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

Bidomain modeling of electrical and mechanical properties of cardiac tissue

Throughout the history of cardiac research, there has been a clear need to establish mathematical models to complement experimental studies. In an effort to create a more complete picture of cardiac phenomena, the bidomain model was established in the late 1970s to better understand pacing and defibrillation in the heart. This mathematical model has seen ongoing use in cardiac research, offering mechanistic insight that could not be obtained from experimental pursuits. Introduced from a historical perspective, the origins of the bidomain model are reviewed to provide a foundation for researchers new to the field and those conducting interdisciplinary research. The interplay of theory and experiment with the bidomain model is explored, and the contributions of this model to cardiac biophysics are critically evaluated. Also discussed is the mechanical bidomain model, which is employed to describe mechanotransduction. Current challenges and outstanding questions in the use of the bidomain model are addressed to give a forward-facing perspective of the model in future studies.

Highly trabeculated structure of the human endocardium underlies asymmetrical response to low-energy monophasic shocks

Novel low-energy defibrillation therapies are thought to be driven by virtual-electrodes (VEs), due to the interaction of applied monophasic electric shocks with fine-scale anatomical structures within the heart. Significant inter-species differences in the cardiac (micro)-anatomy exist, however, particularly with respect to the degree of endocardial trabeculations, which may underlie important differences in response to low-energy defibrillation protocols. Understanding the interaction of monophasic electric fields with the specific human micro-anatomy is therefore imperative in facilitating the translation and optimisation of these promising experimental therapies to the clinic. In this study, we sought to investigate how electric fields from implanted devices interact with the highly trabeculated human endocardial surface to better understand shock success in order to help optimise future clinical protocols. A bi-ventricular human computational model was constructed from high resolution (350 μm) ex-vivo MR data, including anatomically accurate endocardial structures. Monophasic shocks were applied between a basal right ventricular catheter and an exterior ground. Shocks of varying strengths were applied with both anodal [positive right ventricle (RV) electrode] and cathodal (negative RV electrode) polarities at different states of tissue refractoriness and during induced arrhythmias. Anodal shocks induced isolated positive VEs at the distal side of "detached" trabeculations, which rapidly spread into hyperpolarised tissue on the surrounding endocardial surfaces following the shock. Anodal shocks thus depolarised more tissue 10 ms after the shock than cathodal shocks where the propagation of activation from VEs induced on the proximal side of "detached" trabeculations was prevented due to refractory endocardium. Anodal shocks increased arrhythmia complexity more than cathodal shocks during failed anti-arrhythmia shocks. In conclusion, multiple detached trabeculations in the human ventricle interact with anodal stimuli to induce multiple secondary sources from VEs, facilitating more rapid shock-induced ventricular excitation compared to cathodal shocks. Such a mechanism may help explain inter-species differences in response to shocks and help to develop novel defibrillation strategies.

Cardiac strength-interval curves calculated using a bidomain tissue with a parsimonious ionic current

The strength-interval curve plays a major role in understanding how cardiac tissue responds to an electrical stimulus. This complex behavior has been studied previously using the bidomain formulation incorporating the Beeler-Reuter and Luo-Rudy dynamic ionic current models. The complexity of these models renders the interpretation and extrapolation of simulation results problematic. Here we utilize a recently developed parsimonious ionic current model with only two currents—a sodium current that activates rapidly upon depolarization I_Na and a time-independent inwardly rectifying repolarization current I_K—which reproduces many experimentally measured action potential waveforms. Bidomain tissue simulations with this ionic current model reproduce the distinctive dip in the anodal (but not cathodal) strength-interval curve. Studying model variants elucidates the necessary and sufficient physiological conditions to predict the polarity dependent dip: a voltage and time dependent I_Na, a nonlinear rectifying repolarization current, and bidomain tissue with unequal anisotropy ratios.

Effects of premature anodal stimulations on cardiac transmembrane potential and intracellular calcium distributions computed by anisotropic Bidomain models

AIMS

Cardiac unipolar electrode stimulations induce a particular structure of the transmembrane potential distribution (Vm), called virtual electrode polarization (VEP), which plays an important role in the mechanisms of cardiac excitation, reentry induction, and ventricular defibrillation. Recent experimental studies, based on the optical mapping techniques, have shown that premature stimulations also induce significant changes in the intracellular calcium (Cai) spatial distribution. The aim of this work is to investigate and compare by means of numerical simulations the morphology of the Vm and Cai patterns, generated by applying an S1-S2 stimulation protocol with a premature S2 anodal pulse.

