Extracellular (volume conductor) effect on adjoining cardiac muscle electrophysiology

Modelling the interaction between stem cells derived cardiomyocytes patches and host myocardium to aid non-arrhythmic engineered heart tissue design

Application of epicardial patches constructed from human-induced pluripotent stem cell- derived cardiomyocytes (hiPSC-CMs) has been proposed as a long-term therapy to treat scarred hearts post myocardial infarction (MI). Understanding electrical interaction between engineered heart tissue patches (EHT) and host myocardium represents a key step toward a successful patch engraftment. EHT retain different electrical properties with respect to the host heart tissue due to the hiPSC-CMs immature phenotype, which may lead to increased arrhythmia risk. We developed a modelling framework to examine the influence of patch design on electrical activation at the engraftment site. We performed an in silico investigation of different patch design approaches to restore pre-MI activation properties and evaluated the associated arrhythmic risk. We developed an in silico cardiac electrophysiology model of a transmural cross section of host myocardium. The model featured an infarct region, an epicardial patch spanning the infarct region and a bath region. The patch is modelled as a layer of hiPSC-CM, combined with a layer of conductive polymer (CP). Tissue and patch geometrical dimensions and conductivities were incorporated through 10 modifiable model parameters. We validated our model against 4 independent experimental studies and showed that it can qualitatively reproduce their findings. We performed a global sensitivity analysis (GSA) to isolate the most important parameters, showing that the stimulus propagation is mainly governed by the scar depth, radius and conductivity when the scar is not transmural, and by the EHT patch conductivity when the scar is transmural. We assessed the relevance of small animal studies to humans by comparing simulations of rat, rabbit and human myocardium. We found that stimulus propagation paths and GSA sensitivity indices are consistent across species. We explored which EHT design variables have the potential to restore physiological propagation. Simulations predict that increasing EHT conductivity from 0.28 to 1-1.1 S/m recovered physiological activation in rat, rabbit and human. Finally, we assessed arrhythmia risk related to increasing EHT conductivity and tested increasing the EHT Na+ channel density as an alternative strategy to match healthy activation. Our results revealed a greater arrhythmia risk linked to increased EHT conductivity compared to increased Na+ channel density. We demonstrated that our modeling framework could capture the interaction between host and EHT patches observed in in vitro experiments. We showed that large (patch and tissue dimensions) and small (cardiac myocyte electrophysiology) scale differences between small animals and humans do not alter EHT patch effect on infarcted tissue. Our model revealed that only when the scar is transmural do EHT properties impact activation times and isolated the EHT conductivity as the main parameter influencing propagation. We predicted that restoring physiological activation by tuning EHT conductivity is possible but may promote arrhythmic behavior. Finally, our model suggests that acting on hiPSC-CMs low action potential upstroke velocity and lack of IK1 may restore pre-MI activation while not promoting arrhythmia.

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.

Anisotropic Cardiac Conduction

Anisotropy is the property of directional dependence. In cardiac tissue, conduction velocity is anisotropic and its orientation is determined by myocyte direction. Cell shape and size, excitability, myocardial fibrosis, gap junction distribution and function are all considered to contribute to anisotropic conduction. In disease states, anisotropic conduction may be enhanced, and is implicated, in the genesis of pathological arrhythmias. The principal mechanism responsible for enhanced anisotropy in disease remains uncertain. Possible contributors include changes in cellular excitability, changes in gap junction distribution or function and cellular uncoupling through interstitial fibrosis. It has recently been demonstrated that myocyte orientation may be identified using diffusion tensor magnetic resonance imaging in explanted hearts, and multisite pacing protocols have been proposed to estimate myocyte orientation and anisotropic conduction in vivo. These tools have the potential to contribute to the understanding of the role of myocyte disarray and anisotropic conduction in arrhythmic states.

Electrodiffusion phenomena in neuroscience: a neglected companion

The emerging technological revolution in genetically encoded molecular sensors and super-resolution imaging provides neuroscientists with a pass to the real-time nano-world. On this small scale, however, classical principles of electrophysiology do not always apply. This is in large part because the nanoscopic heterogeneities in ionic concentrations and the local electric fields associated with individual ions and their movement can no longer be ignored. Here, we review basic principles of molecular electrodiffusion in the cellular environment of organized brain tissue. We argue that accurate interpretation of physiological observations on the nanoscale requires a better understanding of the underlying electrodiffusion phenomena.

