Effects of Premature Beats on Repolarization of Postextrasystolic Beats
Background A short-long-short sequence of cycle lengths predisposes to reentrant tachyarrhythmias. There is limited information about the effects of premature ventricular contractions (PVCs) on repolarization of postextrasystolic depolarizations (PEDs). Such information would contribute to understanding the mechanism for facilitating reentry with short-long-short cycle lengths.
Methods and Results We introduced PVCs, over a range of coupling intervals and during a range of basic drive cycle lengths (BCLs), and determined PED repolarization. Our results from whole-animal experiments, isolated cell studies, and computer simulations are reported. In the whole-animal experiments, PED refractory periods (RPs) were longer than RPBCL. The greatest difference between RPPED and RPBCL (ΔRPmax) occurred after short coupling interval PVCs and was 4.3±0.8, 4.2±0.8, and 2.1±0.5 ms (mean±SEM) during drives with short, intermediate, and long BCLs, respectively. The diastolic interval preceding the PED (DIPED) was inversely related to the coupling interval between the basic drive beat and the PVC and directly related to RPPED. PED action potential durations (APDs) of isolated canine myocytes were 9.8±4.9 ms (mean±SEM) longer than APD BCL (n=19). The DiFrancesco-Noble membrane equations were used in simulations of action potential propagation in a one-dimensional cable, with stimulation protocols duplicating those in the animal experiments. PVCs prolonged APDPED, and APDPED was prolonged more during short than during long BCL drives. There was a direct relation between DIPED and APDPED. Analysis of the membrane currents over the time course of the PVCs and PEDs suggested that the ionic basis for PED repolarization prolongation was the interaction of Ito and IK. Hyperpolarizing constant-current injections introduced immediately after the spike of isolated myocyte action potentials caused APD prolongation. This observation is consistent with the Ito and IK interaction causing PED repolarization prolongation.
Conclusions PED repolarization prolongation could provide sites for unidirectional block to propagation of PVCs after PEDs and could facilitate initiation of reentrant tachyarrhythmias after short-long-short sequences of cycle lengths.
A short-long-short sequence of cycle lengths frequently precedes spontaneous ventricular tachycardia in patients.1 2 This sequence of cycle lengths has been used to facilitate initiation of tachyarrhythmias in patients undergoing electrophysiological testing.3 4 5 It has also been shown to facilitate induction of ventricular tachycardia in dogs with myocardial infarction.6 In the animal model, differences in the magnitude of RP prolongation in ischemic and nonischemic myocardium caused unidirectional block of premature complexes (PVCs) that followed long cycles.
The effects of cycle length on RP have been studied extensively.7 8 9 10 11 12 13 14 15 16 17 18 19 However, these effects are complex and still have not been completely defined. Changes in cardiac cycle lengths have immediate effects on repolarization properties of the subsequent beat, ie, APD and RP, but a new steady state is not reached immediately.13 14 15 16 17 18 19 The most dramatic cycle length–induced changes in repolarization properties occur during the first cycle. RPs then oscillate over several cycles, and a new steady state is reached only after several hundred cycles. This slow change in RP from the steady-state duration associated with one BCL to the steady-state duration associated with another BCL has been referred to as “cardiac memory.”17
Single PVCs have even more complicated effects on repolarization properties.11 18 20 21 These effects result in part from changes in cycle length, which are nonuniform if the site of origin of the PVC differs from the site of origin of the BCL beat. PVCs usually have shorter repolarization durations than repolarization duration during the BCL drive. This decrease in repolarization duration results because of the short PVC cycle length. However, the effect of PVCs on repolarization duration cannot be explained solely on the basis of either the PVC cycle length or memory for the repolarization properties of the BCL beats. Gettes et al20 reported that the APD of PVCs in porcine Purkinje fibers was shorter than the APD during regular drive at the PVC cycle length. They attributed this phenomenon to the proximity of the PVC upstroke to the downstroke of the preceding beat, ie, the DI preceding the PVC. The relation of repolarization durations over a range of PVC cycle lengths to the preceding DI has also been referred to as electrical restitution.18 12 21 Electrical restitution characterizes repolarization durations of PVCs as a function of the premature cycle lengths, from the earliest possible PVC to the BCL.
The effects of single PVCs on the repolarization properties of the beats immediately after them, ie, the PED, have not been studied in detail. On the one hand, it might be expected that “memory” for the short cycle length of the PVC would shorten repolarization properties of the PED compared with the BCL repolarization properties. On the other hand, the DI preceding a PED is longer than the DI of the BCL beat because of the short repolarization properties of the PVC. Prolongation of the DI would be expected to increase the repolarization duration of the PED compared with the repolarization duration during the BCL. Alternatively, the two mechanisms might act to cancel one another so that repolarization properties of the PED would equal those of the BCL beat.
This report concerns our results from whole-animal experiments, isolated myocyte experiments, and computer simulations in which the effects of PVCs on PED repolarization properties were characterized. The results demonstrate that PVCs cause prolongation of PED repolarization, and the computer simulations and isolated myocyte studies provide insights concerning the likely ionic basis for this prolongation. Prolongation of PED repolarization could contribute to arrhythmogenesis by providing a site for the unidirectional block that is essential for sustained reentrant arrhythmias.
Experiments were done on 21 mongrel dogs. The animals were anesthetized with bolus injections of 30 mg/kg pentobarbital. Anesthesia was maintained with a continuous slow infusion of pentobarbital 120 mg IV in 1000 mL physiological saline solution. Additional bolus injections of 30 to 60 mg of intravenous pentobarbital were given as needed to maintain deep anesthesia, which was evaluated by testing the corneal reflex.
The trachea was intubated, and a fixed-volume pump respirator was used to ventilate the animal with room air. The chest was opened by midline sternotomy, and the heart was suspended in a pericardial cradle. The sinus node was crushed to slow the intrinsic heart rate and permit pacing over a range of BCLs. A bipolar hook stimulating electrode was placed in the right atrial appendage. An array of 1-mm-diameter silver electrodes embedded in an epoxy plaque was sutured to the anterior surface of the right ventricle. All of these electrodes were confined to a 1.4-cm2 region. These unipolar electrodes were used to pace the ventricle simultaneously with the atrium, to measure RPs, and to record electrograms. The atrium and ventricle were paced simultaneously to decrease the likelihood of retrograde atrial activation and echo beats during RP measurements. A needle electrode placed in the chest wall served as the indifferent electrode for ventricular pacing and RP test stimuli. A Wilson central terminal was used as the indifferent recording electrode, and a vertical-lead ECG was displayed on an oscilloscope to monitor the heart’s response to the BCL stimuli (S1), to the premature stimuli (S2), and to the test stimuli used to measure RP.
