INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

THE DURATION of the cardiac action potential is determined in large part by the preceding diastolic interval. This relationship between action potential duration and diastolic interval, known as the action potential duration restitution relation, is an important determinant of cardiac dynamics (17). In particular, if the slope of the restitution relation is 1, an alternation of action potential duration, or electrical alternans, commonly develops during high-frequency pacing (2, 8).

It has been suggested that rate-dependent electrical alternans may be a precursor to the development of ventricular arrhythmias, particularly ventricular fibrillation (VF) (6, 10, 19, 22). In support of this idea, several recent experiments (5, 11, 23) have shown that when the slope of the restitution relation is 1, rapid pacing induces both alternans and fibrillation in isolated ventricles. If the slope of the restitution relation is reduced to <1, neither electrical alternans nor fibrillation occurs (5, 11, 12, 23). Unfortunately, the interventions used to date to suppress alternans and fibrillation [high-dose calcium channel blockers (23), hyperkalemia (12), and bretylium (5)] have limited clinical utility. More effective means of suppressing alternans need to be identified, a process that would be facilitated by a more complete understanding of the ionic basis for alternans.

One approach to determining the ionic basis for alternans is to use a computer model, several of which have been developed. For example, Luo and Rudy (15, 16), using data obtained primarily from guinea pig myocytes, developed a comprehensive ionic model (LR1) that subsequently was updated (LRd) to include formulations for the rapid and slow components of the delayed rectifier K⁺ current (I_Kr and I_Ks, respectively). Recently, Winslow et al. (26) modified the LRd model using data for ionic currents obtained from canine ventricular myocytes (CVM) and a formulation for calcium dynamics developed originally in guinea pig myocardial cells (9). An alternative formulation for calcium dynamics has been proposed by Chudin et al. (1) in their modification of the LR1 model.

Each of the models described above has limitations with respect to the study of the ionic basis for electrical alternans. The Winslow and LRd models do not produce sustained alternans at rapid pacing rates, whereas the Chudin model, which does generate electrical alternans, lacks formulations for repolarizing K⁺ currents likely to contribute importantly to alternans [I_Kr, I_Ks, and the transient outward K⁺ current (I_to)].

Given that a complete ionic model that generates electrical alternans is not currently available, we set out to develop such a model, guided by the results obtained from our experimental studies in the canine ventricle (11, 23). Our initial objectives were to develop an ionic model of the CVM that exhibits stable electrical alternans and to use the model to identify the ionic currents responsible for alternans. Once the relevant ionic currents were identified, we then manipulated these currents to eliminate alternans. Our expectation is that the same ionic manipulations that suppress alternans in the ionic model will suppress fibrillation in vivo, in which case the results of the present study may suggest novel approaches to the prevention of VF.

