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Construction of a Computer Model to Investigate Sawtooth Effects in the Purkinje System
Abstract and Figures
The sawtooth effect refers to how one end of a cardiac cell is depolarized, while the opposite end is hyperpolarized, upon exposure to an exogenous electric field. Although hypothesized, it has not been observed in tissue. The Purkinje system is a one-dimensional (1-D) cable-like system residing on the endocardial surface of the heart and is the most obvious candidate for the manifestation of this phenomenon. This paper describes a computer modeling study of the effect of electric fields on the Purkinje system. Starting with a three-dimensional geometrically realistic, finite element, ventricular description, a Purkinje system is constructed which adheres to general physiological principles. Electrical activity in the Purkinje is described by use of 1-D cubic Hermite finite elements. Such a formulation allows for accurate modeling of loading effects at the Purkinje-myocyte junctions, and for preserving the discrete nature of the system. The response of a strand of Purkinje cells to defibrillation strength shocks is computed under several conditions. Also, the response of the isolated Purkinje network is illustrated. Results indicate that the geometry of the Purkinje system is the greatest determinant for far field excitation of the system. Given parameters within the plausible physiological range, the 1-D nature of the Purkinje system may lead to sawtooth potentials which are large enough to affect excitation. Thus, the Purkinje system is capable of affecting the defibrillation process, and warrants further experimentation to elucidate its role
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... The electrical behaviour of the heart has been extensively studied in various mathematical works [1,5,11,12,17,18]. The Purkinje system, initially identified by physiologist J. Purkinje (1787-1869), plays a significant role in cardiac activity and is represented as a one-dimensional tree-like structure [1,5]. ...
... The electrical behaviour of the heart has been extensively studied in various mathematical works [1,5,11,12,17,18]. The Purkinje system, initially identified by physiologist J. Purkinje (1787-1869), plays a significant role in cardiac activity and is represented as a one-dimensional tree-like structure [1,5]. In a normal heart rhythm, the electrical current originates from the sinus node, travels through the two atria, inducing their contractions, and then converges at the atrioventricular node. ...
... Subsequently, it travels along the His bundle and the Purkinje network to reach the ventricular walls, leading to their contraction. The interaction between the myocardium and the Purkinje system has been explored in various ways [1,5,15]. In [19], the authors introduced a model of the junction between the Purkinje network and the myocardium, presenting a mathematical framework for continuous-level coupling conditions. ...
The objective of this paper is to analyze a coupled problem that describes the propagation of the electric wave in the heart. The problem comprises coupled partial differential equations posed on a three-dimensional domain representing the heart and on a one-dimensional tree representing the Purkinje network. Each system of PDEs is itself coupled to ordinary differential equations that describe the electrical activity at the cellular level. We establish the existence of a unique solution, utilizing a fixed-point approach with a judicious and non-conventional choice of functional spaces and contraction.
2010 Mathematics Subject Classification. Primary 35Q92, Secondary: 35R30.
... The definition of I ion function depends on ionic models (see [61,69]) and the references therein). These models can be kind of phenomenological ( [29,30,20,28]) or physiological deviation( [23,50]). In both cases, the ion current depend on V m and a variable fields that denote ω, so we have I ion = I ion (V m , ω). ...
The main cause of sudden cardiac death has been attributed to cardiac electrical abnormalities, preventing blood circulation to various compartments of the body. Therefore any measure that may help curtail such devastating effects is welcome by the medical staff and public health officials. The obvious difficulty of performing direct measurements on the heart’s surface being highly invasive, has motivated a wide interest in the numerical simulation of cardiac mathematical models. These models are developed in order to understand the mechanisms of the electrical activity of the heart could lead to a more accurate diagnosis and earlier, and in return, reduce the quantity of heart related deaths. Indeed, Electrocardiography (ECG) studies the relationship between the electrical activity of the heart and its induced and measured electric potential on the surface of the torso
... Since the heart's physiology involves multiple physics systems, e.g. electrophysiology, (passive and active) mechanics and hemodynamics, an effective integrated computational model is very challenging the eikonal/monodomain model [44], the bidomain/bidomain model [47], the monodomain/bidomain model [48], and the monodomain/monodomain model [49,50]. Here, the first model refers to the one used for the Purkinje network and the second to that applied for the myocardium. ...