Pacing Protocol and Intervals Measured
The pattern of stimulation and the intervals measured for analysis of the results are diagrammed in Fig 1⇓. Four successive action potentials, marked 1 through 4, are shown at the top of the figure. The stimulus intervals, activation time intervals, RPs, and the DIs preceding each beat are diagrammed at the bottom of the figure. Action potentials 1 and 2 are the last beats in response to a train of regular S1s delivered at a selected BCL. Action potential 3 is a PVC in response to S2. The S1-S2 interval is the coupling interval of the PVC. Action potential 4 is the PED, and it is the first beat in response to a new train of S1. The S2-S1 interval in the animal experiments, isolated myocyte experiments, and computer simulations always equaled the S1-S1 interval, ie, the BCL. ABCL, APVC, and APED are the local activation times of the regularly driven beats, the PVC, and the PED, respectively. RPBCL, RPPVC, and RPPED are the refractory periods of those beats. Similarly, APDBCL, APDPVC, and APDPED represent the respective action potential durations. DIBCL, DIPVC, and DIPED represent the intervals between the activation times of the BCL, PVC, or PED and the RPs (animal experiments) or APDs (isolated myocyte experiments and computer simulations) of the beats preceding them.
Refractory Period Measurement Protocol
Each heart was paced at a fixed S1-S1 interval for at least 5 minutes before RP measurements were initiated. This provided time for RPs to reach their steady-state durations. S1 was delivered simultaneously to the right atrium and one of the electrodes on the surface of the right ventricle. RPBCL was measured by delivery of 2-ms, twice-diastolic-threshold, constant-current, cathodal stimuli to another electrode on the array. The RP test stimuli were delivered after every sixth S1. They were first delivered early in the cycle, at a time when they did not elicit a response. Then the RP test stimuli were delayed in 1-ms increments until a propagated response occurred. This caused the rhythm to remain regular until the RP test stimulus was delayed sufficiently to induce a propagated response. RPBCL was taken as the shortest S1-RP test stimulus interval that induced a propagated response minus the activation time at the RP test site. The activation time was taken as the time of the minimum derivative of the QRS in the electrogram recorded from the test site and was referenced to S1. After RPBCL had been measured, PVCs were induced by S2 stimuli delivered to the basic drive site after every sixth S1. The time of S2 was first set 2 to 3 ms longer than RPBCL at the drive site. In all experiments, the interval between S2 and the next S1, ie, the PED, always equaled the S1-S1 interval (see Fig 1⇑). RPPVC and RPPED were measured by delivery of RP test stimuli to the test site during repolarization of the PVC or the PED as described for the RPBCL. The S1-S2 interval was then incremented by 5 to 10 ms for short S1-S2 intervals and by 20 to 40 ms as the S1-S2 interval approached the BCL. RPPVC and RPPED were measured during drives with each S1-S2 coupling interval. This procedure was repeated during drives at 2 to 4 BCL and drives of 1 to 2 sites in each experiment: 1 drive site and 1 test site (n=12), 1 drive site and 2 test sites (n=4), 2 drive sites and 1 test site (n=3), 2 drive sites and 2 test sites (n=1), and 2 drive sites and 3 test sites (n=1). Measurements were made over a sufficient range of S1-S2 coupling intervals in 18 experiments to construct restitution curves (see “Data Analysis”). In 3 other experiments, RP was measured during regular drive, after PVCs with short coupling intervals, and after the PEDs that followed the short-coupling-interval PVCs. There were insufficient data to construct restitution curves for these 3 experiments, but the data were used in other analyses (see “Data Analysis”).
Isolated Myocyte Studies
Ventricular myocytes were obtained from the epicardial region of the pulmonary conus of three dogs. The animals were anesthetized with pentobarbital (30 mg/kg), and the hearts were rapidly removed via a lateral thoracotomy. The aorta was quickly cannulated, and the coronary circulation was flushed with 140 mL of a cold (≈4°C), nominally Ca2+-free solution to stop contractile activity and reduce metabolic rate. The solution contained (in mmol/L) NaCl 126.0, KCl 4.4, dextrose 22.0, MgCl2 5.0, taurine 20.0, creatine 5.0, sodium pyruvate 5.0, NaH2PO4 1.0, HEPES 24, and NaOH 12.5 (pH 7.4). The left coronary artery was rapidly cannulated via the left coronary ostia, and the heart was placed, anterior surface facing up, in a temperature-controlled chamber (34°C). Flow rate through the cannula was held at 40 mL/min. Perfusion with the above solution was maintained for 10 minutes and was followed by recirculation of the same solution (200 mL) containing 1 mg/mL collagenase (class II, Worthington Biochemical), 0.1 mg/mL protease (type XIV, Sigma), and 0.1 mmol/L CaCl2 for 30 minutes. After the enzyme digestion, the heart was perfused with the same solution, without enzymes, for 10 minutes. Color changes in the heart delineated the epicardial region exposed to the enzymes. This included the area of the pulmonary conus studied in the whole-animal experiments, although other hearts were used for the whole-animal studies. Myocytes were obtained by removing thin portions from the epicardial surface of the conus. The tissue was minced with iris scissors in 0.1 mmol/L Ca2+ and then gently shaken for ≈5 minutes. This procedure yielded ≈60% to 80% rod-shaped, quiescent cells. Cells were dispersed into a glass-bottomed chamber mounted on an inverted phase-contrast microscope (Diaphot, Nikon) and continuously superfused at 36°C with a solution containing (in mmol/L) NaCl 126.0, dextrose 11.0, KCl 4.4, MgCl2 1.0, CaCl2 1.0, and HEPES 24.0 titrated with 13.0 mmol/L NaOH (pH 7.4).