Cardiac bidomain bath-loading effects during arrhythmias: interaction with anatomical heterogeneity

Cardiac tissue is always surrounded by conducting fluid, both in vivo (blood) and in experimental preparations (Tyrode's solution), which acts to increase conduction velocity (CV) close to the tissue-fluid interface, inducing transmural wavefront curvature. Despite its potential importance, computer modeling studies focused on arrhythmia mechanisms have previously not accounted for these bath-loading effects. Here, we investigate the increase in CV and concomitant change in transmural wavefront profiles upon both propagation and arrhythmia dynamics within models of differing anatomical complexity. In simplified slab models, in absence of transmural fiber rotation, bath-loading induced transmural wavefront curvature dominates, significantly increasing arrhythmia complexity compared to no bath. In the presence of fiber rotation, bath-loading effects are less striking and depend upon propagation direction: the bath accentuates natural concave curvature caused by transmurally rotating fibers, but attenuates convex curvature, which negates overall impact upon arrhythmia complexity. Finally, we demonstrate that the high degree of anatomical complexity within whole ventricular models modulates bath-loading induced transmural wavefront curvature. However, key is the increased surface CV that dramatically reduces both arrhythmia inducibility and resulting complexity by increasing wavelength and reducing the available excitable gap. Our findings highlight the importance of including bath-loading effects during arrhythmia mechanism investigations, which could have implications for interpreting and comparing simulation results with experimental data where such effects are inherently present.

Bidomain ECG simulations using an augmented monodomain model for the cardiac source

The electrocardiogram (ECG) is an essential clinical tool for the non-invasive assessment of cardiac function. Computational simulations of ECGs using bidomain models are considered the biophysically most detailed approach, but computational costs are significant. Alternatively, pseudo-bidomain formulations can be used, combining a monodomain model with an infrequent bidomain solve to obtain full extracellular potential (φ(e)) distributions and traces. However, previous attempts at such approaches did not see the expected significant decrease in compute time and did not include important effects of bath-loading on activation wavefront morphology (present in full bidomain models), representing a less accurate source term for φ(e) solution. ECG traces can also be derived from computationally cheaper φ(e) recovery techniques, whereby the time-course of φ(e) is approximated at a particular point using the monodomain transmembrane potential as source term. However, φ(e) recovery methods also assume tissue to be immersed in an unbounded conductive medium; not the case in most practical scenarios. We recently demonstrated how bath-loading effects in bidomain simulations could be replicated using an augmented monodomain model, faithfully reproducing bidomain wavefront shapes and activation patterns. Here, a computationally-efficient pseudobidomain formulation is suggested which combines the advantages of an augmented monodomain method with an infrequent bidomain solve, providing activation sequences, ECG traces and φ(e) distributions in a bounded medium surrounding the heart which closely match those of the full bidomain, but at ≈ 10% the computational cost. We demonstrate the important impact of both bath-loading and a finite surrounding bath on spatiotemporal φ(e) distributions, thus demonstrating the utility of our novel pseudo-bidomain model in ECG computation with respect to previous pseudo-bidomain and φ(e) recovery approaches.

Representing cardiac bidomain bath-loading effects by an augmented monodomain approach: application to complex ventricular models

Although the cardiac bidomain model has been widely used in the simulation of electrical activation, its relatively computationally expensive nature means that monodomain approaches are generally required for long-duration simulations (for example, investigations of arrhythmia mechanisms). However, the presence of a conducting bath surrounding the tissue is known to induce wavefront curvature (surface leading bulk), a phenomena absent in standard monodomain approaches. Here, we investigate the biophysical origin of the bidomain bath-loading induced wavefront curvature and present a novel augmented monodomain-equivalent bidomain approach faithfully replicating all aspects of bidomain wavefront morphology and conduction velocity, but with a fraction of the computational cost. Bath-loading effects are shown to be highly dependent upon specific conductivity parameters, but less dependent upon the thickness or conductivity of the surrounding bath, with even relatively thin surrounding fluid layers (~ 0.1 mm) producing significant wavefront curvature in bidomain simulations. We demonstrate that our augmented monodomain approach can be easily adapted for different conductivity sets and applied to anatomically complex models, thus facilitating fast and accurate simulation of cardiac wavefront dynamics during long-duration simulations, further aiding the faithful comparison of simulations with experiments.

Reentry in survived subepicardium coupled to depolarized and inexcitable midmyocardium: insights into arrhythmogenesis in ischemia phase 1B

BACKGROUND

Delayed ventricular arrhythmias during acute myocardial ischemia (1B arrhythmias) are associated with an increase in tissue impedance and are most likely sustained in a thin subepicardial layer.

