-Recent experimental and theoretical results have stressed the importance of modeling studies of reentrant arrhythmias in cardiac tissue and at the whole heart level. We introduce a sixvariable model obtained by a reformulation of the PriebeBeuckelmann model of a single human ventricular cell. The reformulated model is 4.9 times faster for numerical computations and it is more stable than the original model. It retains the action potential shape at various frequencies, restitution of action potential duration, and restitution of conduction velocity. We were able to reproduce the main properties of epicardial, endocardial, and M cells by modifying selected ionic currents. We performed a simulation study of spiral wave behavior in a two-dimensional sheet of human ventricular tissue and showed that spiral waves have a frequency of 3.3 Hz and a linear core of ϳ50-mm diameter that rotates with an average frequency of 0.62 rad/s. Simulation results agreed with experimental data. In conclusion, the proposed model is suitable for efficient and accurate studies of reentrant phenomena in human ventricular tissue. action potential; computer simulation; mathematical model; reentrant arrhythmia; spiral wave THE HISTORY of modeling biological excitable media such as nerve and heart tissue started 50 years ago with the Hodgkin-Huxley model of the giant squid axon (19). The first model of cardiac tissue (Purkinje fibers) was proposed by Noble in 1962 (30) and consisted of four variables. During the following decades, the experimental techniques for studying the properties of the cell membrane were improved continuously, leading to new cardiac tissue models of increasing accuracy, e.g., the phase- The Priebe-Beuckelmann (PB) model is based on the phase-2 LR model. However, five major ionic currents, including the fast (I Kr ) and slow (I Ks ) components of the delayed rectifier K ϩ current, the L-type Ca 2ϩ current (I Ca ), the transient outward K ϩ current (I to ), and the inward rectifier K ϩ current (I K1 ), are based on experimental data obtained on human myocytes. In addition, parameters of the intracellular Ca 2ϩ concentration ([Ca 2ϩ ] i ) handling were changed in such a way that simulated transients are comparable to observed experimental data on human myocytes (4). The remaining currents have been adjusted from the LR model, with their amplitude scaled to fit human cell data.Priebe and Beuckelmann developed their model to compare the electrophysiological properties of failing and nonfailing ventricular myocytes. It can be used for accurate simulations of the ionic currents and concentrations in a single cell during electrical activity. Our objective, however, is to simulate reentrant sources of arrhythmias in two (2-D) and three dimensions (3-D), which are believed to underlie most ventricular tachycardias and ventricular fibrillation (18,21,43). Although the PB model is based on experimental measurements in human heart tissue, we refrained from using it for the study of reentrant arrhythmias for several reasons. F...
Recent experimental studies show that the restitution curve of cardiac tissue can have a negative slope. We study how the negative slope of the restitution curve can influence basic processes in excitable media, such as periodic forcing of an excitable cell, circulation of a pulse in a ring, and spiral wave rotation in two dimensions. We show that negatively sloped restitution curve can result in instabilities if the slope of the restitution curve is steeper than -1 and report different manifestations of this instability. (c) 2002 American Institute of Physics.
We study numerically the dynamics of spiral waves in an excitable medium with negative restitution. For our study we use two models of the excitable medium: a cellular automaton and a reaction-diffusion model. There are no significant effects of negative restitution as long as the slope of the restitution curve is less steep than Ϫ1. In media with slopes steeper than Ϫ1, the dynamics of spiral waves can change significantly: ͑1͒ the average restitution time jumps to a value where the slope of the restitution curve is about Ϫ1; ͑2͒ spiral waves can break up into turbulent patterns. We discuss a possible connection between such instabilities and fibrillation in atrial tissue. DOI: 10.1103/PhysRevE.63.041912 PACS number͑s͒: 87.18.Ϫh, 82.20.Wt, 82.40.Ck Rotating spiral waves occur in a wide variety of nonlinear excitable media ͓1͔. While spiral waves may be stable under certain conditions, they can also break up into turbulent patterns. In the case of the heart, spiral waves cause tachycardia, a dangerous cardiac arrhythmia associated with a rapid heart beat ͓2,3͔. Tachycardia can deteriorate into fibrillation, which is fatal if it occurs in the ventricles of the heart, or a serious complication if it occurs in the atria. One possible mechanism of such deterioration is spiral breakup ͓4-10͔.It has been shown that spiral breakup heavily depends on a property of the excitable medium called its restitution curve ͓11͔. The restitution curve of a medium is the dependency of some pulse characteristic on the restitution time, which is the interval between the start of a pulse and the end of the previous pulse. Often the measured characteristic of the pulse is its duration ͑called action potential duration or APD in the case of cardiac tissue͒, which yields the APD restitution curve. Another important pulse characteristic is its refractory period ͑RP͒, which is the time interval during which a cell cannot be excited after a previous excitation; this leads to the RP restitution curve. In cardiac tissue, the RP and APD restitution curves are similar. Under normal conditions, longer restitution times lead to longer pulse durations and refractory periods, so that the restitution curve has positive slope everywhere. In this case, it has been shown that instabilities, e.g., spiral breakup, can occur if the slope of the restitution curve is steeper than 1 ͓11͔.Experiments show that the slope of the RP restitution curve of cardiac tissue can become negative ͑we call this negative restitution͒ ͓12͔. In the case of atrial tissue, negative restitution has been observed in healthy animals, and it became even more pronounced under pathological conditions ͑chronic atrial fibrillation͒ ͓14͔. It was also shown in a modeling study that even a small degree of negative restitution can be important for wave propagation in a ring of excitable tissue ͓13͔. In spite of the existence of negative restitution in cardiac tissue, the influence of negative restitution on spiral wave dynamics has not yet been studied.In this paper, we study numerica...
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