ИСТИНА

Waves Propagation in Second-Generation Ionic Models of Cardiac Tissue. Recently, it has been shown that cardiac cell models accounting for changes in intracellular ionic concentrations (“second-generation models”) have widely been used in a way that violates charge conservation. This problem was tackled by directly or indirectly imposing charge conservation for the case of single cells. In this Chapter, we study the effects of charge conservation violation on wave propagation. We show that the effect of charge conservation violation is small in homogeneous media, but may become significant in heterogeneous media. We conclude by proposing a proper method of simulating wave propagation in second-generation models, given the present understanding. Introduction Excitable cells have first been modeled quantitatively in the seminal paper by Hodgkin and Huxley on the giant squid axon. Hodgkin and Huxley viewed the cell as a capacitor and its ionic channels as variable resistors. In the following decades, new ionic channels were discovered and the characterizations became much more precise. A fundamental shift in model development was making intracellular ion concentrations state variables. Models with this feature are called “second generation” models. Second-generation models are now generally accepted to be an important tool for studying the electrophysiology of excitable cells. However, recent findings show that they suffer from a mathematical deficiency. The problems arise from the fact that the equations of second-generation models are linearly dependent, or stated differently, that transmembrane voltage (Vm), although treated as an independent variable, is a function of the intracellular ion concentrations. This defect leads to a number of artifacts. For example, rest states of second-generation models are not unique but depend on the initial conditions of a simulation. A closely related problem is that external pacing may lead to drifts in intracellular concentrations and Vm. As countless studies of single cells and of wave propagation have already been carried out using second-generation models, the question of how reliable their results are needs to be addressed. Also, researchers need to know what is the best way to design future simulation studies. Regarding single cell simulations, new models have been developed recently that enforce charge conservation by eliminating Vm as an independent variable but expressing it as a function of the intracellular ion concentrations. In particular, Endresen et al. showed that this removes the dependency of the limit cycle on initial conditions in a model of the sinusatrial mode. In a different approach, Hund et al. Showed that in the Luo-Rudy dynamic model unphysiological drifts can be avoided if all stimulation currents are actually carried by ions (“specific” stimulation). All studies listed above were performed for the excitation of a single cardiac cell. The most important application of these model is, however, to wave propagation, which we study for the first time in this Chapter. The main new problem is that the equation that has since Hodgkin and Huxley been used for wave propagation contains a voltage diffusion term which results in a violation of charge conservation in second generation models. We show that the effect of these unspecific currents is minimal for homogeneous media, but for heterogeneous media, it can lead to errors of significant size. Finally, we extend the work of to propose an appropriate method for future studies of wave propagation in second generation models of excitable tissue. Methods: For numerical simulations, we use the Luo-Rudy model (LR) of the ventricular mammalian myocyte, if not stated otherwise. In every time step, ionic currents through ion channels, pumps, and exchangers are updated according to model equations. From these ionic current, the changes in ionic concentrations are computed in the following way: If this stimulation is supposed to be nonspecific, we do not adapt the ion concentration. If, on the other hand, we need specific stimulation, we change an ion concentration accordingly. By default, we stimulate using potassium ions, which changes, where D is a diffusion constant. We use a forward Euler scheme with space step 0.1 mm and fixed time step of 0.02 ms. The flow that results from diffusion in each time step was either considered to be nonspecific (nonspecific diffusive current), or to be carried by potassium ions (specific diffusive current), in which case we changed [K+]i accordingly (see Results). For two-dimensional simulations, we used a forward Euler scheme with time step 0.02 ms for the reaction part and an alternating direction scheme with space step 0.2 mm to compute diffusion. Discussion Second-generation models that are stimulated by nonspecific currents, as is common practice, show unphysiological behavior, which has lead to concern among researchers considering such models for simulations. It has been shown that these unphysiological behaviors can be avoided in single cells by reformulating the models in a way that the mathematical deficiency causing these behaviors, a linear dependency in the model equations, is removed, or by stimulating them only with currents that are accompanied by a corresponding ion flow. We showed mathematically that nonspecific currents directly add to the integration constant V0 that relates Vm to the ionic concentrations. The integration constant V0 may be regarded as a model parameter. We studied the dependency of resting potential and ion concentration on V0 in the Luo-Rudy model, finding that only very large changes in V0 (_ 25000 mV) produce significant changes in the rest state. The Courtemanche model of human atrial cells yields similar results. Using the above dependency, we can quantitatively assess the effect of given nonspecific stimulations. In one-dimensional propagation, we showed that there is no effect for a pulse of fixed shape and propagation velocity. If a pulse does change in time, e. g. due to inhomogeneities in the medium, there may be a large distortion in V0. In two dimensions, we showed that V0 permanently changes only along the tip trajectory of the spiral. Although these changes may be sizable, the long-term buildup is only slow, because it is confined to a narrow region around the tip. Besides, if a spiral tip goes along a certain path and later the same path backwards, the initial changes in V0 are essentially neutralized. This is often the case for the widespread regular tip patterns. In three dimensions, the dominating reentrant patterns are scroll waves. The relation of in- and outflow should are similar to those of spiral waves in 2D. We expect no new effects and do not perform simulations. We established a method appropriate for future studies of wave propagation in cardiac tissue, in which each diffusive charge flow is accompanied by a corresponding flow of potassium ions. This results in a buildup of potassium at domain junction, which is not a mathematical defect model, but reflects the fact that ions are not free to diffuse in our model, which is probably not how real cells work. This potassium buildup is not as severe as buildup of V0 because it does not change the equilibrium, so that the potassium concentration would be eventually restored if the cell is given sufficient rest. On the other hand, dynamic equilibria may be significantly different for the (standard) non-diffusing and diffusing ions. We have not yet resolved this issue. A way to avoid potassium buildup even without knowing much about the true mechanisms of ion diffusion between cells is to introduce weak diffusion of potassium ions, which would not significantly affect the behavior of the cell in other respects.