In previous work, Zhang et al. (2021) developed an integrated smoothed particle hydrodynamics (SPH) method to simulate the principle aspects of cardiac function, including electrophysiology, passive and active mechanical response of the myocardium. As the inclusion of the Purkinje network in electrocardiology is recognized as fundamental to accurately describing the electrical activation in the right and left ventricles, in this paper, we present a multi-order SPH method to handle the electrical propagation through the Purkinje system and in the myocardium with monodomain/monodomain coupling strategy. We first propose an efficient algorithm for network generation on arbitrarily complex surface by exploiting level-set geometry representation and cell-linked list neighbor search algorithm. Then, a reduced-order SPH method is developed to solve the monodomain equation to characterize the fast electrical activation through the Purkinje network. Finally, a multi-order coupling paradigm is introduced to capture the coupled nature of potential propagation arising from the interaction between the network and the myocardium. A set of numerical examples are studied to assess the computational performance, accuracy and versatility of the proposed method. In particular, numerical study performed in realistic left ventricle demonstrates that the present method features all the physiological issues that characterize a heartbeat simulation, including the initiation of the signal in the Purkinje network and the systolic and diastolic phases. As expected, the results underlie the importance of using physiologically realistic Purkinje network for modeling cardiac function.
... As regards the cardiac modeling with inclusion of the Purkinje network, numerical studies have been mainly focused on the myocardium electrosphysiology with different coupling strategies, e.g. the eikonal/eikonal model [33], the eikonal/monodomain model [44], the bidomain/bidomain model [47], the monodomain/bidomain model [48], and the monodomain/monodomain model [49], [50]. Here, the first model refers to the one used for the Purkinje network and the second to that applied for the myocardium. ...
In previous work, Zhang et al. (2021) [1] developed an integrated smoothed particle hydrodynamics (SPH) method to simulate the principle aspects of cardiac function, including electrophysiology, passive and active mechanical response of the myocardium. As the inclusion of the Purkinje network in electrocardiology is recognized as fundamental to accurately describing the electrical activation in the right and left ventricles, in this paper, we present a multi-order SPH method to handle the electrical propagation through the Purkinje system and in the myocardium with monodomain/monodomain coupling strategy. We first propose an efficient algorithm for network generation on arbitrarily complex surface by exploiting level-set geometry representation and cell-linked list neighbor search algorithm. Then, a reduced-order SPH method is developed to solve the one-dimensional monodomain equation to characterize the fast electrical activation through the Purkinje network. Finally, a multi-order coupling paradigm is introduced to capture the coupled nature of potential propagation arising from the interaction between the network and the myocardium. A set of numerical examples are studied to assess the computational performance, accuracy and versatility of the proposed methods. In particular, numerical study performed in realistic left ventricle demonstrates that the present method features all the physiological issues that characterize a heartbeat simulation, including the initiation of the signal in the Purkinje network and the systolic and diastolic phases. As expected, the results underlie the importance of using physiologically realistic Purkinje network for modeling cardiac functions.
... Previous modeling studies investigating the effect of Purkinje fibers on the cardiac response to defibrillation shocks found that the one-dimensional cable structure of Purkinje fibers makes them more excitable than the bulk of myocardial tissue (Boyle et al., 2010;Vigmond and Clements, 2007). The geometry of Purkinje fibers thus appears to contribute to their higher sensitivity to stimulation compared to ventricular muscle. ...
