maximum hyperpolarization, an estimate of the ratio of anodal to cathodal threshold, decreases rapidly with increasing cathode-tissue separation. In the parallel case, the depth of the conductive bath significantly affected the transmembrane potential distribution in the tissue. The use of a 3-D model more realistically simulates real-life electrical stimulation (such as stimulation with an implantable pacemaker) and provides insight into the effect of the volume conductor adjacent to the tissue.

Discrete versus syncytial tissue behavior in a model of cardiac stimulation--II: Results of simulation

The research presented here combines mathematical modeling and computer simulation in developing a new model of the membrane polarization induced in the myocardium by the applied electric field. Employing this new model termed the "period" bidomain model, the steady-state distribution of the transmembrane potential is calculated on a slice of cardiac tissue composed of abutting myocytes and subjected to two point-source extracellular current stimuli. The goal of this study is to examine the relative contribution of cellular discreteness and macroscopic syncytial tissue behavior in the mechanism by which the applied electric field alters the transmembrane potential in cardiac muscle. The results showed the existence of oscillatory changes in the transmembrane potential at cell ends owing to the local resistive inhomogeneities (gap-junctions). This low-magnitude sawtooth component in the transmembrane potential is superimposed over large-scale transmembrane potential excursions associated with the syncytial (collective) fiber behavior. The character of the cardiac response to stimulation is determined primarily by the large-scale syncytial tissue behavior. The sawtooth contributes to the overall tissue response only in regions where the large-scale transmembrane potential component is small.

Strength-interval curves for cardiac tissue predicted using the bidomain model

INTRODUCTION

Strength-interval curves are predicted for unipolar anodal and cathodal stimulation of cardiac muscle.

METHODS AND RESULTS

Cardiac tissue is represented by the bidomain model, and the active properties of the membrane are described by the Beeler-Reuter model. Two successive stimuli (S1 and S2) are delivered through a single extracellular electrode. The S2 threshold is determined as a function of the S1-S2 interval, for anodal and cathodal S2 stimuli with 2-, 5-, 10-, and 20-msec durations. Each of the resulting cathodal and anodal strength-interval curves is divided into two parts: one section corresponding to make stimulation (long intervals) and the other section corresponding to break stimulation (short intervals). Generally, the cathodal strength-interval curves are decreasing functions of interval, except for an anomalous section of the 20-msec duration cathodal curve in the interval range from 310 to 318 msec. At short intervals, the anodal strength-interval curve contains a deep dip, which is more prominent for longer S2 durations. The cathodal threshold is less than the anodal threshold for all intervals except those corresponding to the end of the refractory period.

CONCLUSION

The bidomain model predicts complex anodal and cathodal strength-interval curves, with the anodal curve containing a dip (supernormal stimulation). These results resemble the experimental observations of Dekker.

A mathematical model of make and break electrical stimulation of cardiac tissue by a unipolar anode or cathode

Numerical simulations of electrical stimulation of cardiac tissue using a unipolar extracellular electrode were performed. The bidomain model with unequal anisotropy ratios represented the tissue, and the Beeler-Reuter model represented the active membrane properties. Four types of excitation were considered: cathode make (CM), anode make (AM), cathode break (CB), and anode break (AB). The mechanisms of excitation were: for CM, tissue under the cathode was depolarized to threshold; for AM, tissue at a virtual cathode was depolarized to threshold; for CB, a long cathodal pulse produced a steady-state depolarization under the cathode and hyperpolarization at a virtual anode. At the end (break) of the pulse, the depolarization diffused into the hyperpolarized tissue, resulting in excitation. For AB, a long anodal pulse produced a steady-state hyperpolarization under the anode and depolarization at a virtual cathode. At the end (break) of the pulse, the depolarization diffused into the hyperpolarized tissue, resulting in excitation. For AB stimulation, decay of the hyperpolarization faster than that of the depolarization was necessary. The thresholds for rheobase and diastolic CM, AM, CB, and AB stimulation were 0.038, 0.41, 0.49, and 5.3 mA, respectively, for an electrode length of 1 mm and a surface area of 1.5 mm2. Threshold increased as the size of the electrode increased. The strength-duration curves for CM and AM were similar except when the duration was shorter than 0.2 ms, in which case the AM threshold rose more quickly with decreasing duration than did the CM threshold. CM and AM resulted in similar strength-frequency curves. The model agrees qualitatively, but (in some cases) not quantitatively, with experiments.

Determination of current density distributions generated by electrical stimulation of the human cerebral cortex

With the use of a 3-dimensional finite element model of the human brain based on structural data from MRI scans, we simulated patterns of current flow in the cerebral hemisphere with different types of electrical stimulation. Five different tissue types were incorporated into the model based on conductivities taken from the literature. The boundary value problem derived from Laplace's equation was solved with a quasi-static approximation. Transcranial electrical stimulation with scalp electrodes was poorly focussed and required high levels of current for stimulation of the cortex. Direct cortical stimulation with bipolar (adjacent) electrodes was found to be very effective in producing localized current flows. Unipolar cortical stimulation (with a more distant reference electrode) produced higher current densities at the same stimulating current as did bipolar stimulation, but stimulated a larger region of the cortex. With the simulated electrodes resting on the pia-arachnoid, as usually occurs clinically, there was significant shunting of the current (7/8 of the total current) through the CSF. Possible changes in electrodes and stimulation parameters that might improve stimulation procedures are considered.

Initiation of sustained ventricular tachyarrhythmias in a canine model of chronic myocardial infarction: importance of the site of stimulation

The importance of the site of stimulation to the initiation of sustained ventricular tachyarrhythmias was determined in 24 adult mongrel dogs. Studies were performed 3-30 days after two-stage occlusion of the mid- or distal left anterior descending coronary artery, modified by a reperfusion stage. Unipolar cathodal stimuli of twice-threshold intensity and 2 msec duration were introduced at five to 24 sites in each dog in the distribution of occluded and nonoccluded vessels. Strength-interval curves were constructed from 232 measurements at these sites and local properties of excitability and refractoriness were correlated with the ability to initiate arrhythmias. All dogs had sustained ventricular tachyarrhythmias inducible from at least one site. Intramyocardial sites with normal excitability and refractoriness within 2 cm of an area of infarction were most often successful (27 of 44, 61%) in the initiation of sustained arrhythmias. Less successful sites included normal left ventricular plunge electrode sites less than 2 cm from an area of infarction (eight of 32, 25%) (p = 0.002), left ventricular plunge electrode sites within an area of infarction (20 of 103, 19%) (p less than 0.001), normal right ventricular sites (five of 24, 21%) (p less than 0.001), and endocardial catheter sites (six of 29, 21%), (p less than 0.001). These findings suggest that local properties of excitability and refractoriness at the site of stimulation, as well as anatomic and geometric factors, may be critical in the initiation of sustained ventricular tachyarrhythmias using the technique of programmed electrical stimulation.