Suction pipettes were made with borosilicate capillary tubing (Corning number 7052, 1.65-mm OD, 1.2-mm ID) and had resistances of 2 to 4 MΩ when filled. They were filled with a solution containing (in mmol/L) KCl 113, NaCl 10, MgCl2 0.5, K2ATP 5, HEPES 10, dextrose 5, and KOH 11 (pH 7.1). Membrane potential was measured with an Axoclamp 2A amplifier in the bridge mode and digitized at 4 kHz. Depolarizing constant-current pulses (2 to 3 ms in duration, twice diastolic threshold) were delivered via the suction pipette to elicit action potentials at a basic cycle length of either 400 or 500 ms. APD was measured at 90% repolarization (APD90). All myocytes used in this study had a prominent spike-and-dome configuration, indicative of a large Ito22 23 and consistent with previous reports of canine epicardial ventricular myocytes.24 25
Our objective in the myocyte experiments was to determine the maximum effect of the PVC on the duration of the PED by maximizing the diastolic interval between the PVC and PED. This was achieved by initiating PVCs early during repolarization at takeoff potentials that ranged between −54 and −65 mV. To assess the effect of phase 1 amplitude on APD, a brief hyperpolarizing current pulse was delivered to paced cells immediately after the peak of the action potential. This current pulse was delivered via the suction pipette.
ΔRPmax was identified for each set of RP measurements. A set of RP measurements was defined as the measurements acquired during drive of one site at a selected BCL and included RPBCL, RPPVC over the range of S1-S2 intervals used, and RPPED after PVCs over the same range of S1-S2 coupling intervals. As mentioned above, in three whole-animal experiments, RP measurements were taken only during BCL drive and after short coupling PVCs and the PEDs that followed them. The data from these experiments were used for assessing ΔRPmax, but there were insufficient data to construct complete restitution curves for these experiments. Restitution curves of the relation of DI to RP were constructed from data from the other 18 whole-animal experiments. The mean±SEM of ΔRPmax was calculated from all sets of measurements without regard to BCL. Then sets of measurements were grouped into short, intermediate, and long BCLs, and the mean±SEM of ΔRPmax was calculated for each BCL group. The differences between APD90 of regularly driven beats and APD90 of PEDs (ΔAPD) were determined from action potentials recorded from the isolated myocytes, and the mean±SEM of these values was calculated. The paired t test was used to assess the significance of ΔRPmax and ΔAPD in the whole-animal and isolated myocyte studies, respectively. Student’s t test was used to assess differences in ΔRPmax between BCL groups in the whole-animal experiments. Values of P<.05 were considered significant.
A set of computer simulations was performed to further characterize the effects of PVCs on PED repolarization properties and to gain insight into the possible ionic mechanism for the experimental results. The relation between DIPED and APDPED as a function of BCL was evaluated with a one-dimensional propagation model. The one-dimensional model allowed assessment of repolarization properties in response to a range of BCLs and S1-S2 intervals. It also allowed application of stimuli and measurement of APD at different sites. Simulations using a one-dimensional propagation model eliminated the possibility that electrotonic effects of anisotropic action potential propagation affected APD.26
The electrical activity in the myocardium was described by the equation
where Im is the transmembrane current density, Cm the specific membrane capacitance, Vm the transmembrane potential, and Iion the ionic current source density.27 In this one-dimensional fiber model, spatial current flow was characterized by a monodomain representation of tissue structure28 :
where a is the radius of a single myocardial fiber (0.001 cm) and Ri the specific intracellular resistance of the myoplasm (250 Ω-cm). Equations 1 and 2 were combined and then discretized in space and time according to the method of Crank and Nicholson as described in Reference 29. Equation 2 was discretized in space into a set of 101 elements of length dx=0.005 cm to monitor action potential propagation over a 5-mm distance. Equation 1 was discretized in time by a previously described variable-time-step method that ensured fine temporal integration to monitor propagation of the depolarization wave front (dtmin=0.002 ms) while allowing larger time steps to be taken during periods in which variables in the model were changing slowly (dtmax=1.024 ms).30
In Equation 1, Iion was determined from the DiFrancesco and Noble31 membrane equations for sheep Purkinje fibers. All software was written in fortran-77, and calculations were performed on an IBM RS/6000 model 560 workstation at Tulane Computing Services. To ensure that the membrane equations had been coded correctly, we compared our solutions using software written via translation of the program heart32 with solutions using software assembled according to the guidelines described by Cabo and Barr.33 Although the animal-experiment data were from the canine epicardium, we used the DiFrancesco-Noble Purkinje fiber equations because they include contemporary mathematical representations of active ionic transfer processes. Importantly, the DiFrancesco-Noble equations account for changes in intracellular and extracellular ion concentrations, intracellular calcium uptake and release, a sodium-calcium exchange current, and an Ito.
To mimic features of the whole-animal and isolated myocyte pacing protocols, the one-dimensional model was paced over a range of BCLs (400, 500, 600, and 1000 ms). All simulations were initiated with an intracellular current square-wave pulse of 1.0-mA/cm2 intensity and 2-ms duration applied at three nodes at one end of the cable. The model was paced for four to five beats, and then an S2 was introduced at the same nodes at the cable end to initiate a PVC. S2 and S1 had identical magnitudes, and S2 was introduced over a range of S1-S2 intervals (250 to 400 ms, 25-ms steps). Action potentials in response to the stimulus protocol were “recorded” 3.75 mm from the stimulated end of the model. The activation time of the action potential was taken as the time of the maximum temporal derivative during the upstroke. APD was measured from the time of the maximum temporal derivative during the upstroke of the action potential to the time the downstroke returned to −60 mV. DI was taken as the interval between the time of dv/dtmax of the upstroke of one beat and the time the downstroke of the preceding beat reached −60 mV.
Effects of PVCs on PED Repolarization
Effects of PVCs on RPPED were evaluated during drives at 2 to 4 BCLs in each experiment. Data were grouped into short, intermediate, and long BCLs. There were 22 drives in the short-BCL group (300 to 350 ms), 24 drives in the intermediate group (380 to 430 ms), and 27 drives in the long group (450 to 550 ms).
PVCs with short coupling intervals had greater effects on RPPED than PVCs with long coupling intervals. As the coupling interval of the PVC was increased, the magnitude of its effect on RPPED diminished, and the duration of RPPED approached the duration of RPBCL. During short BCL drives, short-coupling-interval PVCs prolonged RPPED in all but one experiment. In that experiment, PVCs also shortened RPPED during intermediate BCL drives. In two other experiments, short-coupling-interval PVCs prolonged RPPED during short BCL drives but shortened RPPED during long BCL drives. In all of the other 18 experiments, short-coupling-interval PVCs prolonged RPPED during all BCL drives.