Magnetic resonance imaging (MRI) employs time-varying magnetic gradient fields, which induce electric fields (E-fields) in the patient that can potentially stimulate the heart. The performance of novel gradient systems is increasingly restricted by regulatory safety limits, thus motivating a deeper understanding of cardiac stimulation (CS) in MRI. This thesis investigates the thresholds and mechanisms underlying CS using an integrative approach of modeling and measurements. First, a numerical modeling framework was developed that combines E-field simulations in computational body models with electrophysiological cardiac fiber models to predict stimulation thresholds and sites in the heart. The CS thresholds predicted for two commercial gradient systems were >10-fold higher than the regulatory limits. Second, cardiac magnetostimulation thresholds were measured in ten healthy pigs using magnetic field pulses created by capacitor discharges into a coil. The average threshold E-field in the porcine heart was 92.9 pm 13.5 V/m. CS thresholds predicted in individualized porcine models derived from MR images reproduced the measurements with deviations of <18%, thus demonstrating the validity of the model for the experimental magnetic field waveform. The numerical and experimental results presented in this thesis inform the derivation of safe operational limits for MRI without unnecessarily restricting gradient performance and thus imaging speed and resolution.
... The bidomain/monodomain electrophysiology model has been widely used to study different components of the cardiac electrical network such as the atrial depolarization also including pathologic atrial fibrillation [36,16] or to model the AV-node depolarization [37,38]. The depolarization in the ventricular myocardium has been investigated in a series of works [17,39,40,41] also including the fast conduction Purkinje network [14,21,42], which is needed to reproduce a realistic ventricular depolarization, especially in the presence of infarction [43] or reentry initiation of arrhythmias [44,45,46]. In these works, the geometry of the Purkinje network is generally obtained by applying a growing algorithm to a one-dimensional (1D) network of fibers, which has to be sufficiently dense in order to correctly activate the 3D myocardium [47,48,49]. ...
In this study we present a novel computational model for unprecedented simulations of the whole cardiac electrophysiology. According to the heterogeneous electrophysiologic properties of the heart, the whole cardiac geometry is decomposed into a set of coupled conductive media having different topology and electrical conductivities: (i) a network of slender bundles comprising a fast conduction atrial network, the AV-node and the ventricular bundles; (ii) the Purkinje network; and (iii) the atrial and ventricular myocardium. The propagation of the action potential in these conductive media is governed by the bidomain/monodomain equations, which are discretized in space using an in-house finite volume method and coupled to three different cellular models, the Courtemanche model [1] for the atrial myocytes, the Stewart model [2] for the Purkinje Network and the ten Tusscher-Panfilov model [3] for the ventricular myocytes. The developed numerical model correctly reproduces the cardiac electrophysiology of the whole human heart in healthy and pathologic conditions and it can be tailored to study and optimize resynchronization therapies or invasive surgical procedures. Importantly, the whole solver is GPU-accelerated using CUDA Fortran providing an unprecedented speedup, thus opening the way for systematic parametric studies and uncertainty quantification analyses.
In this study we present a novel computational model for unprecedented simulations of the whole cardiac electrophysiology. According to the heterogeneous electrophysiologic properties of the heart, the whole cardiac geometry is decomposed into a set of coupled conductive media having different topology and electrical conductivities: (i) a network of slender bundles comprising a fast conduction atrial network, the AV–node and the ventricular bundles; (ii) the Purkinje network; and (iii) the atrial and ventricular myocardium. The propagation of the action potential in these conductive media is governed by the bidomain/monodomain equations, which are discretized in space using an in–house finite volume method and coupled to three different cell models, the Courtemanche model [1] for the atrial myocytes, the Stewart model [2] for the Purkinje Network and the ten Tusscher–Panfilov model [3] for the ventricular myocytes. The developed numerical model correctly reproduces the cardiac electrophysiology of the whole human heart in healthy and pathologic conditions and it can be tailored to study and optimize resynchronization therapies or invasive surgical procedures. Importantly, the whole solver is GPU–accelerated using CUDA Fortran providing an unprecedented speedup, thus opening the way for systematic parametric studies and uncertainty quantification analyses.
