Extracellular discontinuities in cardiac muscle: evidence for capillary effects on the action potential foot

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.

Intercellular Sodium Regulates Repolarization in Cardiac Tissue with Sodium Channel Gain of Function

In cardiac myocytes, action potentials are initiated by an influx of sodium (Na⁺) ions via voltage-gated Na⁺ channels. Na⁺ channel gain of function (GOF), arising in both inherited conditions associated with mutation in the gene encoding the Na⁺ channel and acquired conditions associated with heart failure, ischemia, and atrial fibrillation, enhance Na⁺ influx, generating a late Na⁺ current that prolongs action potential duration (APD) and triggering proarrhythmic early afterdepolarizations (EADs). Recent studies have shown that Na⁺ channels are highly clustered at the myocyte intercalated disk, facilitating formation of Na⁺ nanodomains in the intercellular cleft between cells. Simulations from our group have recently predicted that narrowing the width of the intercellular cleft can suppress APD prolongation and EADs in the presence of Na⁺ channel mutations because of increased intercellular cleft Na⁺ ion depletion. In this study, we investigate the effects of modulating multiple extracellular spaces, specifically the intercellular cleft and bulk interstitial space, in a novel computational model and experimentally via osmotic agents albumin, dextran 70, and mannitol. We perform optical mapping and transmission electron microscopy in a drug-induced (sea anemone toxin, ATXII) Na⁺ channel GOF isolated heart model and modulate extracellular spaces via osmotic agents. Single-cell patch-clamp experiments confirmed that the osmotic agents individually do not enhance late Na⁺ current. Both experiments and simulations are consistent with the conclusion that intercellular cleft narrowing or expansion regulates APD prolongation; in contrast, modulating the bulk interstitial space has negligible effects on repolarization. Thus, we predict that intercellular cleft Na⁺ nanodomain formation and collapse critically regulates cardiac repolarization in the setting of Na⁺ channel GOF.

Properties of cardiac conduction in a cell-based computational model

The conduction of electrical signals through cardiac tissue is essential for maintaining the function of the heart, and conduction abnormalities are known to potentially lead to life-threatening arrhythmias. The properties of cardiac conduction have therefore been the topic of intense study for decades, but a number of questions related to the mechanisms of conduction still remain unresolved. In this paper, we demonstrate how the so-called EMI model may be used to study some of these open questions. In the EMI model, the extracellular space, the cell membrane, the intracellular space and the cell connections are all represented as separate parts of the computational domain, and the model therefore allows for study of local properties that are hard to represent in the classical homogenized bidomain or monodomain models commonly used to study cardiac conduction. We conclude that a non-uniform sodium channel distribution increases the conduction velocity and decreases the time delays over gap junctions of reduced coupling in the EMI model simulations. We also present a theoretical optimal cell length with respect to conduction velocity and consider the possibility of ephaptic coupling (i.e. cell-to-cell coupling through the extracellular potential) acting as an alternative or supporting mechanism to gap junction coupling. We conclude that for a non-uniform distribution of sodium channels and a sufficiently small intercellular distance, ephaptic coupling can influence the dynamics of the sodium channels and potentially provide cell-to-cell coupling when the gap junction connection is absent.

The electrochemical wave traversing the heart during every beat is essential for cardiac pumping function and supply of blood to the body. Understanding the stability of this wave is crucial to understanding how lethal arrhythmias are generated. Despite this importance, our knowledge of the physical determinants of wave propagation are still evolving. One particular challenge has been the lack of accurate mathematical models of conduction at the cellular level. Because cardiac muscle is an electrical syncytium, in which direct charge transfer between cells drives wave propagation, classical bidomain and monodomain tissue models employ a homogenized approximation of this process. This approximation is not valid at the length scale of single cells, and prevents any analysis of how cellular structures impact cardiac conduction. Instead, so-called microdomain models must be used for these questions. Here we utilize a recently developed modelling framework that is well suited to represent small collections of cells. By applying this framework, we show that concentration of sodium channels at the longitudinal borders of myocytes accelerates cardiac conduction. We also demonstrate that when juxtaposed cells are sufficiently close, this non-uniform distribution induces large ephaptic currents, which contribute to intercellular coupling.

