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The Restitution Portrait. Introduction: Electrical restitution, relating action potential duration (APD) to diastolic interval (DI), was believed to determine the stability of heart rhythm. However, recent studies demonstrate that stability also depends on long-term APD changes caused by memory. This study presents a new method for investigation of rate- and memory-dependent aspects of restitution and for assessment of mapping models of APD. Methods and Results: Bullfrog ventricular myocardium was paced with a 'perturbed downsweep protocol.' Starting from a basic cycle length (BCL) of 1,000 ms, the tissue was paced until steady state was achieved, followed by single beats of longer and shorter cycle lengths. BCL was decreased by 50 to 100 ms and the process repeated. All APDs were plotted as a function of the preceding DI, which allowed simultaneous observation of dynamic, S1-S2, and two constant-BCL restitution curves in a 'restitution portrait.' Responses were classified as 1:1 (stimulus:response), transient 2:2, or persistent 2:2 (alternans) and were related to the slopes of the restitution curves. None of these slopes approached unity for the persistent 2:2 response, demonstrating that the traditional restitution condition does not predict alternans. The restitution portrait was used to evaluate three mapping models of APD. The models with no memory and with one-beat memory did not produce restitution portraits similar to the experimental one. A model with two-beat memory produced a qualitatively similar portrait. Conclusion: The restitution portrait allows a more comprehensive assessment of cardiac dynamics than methods used to date. Further study of models with memory may result in a clinical criterion for electrical instability. (J Cardiovasc Electrophysiol, Vol. 15, pp. 1-12, June 2004) [ABSTRACT FROM AUTHOR]

We investigate, both experimentally and theoretically, the period-doubling bifurcation to alternans in heart tissue. Previously, this phenomenon has been modeled with either smooth or border-collision dynamics. Using a modification of existing experimental techniques, we find a hybrid behavior: Very close to the bifurcation point, the dynamics is smooth, whereas further away it is border-collision-like. The essence of this behavior is captured by a model that exhibits what we call an unfolded border-collision bifurcation. This new model elucidates that, in an experiment, where only a limited number of data points can be measured, the smooth behavior of the bifurcation can easily be missed.

Spatial heterogeneity of repolarization can provide a substrate for reentry to occur in myocardium. This heterogeneity may result from spatial differences in action potential duration (APD) restitution. The restitution portrait (RP) measures many aspects of rate-dependent restitution: the dynamic restitution curve (RC), S1-S2 RC, and short-term memory response. We used the RP to characterize epicardial patterns of spatial heterogeneity of restitution that were repeatable across animals. New Zealand White rabbit ventricles were paced from the epicardial apex, midventricle, or base, and optical action potentials were recorded from the same three regions. A perturbed downsweep pacing protocol was applied that measured the RP over a range of cycle lengths from 1,000 to 140 ms. The time constant of short-term memory measured close to the stimulus was dependent on location. In the midventricle the mean time constant was 19.1 +/- 1.1 s, but it was 39% longer at the apex (P < 0.01) and 23% longer at the base (P = 0.03). The S1-S2 RC slope was dependent on pacing site (P = 0.015), with steeper slope when pacing from the apex than from the base. There were no significant repeatable spatial patterns in steady-state APD at all cycle lengths or in dynamic RC slope. These results indicate that transient patterns of epicardial heterogeneity of APD may occur after a change in pacing rate. Thus it may affect cardiac electrical stability at the onset of a tachycardia or during a series of ectopic beats. Differences in restitution with respect to pacing site suggest that vulnerability may be affected by the location of reentry or ectopic foci.

Many features of the sequence of action potentials produced by repeated stimulation of a patch of cardiac muscle can be modeled by a 1D mapping, but not the full behavior included in the restitution portrait. Specifically, recent experiments have found that (i) the dynamic and S1-S2 restitution curves are different (rate dependence) and (ii) the approach to steady state, which requires many action potentials (accommodation), occurs along a curve distinct from either restitution curve. Neither behavior can be produced by a 1D mapping. To address these shortcomings, ad hoc 2D mappings, where the second variable is a "memory" variable, have been proposed; these models exhibit qualitative features of the relevant behavior, but a quantitative fit is not possible. In this paper we introduce a new 2D mapping and determine a set of parameters for it that gives a quantitatively accurate description of the full restitution portrait measured from a bullfrog ventricle. The mapping can be derived as an asymptotic limit of an idealized ionic model in which a generalized concentration acts as a memory variable. This ionic basis clarifies how the present model differs from previous models. The ionic basis also provides the foundation for more extensive cardiac modeling: e.g., constructing a PDE model that may be used to study the effect of memory on propagation. The fitting procedure for the mapping is straightforward and can easily be applied to obtain a mathematical model for data from other experiments, including experiments on different species.