In this study we present a novel computational model for unprecedented simulations of the whole cardiac electrophysiology. According to the heterogeneous electrophysiologic properties of the heart, the whole cardiac geometry is decomposed into a set of coupled conductive media having different topology and electrical conductivities: (i) a network of slender bundles comprising a fast conduction atrial network, the AV–node and the ventricular bundles; (ii) the Purkinje network; and (iii) the atrial and ventricular myocardium. The propagation of the action potential in these conductive media is governed by the bidomain/monodomain equations, which are discretized in space using an in–house finite volume method and coupled to three different cell models, the Courtemanche model [1] for the atrial myocytes, the Stewart model [2] for the Purkinje Network and the ten Tusscher–Panfilov model [3] for the ventricular myocytes. The developed numerical model correctly reproduces the cardiac electrophysiology of the whole human heart in healthy and pathologic conditions and it can be tailored to study and optimize resynchronization therapies or invasive surgical procedures. Importantly, the whole solver is GPU–accelerated using CUDA Fortran providing an unprecedented speedup, thus opening the way for systematic parametric studies and uncertainty quantification analyses.
Background
During ventricular fibrillation (VF), targeting the excitable gap (EG) of reentry throughout the myocardium with low-energy surface stimulation shows promise for painless defibrillation. However, the Purkinje network may provide alternative pathways for reentry to evade termination. This study investigates the role of the Purkinje network in painless defibrillation.
Methods
In a computational human biventricular model featuring a Purkinje network, VF was initiated with 4 Hz epicardial pacing. Defibrillation was attempted by stimulating myocardial surface EG with a low-energy 2 ms duration pulse at 2x stimulus capture, which was administered at coupling intervals incremented by 0.25 s between 0.25 and 5 s after VF initiation. Defibrillation was accomplished if reentry ceased ≤ 1 s after the defibrillation pulse. The protocol was repeated with the Purkinje network and myocardial surface EG stimulated simultaneously, and again after uncoupling the Purkinje network from the myocardium.
Results
VF with the Purkinje network coupled and uncoupled had comparable dominant frequency in the left (3.81 ± 0.44 versus 3.77 ± 0.53 Hz) and right (3.80 ± 0.37 versus 3.76 ± 0.48 Hz) ventricles. When uncoupling the Purkinje network, myocardial surface EG stimulation terminated VF for all defibrillation pulses. When coupled, myocardial EG surface stimulation terminated VF for only 55% of the defibrillation pulses, but improved to 100% when stimulated simultaneously with Purkinje network EG. Defibrillation failures were attributed to EG evading stimulation in the Purkinje network.
Conclusions
Defibrillation that exclusively targets myocardium can fail due to accessory pathways in the Purkinje network that allow for reentrant activity to evade termination and maintain VF. Painless defibrillation strategies should be adapted to include the Purkinje network.
We describe the microscopic and macroscopic level modeling of the cardiac activation in the realistic three-dimensional ventricles of the human heart. At the microscopic level, the subthreshold behaviour of the excitable elements (cells) is governed by a reaction-diffusion equation derived from the bidomain theory, while in the suprathreshold state the elements obey cellular automata rules. Each cell is assigned a principal fiber direction according to the fiber architecture in the human heart. At the macroscopic level, the excitation of the ventricles is controlled by a model of the conduction system based on reports on the anatomy and physiology of the human heart. In the simulations, the activation was allowed to propagate from the His bundle to Purkinje-myocardial junction sites and through the ventricular myocardium. After careful evaluation of primary activation areas and balancing the activation timing between the right and the left ventricle, the final simulated activation sequence agreed with isochrones obtained from an isolated human heart. The calculated body surface potential and magnetocardiographic maps during the QRS complex correlated well with our clinical recordings on normal subjects. Furthermore, the vectorcardiogram produced by the simulated activation showed a correct pattern, although some minor discrepancies were observed in the morphology of the 12-lead electrocardiogram.
Rated among the most widely adopted anatomy texts in the world, this atlas is the only one to fully cover surface anatomy, osteology, clinical/anatomical relationships, and neuroanatomy, as well as general gross anatomy. Graphic line illustrations and modern imaging are included.
2nd edition, by Frank H. Netter, M.D. 525 pp., Summitt, NJ: Novartis Medical Education, 1998. $84.98 (soft cover: $59.95).