Atrial Rotor Dynamics Under Complex Fractional Order Diffusion

The mechanisms of atrial fibrillation (AF) are a challenging research topic. The rotor hypothesis states that the AF is sustained by a reentrant wave that propagates around an unexcited core. Cardiac tissue heterogeneities, both structural and cellular, play an important role during fibrillatory dynamics, so that the ionic characteristics of the currents, their spatial distribution and their structural heterogeneity determine the meandering of the rotor. Several studies about rotor dynamics implement the standard diffusion equation. However, this mathematical scheme carries some limitations. It assumes the myocardium as a continuous medium, ignoring, therefore, its discrete and heterogeneous aspects. A computational model integrating both, electrical and structural properties could complement experimental and clinical results. A new mathematical model of the action potential propagation, based on complex fractional order derivatives is presented. The complex derivative order appears of considering the myocardium as discrete-scale invariant fractal. The main aim is to study the role of a myocardial, with fractal characteristics, on atrial fibrillatory dynamics. For this purpose, the degree of structural heterogeneity is described through derivatives of complex order γ = α + jβ. A set of variations for γ is tested. The real part α takes values ranging from 1.1 to 2 and the imaginary part β from 0 to 0.28. Under this scheme, the standard diffusion is recovered when α = 2 and β = 0. The effect of γ on the action potential propagation over an atrial strand is investigated. Rotors are generated in a 2D model of atrial tissue under electrical remodeling due to chronic AF. The results show that the degree of structural heterogeneity, given by γ, modulates the electrophysiological properties and the dynamics of rotor-type reentrant mechanisms. The spatial stability of the rotor and the area of its unexcited core are modulated. As the real part decreases and the imaginary part increases, simulating a higher structural heterogeneity, the vulnerable window to reentrant is increased, as the total meandering of the rotor tip. This in silico study suggests that structural heterogeneity, described by means of complex order derivatives, modulates the stability of rotors and that a wide range of rotor dynamics can be generated.

On the Order of the Fractional Laplacian in Determining the Spatio-Temporal Evolution of a Space-Fractional Model of Cardiac Electrophysiology

Space-fractional operators have been used with success in a variety of practical applications to describe transport processes in media characterised by spatial connectivity properties and high structural heterogeneity altering the classical laws of diffusion. This study provides a systematic investigation of the spatio-temporal effects of a space-fractional model in cardiac electrophysiology. We consider a simplified model of electrical pulse propagation through cardiac tissue, namely the monodomain formulation of the Beeler-Reuter cell model on insulated tissue fibres, and obtain a space-fractional modification of the model by using the spectral definition of the one-dimensional continuous fractional Laplacian. The spectral decomposition of the fractional operator allows us to develop an efficient numerical method for the space-fractional problem. Particular attention is paid to the role played by the fractional operator in determining the solution behaviour and to the identification of crucial differences between the non-fractional and the fractional cases. We find a positive linear dependence of the depolarization peak height and a power law decay of notch and dome peak amplitudes for decreasing orders of the fractional operator. Furthermore, we establish a quadratic relationship in conduction velocity, and quantify the increasingly wider action potential foot and more pronounced dispersion of action potential duration, as the fractional order is decreased. A discussion of the physiological interpretation of the presented findings is made.

Evaluation of excitation propagation in the rabbit heart: optical mapping and transmural microelectrode recordings

Background

Because of the optical features of heart tissue, optical and electrical action potentials are only moderately associated, especially when near-infrared dyes are used in optical mapping (OM) studies.

Objective

By simultaneously recording transmural electrical action potentials (APs) and optical action potentials (OAPs), we aimed to evaluate the contributions of both electrical and optical influences to the shape of the OAP upstroke.

