A Practical Algorithm for Solving Dynamic Membrane Equations

Rush, Stanley; Larsen, Hugh

doi:10.1109/tbme.1978.326270

Cited by 383 publications

(227 citation statements)

References 4 publications

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“…(1) was discretized by the standard five-point stencil 40 . Since a straightforward explicit Euler time integration requires a rather small time step of 0.001 ms for numerical stability, we used the Rush-Larsen method 41 to integrate the gating variables. Furthermore, we considered the diffusion term implicitly, since for the explicit solution, time steps would have been limited by ∆x 2 /4D.…”

Section: Methodsmentioning

confidence: 99%

Control of electrical turbulence by periodic excitation of cardiac tissue

Buran

Bär

Alonso

et al. 2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Electrical turbulence in cardiac tissue is associated with arrhythmias such as the life-threatening ventricular fibrillation. The application of a high-energy electrical shock constitutes an effective defibrillation, but also causes severe side effects. Recent experimental studies have shown that a sequence of low energy electrical far-field pulses is able to terminate fibrillation more gently than a single pulse. During this low-energy antifibrillation pacing (LEAP) only tissue near sufficiently large conduction heterogeneities, such as large coronary arteries, is activated. In order to understand and potentially optimize LEAP, we performed extensive simulations of cardiac tissue perforated by blood vessels. We checked three alternative cellular models that exhibit qualitatively different electrical turbulence. LEAP may operate if and only if the spectrum of this chaotic activity is characterized by a narrow peak around a dominant frequency. For each of 100 initial conditions, we tested different electrical field strengths, pulse shapes, numbers of pulses, and periods between the pulses. It turned out that the optimal period matches the dominant period of the chaotic activity while both over-and underdrive pacing lead to a considerably smaller success probability and higher field strength for reliable defibrillation. An optimal LEAP protocol, which minimizes the required total energy for successful defibrillation, consists of five or six pulses. Compared to a single bi-phasic defibrillation pulse, it reduces the total energy by about 86%, corresponding to an energy reduction of 97 -98% per pulse.PACS numbers: ... Keywords: cardiac modeling, excitable media, defibrillation, low-energy antifibrillation pacing (LEAP)In ventricular fibrillation, the heart is quivering instead of pumping, a condition which leads to cardiac arrest and ultimately to death. This loss of synchronous contraction stems from disorganized electrical activity in the ventricles. Emergency defibrillation is achieved by the application of a strong electrical shock which is accompanied by severe side effects such as tissue damage and trauma. Recently, an alternative treatment using a sequence of low energy pulses has been suggested in which each electrical pulse excites only localized tissue sites at the large heterogeneities -such as blood vessels -instead of the entire tissue. In laboratory experiments in vitro and in vivo, this so-called low-energy antifibrillation pacing (LEAP) with five pulses achieves defibrillation with an energy reduction of 80 -90 % per pulse in comparison with standard single shock treatment. Here, we perform an extensive numerical study of LEAP employing far field pacing at the major blood vessels. We have tested three alternative electrophysiological models that exhibit qualitatively different spatiotemporally chaotic activity. LEAP operates if and only if the spectrum of this chaotic activity is characterized by a narrow peak around a dominant frequency. In this case, a resonant pacing with the same frequency and a ...

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Section: Methodsmentioning

confidence: 99%

Control of electrical turbulence by periodic excitation of cardiac tissue

Buran

Bär

Alonso

et al. 2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

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“…The methods of Rush and Larsen (1978) and Victorri et al (1985) were used to improve computational efficiency.…”

Section: Model Formulationsmentioning

confidence: 99%

Dynamics of human atrial cell models: Restitution, memory, and intracellular calcium dynamics in single cells