Sawtooth Effect in Cell-Pairs.Introduction: The question of how a defibrillation shock affects the myocardium far (> ~1 mm; the space constant of continuum tissue models) from the electrode is not fully understood. According to a long-standing, yet to be verified, hypothesis, the relatively high-resistance intercellular gap junctions may help in coupling the shock effect to the distant myocardium by redistributing the defibrillation current and creating a sawtooth pattern of polarization in which every cell undergoes hyperpolarization and depolarization. The goal of this study was to conduct an in-depth theoretical and experimental investigation of the sawtooth effect in the simplest coupled system, that of an isolated cell-pair.
Methods and Results: Theoretically, we present a relationship between sawtooth amplitude (STA) and junctional resistance (Rj), and show that, in a cell-pair with two cells of different lengths, the sawtooth effect may not necessarily appear as a reversal in polarization across the junction when Rj is below a critical value. Experimentally, we optically mapped transmembrane potential responses along the lengths of enzymatically isolated guinea pig cell-pairs at 10– or 17–μm resolution, and estimated STA as the magnitude of discontinuity in responses at the intercellular junction. From 14 cell-pairs, STA was estimated to be ~11 mV for a nominal 10 V/cm field. Based on our theoretical results, this value corresponds to an Rj of ~18 M Ω.
Conclusion: The intercellular junction induces a measurable sawtooth effect in the simplest system of an isolated cell-pair. An accounting for the sawtooth effect might be essential for understanding fieldtissue interaction far from the electrode and to accurately predict tissue response during field stimulation.
To obtain information conceming the time course and instantaneous distribution of
the excitatory process of the normal human healt, studies were made on isolated human hearts from seven individuals who died from various cerebral conditions, but who had no history of cardiac disease. Measurements were made from as many as 870 intramural terminals.
In the isolated human hearts three endocardial areas were synchronously excited
0 to 5 msec after the start of the left ventricular activity potential. These areas increased rapidly in si ze dUl'ing the next 5 to 10 msec and became confluent in 15 to 20 msec. The left ventricular areas Rrst excited were (1) high on the anterior paraseptal wall just below the attachment of the mitral valve; (2) central on the left surface of the interventricular septum and (3) posterior paraseptal about one third of the distance from apex to base. The last part of the left ventricle to be activated usually was the posterobasal area. Endocardial activation of the right ventricle was found to start near the insertion of the anterior papillary muscle 5 to 10 msec af ter onset of the left ventricular cavity potential. Septal activation started in the micldle third of the left side of the interventricular septurn, somewhat anteriorly, and at the lower third at the junction of the septum and posterior wall. The epicardial excitation pattem reflected the movements of the intramural excitation wave. Conduction velocity was determined in one heart by an appropriate stimulation technic. Atrial excitation, explored in two hearts, spread more or less according to concentric isochronic lines.
Control studies, carried out on Rve canine hearts, disclosed that the pattem of
ventricular excitation did not change af ter isolation and perfusion. However, total
excitation was completed earlier in the isolated heart, and conduction velocity increased. Careful mapping illustrations are presented.
The passive electrical properties of subendocardial Purkinje fibers surviving in infarcted regions of canine ventricle 24 hours after coronary ligation were studied by using microelectrode techniques and cable theory. In normal hearts, cells within the subendocardial Purkinje fiber strands were found to be well coupled to each other but electrically isolated from neighboring myocardium. Voltage response to intracellular current injection was consistent with one-dimensional cable behavior and yielded estimates of passive electrical properties in general agreement with previous work on free-running Purkinje strands (membrane length constant, 1.2 +/- 0.1 mm; membrane time constant, 7.3 +/- 0.8 msec; input resistance, 67.4 +/- 7.4 K omega; membrane resistance, 8.2 +/- 0.7 K omega.cm; axial resistance, 0.52 +/- 0.06 M omega/cm; membrane capacitance, 960 +/- 102 nF/cm) (n = 21). On the day after coronary ligation, subendocardial Purkinje fiber action potentials were prolonged and slightly depolarized. Significant increases were measured in input resistance (+40.5%), membrane resistance (+43.9%), and axial resistance (+47.5%), whereas membrane capacitance was found to be significantly decreased (-24.3%) (n = 19). Conduction velocity, membrane length constant, membrane time constant, and the time constant and capacitance for the foot of the action potential remained unchanged. These results are consistent with electrical uncoupling between adjacent cells, which will increase internal resistivity, accompanied by changes in cellular phospholipid content, which can increase membrane resistance and alter membrane capacitance. Alternatively, the results can be explained by a simple model in which the apparent electrical structure is altered by changes in electrical coupling alone, with specific electrical properties remaining constant. Although the mechanisms underlying the observed changes remain uncertain, the present study indicates that myocardial infarction is associated with alterations in the passive electrical structure of surviving subendocardial Purkinje fibers, which, together with changes in action potential configuration, may provide a substrate for the generation of ventricular arrhythmias 24 hours after coronary ligation.