OBJECTIVE

The goal of this study was to test the hypothesis that heterogeneous uncoupling between depolarized midmyocardium and surviving subepicardium results in heterogeneous refractoriness in the latter, providing the reentry substrate after a premature beat.

METHODS

A 3-dimensional bidomain slab was constructed comprising a normal subepicardial layer coupled to a slightly depolarized (-80 to -60 mV) but inexcitable midmyocardium. Experimentally measured tissue impedance served as input for the model. Four stages of heterogeneous uncoupling between the 2 layers were simulated, each corresponding to an experimental ischemic impedance value. Effective refractory periods (ERP), conduction velocities, and inducibility of reentry were examined.

RESULTS

Heterogeneous uncoupling resulted in subepicardial ERP dispersion, allowing reentry to occur. The minimum ERP dispersion needed to induce reentry was 28 ms. Reentry induction was only possible in this model at the 2 intermediate stages of uncoupling, and only when midmyocardial resting membrane potential was more negative than -60 mV. Complete uncoupling of the layers resulted in normal subepicardial conduction without arrhythmias. The minimum length of the reentrant pathway was 2.5 cm, comparable to 2.4 cm reported in previous experiments.

CONCLUSION

Heterogeneous uncoupling to a negative sink such as depressed inexcitable midmyocardium may be a substrate for ischemia 1B arrhythmias. Total uncoupling removes the arrhythmogenic substrate.

Effects of cardiac microstructure on propagating electrical waveforms

Electrical waveforms measured during propagation at microscopic level are considerably affected by normal variations in cardiac microstructure as well as by the superfusing fluid. On the basis of evidence we present in this article, we argue that the anisotropic waveform variations discussed here are explained primarily by the associated variations in different microstructural components of myocardial architecture rather than by the effects of the perfusing bath. The results suggest that different components of myocardial architecture have preferential effects on f1.gif" BORDER="0">(max) and on the shape of the foot of the transmembrane action potential (V(m) foot). Resistive discontinuities primarily affect f1.gif" BORDER="0">(max), and an additional capacitive component in the local circuit due to the capillaries in interstitial space primarily affects V(m) foot. Resistive discontinuities also have an important influence on cardiac conduction. These discontinuities include spatial variations in the size of interstitial space (interstitial resistive discontinuities) and the role of cellular scaling (effects of cell size) when changes occur in the cellular and multicellular distribution of gap junctions during remodeling of normal mature myocardium to proarrhythmic structural substrates. The full text of this article is available at http://www.circresaha.org.

Influence of a perfusing bath on the foot of the cardiac action potential

Recently, Spach et al (Circ Res. 1998;83:1144-1164) measured the transmembrane action potential 150 to 200 microm below the tissue surface during longitudinal and transverse propagation. They found that "during longitudinal propagation there was initial slowing of V(m) [action potential] foot that resulted in deviations from a simple exponential. " (p 1144). They attributed this behavior to the effects of capillaries on propagation. The purpose of this commentary is to show that the perfusing bath plays an important role in determining the time course of the action potential foot, even when the transmembrane potential is measured 150 microm below the tissue surface. Using numerical simulations based on the bidomain model, we find that the action potential foot for transverse propagation is nearly exponential (tau(foot)=314 micros). For longitudinal propagation, the action potential foot is not exponential because of an initial slowing (best-fit tau(foot)=483 micros). We conclude that the perfusing bath must be taken into account when interpreting data showing differences in the shape of the action potential foot with propagation direction, even if the transmembrane potential is measured 150 microm below the tissue surface.

Propagation in cardiac tissue adjacent to connective tissue: two-dimensional modeling studies

The conditions for activation transmission across a region of extracellular space was demonstrated in two-dimensional preparations with results consistent with those previously seen in the one-dimensional fiber studies. In addition, one sees changes in action potential morphology which occur in the tissue nearest the connective-tissue border as well as changes in conduction velocity along the border. These results hinge on an adequate representation of the connective-tissue region achieved by careful implementation of the boundary conditions in the intracellular and interstitial spaces and the expansion of the connective-tissue discretization to a "double-tier network" description. Through a series of simulations, a clear dependence on fiber orientation is illustrated in the efficacy to transmit activation. The collision of a front with an embedded connective-tissue region was also examined. The results revealed that fibers aligned normal to a planar stimulus would more greatly disrupt the advancement of a planar front. Such pronounced disruptions have been shown to be proarrhythmic in the literature. The increasing evidence of the ability of connective tissue to transmit activation has implications in understanding spread of activation through infarcted tissues and through the healthy ventricular wall in the presence of connective-tissue sheets.