METHODS AND RESULTS

We perform parallel finite element simulations of a three-dimensional orthotropic Bidomain model on a block of ventricular tissue by using four membrane models of two species (guinea pig and rabbit), that incorporate the phenomenological or more detailed mechanistic descriptions of the calcium dynamics. During the S2 anodal stimulus, the Cai spatial distribution, computed with all the considered models, presents a configuration similar to the typical VEP pattern of Vm, with a minimum inside the virtual anode and two maxima in the virtual cathodes. After the S2 stimulus turns off, the anode break excitation mechanism yields a Vm pattern exhibiting a clearly propagating wavefront. Differently, the Cai patterns do not show a clear separation between the resting and the activated regions, with the exception of one of the phenomenological models considered, but they show warped dog-bone shaped equi-level lines around an elevation in the virtual anode region.

CONCLUSION

The VEP pattern of the Cai spatial distribution during the S2 stimulus is in agreement with the previous experimental studies. Moreover, the Cai minimum in the virtual anode can be mainly attributable to the outflow of calcium ions produced by the sodium-calcium (NCX) exchanger, without a significant contribution of the ICaL current.

Intracellular calcium and the mechanism of the dip in the anodal strength-interval curve in cardiac tissue

BACKGROUND

The strength-interval (SI) curve is an important measure of refractoriness in cardiac tissue. The anodal SI curve contains a "dip" in which the S2 threshold increases with interval. Two explanations exist for this dip: (1) electrotonic interaction between regions of depolarization and hyperpolarization; and (2) the sodium-calcium exchange (NCX) current. The goal of this study is to use mathematical modeling to determine which explanation is correct.

METHODS AND RESULTS

The bidomain model represents cardiac tissue and the Luo-Rudy model describes the active membrane. The SI curve is determined by applying a threshold stimulus at different time intervals after a previous action potential. During space-clamped and equal-anisotropy-ratios simulations, anodal excitation does not occur. During unequal-anisotropy-ratios simulations, electrotonic currents, not membrane currents, are present during the few milliseconds before excitation. The dip disappears with no NCX current, but is present with 50% or 75% reduction of it. The calcium-induced-calcium-release (CICR) current has little effect on the dip.

CONCLUSIONS

These results indicate that neither the NCX nor the CICR current is responsible for the dip in the anodal SI curve. It is caused by the electrotonic interaction between regions of depolarization and hyperpolarization following the S2 stimulus.

The Mechanical Bidomain Model: A Review

The mechanical bidomain model is a new mathematical description of the elastic behavior of cardiac tissue. Its primary advantage over previous models is that it accounts for forces acting across the cell membrane arising form differences in the displacement of the intracellular and extracellular spaces. In this review, I describe the development of the mechanical bidomain model. I emphasize new predictions of the model, such as the existence of boundary layers at the tissue surface where the membrane forces are large, and pressure differences between the intracellular and extracellular spaces. Although the theoretical analysis is quite mathematical, I highlight the types of experiments that could be used to test the model predictions. Finally, I present open questions about the mechanical bidomain model that may be productive future directions for research.

Development of an anatomically detailed MRI-derived rabbit ventricular model and assessment of its impact on simulations of electrophysiological function

Recent advances in magnetic resonance (MR) imaging technology have unveiled a wealth of information regarding cardiac histoanatomical complexity. However, methods to faithfully translate this level of fine-scale structural detail into computational whole ventricular models are still in their infancy, and, thus, the relevance of this additional complexity for simulations of cardiac function has yet to be elucidated. Here, we describe the development of a highly detailed finite-element computational model (resolution: approximately 125 microm) of rabbit ventricles constructed from high-resolution MR data (raw data resolution: 43 x 43 x 36 microm), including the processes of segmentation (using a combination of level-set approaches), identification of relevant anatomical features, mesh generation, and myocyte orientation representation (using a rule-based approach). Full access is provided to the completed model and MR data. Simulation results were compared with those from a simplified model built from the same images but excluding finer anatomical features (vessels/endocardial structures). Initial simulations showed that the presence of trabeculations can provide shortcut paths for excitation, causing regional differences in activation after pacing between models. Endocardial structures gave rise to small-scale virtual electrodes upon the application of external field stimulation, which appeared to protect parts of the endocardium in the complex model from strong polarizations, whereas intramural virtual electrodes caused by blood vessels and extracellular cleft spaces appeared to reduce polarization of the epicardium. Postshock, these differences resulted in the genesis of new excitation wavefronts that were not observed in more simplified models. Furthermore, global differences in the stimulus recovery rates of apex/base regions were observed, causing differences in the ensuing arrhythmogenic episodes. In conclusion, structurally simplified models are well suited for a large range of cardiac modeling applications. However, important differences are seen when behavior at microscales is relevant, particularly when examining the effects of external electrical stimulation on tissue electrophysiology and arrhythmia induction. This highlights the utility of histoanatomically detailed models for investigations of cardiac function, in particular for future patient-specific modeling.

