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Gaur N, Rudy Y, Hool L.Contributions of Ion Channel Currents to Ventriculart Action Potential Changes and Induction of Early Afterdepolarizations During Acute Hypoxia. Circulation Research. 2009 Oct 15. [Epub ahead of print]

Rationale: Variability in delivery of oxygen can lead to electric instability in the myocardium and the generation of
arrhythmias. In addition ischemic heart disease and angina are associated with an increase in circulating
catecholamines that further increases the risk of developing ventricular tachyarrhythmias.
Objective: We investigated the net effects of acute hypoxia and catecholamines on the cardiac action potential.
Methods and Results: We incorporated all published data on the effects of hypoxia on the late Na current (INa-L),
the fast Na urrent (INa), the basal L-type Ca2 hannel current (ICa-L), and the slow (IKs) and rapid
components of the delayed rectifier K-current (IKr) in the absence and presence of beta-adrenergic receptor
(BAR) stimulation into the Luo–Rudy model of the action potential. Hypoxia alone had little effect on the action
potential configuration or action potential duration. However in the presence of BAR stimulation, hypoxia
caused a prolongation of the action potential and early afterdepolarizations (EADs) and spontaneous tachycardia
were induced. Experiments performed in guinea pig ventricular myocytes confirmed the modeling results.
Conclusions: EADs occur predominantly because of the increased sensitivity of ICa-L to BAR stimulation during
hypoxia. BAR stimulation is necessary to induce EADs as EADs are never observed during hypoxia in the
absence of BAR stimulation.

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L. Livshitz, Y. Rudy "Uniqueness and Stability of Action Potential Models during Rest, Pacing, and Conduction Using Problem-Solving Environment". 2009, Biophysical J, 97(5) pp. 1265 - 1276

Matlab code available:LRd_strand2009 (click here)

Development and application of physiologically detailed dynamic models of the action potential (AP) and Ca 2+ cycling in cardiac cells is a rapidly growing aspect of computational cardiac electrophysiology. Given the large scale of the nonlinear system involved, questions were recently raised regarding reproducibility, numerical stability, and uniqueness of model solutions, as well as ability of the model to simulate AP propagation in multicellular configurations. To address these issues, we reexamined ventricular models of myocyte AP developed in our laboratory with the following results. 1), Recognizing that the model involves a system of differential-algebraic equations, a procedure is developed for estimating consistent initial conditions that insure uniqueness and stability of the solution. 2), Model parameters that can be used to modify these initial conditions according to experimental values are identified. 3), A convergence criterion for steady-state solution is defined based on tracking the incremental contribution of each ion species to the membrane voltage. 4), Singularities in state variable formulations are removed analytically. 5), A biphasic current stimulus is implemented to completely eliminate stimulus artifact during long-term pacing over a broad range of frequencies. 6), Using the AP computed based on 15 above, an efficient scheme is developed for computing propagation in multicellular models.

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Decker KF, Heijman J, Silva JR, Hund TJ, Rudy Y. Properties and Ionic Mechanisms of Action Potential Adaptation, Restitution and Accommodation in Canine Epicardium. Am J Physiol Heart Circ Physiol . 2009 Jan 23. [Epub ahead of print]

Computational models of cardiac myocytes are important tools for understanding ionic mechanisms of arrhythmia. This work presents a new model of the canine epicardial myocyte that reproduces a wide range of experimentally observed rate dependent behaviors in cardiac cell and tissue, including action potential duration (APD) adaptation, restitution and accommodation. Model behavior depends on updated formulations for the 4-AP sensitive transient outward current (Ito1), the slow component of the delayed rectifier potassium current (IKs), the L-type Ca2+ channel (ICa,L) and the sodium-potassium pump (INaK) fit to data from canine ventricular myocytes. We find that Ito1 plays a limited role in potentiating peak ICa,L and sarcoplasmic reticulum Ca2+ release for propagated APs, but modulates the time course of APD restitution. IKs plays an important role in APD shortening at short diastolic intervals, despite a limited role in AP repolarization at longer cycle lengths. In addition, we find that ICa,L plays a critical role in APD accommodation and rate dependence of APD restitution through its indirect role in intracellular Na+ accumulation and increased outward INaK at rapid heart rates. Our simulation results provide valuable insight into the mechanistic basis of rate-dependent phenomena important for determining the heart's response to rapid and irregular pacing rates (e.g. arrhythmia). Accurate simulation of rate dependent phenomena and increased understanding of their mechanistic basis will lead to more realistic multicellular simulations of arrhythmia and identification of molecular therapeutic targets.

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Rudy Y, Ackerman MJ, Bers DM, Clancy CE, Houser SR, London B, McCulloch AD, Przywara DA, Rasmusson RL, Solaro RJ, Trayanova NA, Van Wagoner DR, Varró A, Weiss JN, Lathrop DA.Systems approach to understanding electromechanical activity in the human heart: a national heart, lung, and blood institute workshop summary. Circulation . 2008 Sep 9;118(11):1202-11.