The results from all 21 experiments are summarized in Fig 2A⇓. ΔRPmax was identified for each set of measurements, and the means±SEM of these values were calculated. As mentioned above, ΔRPmax occurred after PVCs with short coupling intervals. ΔRPmax averaged 4.3±0.8 ms for the short-BCL group, 4.2±0.8 ms for the intermediate-BCL group, and 2.1±0.5 ms for the long-BCL group. When all BCLs were considered together, ΔRPmax averaged 3.5±0.4 ms. The mean ΔRPmax was significant for all groups of cycle lengths. In addition, the mean ΔRPmax for the short- and intermediate-BCL groups were significantly greater than the mean ΔRPmax for the long-BCL group. The medians for ΔRPmax were 5.5, 3.5, and 2.0 ms for the short-, intermediate-, and long-BCL groups, respectively.
ΔAPDmax over a range of S1-S2 coupling intervals was also determined for three BCLs in the computer simulations. As shown in Fig 2B⇑, APDPED was longer than APDBCL during all BCLs. In addition, APDPED prolongation was greatest after short-coupling-interval PVCs and was greatest during the shortest BCL drive. ΔAPDmax was 16 ms during the 400-ms BCL drive, 13 ms during the 500-ms drive, and 9 ms during the 600-ms drive. Although quantitatively larger, the direction of changes in APD in the model corresponded to the results of the animal experiments.
The effects of PVCs on PED repolarization were also evaluated with studies of isolated epicardial myocytes. Action potentials recorded from one myocyte are shown in Fig 3⇓. The BCL was 400 ms. A PVC delivered early during repolarization prolonged the APDPED by 16 ms from 234 to 250 ms and increased the magnitude of phase 1 repolarization. The mean±SEM of ΔAPD for the 11 cells stimulated at a 400-ms BCL was 10.7±1.6 ms, P<.001 (239±2.4 ms compared with 250±3.0 ms). For the 8 cells stimulated at a 500-ms BCL, ΔAPD was 8.5±1.4 ms, P<.005 (245±6.1 ms compared with 253±5.7 ms). The medians for ΔAPD were 8.5 ms and 9 ms, respectively, during the 500-ms and 400-ms BCL drives. The ΔAPDs during the 400-ms and 500-ms BCL drives were not statistically significantly different from each other. However, there was considerable variability in APD between cells, and only 1 cell was paced at both BCLs.
Relation of S1-S2 to PED Repolarization
Fig 4A⇓ shows the relation between the S1-S2 coupling interval and RPPED during drives at 480-, 400-, and 320-ms BCLs in one experiment. During drives at all BCLs, short S1-S2 coupling intervals increased RPPED. There was an inverse relation between RPPED and the S1-S2 coupling interval. That is, short-coupling-interval PVCs prolonged RPPED more than PVCs with long coupling intervals. In addition, as noted above, the effects of PVCs on RPPED were greatest during the shortest-BCL drive.
The relation between S1-S2 and APDPED during simulations with drives at three BCLs is shown in Fig 4B⇑. As in the animal experiments, APDPED was inversely related to the S1-S2 coupling interval. Thus, APDPED was long when the S1-S2 coupling interval was short and decreased as the S1-S2 coupling interval approached the BCL. In accord with the animal experiments (Fig 4A⇑), the effect of PVCs on APDPED was greatest during drive at the shortest BCL.
Electrical Restitution and the Relation of S1-S2 to DIPED
Prolongation of PED repolarization cannot be explained on the basis of “memory” for the short PVC cycle. We therefore examined whether PED repolarization prolongation was due to proximity effects. Data for this analysis are shown in Figs 5⇓ and 6⇓. These data are from the same experiment as illustrated in Fig 4A⇑ and the same simulations as illustrated in Fig 4B⇑.
Fig 5A⇑ shows restitution curves of the relation between the DIPVC and RPPVC during short, intermediate, and long BCLs. RPPVC was always shorter than RPBCL in this and all experiments. RPPVC became progressively longer as DIPVC increased. DIPVC increased because we systematically increased the S1-S2 interval. In this and in other experiments, RPPVC of the earliest possible PVC was often 1 to 2 ms longer than the RPPVC of a PVC occurring a few milliseconds later. This characteristic of early restitution has been previously noted.21 In addition, large changes in RPPVC occurred with small changes in DIPVC when DIPVC was short. When DIPVC was more than ≈50 ms, however, further increases in DIPVC had less effect on RPPVC. The forms of the restitution curves shown in Fig 5A⇑ are consistent with previous reports.8 12 21 Restitution curves of the relation of DIPVC to APDPVC constructed from data acquired from the 400-, 500-, and 600-ms BCL simulations are shown in Fig 5B⇑. As in the animal experiments, APDPVC increased with DIPVC, and the expected cycle-length dependence in the restitution curves was observed. Although the characteristic flattening of the restitution curves at long DIPVC was absent from the simulations at 400-, 500-, and 600-ms BCLs (Fig 5B⇑), flattening of the curve was apparent in data from a 1000-ms BCL simulation (Fig 5C⇑). This occurred because APD from the DiFrancesco-Noble membrane equations is longer than epicardial myocyte APD. As a result, a DIPVC sufficiently long to recreate the flat portion of the restitution curve occurred only with simulations at BCL longer than 600 ms.
The relations of DIPED to S1-S2 for the animal experiment and the computer simulations are shown in Fig 6A⇑ and 6B⇑. DIPED increased as the S1-S2 interval decreased during drives at all three BCLs. This occurred because RPPVC and APDPVC decreased as the S1-S2 interval decreased (see Fig 5⇑). These analyses suggested that prolongation of PED repolarization was most likely due to effects of DIPED.
Relation of DI and BCL to PED Repolarization
Fig 7A⇓ shows that the refractory period of the PED increased as DIPED increased. In addition, the slope of the RPPED versus DIPED relation was less at long than at short BCLs. ΔRPmax was 2 ms for the 480-ms BCL drive, 3 ms for the 400-ms BCL drive, and 5 ms for the 320-ms BCL drive. The slopes (m) of the relation were m=0.029, m=0.058, and m=0.131 for the 480-, 400-, and 320-ms BCLs, respectively. The relation of DIPED to RPPED was analyzed in terms of its slope because these data were confined to the linear portion of the restitution curve (see Fig 5A⇑).
Analysis of data from the simulations during drives at the three BCLs also demonstrated a direct relation between DIPED and APDPED (Fig 7B⇑). As in the animal experiments, the slope of the relation of DIPED to APDPED increased as the BCL decreased. The slopes of the relation were m=0.079, m=0.138, and m=0.205 for the 600-, 500-, and 400-ms BCL drives, respectively. Thus, in both the animal experiments and simulations, the slope of the relation of RPPED and APDPED versus DIPED decreased as the BCL increased.