A mechanism for the upper limit of vulnerability

BACKGROUND

The strongest shock that induces reentry in the heart is the upper limit of vulnerability (ULV). In order to understand defibrillation, one must know what causes the ULV.

OBJECTIVE

The goal of this study was to examine the mechanism of the upper limit of vulnerability.

METHODS

Numerical simulations of cardiac tissue were performed using the bidomain model. An S2 shock was applied during the refractory period of the S1 action potential, and results using a smooth curving fiber geometry were compared with results using a smooth plus random fiber geometry.

RESULTS

When using a smooth fiber geometry only, no ULV was observed. However, when a random fiber geometry was included, the ULV was present. The difference arises from the fate of the shock-induced break wave front when it reaches the edge of the tissue hyperpolarized by the shock (the virtual anode).

CONCLUSION

Our numerical simulations suggest that local heterogeneities throughout the tissue may be crucial for determining the fate of the shock-induced wave front at the edge of the virtual anode, and therefore play an important role in the mechanism underlying the ULV.

Calculation of optical signal using three-dimensional bidomain/diffusion model reveals distortion of the transmembrane potential

Optical mapping experiments allow investigators to view the effects of electrical currents on the transmembrane potential, V(m), as a shock is applied to the heart. One important consideration is whether the optical signal accurately represents V(m). We have combined the bidomain equations along with the photon diffusion equation to study the excitation and emission of photons during optical mapping of cardiac tissue. Our results show that this bidomain/diffusion model predicts an optical signal that is much smaller than V(m) near a stimulating electrode, a result consistent with experimental observations. Yet, this model, which incorporates the effect of lateral averaging, also reveals an optical signal that overestimates V(m) at distances >1 mm away from the electrode. Although V(m) falls off with distance r from the electrode as exp(-r/lambda)/r, the optical signal decays as a simple exponential, exp(-r/lambda). Moreover, regions of hyperpolarization adjacent to a cathode are emphasized in the optical signal compared to the region of depolarization under the cathode. Imaging methods utilizing optical mapping techniques will need to account for these distortions to accurately reconstruct V(m).

The role of photon scattering in optical signal distortion during arrhythmia and defibrillation

Optical mapping of arrhythmias and defibrillation provides important insights; however, a limitation of the technique is signal distortion due to photon scattering. The goal of this experimental/simulation study is to investigate the role of three-dimensional photon scattering in optical signal distortion during ventricular tachycardia (VT) and defibrillation. A three-dimensional realistic bidomain rabbit ventricular model was combined with a model of photon transport. Shocks were applied via external electrodes to induce sustained VT, and transmembrane potentials (V(m)) were compared with synthesized optical signals (V(opt)). Fluorescent recordings were conducted in isolated rabbit hearts to validate simulation results. Results demonstrate that shock-induced membrane polarization magnitude is smaller in V(opt) and in experimental signals as compared to V(m). This is due to transduction of potentials from weakly polarized midmyocardium to the epicardium. During shock-induced reentry and in sustained VT, photon scattering, combined with complex wavefront dynamics, results in optical action potentials near a filament exhibiting i), elevated resting potential, ii), reduced amplitude relative to pacing, and iii), dual-humped morphologies. A shift of up to 4 mm in the phase singularity location was observed in V(opt) maps when compared to V(m). This combined experimental/simulation study provides an interpretation of optical recordings during VT and defibrillation.

Simulations of optical mapping during electroporation

Experiments using optical mapping suggest that electroporation occurs in cardiac tissue when the transmembrane potential, Vm, is observed to be significantly less than +/- 400 mV. Our hypothesis, which we test by numerical simulation, is that Vm is greater than +/- 400 mV at the tissue surface, but optical mapping underestimates Vm because it averages over depth. Results indicate a significant underestimation of Vm. Experimental studies indicate a depolarization of the resting transmembrane potential, Vrest, after a strong shock. In a homogeneous model, electroporation only occurs near the tissue surface. Just as Vm during the stimulus is underestimated due to averaging, we hypothesize that the depolarization of Vrest is also underestimated.

Examination of depth-weighted optical signals during cardiac optical mapping: A simulation study

Optical mapping has become a powerful tool to explore complex cardiac propagation. Many experiments and studies claimed that the fluorescence obtained from tissue surface is the averaged response of the transmembrane potential upon probing depth rather than only on the surface. With the electrical propagation model and the photon transport model, the effects of depth-weighted optical signals are examined both during a normal excitation wave and a spiral wave. Our results indicate that depth-weighted optical signals may infer cardiac activation dynamics, such as the mode and the direction of the propagation, the spatial distribution of depolarization or repolarization.