Electroporation uses electric pulses to promote delivery of DNA and drugs into cells. This study presents a model of electroporation in a spherical cell exposed to an electric field. The model determines transmembrane potential, number of pores, and distribution of pore radii as functions of time and position on the cell surface. For a 1-ms, 40 kV/m pulse, electroporation consists of three stages: charging of the cell membrane (0-0.51 micros), creation of pores (0.51-1.43 micros), and evolution of pore radii (1.43 micros to 1 ms). This pulse creates approximately 341,000 pores, of which 97.8% are small ( approximately 1 nm radius) and 2.2% are large. The average radius of large pores is 22.8 +/- 18.7 nm, although some pores grow to 419 nm. The highest pore density occurs on the depolarized and hyperpolarized poles but the largest pores are on the border of the electroporated regions of the cell. Despite their much smaller number, large pores comprise 95.3% of the total pore area and contribute 66% to the increased cell conductance. For stronger pulses, pore area and cell conductance increase, but these increases are due to the creation of small pores; the number and size of large pores do not increase.

Novel methods need to be developed to detect electrical instability in children. The dynamical properties of action potential restitution play an important role in the development of instability leading to arrhythmias. A new method, the Restitution Portrait (RP), was developed to visualize and quantify these properties at the action potential level. Here, we apply the RP method using the activation-recovery interval (ARI) from the ECG to detect, in vitro, repolarization abnormalities in neonatal and preadolescent rabbit myocardium with drug-induced Long QT Syndrome (LQTS Type 1 or Type 2). The ECG was recorded during programmed endocardial pacing to record the RP. The ECG RP demonstrated significant changes in dynamical restitution components during drug-induced LQTS compared to baseline. This study shows that the ECG RP may be an important noninvasive diagnostic tool for detecting electrical instability in the young.

Electroporation, in which electric pulses create transient pores in the cell membrane, is an important technique for drug and DNA delivery. Electroporation kinetics is mathematically described by an advection-diffusion boundary value problem. This study uses singular perturbation to derive a reduced description of the pore creation transient in the form of a single integrodifferential equation for the transmembrane voltage Vt. The number of pores and the distribution of their radii are computed from Vt. The analysis contains two nonstandard features: the use of the voltage deviation to peel away the strong exponential dependence of pore creation upon the transmembrane potential, and the autonomous approximation of the pore evolution. Comparing the predictions of the reduced equation with the simulations of the original problem demonstrates that this analysis allows one to predict with good accuracy the number and distribution of pores as a function of the electric pulse strength.

Modern models of cardiac membranes are often highly complex and computationally expensive, particularly when used in long simulations of spatially extended models of cardiac tissue. Therefore, there is a need for simpler membrane models that preserve the features of the complex models deemed important. This communication describes an empirical procedure that was used to choose the parameters of the three-variable Fenton-Karma (FK3V) model to reproduce the restitution properties of the Courtemanche-Ramirez-Nattel model of atrial tissue. The resulting parameter values for the FK3V model and the sensitivity table for all its parameters are provided. Thus, this study gives insight into the behavior of the FK3V model and the effect of its parameters on restitution properties.

Restitution, the characteristic shortening of action potential duration (APD) with increased heart rate, has been studied extensively because of its purported link to the onset of fibrillation. Restitution is often represented in the form of mapping models where APD is a function of previous diastolic intervals (DIs) and/or APDs, A(n+1)=F(D(n),A(n),D(n-1),A(n-1),...), where A(n+1) is the APD following a DI given by D(n). The number of variables previous to D(n) determines the degree of memory in the mapping model. Recent experiments have shown that mapping models should contain at least three variables (D(n),A(n),D(n-1)) to reproduce a restitution portrait (RP) that is qualitatively similar to that seen experimentally, where the RP shows three different types of restitution curves (RCs) [dynamic, S1-S2, and constant-basic cycle length (BCL)] simultaneously. However, an interpretation of the different RCs has only been presented in detail for mapping models of one and two variables. Here we present an analysis of the different RCs in the RP for mapping models with an arbitrary amount of memory. We determine the number of variables necessary to represent the different RCs in the RP. We also present a graphical visualization of these RCs. Our analysis reveals that the dynamic and S1-S2 RCs reside on two-dimensional surfaces, and therefore provide limited information for mapping models with more than two variables. However, constant-BCL restitution is a feature of the RP that depends on higher dimensions and can possibly be used to determine a lower bound on the dimensionality of cardiac dynamics.

Rapid pacing is an important tool for understanding cardiac arrhythmias. A recent experiment involving rapid pacing of sheep atria indicated that the initiation of atrial arrhythmias may be related to the 1:1/2:1 bistability. To elucidate the mechanism of this relation, this study applied the pacing protocol from the sheep study to an idealized model of the right atrium. The model included all major anatomical features, the sino-atrial node, and the regional differences in the action potential duration (APD). A pacing protocol was applied, in which the basic cycle length (BCL) was decreased in steps of 10 ms until the response switched to 2:1, then BCL was increased. The 1:1-to-2:1 transitions occurred at shorter BCLs than the 2:1-to-1:1 transitions yielding a global bistability window of 60ms. As in the sheep study, idiopathic waves were observed at BCLs within or near the bistability window. The model was used to quantify the types, prevalence, and persistence of idiopatic waves, study their initiation and termination, and relate them to the model components. The results demonstrate that idiopatic waveforms move with the shift of the bistability window and that they disappear when bistability is eliminated. Thus, this modeling study supports causal relationship between the 1:1/2:1 bistability and the initiation of arrhythmias.