Fig 8⇓ summarizes data from all experiments and further documents the influence of BCL on RPPED prolongation. The slope of the relation of DIPED to RPPED is plotted as a function of the change in BCL. The shortest BCL in each experiment is plotted at zero. There was a positive slope for the relation of DIPED to RPPED for all but three observations made during drive at the shortest BCL and for most observations made during drives at longer BCLs. Since DIPED was inversely related to the S1-S2 coupling interval (see Fig 6⇑), the data indicate that RPPED was longest when the S1-S2 interval was the shortest and RPPED decreased as the S1-S2 interval increased. In addition, the slope of the relation of DIPED to RPPED usually decreased as the BCL increased (Fig 8A⇓). This indicates that PVCs had greater effects on RPPED during drives at short BCLs than at long BCLs.
Fig 8B⇑ shows data from experiments in which the slopes of the relation of DIPED to RPPED were negative or close to zero during drive at the shortest BCL. The two curves shown as solid lines are from one experiment and represent data acquired during drives at two BCLs and drives of two sites. In this experiment, there was a negative slope to the relation of DIPED to RPPED during drives at both BCLs. This indicates that in this experiment, RPPED increased as DIPED decreased. The magnitude of RPPED shortening that occurred in this experiment was less during long- than short-BCL drives. The dashed curve in Fig 8B⇑ is from another experiment. The slope of the relation of DIPED to RPPED was close to 0 during the short-BCL drive, and there was a direct relation between DIPED and RPPED during drive at the longer BCL. The reasons for these exceptions to the usual relation of DIPED to RPPED that we observed are uncertain. However, the positive slopes of the relation of DIPED to RPPED that were present in the great majority of the experiments indicated that DIPED was an important determinant of RPPED. The decrease in the slope of the relation during long-BCL drives indicated that PVCs had greater effects on RPPED during drives at short BCLs than during drive at long BCLs.
Ionic Mechanisms for PED Repolarization Prolongation
It was not possible to determine the ionic mechanism for RPPED prolongation in the intact-heart experiments. However, observations from the computer simulations and isolated myocyte studies provided insights into the mechanism.
The time courses for selected variables from the DiFrancesco-Noble equations in simulations during PVCs and the PEDs that followed them were analyzed. Because the magnitude of PED prolongation was greatest during drives at short BCLs, all records presented here are from the shortest-BCL (400-ms) simulations. Fig 9A⇓ shows the transmembrane potentials for PVCs at a number of S1-S2 coupling intervals, and Fig 9B⇓ shows the transmembrane potentials for the PEDs after those PVCs. As expected, the PVC action potential was shortest at an S1-S2 coupling interval of 250 ms and longest at an S1-S2 coupling interval that equaled the BCL, ie, 400 ms. Consistent with the data in Fig 7⇑, the PED action potential was longest after the PVC with the 250-ms S1-S2 coupling interval, and it was shortest when the S1-S2 coupling interval equaled the BCL. At intermediate S1-S2 intervals of 300 and 350 ms, the PED action potentials had durations intermediate between the APDs of PEDs that followed PVCs, with coupling intervals of 250 and 400 ms.
In simulations for each S1-S2 coupling interval, all membrane equation variables at the observation point in the one-dimensional model were stored. Those variables were then monitored over the time course of the PVC and PED. The main determinants of APD in the DiFrancesco-Noble membrane equations are the IK, the IK1, the Ito, and the ICa. Fig 10A⇓ illustrates the time course of IK, Ito, and ICa during the PVCs shown in Fig 9A⇑. As the coupling interval decreased, the magnitude of IK increased early in the plateau of the PVC, causing a more rapid rate of repolarization and decreased APD. This increased DIPED. Part of the increased IK resulted from incomplete deactivation as DIPVC was reduced.34 Also contributing to the increase in IK during the PVC was the decrease in Ito, which made Vm more positive immediately after the peak of the action potential. Increased Vm caused greater activation of IK. Shown in Fig 10B⇓ is the time course of these currents and the inward rectifier IK1 during PEDs that followed PVCs with S1-S2 coupling intervals of 400 and 250 ms. As in the case of the PVC, IK was smallest when DIPED was longest, ie, after the PVC with a 250-ms S1-S2 coupling interval. This reduction in IK early in the plateau of the PED slowed the rate of repolarization, resulting in delayed rise in IK1 and thus increased APDPED. In both the PVC and PED, ICa remained relatively constant throughout most of the action potential and thus had little effect on APD.
Action potentials recorded from an isolated myocyte are shown in Fig 3⇑ and demonstrate that in addition to APDPED prolongation, phase 1 amplitude increased in the PED. This was presumably due to greater activation of Ito.22 25 To assess the possibility that a less positive Vm at the start of the plateau contributes to PED prolongation, phase 1 amplitude was artificially increased by application of a brief hyperpolarizing current pulse immediately after the peak of the action potential. The effects of current injection on the action potential of one cell paced at a 400-ms BCL are shown in Fig 11⇓. The action potential associated with current injection was 22 ms longer than the one without current injection. Qualitatively, the same results were observed in each of the five other cells examined.
The effects of changes in steady-state cycle lengths on myocardial repolarization properties have been studied extensively, and the direct relation between cycle length and repolarization properties has been established.7 8 9 10 11 12 13 14 15 16 17 18 19 However, the relation is complicated. Most of the effect of a new cycle length occurs during the first beat. Repolarization properties then oscillate for several cycles, and slow changes in repolarization duration continue for up to several hundred cycles before a new steady state is reached. This slow change in repolarization properties has been attributed to “memory” for the previous BCL drive.
The effects of single premature beats on myocardial repolarization properties have also been studied extensively and are even more complex than the effects due to changes in BCL. Repolarization usually shortens during PVCs. However, some studies have reported35 36 that premature action potentials actually had longer durations than action potentials with longer cycle lengths. The reasons for these exceptions are uncertain, but they may be related to specific characteristics of the pacing protocols used or to differences in the experimental preparations. The effects of PVCs on repolarization differ from species to species and even from location to location in the same species.20 35 37 Gettes et al20 reported that APDPVC cannot be explained solely on the basis of cycle length. They found that porcine Purkinje fiber APDs were shorter during PVCs with short coupling intervals than during regular drive at the cycle length of the premature coupling interval. They referred to this phenomenon as cycle length–independent shortening of the APD. In myocardial cells, on the other hand, they found a cycle length–independent lengthening of premature APD. In both tissues, the magnitude of the cycle length–independent change in the premature APD was related to the proximity of the upstroke of the premature action potential to the downstroke of the preceding action potential and to the APD of the preceding action potential. That is, the magnitude of cycle length–independent change in the premature APD was related to the preceding DI (electrical restitution).