Methods and Results

A standard glass microelectrode and OM, using an near-infrared fluorescent dye (di-4-ANBDQBS), were used to simultaneously record transmural APs and OAPs in a Langendorff-perfused rabbit heart during atrial, endocardial, and epicardial pacing. The actual profile of the transmural AP upstroke across the LV wall, together with the OAP upstroke, allowed for calculations of the probing-depth constant (k ~2.1 mm, n = 24) of the fluorescence measurements. In addition, the transmural AP recordings aided the quantitative evaluation of the influences of depth-weighted and lateral-scattering components on the OAP upstroke. These components correspond to the components of the propagating electrical wave that are transmural and parallel to the epicardium. The calculated mean values for the depth-weighted and lateral-scattering components, whose sum comprises the OAP upstroke, were (in ms) 10.18 ± 0.62 and 0.0 ± 0.56 for atrial stimulation, 9.37 ± 1.12 and 3.01 ± 1.30 for endocardial stimulation, and 6.09 ± 0.79 and 8.16 ± 0.98 for epicardial stimulation; (n = 8 for each). For this dye, 90% of the collected fluorescence originated up to 4.83 ± 0.18 mm (n = 24) from the epicardium.

Conclusions

The co-registration of OM and transmural microelectrode APs enabled the probing depth of fluorescence measurements to be calculated and the OAP upstroke to be divided into two components (depth-weighted and lateral-scattering), and it also allowed the relative strengths of their effects on the shape of the OAP upstroke to be evaluated.

Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization

Impulse propagation in biological tissues is known to be modulated by structural heterogeneity. In cardiac muscle, improved understanding on how this heterogeneity influences electrical spread is key to advancing our interpretation of dispersion of repolarization. We propose fractional diffusion models as a novel mathematical description of structurally heterogeneous excitable media, as a means of representing the modulation of the total electric field by the secondary electrical sources associated with tissue inhomogeneities. Our results, analysed against in vivo human recordings and experimental data of different animal species, indicate that structural heterogeneity underlies relevant characteristics of cardiac electrical propagation at tissue level. These include conduction effects on action potential (AP) morphology, the shortening of AP duration along the activation pathway and the progressive modulation by premature beats of spatial patterns of dispersion of repolarization. The proposed approach may also have important implications in other research fields involving excitable complex media.

Phase Relationship between Alternans of Early and Late Phases of Ventricular Action Potentials

BACKGROUND

Alternans of early phase and of duration of action potential (AP) critically affect dispersion of refractoriness through their influence on conduction and repolarization. We investigated the phase relationship between the two alternans and its effect on conduction.

METHODS AND RESULTS

Transmembrane potentials recorded from ventricles of eight swine and three canines during paced activation intervals of ≤300 ms were used to quantify alternans of maximum rate of depolarization (|dv/dt|(max)) and of action potential duration (APD). Incidence of APD alternans was 62 and 76% in swine and canines. Alternans of APD was frequently accompanied with alternans of |dv/dt|(max). Of these, 4 and 26% were out of phase in swine and canines, i.e., low |dv/dt|(max) preceded long APD. Computer simulations show that out of phase alternans attenuate variation of wavelength and thus minimize formation of spatially discordant alternans.

CONCLUSION

The spontaneous switching of phase relationship between alternans of depolarization and repolarization suggests that mechanisms underlying these alternans may operate independent of each other. The phase between these alternans can critically impact spatial dispersion of refractoriness and thus stability of conduction, with the in phase relation promoting transition from concord to discord while out of phase preventing formation of discord.

Mechanisms of defibrillation

Electrical shock has been the one effective treatment for ventricular fibrillation for several decades. With the advancement of electrical and optical mapping techniques, histology, and computer modeling, the mechanisms responsible for defibrillation are now coming to light. In this review, we discuss recent work that demonstrates the various mechanisms responsible for defibrillation. On the cellular level, membrane depolarization and electroporation affect defibrillation outcome. Cell bundles and collagenous septae are secondary sources and cause virtual electrodes at sites far from shocking electrodes. On the whole-heart level, shock field gradient and critical points determine whether a shock is successful or whether reentry causes initiation and continuation of fibrillation.