The effect of the cut surface during electrical stimulation of a cardiac wedge preparation

Optical mapping from the cut surface of a "wedge preparation" allows observation inside the heart wall, below the epicardium or endocardium. We use numerical simulations based on the bidomain model to illustrate how the transmembrane potential is influenced by the cut surface. The distribution of transmembrane potential around a unipolar cathode depends on the fiber angle. For intermediate angles, hyperpolarization appears on only one side of the electrode, and is large and widespread.

The effect of the fiber curvature gradient on break excitation in cardiac tissue

BACKGROUND

Break excitation has been hypothesized as a mechanism for the initiation of reentry in cardiac tissue. One way break excitation can occur is by virtual electrodes formed due to a curving fiber geometry. In this article, we are concerned with the relationship between the peak gradient of fiber curvature and the threshold for break stimulation and the initiation of reentry.

METHODS

We calculate the maximum gradient of fiber curvature for different scales of fiber geometry in a constant tissue size (20x20 mm), and also examine the mechanisms by which reentry initiation fails.

RESULTS

For small peak gradients, reentry fails because break excitation does not occur. For larger peak gradients, reentry fails because break excitation fails to develop into full-scale reentry. For strong stimuli above the upper limit of vulnerability, reentry fails because the break excitation propagates through the hyperpolarized region and then encounters refractory tissue, causing the wave front to die.

Two-Photon Excitation of di-4-ANEPPS for Optical Recording of Action Potentials in Rabbit Heart

Cardiac action potentials have been measured with single-photon excitation (SPE) of transmembrane voltage-sensitive fluorescent dye. Two-photon excitation (TPE) may have advantages for localization and depth of the tissue region from which the action potential is measured. However measurements of action potentials with SPE have not been demonstrated. We sought to develop a method for TPE of di-4-ANEPPS and test whether the method yields voltage-dependent fluorescence in cardiac tissue. We modified our SPE and ratio-metric fluorescence recording system to use a femtosecond pulsed near-infrared laser. Modifications were made to enhance fluorescence collection efficiency and to block infrared laser light from entering the fluorescence collection system. Fluorescence was collected simultaneously in green (510-570 nm) and red (590-700 nm) wavelength bands. Action potentials were observed in the ratio of the green signal to the red signal, but were not observed above the noise level in either of the individual signals. Incorporation of a common-mode noise subtraction method revealed action potentials in green and red signals. We also found that the di-4-ANEPPS fluorescence emission spectrum for TPE at 930 nm was similar to the emission spectrum for SPE at 488 nm. The multiphoton method may be beneficial for highly localized cardiac optical measurements.

Examination of stimulation mechanism and strength-interval curve in cardiac tissue

Understanding the basic mechanisms of excitability through the cardiac cycle is critical to both the development of new implantable cardiac stimulators and improvement of the pacing protocol. Although numerous works have examined excitability in different phases of the cardiac cycle, no systematic experimental research has been conducted to elucidate the correlation among the virtual electrode polarization pattern, stimulation mechanism, and excitability under unipolar cathodal and anodal stimulation. We used a high-resolution imaging system to study the spatial and temporal stimulation patterns in 20 Langendorff-perfused rabbit hearts. The potential-sensitive dye di-4-ANEPPS was utilized to record the electrical activity using epifluorescence. We delivered S1-S2 unipolar point stimuli with durations of 2-20 ms. The anodal S-I curves displayed a more complex shape in comparison with the cathodal curves. The descent from refractoriness for anodal stimulation was extremely steep, and a local minimum was clearly observed. The subsequent ascending limb had either a dome-shaped maximum or was flattened, appearing as a plateau. The cathodal S-I curves were smoother, closer to a hyperbolic shape. The transition of the stimulation mechanism from break to make always coincided with the final descending phase of both anodal and cathodal S-I curves. The transition is attributed to the bidomain properties of cardiac tissue. The effective refractory period was longer when negative stimuli were delivered than for positive stimulation. Our spatial and temporal analyses of the stimulation patterns near refractoriness show always an excitation mechanism mediated by damped wave propagation after S2 termination.

How the spatial frequency of polarization influences the induction of reentry in cardiac tissue

Influences of spatial frequency of polarization.