Although considerable attention has been paid to the effects of changes in steady-state cycle length and the effects of single PVCs on repolarization properties, little attention has been paid to the effects of PVCs on repolarization properties of PEDs. In a study directed at evaluating RPPVC over a range of coupling intervals in ischemic and nonischemic myocardium, we noted that RPPED was often a few milliseconds longer than the RP during regular drive.38 However, we did not systematically characterize the effects of PVCs on PED repolarization properties in that study.
Spontaneous ventricular tachyarrhythmias are frequently initiated by a short-long-short sequence of cycle lengths.1 2 The first short cycle is usually due to a PVC, and the long cycle is due to the compensatory pause after the PVC. The next short cycle is due to a PVC after the PED, and this PVC initiates the tachycardia. Demonstration of the effects of PVCs on PED repolarization properties should therefore contribute to understanding the mechanism by which short-long-short sequences of cycle lengths facilitate the initiation of tachyarrhythmias.
El Sherif et al6 found that a long cycle preceding premature stimulation facilitated induction of ventricular tachycardia in a canine myocardial infarction model. The long cycle prolonged the RP of ischemic myocardium more than the RP of the nonischemic tissue. They suggested that this differential effect of the long cycle on RPs of ischemic and nonischemic myocardium produced a gradient of RPs. This gradient of RPs resulted in an arc of functional conduction block that permitted initiation of reentry when a premature stimulus was delivered during the subsequent cycle. They also suggested that the mechanism for the differential effect of the long cycle length in ischemic and nonischemic tissue was likely due to ischemic myocardium having a shorter “memory” for the preceding cycle lengths. Short-long-short patterns of stimulation have also been proposed to facilitate induction of ventricular tachycardia in patients undergoing electrophysiological testing.3 4 5 The pattern of stimulation that was suggested consisted of a train of regularly driven beats followed by a long cycle that was in turn followed by two consecutive premature stimuli.
In the study reported here, we used whole-animal and isolated myocyte experiments and computer simulations to investigate the effects of single PVCs on the repolarization properties of PEDs. The stimulus protocol was selected to be similar to the pattern of cycle lengths associated with spontaneous tachycardias. However, the timing parameters we used were comparable to the conditions associated with spontaneous premature atrial complexes. Namely, the compensatory pause was incomplete, and the activation sequences of the premature beats and the regularly driven beats were the same. Keeping the activation sequences of the regularly driven beats and the premature beats the same accomplished two things. First, the premature cycle lengths remained equal throughout the heart. If the premature stimulus had been delivered to a site other than the basic drive site, the PVC and PED cycle lengths would have differed throughout the ventricles because of differences in the activation times at individual sites during the two drives. Second, activation sequence–induced changes in the magnitude of electrotonic effects on repolarization properties were avoided by keeping the activation sequences of the regularly driven beats and the premature beats the same.26 In addition, the interval between the premature beat and the next regularly driven beat was kept the same as the BCL. This diminished the occurrence of escape beats during slow basic drive pacing rates. The complexity of the effects of a cycle length longer than the BCL on PED repolarization was also avoided. The RP test pulse was first delivered early in the cardiac cycle, at a time when the ventricle was still refractory. The test pulse was then delayed in 1-ms increments until a propagated response occurred. The rhythm being evaluated was therefore uninterrupted by responses to the RP test pulse until it was delayed sufficiently to induce a response. Thus, the complexity of variations in cycle lengths that would have occurred if the RP test stimuli had first been delivered late in the cycle and then moved earlier was averted.
Our pacing protocol, the short-long-short protocol suggested for facilitating tachyarrhythmias during electrophysiological testing,3 4 5 and the short-long-short pacing protocol El Sherif et al6 used all differ from the pattern associated with spontaneous ventricular tachyarrhythmias.1 2 However, all of these short-long-short patterns have one thing in common: prolongation of the DI preceding the last short cycle. In the case of spontaneous ventricular tachycardia, the first short cycle is due to a PVC. The DI preceding the next beat, ie, DIPED, is prolonged in this situation both because of the compensatory pause after the PVC and because of the short RPPVC. The sequence of cycle lengths that has been proposed for initiating clinical and experimental tachycardias consisted of a single long cycle introduced after a train of regular beats. Premature beats were then initiated in the next cycle. In this situation, DIPED was prolonged solely because of the long cycle preceding the cycle in which the PVC was introduced. With our pacing protocol, the S2-S1 interval equaled the BCL, and DIPED was prolonged solely because RPPVC and APDPVC were short. For PVCs with comparable coupling intervals, DIPED would be longer after spontaneous PVCs than after PVCs initiated with our stimulation protocol, with the protocols suggested for electrophysiological testing, or with those used to initiate tachyarrhythmias in animal experiments. The longer DIPED associated with spontaneously occurring PVCs would be expected to prolong RPPED even more than the prolongation that occurred with our pacing protocol.
Since cycle length effects on RPs dissipate slowly, our pacing protocol, in which PVCs were introduced after every sixth beat, may have had an effect on RPPED. However, frequent PVC cycle lengths would be expected to shorten RPPED. RPPED measured after rare PVCs would be expected to be longer than those measured after more frequent PVCs. Thus, although there were differences between the pattern of stimulation in our pacing protocol and the pattern of cycle lengths associated with spontaneous PVCs, these differences cannot account for the RPPED prolongation we demonstrated. In fact, RPPED prolongation would be expected to be even greater if the PVCs had been less frequent and the compensatory pauses after the PVCs had been complete.
Our whole-animal experiments, isolated myocyte studies, and computer simulations all demonstrated that PED repolarization is longer than repolarization during regular drive. Prolongation of APD of PEDs recorded from isolated feline ventricular myocytes has also been demonstrated by Rubenstein and Lipsius.39 The emphasis of their study was the ability of PVCs to reverse the phase of action potential alternans and the possible relation of that reversal to reentrant arrhythmias. In our study, the magnitude of the effects of PVCs on PED repolarization was greatest for PVCs with short coupling intervals and was greatest during short-BCL drives. The effects of PVCs on PED repolarization were qualitatively the same in both of the experimental preparations and in the computer simulations. However, the magnitude of the effects was greater in the computer simulations and isolated epicardial myocyte studies than in the whole-animal studies. The reasons for this difference are uncertain. However, previous work has shown that increased DI prolonged APD more in the epicardium than in the endocardium.24 Thus, our epicardial RPPED determinations may have been electrotonically attenuated by underlying endocardial cells. In addition, although APD and RP, in general, track each other in normal myocardium, the values are not equal. It is possible that a given change in APD was less detectable with RP measurements.
