The experimental and clinical possibilities for studying cardiac arrhythmias in human ventricular myocardium are very limited. Therefore, the use of alternative methods such as computer simulations is of great importance. In this article we introduce a mathematical model of the action potential of human ventricular cells that, while including a high level of electrophysiological detail, is computationally cost-effective enough to be applied in large-scale spatial simulations for the study of reentrant arrhythmias. The model is based on recent experimental data on most of the major ionic currents: the fast sodium, L-type calcium, transient outward, rapid and slow delayed rectifier, and inward rectifier currents. The model includes a basic calcium dynamics, allowing for the realistic modeling of calcium transients, calcium current inactivation, and the contraction staircase. We are able to reproduce human epicardial, endocardial, and M cell action potentials and show that differences can be explained by differences in the transient outward and slow delayed rectifier currents. Our model reproduces the experimentally observed data on action potential duration restitution, which is an important characteristic for reentrant arrhythmias. The conduction velocity restitution of our model is broader than in other models and agrees better with available data. Finally, we model the dynamics of spiral wave rotation in a two-dimensional sheet of human ventricular tissue and show that the spiral wave follows a complex meandering pattern and has a period of 265 ms. We conclude that the proposed model reproduces a variety of electrophysiological behaviors and provides a basis for studies of reentrant arrhythmias in human ventricular tissue.

According to one hypothesis, the chaotic excitation dynamics during VF are the result of dynamical instabilities in action potential duration (APD) the occurrence of which requires that the slope of the APD restitution curve exceeds 1. Other factors such as electrotonic coupling and cardiac memory also determine whether these instabilities can develop. In this paper we study the conditions for alternans and spiral breakup in human cardiac tissue. Therefore, we develop a new version of our human ventricular cell model, which is based on recent experimental measurements of human APD restitution and includes a more extensive description of intracellular calcium dynamics. We apply this model to study the conditions for electrical instability in single cells, for reentrant waves in a ring of cells, and for reentry in two-dimensional sheets of ventricular tissue. We show that an important determinant for the onset of instability is the recovery dynamics of the fast sodium current. Slower sodium current recovery leads to longer periods of spiral wave rotation and more gradual conduction velocity restitution, both of which suppress restitution-mediated instability. As a result, maximum restitution slopes considerably exceeding 1 (up to 1.5) may be necessary for electrical instability to occur. Although slopes necessary for the onset of instabilities found in our study exceed 1, they are within the range of experimentally measured slopes. Therefore, we conclude that steep APD restitution-mediated instability is a potential mechanism for VF in the human heart. reentrant arrhythmias; human ventricular myocytes; restitution properties; spiral waves; computer simulation ONE OF THE MOST extensively investigated hypotheses for ventricular fibrillation (VF) is the so-called restitution hypothesis. In its initial form the hypothesis stated that if the action potential duration (APD) restitution curve has a maximum slope steeper than 1, it will lead to APD alternans (16,41). In tissue this APD alternans can result in the fragmentation of a spiral wave, leading to fibrillation-like excitation patterns (21,22,45,48). Modeling studies have confirmed that a steep restitution curve indeed promotes instability. However, modeling studies have also shown that the criterion of an APD restitution slope Ͼ1 is an oversimplification that only holds for very simple models. In more realistic and complex models, it has been shown that other factors such as short-term cardiac memory, electrotonic interactions between cells, conduction velocity (CV) restitution, the range of diastolic intervals (DIs) over which restitution is steeper than 1, and the range of DIs visited during spiral wave rotation all play an important role in determining whether alternans and spiral breakup will occur (5,7,10,11,42,48,64). In an extensive study of restitutioninduced instability, Cherry and Fenton (6), for example, showed that because of strong electrotonic interactions and gradual CV restitution, spiral breakup may not occur even if the APD restitution curve has ...

In this paper, we formulate a model for human ventricular cells that is efficient enough for whole organ arrhythmia simulations yet detailed enough to capture the effects of cell level processes such as current blocks and channelopathies. The model is obtained from our detailed human ventricular cell model by using mathematical techniques to reduce the number of variables from 19 to nine. We carefully compare our full and reduced model at the single cell, cable and 2D tissue level and show that the reduced model has a very similar behaviour. Importantly, the new model correctly produces the effects of current blocks and channelopathies on AP and spiral wave behaviour, processes at the core of current day arrhythmia research. The new model is well over four times more efficient than the full model. We conclude that the new model can be used for efficient simulations of the effects of current changes on arrhythmias in the human heart.