Sympathetic neuromuscular transmission at a varicosity in a syncytium

The autonomic neuromuscular junction at a varicosity in the vas deferens is defined by the localization of the vesicle-associated protein syntaxin in high concentrations in the axolemma and a high density of P2x1 receptors in a cluster beneath the varicosity. Calcium fluxes have been observed in all individual varicosities of a nerve terminal on the arrival of an impulse even though recordings made from these varicosities of the electrical signs of transmission with loose-patch electrodes over the varicosities show that they have very different probabilities for the secretion of a quantum. The fact that some varicosities seldom release a quantum on the arrival of an impulse is supported by the observation that antibodies against the N-terminus of synaptotagmin, which uniquely label the inside of synaptic vesicles when they undergo exocytosis, fail to do so in some varicosities during nerve stimulation whereas they do in others. It is suggested that the probability for secretion from a varicosity depends on the number of secretosomes that the varicosity possesses, where a secretosome is a complex of syntaxin, synaptotagmin, an N-type calcium channel, and a synaptic vesicle.

Quantal currents and potential in the three-dimensional anisotropic bidomain model of smooth muscle

The potential generated in the smooth muscle of the vas deferens on release of a quantum of transmitter from a varicosity was analyzed using a three-dimensional bidomain continuum model. Current was injected at the origin of the bidomain; this current had the temporal characteristics of the junctional current. The membrane potential, intracellular potential, and extracellular potential, as well as the extracellular current, were then calculated throughout the bidomain at different times. Calculations were performed to show the effect of changing the anisotropy ratios of the intracellular and extracellular conductivities on the spread of current and potential in each of the three dimensions. These results provide a theoretical framework for ascertaining the time course of transmitter interaction at a varicosity following the secretion of a quantum of transmitter.

Autonomic neuromuscular transmission at a varicosity

This review attempts to clarify the definition of what constitutes an autonomic neuromuscular function formed by a varicosity. Ultrastructural studies of serial sections through varicosities, partly or wholly bare of Schwann cell covering, show that areas of close apposition occur between varicosities and muscle cell membrane that vary between 20 and 150 nm, depending on the muscle considered. Consideration of the diffusion of purine transmitters and their receptor kinetics after secretion in a packet show that the number of purinergic receptor channels opened at a site of 150 nm apposition by a varicosity is about 15% of that at a site of 50 nm apposition. These results, together with the analysis of the stochastic fast component and the deterministic slow components of the rising phase of the EJP suggest that the stochastic fast component is due to varicosities that form especially close appositions (20-50 nm), whereas the deterministic slow component is due to the large number of varicosities at distances up to about 150 nm. Varicosities forming appositions of 20-150 nm with muscle cells several hundred micrometers long possess junctional receptor types distinct from extrajunctional receptors. According to this argument, then, there are two different classes of varicosities: one that gives rise to a relatively large junctional current and another that is responsible for a very small junctional current. Present evidence suggests that two subclasses of varicosities can be discerned amongst the varicosities that generate large junctional currents. One of these subclasses of varicosity possesses relatively few post-junctional receptors compared with the amount of transmitter reaching the receptors from the varicosity, so that the junctional current generated is determined by the size of the receptor population; in this case, the size of the transmitter packages released from these varicosities is unknown and the size of the junctional current is relatively constant. The other subclass of varicosity possesses large receptor patches, sufficient to accommodate the largest amounts of transmitter released from the varicosities: in this case, the size of the transmitter packages is shown to be highly non-uniform. These speculations await confirmation by direct labelling of the receptor patches beneath varicosities, a possibility that is likely to be realized in the near future.