Effect of cell geometry on conduction velocity in a subcellular model of myocardium

We have studied the effect of cell geometry on propagation velocity of the cardiac impulse using a subcellular computer model of myocardium. Variation of cell size has only small effects on longitudinal and transverse conduction velocities, when the ratio of cell length/width is constant, for cell sizes (length x width) between (60 microm x 20 microm) and (120 microm x 40 microm). The results were not dependent on gap-junction conductance (range 0.25-1 microS), gap-junction distribution, or the specific tissue architecture. Longitudinal conduction velocity increased with the cell length/width ratio and transverse velocity decreased. The cell length/width ratio was a good estimator of the anisotropic ratio. In conclusion, cell length/width ratio is more important than cell size in determining conduction velocity.

Towards modeling of cardiac micro-structure with catheter-based confocal microscopy: a novel approach for dye delivery and tissue characterization

This work presents a methodology for modeling of cardiac tissue micro-structure. The approach is based on catheter-based confocal imaging systems, which are emerging as tools for diagnosis in various clinical disciplines. A limitation of these systems is that a fluorescent marker must be available in sufficient concentration in the imaged region. We introduce a novel method for the local delivery of fluorescent markers to cardiac tissue based on a hydro-gel carrier brought into contact with the tissue surface. The method was tested with living rabbit cardiac tissue and applied to acquire three-dimensional image stacks with a standard inverted confocal microscope and two-dimensional images with a catheter-based confocal microscope. We processed these image stacks to obtain spatial models and quantitative data on tissue microstructure. Volumes of atrial and ventricular myocytes were 4901 +/- 1713 and 10 299 +/-3598 mum (3) (mean+/-sd), respectively. Atrial and ventricular myocyte volume fractions were 72.4 +/-4.7% and 79.7 +/- 2.9% (mean +/-sd), respectively. Atrial and ventricular myocyte density was 165 571 +/- 55 836 and 86 957 +/- 32 280 cells/mm (3) (mean+/-sd), respectively. These statistical data and spatial descriptions of tissue microstructure provide important input for modeling studies of cardiac tissue function. We propose that the described methodology can also be used to characterize diseased tissue and allows for personalized modeling of cardiac tissue.

Extracellular space attenuates the effect of gap junctional remodeling on wave propagation: a computational study

Ionic channels and gap junctions are remodeled in cells from the 5-day epicardial border zone (EBZ) of the healing canine infarct. The main objective of the study was to determine the effect of gap junctional conductance (Gj) remodeling and Cx43 redistribution to the lateral membrane on conduction velocity (theta) and anisotropic ratio, and how gap junctional remodeling is modulated by the extracellular space. We first implemented subcellular monodomain and two-domain computer models of normal epicardium (NZ) to understand how extracellular space modulates the relationship between Gj and theta in NZ. We found that the extracellular space flattens the Gj-theta relationship, thus theta becomes less sensitive to changes in Gj. We then investigated the functional consequences of Gj remodeling and Cx43 distribution in subcellular computer models of cells of the outer pathway (IZo) and central pathway (IZc) of reentrant circuits. In IZo cells, side-to-side (transverse) Gj is 10% the value in NZ cells. Such Gj remodeling causes a 45% decrease in transverse theta (theta(T)). Inclusion of an extracellular space reduces the decrease in theta(T) to 31%. In IZc cells, Cx43 redistribution along the lateral membrane results in a 29% increase in theta(T). That increase in theta(T) is a consequence of the decrease in access resistance to the Cx43 plaques that occur with the Cx43 redistribution. Extracellular space reduces the increase in theta(T) to 10%.

IN CONCLUSION

1), The extracellular space included in normal epicardial simulations flattens the Gj-theta relationship with theta becoming less sensitive to changes in Gj. 2), The extracellular space attenuates the effects of gap junction epicardial border zone remodeling (i.e., Gj reduction and Cx43 lateralization) on theta(T).