Abstract-Sudden cardiac death is a major cause of death in the industrialized world, claiming approximately 300 000 victims annually in the United States alone. In most cases, sudden cardiac death is caused by ventricular fibrillation (VF). Experimental studies in large animal hearts have shown that the uncoordinated contractions during VF are caused by large numbers of chaotically wandering reentrant waves of electrical activity. However, recent clinical data on VF in the human heart seem to suggest that human VF may have a markedly different organization. Here, we use a detailed model of the human ventricles, including a detailed description of cell electrophysiology, ventricular anatomy, and fiber direction anisotropy, to study the organization of human VF. We show that characteristics of our simulated VF are qualitatively similar to the clinical data. Furthermore, we find that human VF is driven by only approximately 10 reentrant sources and thus is much more organized than VF in animal hearts of comparable size, where VF is driven by approximately 50 sources. We investigate the influence of anisotropy ratio, tissue excitability, and restitution properties on the number of reentrant sources driving VF. We find that the number of rotors depends strongest on minimum action potential duration, a property that differs significantly between human and large animal hearts. Based on these findings, we suggest that the simpler spatial organization of human VF relative to VF in large animal hearts may be caused by differences in minimum action potential duration. Both the simpler spatial organization of human VF and its suggested cause may have important implications for treating and preventing this dangerous arrhythmia in humans. (Circ Res. 2007;100:e87-e101.)Key Words: ventricular fibrillation Ⅲ computer simulation Ⅲ spatial organization V entricular fibrillation (VF) is the single most common cause of sudden cardiac death, the largest cause of death in the Western world. During VF, the contraction of the ventricles becomes rapid, uncoordinated, and highly ineffective, causing this condition to be lethal within minutes, unless halted by defibrillation. The highly disorganized contractions during VF are caused by a severely disturbed, turbulent conduction of the electrical excitation wave.Experimental studies in animal hearts and tissue 1-6 have shown that the turbulent electrical activity typical of VF is caused by the presence of multiple reentrant waves of electrical excitation. Because of their reentrant behavior and high frequency, these rotors act as self-perpetuating, independent sources of excitation that take over control from the slower sinus node. The number of rotors present during VF is a good quantifier of the complexity and amount of disorganization of the excitation pattern. Results in animal hearts suggest that the number of reentrant sources present during VF increases as a function of heart size. For example, in rabbit hearts, VF can be driven by just 1 or 2 sources, 2,7 whereas in sheep hearts, VF...

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Myocardial fibrosis is an important risk factor for cardiac arrhythmias. Previous experimental and numerical studies have shown that the texture and spatial distribution of fibrosis may play an important role in arrhythmia onset. Here, we investigate how spatial heterogeneity of fibrosis affects arrhythmia onset using numerical methods. We generate various tissue textures that differ by the mean amount of fibrosis, the degree of heterogeneity and the characteristic size of heterogeneity. We study the onset of arrhythmias using a burst pacing protocol. We confirm that spatial heterogeneity of fibrosis increases the probability of arrhythmia induction. This effect is more pronounced with the increase of both the spatial size and the degree of heterogeneity. The induced arrhythmias have a regular structure with the period being mostly determined by the maximal local fibrosis level. We perform ablations of the induced fibrillatory patterns to classify their type. We show that in fibrotic tissue fibrillation is usually of the mother rotor type but becomes of the multiple wavelet type with increase in tissue size. Overall, we conclude that the most important factor determining the formation and dynamics of arrhythmia in heterogeneous fibrotic tissue is the value of maximal local fibrosis.

A key characteristic of plant development is its plasticity in response to various and dynamically changing environmental conditions. Tropisms contribute to this flexibility by allowing plant organs to grow from or towards environmental cues. Halotropism is a recently described tropism in which plant roots bend away from salt. During halotropism, as in most other tropisms, directional growth is generated through an asymmetric auxin distribution that generates differences in growth rate and hence induces bending. Here, we develop a detailed model of auxin transport in the Arabidopsis root tip and combine this with experiments to investigate the processes generating auxin asymmetry during halotropism. Our model points to the key role of root tip architecture in allowing the decrease in PIN2 at the salt-exposed side of the root to result in a re-routing of auxin to the opposite side. In addition, our model demonstrates how feedback of auxin on the auxin transporter AUX1 amplifies this auxin asymmetry, while a salt-induced transient increase in PIN1 levels increases the speed at which this occurs. Using AUX1-GFP imaging and pin1 mutants, we experimentally confirmed these model predictions, thus expanding our knowledge of the cellular basis of halotropism.

Heterogeneity of cardiac tissue is an important factor determining the initiation and dynamics of cardiac arrhythmias. In this paper, we studied the effects of gradients of electrophysiological heterogeneity on reentrant excitation patterns using computer simulations. We investigated the dynamics of spiral waves in a two-dimensional sheet of cardiac tissue described by the Luo-Rudy phase 1 (LR1) ventricular action potential model. A gradient of action potential duration (APD) was imposed by gradually varying the local current density of K(+) current or inward rectifying K(+) current along one axis of the tissue sheet. We show that a gradient of APD resulted in spiral wave drift. This drift consisted of two components. The longitudinal (along the gradient) component was always directed toward regions of longer spiral wave period. The transverse (perpendicular to the gradient) component had a direction dependent on the direction of rotation of the spiral wave. We estimated the velocity of the drift as a function of the magnitude of the gradient and discuss its implications.