Effect of a perfusing bath on the rate of rise of an action potential propagating through a slab of cardiac tissue

Experiments show that the rate of rise of the action potential depends on the direction of propagation in cardiac tissue. Two interpretations of these experiments have been presented: (i) the data are evidence of discrete propagation in cardiac tissue, and (ii) the data are an effect of the perfusing bath. In this paper we present a mathematical model that supports the second interpretation. We use the bidomain model to simulate action potential propagation through a slab of cardiac tissue perfused by a bath. We assume an intracellular potential distribution and solve the bidomain equations analytically for the transmembrane and extracellular potentials. The key assumption in our model is that the intracellular potential is independent of depth within the tissue. This assumption ensures that all three boundary conditions at the surface of a bidomain are satisfied simultaneously. One advantage of this model over previous numerical calculations is that we obtain an analytical solution for the transmembrane potential. The model predicts that the bath reduces the rate of rise of the transmembrane action potential at the tissue surface, and that this reduction depends on the direction of propagation. The model is consistent with the hypothesis that the perfusing bath causes the observed dependence of the action-potential rate of rise on the direction of propagation, and that this dependence has nothing to do with discrete properties of cardiac tissue.

Unidirectional block in a computer model of partially coupled segments of cardiac Purkinje tissue

The initiation of a reentrant circuit requires a zone of slow conduction and a zone of unidirectional block. This study used computer model conditions under which partial coupling between segments of cardiac Purkinje tissue resulted in unidirectional block. The structure used was one-dimensional and divided into three segments: a middle segment of variable length coupled to two long (semi-infinite in concept) segments. The DiFrancesco-Noble equations represented the ionic currents of the membrane. The results show that the possibility of unidirectional block depends on the size of the middle segment and the coupling resistances between the segments. No combination of coupling resistances allowed unidirectional block for middle segments with a length of two space constants (4 mm) or longer. Unidirectional block occurred for many combinations of coupling resistances as the length of the middle segment decreased to around half a space constant (1 mm). The number of length combinations that caused unidirectional block decreased again as segment length further decreased. These results provide a possible mechanism of unidirectional block for situations where islands of viable tissue are connected through nonviable tissue, such as in a healed myocardial infarction.

Reflection after delayed excitation in a computer model of a single fiber

Reflection (reflected reentry) is a case of reentry in a one-dimensional structure, divided into proximal and distal segments, in which tissue excited by a wave front propagating in a forward direction is reexcited by electrical activity coming backward from the original direction of propagation. Cases of reflection have been demonstrated in Purkinje fibers and in ventricular muscle preparations containing multiple fibers. Several mechanisms possibly responsible for reflected reentry have been proposed. However, the difficulty in the interpretation of the experimental results, as well as the limited number of different conditions in which reflection was obtained, has kept open the question about conditions and mechanisms for reflection. We have developed a computer model in which reflection occurs. The model involves a single fiber and uses the DiFrancesco-Noble equations for the Purkinje fiber to model the ionic currents. The results show that reflection is possible in a single fiber and that diastolic depolarization (automaticity) is not a requirement for reflection. Active membrane responses to a just-above-threshold stimulus were important for achieving the necessary time delay. Systematic simulations showed further that reflection occurred only when the right coupling conditions linked a short or long proximal fiber to a short distal segment.

Electrophysiological interaction through the interstitial space between adjacent unmyelinated parallel fibers

The influence of interstitial or extracellular potentials on propagation usually has been ignored, often through assuming these potentials to be insignificantly different from zero, presumably because both measurements and calculations become much more complex when interstitial interactions are included. This study arose primarily from an interest in cardiac muscle, where it has been well established that substantial interstitial potentials occur in tightly packed structures, e.g., tens of millivolts within the ventricular wall. We analyzed the electrophysiological interaction between two adjacent unmyelinated fibers within a restricted extracellular space. Numerical evaluations made use of two linked core-conductor models and Hodgkin-Huxley membrane properties. Changes in transmembrane potentials induced in the second fiber ranged from nonexistent with large intervening volumes to large enough to initiate excitation when fibers were coupled by interstitial currents through a small interstitial space. With equal interstitial and intracellular longitudinal conductivities and close coupling, the interaction was large enough (induced Vm approximately 20 mV peak-to-peak) that action potentials from one fiber initiated excitation in the other, for the 40-microns radius evaluated. With close coupling but no change in structure, propagation velocity in the first fiber varied from 1.66 mm/ms (when both fibers were simultaneously stimulated) to 2.84 mm/ms (when the second fiber remained passive). Although normal propagation through interstitial interaction is unlikely, the magnitudes of the electrotonic interactions were large and may have a substantial modulating effect on function.

Steady-state point-source stimulation of a nerve containing axons with an arbitrary distribution of diameters

The paper extends a mathematical model for point-source electrical stimulation of a nerve. In the original model, it was assumed that all the axons in the nerve have the same diameter. In this paper the model is extended to represent a nerve with an arbitrary distribution of axon diameters. It is shown that the assumption of identical axons is justified for a typical human nerve if the 'representative' axon diameter is taken as the area-weighted average of the diameter distribution.