Equations have been developed to describe cardiac action potentials and pacemaker activity. The model takes account of extensive developments in experimental work since the formulation of the M.N.T. (R. E. McAllister, D. Noble and R. W. Tsien, J. Physiol., Lond. 251, 1-59 (1975)) and B.R. (G. W. Beeler and H. Reuter, J. Physiol., Lond. 268, 177-210 (1977)) equations. The current mechanism iK2 has been replaced by the hyperpolarizing-activated current, i_f. Depletion and accumulation of potassium ions in the extracellular space are represented either by partial differential equations for diffusion in cylindrical or spherical preparations or, when such accuracy is not essential, by a three-compartment model in which the extracellular concentration in the intercellular space is uniform. The description of the delayed K current, i_K, remains based on the work of D. Noble and R. W. Tsien (J. Physiol., Lond. 200, 205-231 (1969a)). The instantaneous inward-rectifier, iK1, is based on S. Hagiwara and K. Takahashi's equation (J. Membrane Biol. 18, 61-80 (1974)) and on the patch clamp studies of B. Sakmann and G. Trube (J. Physiol., Lond. 347, 641-658 (1984)) and of Y. Momose, G. Szabo and W. R. Giles (Biophys. J. 41, 311a (1983)). The equations successfully account for all the properties formerly attributed to iK2, as well as giving more complete descriptions of iK1 and i_K. The sodium current equations are based on experimental data of T. J. Colatsky (J. Physiol., Lond. 305, 215-234 (1980)) and A. M. Brown, K. S. Lee and T. Powell (J. Physiol., Lond. 318, 479-500 (1981)). The equations correctly reproduce the range and magnitude of the sodium `window' current. The second inward current is based in part on the data of H. Reuter and H. Scholz (J. Physiol., Lond. 264, 17-47 (1977)) and K. S. Lee and R. W. Tsien (Nature, Lond. 297, 498-501 (1982)) so far as the ion selectivity is concerned. However, the activation and inactivation gating kinetics have been greatly speeded up to reproduce the very much faster currents recorded in recent work. A major consequence of this change is that Ca current inactivation mostly occurs very early in the action potential plateau. The sodium-potassium exchange pump equations are based on data reported by D. C. Gadsby (Proc. natn. Acad. Sci. U.S.A. 77, 4035-4039 (1980)) and by D. A. Eisner and W. J. Lederer (J. Physiol., Lond. 303, 441-474 (1980)). The sodium-calcium exchange current is based on L. J. Mullins' equations (J. gen. Physiol. 70, 681-695 (1977)). Intracellular calcium sequestration is represented by simple equations for uptake into a reticulum store which then reprimes a release store. The repriming equations use the data of W. R. Gibbons & H. A. Fozzard (J. gen. Physiol. 65, 367-384 (1975b)). Following Fabiato & Fabiato's work (J. Physiol., Lond. 249, 469-495 (1975)), Ca release is assumed to be triggered by intracellular free calcium. The equations reproduce the essential features of intracellular free calcium transients as measured with aequorin. The explanatory range of the model entirely includes and greatly extends that of the M.N.T. equations. Despite the major changes made, the overall time-course of the conductance changes to potassium ions strongly resembles that of the M.N.T. model. There are however important differences in the time courses of Na and Ca conductance changes. The Na conductance now includes a component due to the hyperpolarizing-activated current, i_f, which slowly increases during the pacemaker depolarization. The Ca conductance changes are very much faster than in the M.N.T. model so that in action potentials longer than about 50 ms the primary contribution of the fast gated calcium channel to the plateau is due to a steady-state `window' current or non-inactivated component. Slower calcium or Ca-activated currents, such as the Na-Ca exchange current, or Ca-gated currents, or a much slower Ca channel must then play the dynamic role previously attributed to the kinetics of a single type of calcium channel. This feature of the model in turn means that the repolarization process should be related to the