Glossary

h	Voltage-dependent h gate parameter
j	Voltage-dependent j gate parameter
m	Voltage-dependent m gate parameter
Xto	Voltage-dependent Xto gate parameter
h	Voltage-dependent h gate parameter
i	Myoplasmic buffering factor
j	Voltage-dependent j gate parameter
m	Voltage-dependent m gate parameter
SR	Sarcoplasmic reticulum buffering factor
Xto	Voltage-dependent Xto gate parameter
	Sarcoplasmic reticulum Ca2+-dependent Jrel factor
	Controls voltage dependence of INaCa
	Extracellular Na+ INaK factor
d	ICa activation time constant
f	ICa inactivation time constant
fCa	Ca2+-dependent ICa inactivation time constant
Kr	IKr activation time constant
Ks	IKs activation time constant
Acap	Capacitive membrane area
APD	Action potential duration
BCL	Basic cycle length
Csc	Specific membrance capacity
Camax	Maximum change in Ca2+
Camin	Minimum change in Ca2+
[Ca2+]i	Intracellular Ca2+ concentration
[Ca2+]o	Extracellular Ca2+ concentration
[Ca2+]SR	Sarcoplasmic reticulum Ca2+ concentration
[CMDN]tot	Total calmodulin concentration
[CSQN]tot	Total calsequestrin concentration
CVM	Canine ventricular myocyte
d	ICa activation gate
d	Steady-state ICa activation
DI	Diastolic interval
ECa	Ca2+ equilibrium potential
EK	K+ equilibrium potential
EKs	IKs equilibrium potential
ENa	Na+ equilibrium potential
f	ICa inactivation gate
f	Steady-state ICa inactivation
f	Steady-state Ca2+-dependent ICa inactivation
fCa	Ca2+-dependent ICa inactivation gate
fNaK	Voltage-dependent INaK factor
F	Faraday constant
Cab	Peak ICab conductance
K1	Peak IK1 conductance
Kp	Peak IKp conductance
Kr	Peak IKr conductance
Ks	Peak IKs conductance
Na	Peak INa conductance
to	Peak Ito conductance
h	Fast INa inactivation gate
ICa	L-type Ca2+ channel current
Ca	Maximal ICa
ICab	Ca2+ background current
ICahalf	Ca level that reduces CaK by one-half
ICaK	K+ current through the L-type Ca2+ channel
IK1	Inward rectifier K+ current
IKp	Plateau K+ current
IKr	Rapid component of the delayed rectifier K+ current
IKs	Slow component of the delayed rectifier K+ current
INa	Na+ current
INab	Na+ background current
INaCa	Na+/Ca2+ exchange current
INaK	Na+-K+ pump current
NaK	Maximal INaK
IpCa	Sarcolemmal Ca2+ pump current
pCa	Maximal IpCa
Istim	Stimulus current
Ito	Transient outward K+ current
j	Slow INa inactivation gate
Jleak	Leakage Ca2+ flux from the sarcoplasmic reticulum
Jrel	Release Ca2+ flux from the sarcoplasmic reticulum
Jup	Uptake Ca2+ flux to the sarcoplasmic reticulum
JSR	Junctional sarcoplasmic reticulum
kNaCa	Scaling factor for INaCa
ksat	INaCa saturation factor for INaCa
K	Steady-state IK1 activation
KKp	IKp activation
K	Ca2+ half-saturation constant for calmodulin
K	Ca2+ half-saturation constant for calsequestrin
KmCa	Ca2+ half-saturation constant for INaCa
KmfCa	Ca2+ half-saturation constant for fCa
KmK1	K+ half-saturation constant for IK1
KmKo	K+ half-saturation constant for INaK
KmNa	Na+ half-saturation constant for INaCa
KmNai	Na+ half-saturation constant for INaK
KmpCa	Half-saturation constant for IpCa
Kmup	Ca2+ half-saturation constant for Jup
[K+]i	Intracellular K+ concentration
[K+]o	Extracellular K+ concentration
LR1	Luo and Rudy model
LRd	Updated Luo and Rudy model
m	INa activation gate
[Na+]i	Intracellular Na+ concentration
[Na+]o	Extracellular Na+ concentration
NSR	Nonjunctional sarcoplasmic reticulum
Ca	L-type Ca2+ channel permeability to Ca2+
CaK	L-type Ca2+ channel permeability to K+
leak	Ca2+ leakage permability between the sarcoplasmic reticulum and the myoplasm
rel	Ca2+ maximal release permeability from the sarcoplasmic reticulum
R	Ideal gas constant
SR	Sarcoplasmic reticulum
t	Time
T	Temperature
V	Voltage
Vmax	Maximum change in voltage
Vmin	Minimum change in voltage
Vmyo	Myoplasmic volume
VSR	Sarcoplasmic reticulum volume
Vup	Maximal Ca2+ uptake to the sarcoplasmic reticulum
VF	Ventricular fibrillation
XKr	IKr activation gate
X	Steady-state IKr activation
XKs	IKs activation gate
X	Steady-state IKs activation
Xto	Ito activation gate
Yto	Ito inactivation gate

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

To study the ionic mechanism of electrical alternans in canine myocytes, we constructed a CVM model using appropriate formulations of ionic currents from the LRd, Winslow, and Chudin models, altered as necessary to fit experimental voltage-clamp data from CVM. It has been well established that cellular electrical properties in the canine ventricle vary, both between right and left ventricles and within a given ventricle, according to whether a cell resides in the epicardium, endocardium, or midmyocardium (13, 14). Because the Winslow model is the only existing ionic model based on the electrical properties of the canine ventricle, we elected to use that model as the basis for the CVM model. Consequently, the CVM model, like its predecessor, recreates the midmyocardial or M cell action potential. Further alterations of various currents, including I_Ks, I_to and I_NaCa, would be required to model the electrical activity of canine endocardial and epicardial myocytes (13, 29).