The National Heart, Lung, and Blood Institute (NHLBI) convened a workshop of cardiologists, cardiac electrophysiologists, cell biophysicists, and computational modelers on August 20 and 21, 2007, in Washington, DC, to advise the NHLBI on new research directions needed to develop integrative approaches to elucidate human cardiac function. The workshop strove to identify limitations in the use of data from nonhuman animal species for elucidation of human electromechanical function/activity and to identify what specific information on ion channel kinetics, calcium handling, and dynamic changes in the intracellular/extracellular milieu is needed from human cardiac tissues to develop more robust computational models of human cardiac electromechanical activity. This article summarizes the workshop discussions and recommendations on the following topics: (1) limitations of animal models and differences from human electrophysiology, (2) modeling ion channel structure/function in the context of whole-cell electrophysiology, (3) excitation-contraction coupling and regulatory pathways, (4) whole-heart simulations of human electromechanical activity, and (5) what human data are currently needed and how to obtain them. The recommendations can be found on the NHLBI Web site at

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Hund TJ, Decker KF, Kanter E, Mohler PJ, Boyden PA, Schuessler RB, Yamada KA, Rudy Y. Role of activated CaMKII in abnormal calcium homeostasis and I(Na) remodeling after myocardial infarction: Insights from mathematical modeling. J Mol Cell Cardiol . 2008 Jun 28. [Epub ahead of print]

Ca2+/calmodulin-dependent protein kinase II is a multifunctional serine/threonine kinase with diverse cardiac roles including regulation of excitation contraction, transcription, and apoptosis. Dynamic regulation of CaMKII activity occurs in cardiac disease and is linked to specific disease phenotypes through its effects on ion channels, transporters, transcription and cell death pathways. Recent mathematical models of the cardiomyocyte have incorporated limited elements of CaMKII signaling to advance our understanding of how CaMKII regulates cardiac contractility and excitability. Given the importance of CaMKII in cardiac disease, it is imperative that computer models evolve to capture the dynamic range of CaMKII activity. In this study, using mathematical modeling combined with biochemical and imaging techniques, we test the hypothesis that CaMKII signaling in the canine infarct border zone (BZ) contributes to impaired calcium homeostasis and electrical remodeling. We report that the level of CaMKII autophosphorylation is significantly increased in the BZ region. Computer simulations using an updated mathematical model of CaMKII signaling reproduce abnormal Ca2+ transients and action potentials characteristic of the BZ. Our simulations show that CaMKII hyperactivity contributes to abnormal Ca2+ homeostasis and reduced action potential upstroke velocity due to effects on I(Na) gating kinetics. In conclusion, we present a new mathematical tool for studying effects of CaMKII signaling on cardiac excitability and contractility over a dynamic range of kinase activities. Our experimental and theoretical findings establish abnormal CaMKII signaling as an important component of remodeling in the canine BZ.

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Livshitz LM, Rudy Y. Regulation of Ca2+ and electrical alternans in cardiac myocytes: Role of CaMKII and repolarizing currents Am J Physiol Heart Circ Physiol . 2007 Jun;292(6):H2854-66

Matlab code available: LRd07 (click here) and HRd07 (click here)

Alternans of cardiac repolarization is associated with arrhythmias and sudden death. At the cellular level, alternans involves beat-to-beat oscillation of the action potential (AP) and possibly Ca2+ transient (CaT). Because of experimental difficulty in independently controlling the Ca2+ and electrical subsystems, mathematical modelling provides additional insights into mechanisms and causality. Pacing protocols were conducted in a canine ventricular myocyte model with the following results: (I) CaT alternans results from refractoriness of the SR Ca2+ release system; alternation of the L-type Ca2+ current (ICa(L)) has a negligible effect; (II) CaT-AP coupling during late AP occurs through the Na+/Ca2+ exchanger (INaCa) and underlies APD alternans; (III) Increased Ca2+/calmodulin-dependent protein kinase II (CaMKII) activity extends the range of CaT and APD alternans to slower frequencies and increases alternans magnitude; its decrease suppresses CaT and APD alternans, exerting an antiarrhythmic effect; (IV). Increase of the rapid delayed rectifier current (IKr) also suppresses APD alternans, but without suppressing CaT alternans. Thus, CaMKII inhibition eliminates APD alternans by eliminating its cause (CaT alternans), while IKr enhancement does so by weakening CaT-APD coupling. The simulations identify combined CaMKII inhibition and IKr enhancement as a possible antiarrhythmic intervention.

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Hund, T. J. and Y. Rudy (2000). "Determinants of excitability in cardiac myocytes: mechanistic investigation of memory effect." Biophys J 79(6): 3095-104.

The excitability of a cardiac cell depends upon many factors, including the rate and duration of pacing. Furthermore, cell excitability and its variability underlie many electrophysiological phenomena in the heart. In this study, we used a detailed mathematical model of the ventricular myocyte to investigate the determinants of excitability and gain insight into the mechanism by which excitability depends on the rate and duration of pacing (the memory effect).

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Zeng, J. and Y. Rudy (1995). "Early afterdepolarizations in cardiac myocytes: mechanism and rate dependence." Biophys J 68(3): 949-64.