INTRODUCTION

The mechanism by which an electric field induces a rotor during cross-field stimulation of cardiac tissue is not entirely known. Different heterogeneous aspects of cardiac tissue have been offered as possible theories, a prominent one being fiber curvature. The polarization produced when an electric field is applied to a sheet of tissue is varied over many spatial frequencies, depending upon the fiber angle. This article compares the effect of high and low spatial frequencies of polarization on reentry induction.

METHODS AND RESULTS

We incorporate a randomized fiber angle geometry into a two-dimensional active cardiac tissue model with unequal anisotropy ratios already exhibiting smooth, curving fibers. We simulate cross-field stimulation to initiate reentry in the tissue model, and compare the electric field thresholds at different S1-S2 intervals for tissue with randomized fiber angles, tissue with a smooth fiber geometry, and tissue with randomized fiber angles plus smooth, curving fibers. The tissue with both small, random fiber angles and curving fibers has a significantly lower threshold for reentry at certain intervals on the strength-interval curve than for the two cases individually.

CONCLUSION

Cardiac tissue exhibiting a random fiber geometry in conjunction with a smooth fiber geometry includes high and low spatial frequencies of polarization that may have an effect on the mechanism for reentry at certain S1-S2 intervals. Low spatial frequency regions of hyperpolarization carve out excitable pathways, and high spatial frequency regions provide the large gradient of transmembrane potential required to initiate break excitation.

Examination of optical depth effects on fluorescence imaging of cardiac propagation

Optical mapping with voltage-sensitive dyes provides a high-resolution technique to observe cardiac electrodynamic behavior. Although most studies assume that the fluorescent signal is emitted from the surface layer of cells, the effects of signal attenuation with depth on signal interpretation are still unclear. This simulation study examines the effects of a depth-weighted signal on epicardial activation patterns and filament localization. We simulated filament behavior using a detailed cardiac model, and compared the signal obtained from the top (epicardial) layer of the spatial domain with the calculated weighted signal. General observations included a prolongation of the action upstroke duration, early upstroke initiation, and reduction in signal amplitude in the weighted signal. A shallow filament was found to produce a dual-humped action potential morphology consistent with previously reported observations. Simulated scroll wave breakup exhibited effects such as the false appearance of graded potentials, apparent supramaximal conduction velocities, and a spatially blurred signal with the local amplitude dependent upon the immediate subepicardial activity; the combination of these effects produced a corresponding change in the accuracy of filament localization. Our results indicate that the depth-dependent optical signal has significant consequences on the interpretation of epicardial activation dynamics.

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.

Effects of Elevated Extracellular Potassium Ion Concentration on Anodal Excitation of Cardiac Tissue

INTRODUCTION

Anodal excitation of cardiac tissue occurs by two mechanisms: "make" and "break." Anodal strength-interval curves are divided into two sections, with break excitation occurring at short intervals and make at long intervals. Our goal is to determine how an elevated extracellular potassium ion concentration, [K]o, affects the mechanism of anodal excitation and influences the anodal strength-interval curve.

METHODS AND RESULTS

Computer simulations of unipolar stimulation were performed using the bidomain model, with membrane kinetics governed by the Luo-Rudy model. The diastolic threshold for anodal stimulation first decreased and then increased with increasing [K]o, reaching a minimum value at [K]o = 12 mM. The mechanism for diastolic anodal excitation was make for all [K]o values except 13.3 mM, in which case it was break. For low [K]o (4 and 8 mM) the break section of the anodal strength-interval contained a "dip," but for high [K]o (12 and 13 mM), the dip disappeared.

CONCLUSION

High [K]o predisposes cardiac tissue to break excitation, which is thought to play an important role in reentry induction and defibrillation. Because fibrillation raises extracellular [K]o levels, break excitation may play a more important role in defibrillation than is suggested by simulations and experiments using normal [K]o values.

Introduction: Mapping and control of complex cardiac arrhythmias

This paper serves as an introduction to the Focus Issue on mapping and control of complex cardiac arrhythmias. We first introduce basic concepts of cardiac electrophysiology and describe the main clinical methods being used to treat arrhythmia. We then provide a brief summary of the main themes contained in the articles in this Focus Issue. In recent years there have been important advances in the ability to map the spread of excitation in intact hearts and in laboratory settings. This work has been combined with simulations that use increasingly realistic geometry and physiology. Waves of excitation and contraction in the heart do not always propagate with constant velocity but are often subject to instabilities that may lead to fluctuations in velocity and cycle time. Such instabilities are often treated best in the context of simple one- or two-dimensional geometries. An understanding of the mechanisms of propagation and wave stability is leading to the implementation of different stimulation protocols in an effort to modify or eliminate abnormal rhythms. (c) 2002 American Institute of Physics.