inotropic state, as indicated by experimental work. The model successfully reproduces intracellular sodium concentration changes produced by variations in [Na]_o, or Na-K pump block. The sodium dependence of the overshoot potential is well reproduced despite the fact that steady state intracellular Na is proportional to extracellular Na, as in the experimental results of D. Ellis J. Physiol., Lond. 274, 211-240 (1977)). The model reproduces the responses to current pulses applied during the plateau and pacemaker phases. In particular, a substantial net decrease in conductance is predicted during the pacemaker depolarization despite the fact that the controlling process is an increase in conductance for the hyperpolarizing-activated current. The immediate effects of changing extracellular [K] are reproduced, including: (i) the shortening of action potential duration and suppression of pacemaker activity at high [K]; (ii) the increased automaticity at moderately low [K]; and (iii) the depolarization to the plateau range with premature depolarizations and low voltage oscillations at very low [K]. The ionic currents attributed to changes in Na-K pump activity are well reproduced. It is shown that the apparent K_m for K activation of the pump depends strongly on the size of the restricted extracellular space. With a 30% space (as in canine Purkinje fibres) the apparent K_m is close to the assumed real value of 1 mM. When the extracellular space is reduced to below 5%, the apparent K_m increases by up to an order of magnitude. A substantial part of the pump is then not available for inhibition by low [K]_b. These results can explain the apparent discrepancies in the literature concerning the K_m for pump activation.
Electrotonic coupling of cardiac myocytes at gap junctions may influence patterns of conduction in myocardium. To delineate the three-dimensional structure and distribution of intercellular junctions, we analyzed serial ultrathin sections of canine myocardium with transmission electron microscopy and disaggregated myocytes with scanning electron microscopy. Morphometric analysis of left ventricular myocardium sectioned in three orthogonal planes revealed that 80% of total gap junctional membrane occurred in large, ribbon-like gap junctions oriented transversely at cell end processes. The remaining 20% of gap junctional membrane was contained in small gap junctions located within plicate segments (interdigitating regions of cell-to-cell adhesion) of intercalated disks. In serial ultrathin sections, all gap junctions were contiguous with plicate segments. Thus, true "lateral" gap junctions do not exist in working ventricular myocytes and would not likely be able to withstand shear forces created by laterally sliding cells. Examination of serial plastic sections with light microscopy revealed complex overlapping of myocytes such that individual myocytes were connected at intercalated disks to an average of 9.1 +/- 2.2 other myocytes. These observations provide an improved understanding of the extent and distribution of cell junctions and should facilitate experimental and model studies of conduction in myocardium.
This study examines the distribution of the transmembrane potential in the periodic strand of cardiac muscle established by configurations of sources similar to those arising during extracellular stimulation and defibrillation, during intracellular stimulation, and during propagation of action potential. The closed-form solution indicates that during extracellular stimulation with large current and during defibrillation, the periodic component of the transmembrane potential is very important. We postulate that this periodic component causes the depolarization or defibrillation in cardiac muscle, which is different from the depolarization mechanism for a continuous fiber. On the other hand, during propagation and intracellular stimulation, the periodic component only slightly modifies the monotonic decrease of the transmembrane potential, which suggests that the mechanism of propagation in discrete structures may be similar to that of the continuous fiber.