A model of the cardiac ventricular action potential that accounts for dynamic changes in ionic concentrations was used to study the mechanism, characteristics, and rate dependence of early after depolarizations (EADs). A simulation approach to the study of the effects of pharmacological agents on cellular processes was introduced. The simulation results are qualitatively consistent with experimental observations and help resolve contradictory conclusions in the literature regarding the mechanism of EADs.

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Hund, T. J., J. P. Kucera, et al. (2001). "Ionic charge conservation and long-term steady state in the Luo-Rudy dynamic cell model." Biophys J 81(6): 3324-31.

It has been reported in the literature that dynamic cell models, which account for changes in intracellular ion concentrations, show a nonphysiological drift in computed parameters (Yehia et al., 1999; Endresen et al., 2000; Rappel, 2001). This drift occurs during rapid pacing for prolonged periods of time (Yehia et al., 1999; Rappel, 2001). The results of the present study establish that such drift is due to a nonconservative implementation of the stimulus, and not to an intrinsic property of the LRd model. The drift disappears when ions carried by the stimulus current are accounted for in the computation of ion concentrations. In general, when using a dynamic model, any source of charge (such as the stimulus) must also be considered a source of ions. Failure to do so violates conservation and may produce a nonphysiological behavior due to drift of model parameters. In this study, the problem is easily corrected by incorporating the stimulus current into the total K+ current in the LRd formulation. Assuming that other ions in the system (Na+ and Ca2+) to be the stimulus charge carrier is also consistent with the conservation principle, as long as these ions are accounted for in the formulation.
FIGURE 5 (A) Vm,dia as a function of time during pacing with a current stimulus at a BCL of 300 ms using the algebraic method (solid line) and the differential method (dashed line indicated with arrow). In both cases, the stimulus current carries K+ ions into the cell and contributes directly to computed changes in intracellular ion concentrations. Note the lack of drift and identical results for both methods. An additional simulation is shown using the differential method and a current stimulus that does not carry a particular ion species into the cell (dash-dot line). Notice that computed parameters drift if charges carried by the stimulus current are not taken into account in the computation of ion concentration changes.

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We investigate effects of elevated intracellular sodium on the cardiac action potential (AP) and on intracellular calcium using the Luo-Rudy model of a mammalian ventricular myocyte. By slowing AP depolarization (hence velocity) and shortening APD, Na+-overload acts to enhance inducibility of reentrant arrhythmias. Shortened APD with elevated [Ca2+]i (secondary to Na+-overload) also predisposes the myocardium to arrhythmogenic delayed afterdepolarizations.

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Zeng, J., K. R. Laurita, et al. (1995). "Two components of the delayed rectifier K+ current in ventricular myocytes of the guinea pig type. Theoretical formulation and their role in repolarization." Circ Res 77(1): 140-52.

Two distinct delayed rectifier K+ currents, IKr and IKs, were found recently in ventricular cells. We formulated these currents theoretically and investigated their roles in action potential repolarization and the restitution of action potential duration (APD). The Luo-Rudy (L-R) model of the ventricular action potential was used in the simulations. The single delayed rectifier K+ current in the model was replaced by IKr and IKs.

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Luo, C. H. and Y. Rudy (1994). "A dynamic model of the cardiac ventricular action potential. II. Afterdepolarizations, triggered activity, and potentiation." Circ Res 74(6): 1097-113.

Delayed afterdepolarizations (DADs) are induced by spontaneous Ca2+ release from the sarcoplasmic reticulum (SR), which, in turn, activates both the Na(+)-Ca2+ exchanger (INaCa) and a nonspecific Ca(2+)-activated current (Ins(Ca)). The relative contributions of INaCa and of Ins(Ca) to the generation of DADs are different under different degrees of Ca2+ overload. Spontaneous rhythmic activity and triggered activity are caused by spontaneous Ca2+ release from the SR under conditions of Ca2+ overload. Postextrasystolic potentiation reflects the time delay associated with translocation of Ca2+ from network SR to junctional SR. The cell is paced at high frequencies to investigate the long-term effects on the intracellular ionic concentrations.

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Luo, C. H. and Y. Rudy (1994). "A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes." Circ Res 74(6): 1071-96.

Our phase-2 cardiac ventricular action potential model is presented in this paper. The article focuses on processes that regulate intracellular calcium and depend on its concentration. This model for the mammalian ventricular action potential is based mostly on the guinea pig ventricular cell. However, it provides the framework for modeling other types of ventricular cells with appropriate modifications made to account for species differences. The model provides the basis for the study of arrhythmogenic activity of the single myocyte including afterdepolarizations and triggered activity. It can simulate cellular responses under different degrees of calcium overload.

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Luo, C. H. and Y. Rudy (1991). "A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction." Circ Res 68(6): 1501-26.

A mathematical model of the membrane action potential of the mammalian ventricular cell is introduced. The model is based, whenever possible, on recent single-cell and single-channel data and incorporates the possibility of changing extracellular potassium concentration [K]o. Physiological simulations focus on the interaction between depolarization and repolarization (i.e., premature stimulation).

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