The CVM model contains the following ionic current formulations

Stimulus current. I_stim used to drive the model was a square wave pulse consisting of 80 µA/µF of current for 1 ms.

Sodium current. I_Na was the same as that used in the Winslow model (26) except that the discontinuities in the h and j gate formulations were removed.

Inward rectifier K+ current. I_K1 was formulated to agree with data from Freeman et al. (4). These data indicate a smaller outward current at depolarized potentials than is seen in the Winslow model

Rapid component of the delayed rectifier K+ current. I_Kr was fit to the data from Gintant (7). In particular, we reproduced the voltage-clamp experiment used to generate Fig. 2 in his paper. The Winslow formulation of the current was altered to increase rectification, slow kinetics at depolarized potentials, and increase maximum conductance

Slow component of the delayed rectifier K+ current. I_Ks was fit to data from Varro et al. (25), specifically the results shown in Fig. 2 of their paper. The Winslow model was altered to increase the magnitude of the current and shift activation to less positive voltages

Transient outward K+ current. I_to in the model was the same as that in the Winslow model

Plateau K⁺ current. I_Kp was the same as that in the Winslow model

Na+-K⁺ pump current. I_NaK was the same as that in the LRd model

Na+/Ca²⁺ exchange current, sarcolemmal pump current, and Ca²⁺ and Na⁺ background currents. I_NaCa, I_pCa, I_Cab, and I_Nab were the same as those in the Winslow model

L-type Ca²+ channel current. I_Ca in the model was a modified version of that found in the LRd model. A time-dependent, enhanced Ca²⁺-induced inactivation was used, as well as a decrease in the current magnitude. These changes produced a smaller, more rapidly inactivating Ca²⁺ current, in agreement with experimental observations by A. C. Zygmunt (personal communication)

K+ current through the L-type Ca²⁺ channel. I_CaK was also a modified version of the LRd formulation

Calcium handling. A modified form of the intracellular calcium dynamics from Chudin et al. (1) was used. We included buffering from calmodulin in the cytoplasm and calsequestrin in the SR, omitted spontaneous release of calcium from the SR, and combined the concentrations of calcium in the JSR and NSR into a single variable

Numerical methods. The equations listed above were solved using parameter values and initial conditions found in Table 1. The simulations were run on Macintosh G3 and G4 computers using a program written in C. The numerical integration scheme was similar to that used in Luo and Rudy (15, 16) and in Rush and Larsen (24). Briefly, the time steps of integration were made small enough so that the changes in voltage and in calcium concentrations remained below maximum values, V_max and Ca_max. If the changes in voltage and calcium concentration were below a minimum value (V_min and Ca_min), the time step was increased. By keeping the changes in voltage small, we could solve the linear gate variable equations exactly during each time step. We used the following values: V_max = 0.8 mV, V_min = 0.2 mV, Ca_max = 1.067 × 10² µM, and Ca_min=2.67 × 10³ µM. (See Refs. 15, 16, and 24 for more details.) The other time-dependent variables in the model were solved using the adaptive fourth-order Runge-Kutta method (21). The errors were normalized as described in Jafri et al. (9). We used a maximum error of 1 × 10⁶, a minimum time step of 0.005 ms, and a maximum time step of 0.5 ms. During the stimulus, the step size was fixed at 0.005 ms.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Action potential and ionic currents. Figure 1 illustrates the action potentials, ionic currents, and Ca²⁺ transients generated by the CVM model at a pacing cycle length of 400 ms. The action potential (Fig. 1A) was characterized by the familiar spike-and-dome morphology of canine midmyocardial cells. I_Ca (Fig. 1B) was of smaller magnitude and inactivated more rapidly than I_Ca in previous models, in agreement with the recent experimental observations of A. C. Zygmunt (private communication). The time course and magnitude of [Ca²⁺]_i (Fig. 1D) was similar to experimental results reported previously (1, 26), indicating that the simplified calcium handling in the CVM model generated realistic Ca²⁺ transients.

View larger version (26K): [in this window] [in a new window]   Fig. 1.   Action potentials, ionic currents, and Ca2+ transients generated by the CVM model after 50 beats at a cycle length of 400 ms. A: action potentials; B: ICa; C: fCa; D: [Ca2+]i; E: INaCa; F: IKr; G: IKs. See Glossary for abbreviations.

