Yuri B. Chernyak

The role of a critical excitation length scale in dynamics of reentrant cardiac arrhythmias

Summary We discuss computer simulations of one- and two-dimensional excitation waves corresponding to normal and abnormal rhythms in a model of myocardium. Our studies are aimed at finding the major physiologic parameters governing such transient processes as the formation of a reentrant wave and its subsequent degradation into the malignant cardiac arrhythmia – ventricular fibrillation. Our results demonstrate that in both the one- and two-dimensional cases the stability of a periodic process (representing a regular or VT rhythm) is determined by the same two physiologic parameters: the wavewidth , which is approximately is the width of depolarized zone, and ratio of to the critical length , which is defined as the wavewidth of an action potential propagating with the minimum possible speed. The amazing feature of general excitable medium is that these two numbers play predominate role in determining the behavior of the system. The parameter determines the length scale (size) of an ectopic region that may initiate a wave. It also determines the duration of the vulnerable window for initiating the unidirectional block as well as a minimum permissible length of the reentrant circuit. In two dimensions the relation between the diameter of the spiral core and the value of determines the pattern of spiral tip motion (near circular versus meandering) as it represents the outcome of the balance between electrical sources from depolarizing membrane and the electrical sinks (diffusive, i.e. Ohmic fluxes). This balance controls in a similar way two important electrophysiologic processes, the separation of a spiral tip from unexcitable obstacle (scar tissue) and the transition to meandering of the spiral tip in the homogeneous 2D medium. We also present some evidence that depending on the value of the waveforms of the simulated ECGs for reentrant activity vary from monomorphic to polymorphic.

Cellular automata model of cardiac excitation waves

Summary We present a new computer simulation technology suitable for rapid and quantitatively reliable simulation of propagating excitation waves in anisotropic myocardium. Our model utilizes a finite element or cellular automata (CA) approach in which the elements undergo transitions between a finite number of states (e.g., excited, refractory) according to specific rules. The transition parameter values for the CA elements at each location are computed using the characteristic relations governing propagation at the given point in the tissue, such as the anisotropy ratio and the dependence of the plane wave speed on diastolic interval. The model is well-suited for analysis of arrhythmogenesis and hypothetical therapeutic interventions. Once the effects of an antiarrhythmic drug or disease process on the characteristic relationships have been determined for different cardiac cell types, the electrical activity in tissue with the modified properties can be simulated by our model. In this article, we discuss the basic structure of the model and use it to demonstrate wavelet formation in myocardium with a fixed scar (infarct) in the presence of a sodium channel blocker. This mechanism may help explain the proarrhythmic effects of these agents.

Spiral waves are stable in discrete element models of two-dimensional homogeneous excitable media

The spontaneous breakup of a single spiral wave of excitation into a turbulent wave pattern has been observed in both discrete element models and continuous reaction-diffusion models of spatially homogeneous 2D excitable media. These results have attracted considerable interest, since spiral breakup is thought to be an important mechanism of transition from the heart rhythm disturbance ventricular tachycardia to the fatal arrhythmia ventricular fibrillation. It is not known whether this process can occur in the absence of disease-induced spatial heterogeneity of the electrical properties of the ventricular tissue. Candidate mechanisms for spiral breakup in uniform 2D media have emerged, but the physical validity of the mechanisms and their applicability to myocardium require further scrutiny. In this letter, we examine the computer simulation results obtained in two discrete element models and show that the instability of each spiral is an artifact resulting from an unphysical dependence of wave speed on wave front curvature in the medium. We conclude that spiral breakup does not occur in these two models at the specified parameter values and that great care must be exercised in the representation of a continuous excitable medium via discrete elements.Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps (∆x and ∆t) are varied independently. On the other hand, anisotropy effects due to the symmetries of the discretization lattice prevent the quantitative calibration of models. We introduce a new class of explicit difference methods for numerical integration of diffusion and reaction-diffusion equations, where the dependence on space and time scales occurs naturally. Numerical solutions approach the exact solution of the continuous diffusion equation for finite ∆x and ∆t, if the parameter γN = D∆t/(∆x) 2 assumes a fixed constant value, where N is an odd positive integer parametrizing the alghorithm. The error between the solutions of the discrete and the continuous equations goes to zero as (∆x) and the values of γN are dimension independent. With these new integration methods, anisotropy effects resulting from the finite differences are minimized, defining a standard for validation and calibration of numerical solutions of diffusion and reaction-diffusion equations. Comparison between numerical and analytical solutions of reaction-diffusion equations give global discretization errors of the order of 10 in the sup norm. Circular patterns of travelling waves have a maximum relative random deviation from the spherical symmetry of the order of 0.2%, and the standard deviation of the fluctuations around the mean circular wave front is of the order of 10. 1 rui@sd.ist.utl.pt 2 jsainhas@sd.ist.utl.pt Phone: (351)-(1)-8417617, FAX: (351)-(1)-8419123