Cherry

Hastings

Evans

2008

Progress in Biophysics and Molecular Biology

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Mathematical models of cardiac cells have become important tools for investigating the electrophysiological properties and behavior of the heart. As the number of published models increases, it becomes more difficult to choose a model appropriate for the conditions to be studied, especially when multiple models describing the species and region of the heart of interest are available. In this paper, we will review and compare two detailed ionic models of human atrial myocytes, the Nygren et al. model (NM) and the Courtemanche et al. model (CM). Although both models include the same transmembrane currents and are largely based on the same experimental data from human atrial cells, the two models exhibit vastly different properties, especially in their dynamical behavior, including restitution and memory effects. The CM produces pronounced rate adaptation of action potential duration (APD) with limited memory effects, while the NM exhibits strong rate dependence of resting membrane potential (RMP), limited APD restitution, and stronger memory, as well as delayed afterdepolarizations and auto-oscillatory behavior upon cessation of rapid pacing. Channel conductance modifications based on experimentally measured changes during atrial fibrillation modify rate adaptation and memory in both models, but do not change the primary rate-dependent properties of APD and RMP for the CM and NM, respectively. Two sets of proposed changes to the NM that yield a spike-and-dome action potential morphology qualitatively similar to the CM at slow pacing rates similarly do not change the underlying dynamics of the model. Moreover, interchanging the formulations of all transmembrane currents between the two models while leaving calcium handling and ionic concentrations intact indicates that the currents strongly influence memory and the rate adaptation of RMP, while intracellular calcium dynamics primarily determine APD rate adaptation. Our results suggest that differences in intracellular calcium handling between the two human atrial myocyte models are responsible for marked dynamical differences and may prevent reconciliation between the models by straightforward channel conductance modifications.

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“…For simple models based on HodgkinHuxley formulation this problem is normally overcome by using the Rush-Larsen (RL) method [18]. However, for the most modern and complex models that are highly based on Markov Chains, the RL method seems to be ineffective in terms of allowing larger time steps during the numerical integration.…”

Section: Methodsmentioning

confidence: 99%

“…The Extended Rush-Larsen Method (SAST1) The method proposed by Rush-Larsen (RL) [19] is very popular in the community of cardiac electrophysiology and focus on the gating variables from Hodgkin and Huxley's formulation [11]. This method is based on a local linearization of the equations in the form…”

Section: Methodsmentioning

confidence: 99%

Reactive Interstitial and Reparative Fibrosis as Substrates for Cardiac Ectopic Pacemakers and Reentries

Oliveira

Barros

Gomes

et al. 2016

Bioinformatics and Biomedical Engineering

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Abstract. Dangerous cardiac arrhythmias have been frequently associated with focal sources of fast pulses, i.e. ectopic pacemakers. However, there is a lack of experimental evidences that could explain how ectopic pacemakers could be formed in cardiac tissue. In recent studies, we have proposed a new theory for the genesis of ectopic pacemakers in pathological cardiac tissues: reentry inside microfibrosis, i.e., a small region where excitable myocytes and non-conductive material coexist. In this work, we continue this investigation by comparing different types of fibrosis, reparative and reactive interstitial fibrosis. We use detailed and modern models of cardiac electrophysiology that account for the micro-structure of cardiac tissue. In addition, for the solution of our models we use, for the first time, a new numerical algorithm based on the Uniformization method. Our simulation results suggest that both types of fibrosis can support reentries, and therefore can generate in-silico ectopic pacemakers. However, the probability of reentries differs quantitatively for the different types of fibrosis. In addition, the new Uniformization method yields 20-fold increase in cardiac tissue simulation speed and, therefore, was an essential technique that allowed the execution of over a thousand of simulations.

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A Practical Algorithm for Solving Dynamic Membrane Equations

Cited by 383 publications

References 4 publications

Control of electrical turbulence by periodic excitation of cardiac tissue

Control of electrical turbulence by periodic excitation of cardiac tissue

Dynamics of human atrial cell models: Restitution, memory, and intracellular calcium dynamics in single cells

Reactive Interstitial and Reparative Fibrosis as Substrates for Cardiac Ectopic Pacemakers and Reentries

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