Electrical alternans. The CVM model generated electrical alternans at physiologically relevant pacing cycle lengths. Figure 2 shows the action potential and selected plateau currents at a cycle length of 180 ms, where the CVM model produced stable alternans of large magnitude. Note that I_Ca, f_Ca, and the Ca²⁺ transient were significantly different between the long and short action potentials, whereas peak I_Kr and peak inward I_NaCa were not. I_Ks varied in magnitude between the long and short action potentials, but the peak current magnitude remained small.

View larger version (25K): [in this window] [in a new window]   Fig. 2.   Action potentials, ionic currents, and Ca2+ transients generated by the CVM model after 50 beats at a cycle length of 180 ms. A: action potentials; B: ICa; C: fCa; D: [Ca2+]i; E: INaCa; F: IKr; G: IKs.

View larger version (23K): [in this window] [in a new window]   Fig. 3.   Action potentials generated by the CVM model at pacing cycle lengths of 8,000-90 ms. Two-to-one block occurred at a cycle length of 80 ms. A: action potential duration (APD) plotted as a function of the basic cycle length (BCL) of pacing over a BCL range of 90-400 ms. B: APD restitution, where APD is plotted as a function of the diastolic interval (DI) for DI <210 ms. The solid line has a slope of 1. Note that alternans occurred where the slope of the restitution relation was 1. C: APD as a function of BCL over a BCL range of 90-8,000 ms. D: examples of action potentials at BCL = 300, 500, 1,000, 2,000, 4,000, and 8,000 ms. Over this range of BCL, resting membrane potential = 94 mV, action potential amplitude = 139 mV, overshoot = 45 mV, and maximum dV/dt = 278-280 V/s. See Glossary for abbreviations.

Role of plateau Na+ and Ca²⁺ currents in alternans. The large difference in I_Ca between the long and short action potentials shown in Fig. 2 suggests that I_Ca contributes significantly to the development of alternans. Experiments using calcium channel blockers also have indicated that I_Ca may mediate alternans (23). To simulate the effects of a generic calcium channel blocker in the model, we decreased the magnitude of I_Ca by 20%. Figure 4 shows the action potential and plateau currents in the decreased I_Ca model at a pacing cycle length of 180 ms. No alternans of I_Ca or action potential duration occurred at this or any other pacing cycle length. As expected, the restitution relation lacked a region of slope equal to 1 (Fig. 5A).

View larger version (26K): [in this window] [in a new window]   Fig. 4.   Action potentials, ionic currents, and Ca2+ transients generated by the reduced ICa CVM model at a pacing cycle length of 180 ms. A: action potentials; B: ICa; C: fCa; D: [Ca2+]i; E: INaCa; F: IKr; G: IKs.

View larger version (19K): [in this window] [in a new window]   Fig. 5.   Relationship between APD and DI in the CVM model after reducing Ca by 20% (A), increasing K1 by 7% (B), increasing Kr by 62% (C), and increasing Ks by 14.3-fold (D). The solid line has a slope of 1. See Glossary for abbreviations.

View larger version (13K): [in this window] [in a new window]   Fig. 6.   Relationship among the kinetics of the calcium-induced inactivation gate (fCa), APD, DI, and the time course of ICa in the normal CVM model (A) and in the reduced ICa model (B) at a pacing cycle length of 180 ms. See text for discussion and Glossary for abbreviations.

View larger version (29K): [in this window] [in a new window]   Fig. 7.   Dose-response relationships between ionic current magnitude and alternans magnitude in the CVM model. Shown are the maximum magnitudes of APD alternans as a function of a particular model parameter. , Control parameter value. Left (from top to bottom): maximum INa conductance, INaCa, maximum ICa permeability, time constant for fCa, and maximum conductance for Ito. Right (from top to bottom): maximum conductance for IKr, time constant for IKr, maximum conductance for IKs, maximum conductance for IK1, and maximum conductance for IKp. See Glossary for abbreviations.