An introduction to mathematical modeling of electrophysiological processes in the myocardium

Summary We discuss major features of modeling cardiac electrophysiology based on the modern concept of an excitable medium such as: general physical mechanisms and energetics of excitability, discrete and continuous aspects of cardiac conduction stemming from its fibrous structure, and anisotropy as another feature of such myocardial structure. We use the propagation velocity as a certain integral measure of the medium excitability and show that the expression for its value always consists of three factors, the scaling factor built out of dimensional constants of the myocardium, and two dimensionless factors, a universal directional factor taking full account of the medium anisotropy, and a dynamical factor that represents balances of all electrical sources and sinks. We describe the minimum, two variable models of an excitable cellular membrane. We show that in the first approximation the effect of the slow inactivation/recovery processes on the propagation velocity can be neglected. The excitation wave becomes in such an approximation a trigger wave of transitions from the resting to the exciting state. Then we discuss the formation of the conduction block at nonzero propagation speed due to the effect of inhibitive (recovery) processes.

Statistical dynamics of the membrane channel gating charge system

The membrane channels gating charge system is well represented as a set of strongly interacting dipoles. The equilibrium and kinetics of the system are described by the corresponding Boltzmann distribution and time dependent solutions of the Fokker-Planck equation. The theory explains very well various experiments on gating current and allows new insight in channel gating.<<ETX>>

Electrodynamic mechanism of voltage dependent gating

A new concept of channel gating is proposed. The concept is based on the idea that the polarization of the channel, due to the presence of gating charges, modifies the effective potential through which the ions migrate. The channel gate is therefore an up or down shift in the potential function seen by the ion in the channel.<<ETX>>

Body surface charge mapping and its application to electrocardiography

We have developed a new surface-source model to account for the bioelectrical potential on the body surface. A two-dimensional representation of three-dimensional bioelectrical sources has been developed using an equivalent surface charge model. The proposed equivalent surface charge mapping technique has been evaluated by computer simulations and applied to body surface Laplacian mapping data.

Toward the Constructive Theory of Human Social Behavior. III. Labor Behavior with Due Regard for Consumption

Our previous papers described a general constructive theory of developed human communities.lJ With a view toward applying the theory to problems concerning the whole community, one should develop in more detail the description of those aspects of individ social behavior that are particularly important for the life of the whole community. Individual consumption and labor behavior represent the basic aspects of individ social behavior. Individual consumption was considered in terms of our general theory3 and more comprehensively in our paper on consumer b e h a ~ i o r . ~ It was shown that our approach does give a reasonable description for this aspect of human social behavior. Because the major part of situations provided for each community element is created by the labor of its members, labor behavior becomes a most important part of the whole theory. In fact, under given material and financial resources, and under a given control structure, labor behavior determines the whole functioning of the community.d It is the purpose of this paper to consider, in general terms, the labor behavior of a community element, an individ. Using a somewhat simplified approach, we will study a separate, independent individ’s behavior and neglect h idher interactions with the whole community. It implies that the contribution of each individ to the balance relations of the whole community is vanishingly small, so one can disconnect the set of equationsl~* corresponding to the whole community and consider behavior of separated individs. We will thus assume that each working

Iterative restitution effects from heart rate variability

Action potential duration (APD) in the heart depends on the timing of the stimuli from SA node and the preceding diastolic interval (DI), the time it rested since the previous excitation. Such effects can be described by a random iterative map involving a heart rate dependent restitution function. In a steady state the stimuli form a stationary random process and iterative maps converge to stationary stochastic APD and DI sequences. We derive analytical expressions for major stochastic characteristics of such sequences (mean value, variance, etc). The results reveal a remarkable role of the slope of the restitution curve for the properties of the stationary output sequences. Our tentative computer simulations of the process corroborate our analytical results for relatively small heart rate variability.

Speed-curvature relation for a discrete model of myocardium

The authors derive the curvature relation (the dependence of the wave speed on wavefront curvature) for a discrete model of an excitable medium allowing inhomogeneities in the limiting case of a medium with no recovery process. The model incorporates an element weighting distribution w that is varied locally to match the required values of the local plane wave speed, critical curvature, and effective diffusion constant. The authors successfully linked their discrete model to myocardium using valves obtained from the Luo-Rudy model.