We cannot comment on whether PVCs would have the same effects on PED repolarization of regions of the myocardium other than the right ventricular epicardial tissue we studied. Additionally, we cannot comment on whether PED repolarization would prolong more in diseased than in normal myocardium. As mentioned above, the greater magnitude of effect we observed in the isolated cells and computer simulations may have been due to the lack of electrotonic interaction in the isolated cells and minimal electrotonic interactions in the cable model of action potential propagation. PVCs might therefore have greater effects on PED repolarization in diseased than in normal myocardium because of the electrical uncoupling that occurs in diseased tissue. Further studies are needed to support or refute this possibility.
The possible effect of PED repolarization prolongation on arrhythmogenesis should be considered in terms of the conditions that result in unidirectional block rather than in terms of the absolute magnitude of PED repolarization prolongation. For example, if the refractory period of a region of the myocardium were just 1 ms less than a PVC cycle length, the PVC could be conducted through that region. However, even a 1-ms prolongation of RPPED would result in block if a second PVC with the same coupling interval followed the PED. Thus, an increase in refractoriness of even a few milliseconds could produce the conditions essential for reentrant arrhythmias.
Use of the DiFrancesco-Noble membrane equations31 is a possible limitation of the computer simulations. These equations describe the ionic transfer processes for sheep Purkinje strands. RP measurements in the animal experiments were taken from the canine epicardium, and the isolated myocytes were harvested from the epicardium. One could argue that the results from the computer simulations might be qualitatively distinct from those of the experiments because differences between the ionic currents in ventricular myocardium and the specialized conduction system tissue are well known. Nevertheless, the results of the computer simulations were qualitatively the same as the animal experiment data. Although there are membrane equation sets that attempt to describe the ionic transfer processes for ventricular myocardium, none of them are better suited to represent our experimental studies than those from DiFrancesco and Noble. The Beeler and Reuter membrane equations40 were considered inappropriate because a number of the component currents that form the basis of action potential initiation and maintenance are inaccurate from a contemporary electrophysiological perspective. The Luo and Rudy41 membrane equations were not used because they have no representation of the sodium-calcium exchange current or of the Ito. Both of these currents are involved in rate-dependent tissue response. The more recent Luo and Rudy42 43 membrane equations clearly provide a contemporary description. However, many of the data for assembly of their component equations were obtained from isolated guinea pig myocytes, which do not have an Ito. Ito is prominent in canine epicardial myocytes,24 and our RP measurements and AP recordings were obtained from canine right ventricular epicardial tissue. We therefore considered representation of Ito important for our analyses. Also, the APD response to plateau voltage in the DiFrancesco-Noble membrane equations is similar to that in epicardial ventricular cells. In view of these considerations, the DiFrancesco-Noble membrane equations seemed to be the best choice for our simulations.
Analysis of the membrane variables in the computer simulations provided indirect evidence of the possible ionic basis for prolongation of APDPED. The prolongation of DIPED associated with short S1-S2 coupling intervals allowed greater time for more complete deactivation of IK.34 Thus, IK was diminished at the beginning of the plateau of the PED, and this caused prolongation of the plateau and increased APD. Another factor contributing to APD prolongation was that the increased DIPED allowed time for more complete reactivation of Ito.22 25 The result was a more rapid rate of phase 1 repolarization and thus a more negative Vm at the beginning of the plateau. The increased negativity reduced the magnitude of IK and prolonged APD. The observation that a more negative Vm at the initiation of the plateau results in action potential prolongation has been described for cardiac Purkinje strands44 45 and canine ventricular epicardium.24 Both preparations have a prominent Ito that mediates phase 1 repolarization. The sensitivity of APD to plateau voltage can be seen by injection of brief constant-current pulses early in the plateau. In Purkinje strands, early hyperpolarization of the plateau prolongs APD.44 45 As shown in Fig 11⇑, we have obtained the same response in isolated canine ventricular epicardial myocytes. The behavior induced by artificial phase 1 hyperpolarization is consistent with the influence of Ito on action potential repolarization. Using canine papillary muscles, trabeculae, and epicardial strips, Litovsky and Antzelevitch24 demonstrated that inhibition of Ito with 4-aminopyridine shortened APD in all preparations. However, the magnitude of shortening was most pronounced in the epicardial preparations. Ito magnitude in epicardial myocytes is five times higher than that in endocardial myocytes.25
In conclusion, the results of our study demonstrate that (1) PED repolarization is prolonged with respect to repolarization during regular drive; (2) there is an inverse relation between PVC coupling intervals and PED repolarization duration; (3) there is a direct relation between DIPED and PED repolarization duration; (4) the magnitude of prolongation of PED repolarization is cycle length–dependent, with greater prolongation of PED repolarization associated with short-BCL drives; (5) the computer simulations and isolated myocyte studies suggested that the likely ionic basis for prolongation of PED repolarization is the interaction of Ito and IK; and (6) the results of our study suggest a possible mechanism for initiation of ventricular tachycardia in the setting of a short-long-short pattern of cycle lengths. Namely, prolongation of repolarization properties of a PED could result in an arc of refractory tissue that could then cause unidirectional block of PVCs after PEDs.
Selected Abbreviations and Acronyms
|ΔRPmax||=||maximum difference between RPPED and RPBCL|
|A||=||local activation time|
|APD||=||action potential duration|
|BCL||=||basic cycle length|
|ICa||=||inward calcium current|
|IK||=||delayed rectifier current|
|IK1||=||inward rectifier current|
|Ito||=||transient outward current|
|PVC||=||premature ventricular contraction|
This study was supported by National Institutes of Health, National Heart, Lung, and Blood Institute grants HL-34288, HL-42873, and R29-HL-54024; the Nora Eccles Treadwell Foundation; the Richard A. and Nora Eccles Harrison Fund for Cardiovascular Research; National Science Foundation National Young Investigator Award BES-945-7212; the Whitaker Foundation; and Louisiana Board of Regents grants RCS-A29 and ENG-B29.