Role of repolarizing K+ currents in alternans. The effects of altering I_to, I_Kp, I_K1, I_Kr, and I_Ks on alternans also were determined (Fig. 7). The magnitude of each of the currents was increased individually until alternans no longer occurred during pacing at any cycle length. Elimination of alternans occurred after increasing I_to by 10%, I_K1 by 7%, or I_Kr by 62%. A substantially greater increase in the magnitude of I_Ks or I_Kp was required to eliminate alternans. Decreasing the magnitude each of the K⁺ currents increased the magnitude of action potential duration alternans with the exception of I_to, where decreasing the magnitude of the current decreased the alternans magnitude.

View larger version (29K): [in this window] [in a new window]   Fig. 8.   [Ca2+]i (left) and ICa (right) in the CVM model after increasing IK1 (A and B), IKr (C and D), or IKs (E and F). See Glossary for abbreviations.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

We developed an ionic model of the canine ventricular muscle cell that generates physiologically realistic action potential duration alternans characterized by a large magnitude and a wide range of pacing cycle lengths over which they appear. Action potential duration alternans was caused primarily by an alternans of I_Ca, where the latter resulted from the time-dependent behavior of the calcium-induced inactivation gate, f_Ca. Alternans was suppressed by reducing the magnitude of I_Ca as well as by increasing the magnitude of selected repolarizing K⁺ currents. Although the CVM model has some limitations, as discussed below, it is the first ionic model of the CVM that reproduces physiological alternans at rapid pacing rates. As such, it provides a useful simulation tool for studying the complicated interactions of cardiac membrane currents.

Role of I_Ca in alternans. The development of action potential duration alternans required that 1) the duration of the action potential have a sensitive dependence on I_Ca and 2) the recovery of I_Ca have a sensitive dependence on diastolic interval. The first condition applied so long as there was a relative balance of repolarizing K⁺ current and I_Ca during the action potential plateau. The second condition was manifest during pacing at short cycle lengths, where partial recovery of I_Ca after short diastolic intervals resulted in short action potential durations, followed by long diastolic intervals. Nearly complete recovery of I_Ca after long diastolic intervals produced action potentials with long durations, followed by short diastolic intervals. By this mechanism, a self-perpetuating sequence of long-short action potential durations was established. A similar mechanism likely contributed to action potential duration alternans in previously published ionic models (1, 22), although alternans of I_Ca was not specifically reported in those studies.

Role of repolarizing K+ currents in alternans. Beat-to-beat alterations of I_K1, I_Kr, and I_Ks appeared to play a minor role in mediating alternans. As expected, I_K1, which has no time dependence, displayed no beat-to-beat variations in magnitude, whereas the beat-to-beat changes in I_Ks were too small to affect action potential duration appreciably at short pacing cycle lengths. Total I_Kr also alternated during alternans; however, peak I_Kr did not, suggesting that alternation of I_Kr resulted from alternans of action potential duration rather than vice versa.

Implications. Decreasing the magnitude of I_Ca, either experimentally (23) or in an ionic model (22), has been shown to eliminate alternans and to convert VF into a periodic rhythm. However, this approach clearly is not useful clinically because decreasing I_Ca decreases the Ca²⁺ transient, thereby reducing contractile force. With the use of the CVM model to explore other methods for eliminating alternans, we found that alternans was suppressed by increasing the magnitude of three repolarizing K⁺ currents: I_K1, I_Kr, and I_Ks.

Limitations. While the CVM model successfully reproduces alternans, it has several limitations. First, the formulation of I_Ca is based solely on the qualitative characteristics of I_Ca. To improve the model, I_Ca should conform to the results of quantitative voltage-clamp experiments, where the latter ideally have been conducted under circumstances that preserve the native behavior of I_Ca during pacing at short cycle lengths (e.g., no buffering of [Ca²⁺]_i or washout of the intracellular space). Second, although the simplified calcium handling in the model reproduces physiological Ca²⁺ transients, it ignores several of the details of calcium release from the SR. Further work needs to be done to incorporate detailed calcium handling mechanisms such as those in the Winslow model (26). Third, the model does not include the late Na⁺ current, which may contribute significantly to plateau duration (28). A formulation of this current that agrees with voltage-clamp experiments also needs to be included to complete the model. Finally, it has been shown that transmural heterogeneity of the heart is caused by differences in I_to, I_Ks, I_NaCa, and the late Na⁺ current in endocardial, midmyocardial, and epicardial canine heart cells (13, 14, 27, 29). We hope in the future to develop specific models for canine endocardium, midmyocardium, and epicardium cells that will take these differences into account.