Macroscopic optical mapping of excitation in cardiac cell networks with ultra-high spatiotemporal resolution

Optical mapping of cardiac excitation using voltage- and calcium-sensitive dyes has allowed a unique view into excitation wave dynamics, and facilitated scientific discovery in the cardiovascular field. At the same time, the structural complexity of the native heart has prompted the design of simplified experimental models of cardiac tissue using cultured cell networks. Such reduced experimental models form a natural bridge between single cells and tissue/organ level experimental systems to validate and advance theoretical concepts of cardiac propagation and arrhythmias. Macroscopic mapping (over >1cm(2) areas) of transmembrane potentials and intracellular calcium in these cultured cardiomyocyte networks is a relatively new development and lags behind whole heart imaging due to technical challenges. In this paper, we review the state-of-the-art technology in the field, examine specific aspects of such measurements and outline a rational system design approach. Particular attention is given to recent developments of sensitive detectors allowing mapping with ultra-high spatiotemporal resolution (>5 megapixels/s). Their interfacing with computer platforms to match the high data throughput, unique for this new generation of detectors, is discussed here. This critical review is intended to guide basic science researchers in assembling optical mapping systems for optimized macroscopic imaging with high resolution in a cultured cell setting. The tools and analysis are not limited to cardiac preparations, but are applicable for dynamic fluorescence imaging in networks of any excitable media.

Cell size and communication: role in structural and electrical development and remodeling of the heart

With the advent of new information about alterations of cardiac gap junctions in disease conditions associated with arrhythmias, there have been major advances in the genetic and metabolic manipulation of gap junctions. In contrast, in naturally occurring cardiac preparations, little is known about cell-to-cell transmission and the subcellular events of propagation or about structural mechanisms that may affect conduction events at this small size scale. Therefore, the aim of this article is to review results that produce the following unifying picture: changes in cardiac conduction due to remodeling cardiac morphology ultimately are limited to changes in three morphologic parameters: (1) cell geometry (size and shape), (2) gap junctions (distribution and conductivity), and (3) interstitial space (size and distribution). In this article, we consider changes in conduction that result from the remodeling of cell size and gap junction distribution that occurs with developmental ventricular hypertrophy from birth to maturity. We then go on to changes in longitudinal and transverse propagation in aging human atrial bundles that are produced by remodeling interstitial space due to deposition of collagenous septa. At present, experimental limitations in naturally occurring preparations prevent measurement of the conductance of individual gap junctional plaques, as well as the delays in conduction associated with cell-to-cell transmission. Therefore, we consider the development of mathematical electrical models based on documented cardiac microstructure to gain insight into the role of specific morphologic parameters in generating the changes in anisotropic propagation that we measured in the tissue preparations. A major antiarrhythmic implication of the results is that an "indirect" therapeutic target is interstitial collagen, because regulation of its deposition and turnover to prevent or alter microfibrosis can enhance side-to-side electrical coupling between small groups of cells in aging atrial bundles.

Use of translucent indium tin oxide to measure stimulatory effects of a passive conductor during field stimulation of rabbit hearts

Biomathematical models and experiments have indicated that passive extracellular conductors influence field stimulation. Because metallic conductors prevent optical mapping under the conductor, we have evaluated a passive translucent indium tin oxide (ITO) thin-film conductor to allow mapping of transmembrane potential (V(m)) and stimulatory current under the conductor. A 1-cm ITO disk was patterned photolithographically and positioned between 0.3-cm(2) mesh shock electrodes on the ventricular epicardium of isolated perfused rabbit hearts stained with 4-{2-[6-(dibutylamino)-2-naphthylenal]ethenyl}-1-(3-sulfopropyl)-, hydroxide, inner salt (di-4-ANEPPS). For a 1-A, 10-ms shock during the action potential plateau, optical maps from fluorescence collected using emission ratiometry (excitation at 488 nm and emissions at 510-570 and >590 nm) indicated that the disk altered V(m) by as much as the height of an action potential. DeltaV(m) became more positive near the edge of the disk, where the ITO conductance gradient was parallel to applied current, and more negative near the opposite edge, where the gradient was not parallel to current. For diastolic shocks, the disk expedited membrane excitation at the sites of positive DeltaV(m) in the heart and in a cardiac model with realistic ITO disk surface and interfacial conductances. Optical maps of ITO transmittance and the model indicated that the disk introduced anodal and cathodal stimulatory current at opposite edges of the disk. Thus ITO allows study of the stimulatory effects of a passive conductor in an electric field.

Basic mechanisms of cardiac impulse propagation and associated arrhythmias

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