Reprint requests to Mary Jo Burgess, MD, CVRTI, Bldg 500, University of Utah, Salt Lake City, UT 84112. E-mail firstname.lastname@example.org.
- Received March 6, 1995.
- Revision received April 24, 1995.
- Accepted April 25, 1995.
- Copyright © 1995 by American Heart Association
Kay GN, Plumb VJ, Arciniegas JG, Henthorn RW, Waldo AL. Torsades de pointes: the long-short initiating sequence and other clinical features: observations in 32 patients. J Am Coll Cardiol. 1990;2:806-817.
Denker S, Lehman M, Mahmua R, Gilbert C, Akhtar M. Divergence between refractoriness of His-Purkinje system and ventricular muscle with abrupt changes in cycle length. Circulation. 1983;68:1212-1221.
Denker S, Lehman M, Mahmua R, Gilbert C, Akhtar M. Facilitation of macroreentry within the His-Purkinje system with abrupt changes in cycle length. Circulation. 1984;69:26-31.
El-Sherif N, Gough WB, Restivo M. Reentrant ventricular arrhythmias in the late myocardial infarction period: mechanisms by which a short-long-short cardiac sequence facilitates the induction of reentry. Circulation. 1991;83:268-278.
Hoffman BF, Suckling EE. Effect of heart rate on cardiac membrane potentials and the unipolar electrogram. Am J Physiol. 1954;179:123-130.
Mendez C, Gruhzit CC, Moe GK. Influence of cycle length upon refractory period of auricles, ventricles and A-V node in the dog. Am J Physiol. 1956;184:287-295.
Greenspan K, Edmands RE, Fisch C. Effects of cycle length alteration on canine cardiac action potentials. Am J Physiol. 1967;212:1416-1420.
Janse MJ, van der Steen ABM, van Dam RT, Durrer D. Refractory period of the dog’s ventricular myocardium following sudden changes in frequency. Circ Res. 1969;24:251-262.
Han J, Moe GK. Cumulative effects of cycle length on refractory periods of cardiac tissue. Am J Physiol. 1969;217:106-109.
vanDam RT, Janse MJ. The effect of changes in rate and rhythm on the refractory period of the ventricular myocardium and specialized conduction system of the canine heart. In: Kao FF, Koizumi K, Vasalle M, eds. Research in Physiology: A Liber Memorialis in Honor of Professor Chandler McCuskey Brooks. Bologna, Italy: Auto Gazzi; 1971:161-177.
Franz MR, Swerdlow CD, Liem LB, Schaefer J. Cycle length dependence of human action potential duration in vivo. Effects of single extrastimuli, sudden sustained rate acceleration and deceleration, and different steady-state frequencies. J Clin Invest. 1988;82:972-979.
Elharrar V, Atarashi H, Surawicz B. Cycle length memory dependent action potential duration in canine cardiac Purkinje fibers. Am J Physiol. 1984;247:H936-H945.
Saitoh H, Bailey JC, Surawicz B. Alternans of action potential duration after abrupt shortening of cycle length: differences between dog Purkinje and ventricular muscle fibers. Circ Res. 1988;62:1027-1040.
Saitoh H, Bailey JC, Surawicz B. Action potential alternans in dog Purkinje and ventricular muscle fibers: further evidence in support of two different mechanisms. Circulation. 1989;80:1421-1431.
Gettes LS, Morehouse N, Surawicz B. Effect of premature depolarization on the duration of action potentials in Purkinje and ventricular fibers of the moderator band of the pig heart: role of proximity and the duration of the preceding action potential. Circ Res. 1972;30:55-66.
Elharrar V, Surawicz B. Cycle length effect on restitution of action potential duration in dog cardiac fibers. Am J Physiol. 1983;244:H782-H792.
Tseng G, Hoffman B. Two components of transient outward current in canine ventricular myocytes. Circ Res. 1989;64:633-647.
Liu DW, Gintant GA, Antzelevitch C. Ionic bases for electrophysiological distinctions among epicardial, midmyocardial and endocardial myocytes from the free wall of the canine left ventricle. Circ Res. 1993;72:671-687.
Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol. 1952;117:500-544.
Crank J, Nicholson P. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc Cambr Phil Soc (Math Phys Sci). 1947;43:50-67.
Pollard AE, Hooke NF, Henriquez CS. Cardiac propagation simulation. Crit Rev Biomed Eng. 1992;20:319-358.
DiFrancesco D, Noble D. A model of cardiac electrical activity incorporating ionic pumps and concentration changes. Philos Trans R Soc Lond [Biol]. 1985;B-307:353-398.
Oxsoft: Heart Program Manual, Version 3.2. Oxford, UK: Oxsoft Ltd; 1990:1-17.
Gintant GA, Cohen IS, Datyner NB, Kline RP. Time-dependent outward currents in the heart. In: Fozzard HA, Haber E, Jennings RB, Katz AM, Morgan HE, eds. The Heart and Cardiovascular System. 2nd ed. New York, NY: Raven Press Ltd; 1992:1121-1169.
Kobayashi Y, Peters W, Khan SS, Mandel WJ, Kargueuzian HS. Cellular mechanisms of differential action potential duration restitution in canine ventricular muscle cells during single versus double premature stimuli. Circulation. 1993;86:955-967.
Hiraoka M, Kawano S. Mechanism of increased amplitude and duration of the plateau with sudden shortening of diastolic intervals in rabbit ventricular cells. Circ Res. 1987;60:14-26.
Rubenstein DS, Lipsius SL. Premature beats elicit phase reversal of mechanoelectrical alternans in cat ventricular myocytes: a possible mechanism for reentrant arrhythmias. Circulation. 1995;91:201-214.
Luo CH, Rudy Y. A model of the ventricular cardiac action potential. Circ Res. 1991;68:1501-1526.
Luo CH, Rudy Y. A dynamic model of the ventricular cardiac action potential, I: simulations of ionic currents and concentration changes. Circ Res. 1994;74:1071-1096.
Luo CH, Rudy Y. A dynamic model of the ventricular cardiac action potential, II: afterdepolarizations, triggered activity, and potentiation. Circ Res. 1994;74:1097-1113.
Weidman S. Effect of current flow on the membrane potential of cardiac muscle. J Physiol. 1951;115:227-234.
Kass RS, Tsien RW. Control of action potential duration by calcium ions in cardiac Purkinje fibers. J Gen Physiol. 1976;67:599-617.