Modification of a cylindrical bidomain model for cardiac tissue

Previous models based on a cylindrical bidomain assumed either that the ratio of intracellular and interstitial conductivities in the principal directions were the same or that there was no radial variation in potential (i.e., a planar front, delta Vm/delta rho = 0). This paper presents a formulation and the expressions for the intracellular, interstitial, extracellular, and transmembrane potentials arising from nonplanar propagation along a cylindrical bundle of cardiac tissue represented as a bidomain with arbitrary anisotropy. For unequal anisotropy, the transmembrane current depends not only on the local change of the transmembrane potential but also on the nature of the transmembrane potential throughout the volume.

Action potential propagation in a thick strand of cardiac muscle

A theoretical model of action potential propagation in a thick strand of cardiac muscle is presented. The calculation takes into account the anisotropic and syncytial properties of the tissue, the presence of the interstitial space, the effect of the surrounding tissue bath, and the variation of the potential both along the strand length and across the strand cross section. The bidomain model is used to represent the electrical properties of the tissue, and the Ebihara-Johnson model is used to represent the properties of the active sodium channels. The calculated wave front is curved, with the action potential at the surface of the strand leading that at the center. The rate of rise of the action potential and the time constant of the action potential foot vary with depth into the tissue. The velocity of the wave front is nearly independent of strand radius for radii greater than 0.5 mm. The conduction velocity decreases as the volume fraction of the interstitial space decreases. In the limit of tightly packed cells, an action potential propagates quickly over the surface of the strand; the bulk of the tissue is then excited by a slow inward wave front initiated on the surface. This model does not predict an increase in conduction velocity when cells are tightly packed, a hypothesis that has been proposed previously to explain the fast conduction velocity in Purkinje fibers of some species.

Simulation of propagation along a cylindrical bundle of cardiac tissue--II: Results of simulation

Previous evaluations of the cylindrical bidomain model of a bundle of cardiac tissue, have been obtained by using an analytic function for the transmembrane potential and assuming the activating wavefront through the bundle cross section is planar. In this paper, nonlinear membrane kinetics are introduced into the bidomain membrane and equal anisotropy ratios are assumed, permitting the transmembrane potential to be computed and its behavior examined at different depths in the bundle and for different values of conductivity and bundle diameters. In contrast with single fiber models, the bundle model reveals that the shape of the action potential is influenced by tissue resistivities. In addition, the steady-state activation wavefront through the cross-section perpendicular to the long axis of the bundle is not planar and propagates with a velocity that lies between that of a single fiber in an unbounded volume and a single fiber in a restricted extracellular space. In general, the bundle model is shown to be significantly better than the classical single fiber model in describing the behavior of real cardiac tissue.

Limitations of approximate solutions for computing the extracellular potential of single fibers and bundle equivalents

The mathematical description of the extracellular field generated by activity in an excitable fiber in an unbounded volume conductor will depend on assumptions made about the sources and the source-field relationship. This paper examines and compares the rigorous and conventional approximate solutions of Laplace's equation used to evaluate the extracellular potential of a single, cylindrical fiber. The single fiber is considered as both a prototypical element (such as a nerve or muscle fiber) and an elementary model of an entire multicellular preparation (e.g., nerve bundle or Purkinje strand). The effects of the fiber radius, the intracellular and extracellular conductivities, and the shape and extent of the source function (either the transmembrane potential or the intracellular potential) on the solutions are discussed. The results show that, in general, the approximate solutions are unsatisfactory for computing the surface extracellular potential when the single fiber is used to represent a large bundle (greater than 300 microns).

Bioelectric sources arising in excitable fibers (ALZA lecture)

This paper reviews the evaluation of bioelectric source strength and source field relationships for excitable fibers. For the single fiber, quantitative expressions describing the source may be derived which are independent of the fields produced by the sources. Rigorous expressions describe the equivalent elemental sources as discs, while the approximate line source is frequently satisfactory under physiological conditions. For fiber bundles the source associated with each fiber cannot be evaluated by the approximate isolated fiber expressions. However, when the bundle can be approximated as a bidomain and if the activation is planar, then mathematical expressions can be obtained. The resulting field behaves as if its origin was an equivalent single fiber. When the bidomain simplification and equal anisotropy ratio approximation is made, the planar waveform assumption can be removed and the resultant source and field can be evaluated. However, the latter are no longer independent.