## Abstract

Rapidly activating K^{+} current (*I*_{Kr}) blockers prolong action potential (AP) duration (APD) in a reverse-frequency-dependent manner and may induce arrhythmias, including torsade de pointes in the ventricle. The *I*_{Kr} blocker dofetilide has been approved for treatment of atrial arrhythmias, including fibrillation. There are, however, a limited number of studies on the action of *I*_{Kr} blockers on atrial AP. When we tested a mathematical model of the human atrial AP (M Courtemanche, RJ Ramirez, S Nattel. *Am J Physiol Heart Circ Physiol* 275: H301–H321, 1998) to examine the effects of dofetilide-type *I*_{Kr} blockade, this model could not reproduce the reverse-frequency-dependent nature of *I*_{Kr} blockade on atrial APD. We modified the model by introducing a slowly activating K^{+} current activation parameter. As the slow time constant was increased, dofetilide-type blockade induced more prominent reverse-frequency-dependent APD prolongation. Using the modified model, we also examined the effects of two more types of *I*_{Kr} blockade similar to those of quinidine and vesnarinone. Voltage- and time-dependent block of *I*_{Kr} through the onset of inhibition by quinidine is much faster than by vesnarinone. When we incorporated the kinetics of the effects of these drugs on *I*_{Kr} into the model, we found that quinidine-type blockade caused a reverse-frequency-dependent prolongation of APD that was similar to the effect of dofetilide-type blockade, whereas vesnarinone-type blockade did not. This finding coincides with experimental observations. The lack of the reverse frequency dependence in vesnarinone-type blockade was accounted for by the slow development of *I*_{Kr} blockade at depolarized potentials. These results suggest that the voltage- and time-dependent nature of *I*_{Kr} blockade by drugs may be critical for the phenotype of the drug effect on atrial AP.

- rapidly activating potassium current
- slowly activating potassium current
- atrial action potential
- reverse frequency dependence
- computer simulation

the delayed rectifier K^{+} current (*I*_{K}) contributes to the phase 2 and phase 3 repolarization of the cardiac action potential (AP) and, thus, controls AP duration (APD). The rapidly activating K^{+} current (*I*_{Kr}) and the slowly activating K^{+} current (*I*_{Ks}) are two distinct components of *I*_{K} in many mammals, including humans (28, 29, 37). *I*_{Kr} inhibition by a number of cardiac and noncardiac drugs prolongs the QT interval and causes arrhythmias, including torsades de pointes in the ventricle (1). The contribution of *I*_{Kr} relative to *I*_{Ks} in AP repolarization is greater during bradycardia than during tachycardia: during tachycardia, the amplitude of *I*_{Ks} increases because of its slow deactivation, whereas the amplitude of *I*_{Kr} remains nearly the same (12). As a result, drugs such as dofetilide and E-4031, which specifically inhibit *I*_{Kr}, prolong APD more prominently during bradycardia (17, 18, 32). This phenomenon is called “reverse frequency dependence.” The reverse-frequency-dependent nature of a drug's effect is considered, at least in part, to be responsible for inducing excessive prolongation of APD and torsades de pointes during bradycardia (11). It is a critical indicator of the arrhythmogenic potential of a drug (34).

Drugs such as dofetilide and E-4031 block *I*_{Kr} in a voltage- and time-independent manner (32, 33). On the other hand, we showed that the positive inotropic agent vesnarinone and the widely used class Ia antiarrhythmic drug quinidine inhibit current flowing through the pore-forming subunit of *I*_{Kr}, HERG (human *ether-a-go-go-*related gene), in a voltage- and time-dependent manner (13, 33). Although both drugs inhibit the current more potently and rapidly as the membrane potential is depolarized, the development of blockade at the same potential is much faster with quinidine than with vesnarinone (13, 33). Accordingly, *I*_{Kr} blockade by drugs can be classified into at least three representative groups: voltage- and time-independent blockade (e.g., by dofetilide and E-4031), fast voltage- and time-dependent blockade (e.g., by quinidine), and slow voltage- and time-dependent blockade (e.g., by vesnarinone). Previous studies have shown that although dofetilide and quinidine cause reverse-frequency-dependent prolongation of APD, vesnarinone does not (10, 12, 32) and that, clinically, dofetilide and quinidine often cause torsades de pointes, but vesnarinone does not (7, 30). These different effects among the drugs may be due to the differences in their kinetics of HERG blockade (13, 33), but the underlying mechanism has not been fully clarified.

*I*_{Kr} and *I*_{Ks} were also found in human atrial myocytes (37), and, recently, the *I*_{Kr}-specific blocker dofetilide was approved for treatment of atrial fibrillation (AF) (6). Therefore, the action of *I*_{Kr} blockers on atrial, as well as ventricular, APs is of interest. The aims of this study were *1*) to establish a human atrial AP model that can differentiate the effects of drugs with the three distinct *I*_{Kr} blockade kinetics, *2*) to elucidate the underlying mechanism of the different effect of *I*_{Kr} blockade on APD prolongation, and *3*) to predict the requirement for non-reverse-frequency-dependent prolongation of APD in *I*_{Kr} blockade. To examine the effects of pure *I*_{Kr} blockade on APD, other possible effects by the *I*_{Kr} blockers [e.g., blockade of ultrarapid delayed rectifier K^{+} current (*I*_{Kur}) and inactivating transient outward K^{+} current (*I*_{to}) by quinidine] needed to be excluded. This is very difficult, if not impossible, in vitro. However, in silico approaches can realize such hypothetical experimental conditions. Therefore, we have developed an applicable human atrial AP model.

We first examined the effect of *I*_{Kr} blockade caused by dofetilide on the model of human atrial AP that was developed by Courtemanche et al. (2). We found that this model could not reproduce dofetilide's reverse-frequency-dependent prolongation of atrial APD. To correct this defect, we incorporated the slow component of *I*_{Ks} (27, 31) into the model. The modified model reproduced and accounted for the different frequency-dependent responses of APD due to three kinetically distinctive *I*_{Kr} blockade types. These results suggest that the voltage and time dependence of *I*_{Kr} blockade by drugs may be critical for the phenotype of drug effect on atrial AP.

## MATERIALS AND METHODS

#### AP simulation.

To simulate the human atrial AP, we used the mathematical model developed by Courtemanche et al. (2). Briefly, this model has passive properties (e.g., membrane capacitance and membrane resistance) and 12 voltage-dependent and carrier-mediated ionic currents as follows: fast Na^{+} current (*I*_{Na}), inward rectifier K^{+} current (*I*_{K1}), *I*_{to}, *I*_{Kur}, *I*_{Kr}, *I*_{Ks}, L-type Ca^{2+} current (*I*_{Ca,L}), sarcolemmal Ca^{2+} pump current (*I*_{p,Ca}), Na^{+}-K^{+} pump current (*I*_{NaK}), Na^{+}/Ca^{2+} exchanger current (*I*_{NaCa}), background Na^{+} current (*I*_{b,Na}), and background Ca^{2+} current (*I*_{b,Ca}). Hodgkin-Huxley-type equations are used to calculate the forward and backward kinetics of the activation and inactivation processes of each voltage-gated ion channel. The program was coded in C/C++, and simulations were run on an IBM-compatible computer with a C++ compiler.

*I*_{Ks} in the original atrial AP model of Courtemanche et al. (2) is expressed as follows (1) where *g*_{Ks} is the maximum conductance for *I*_{Ks},*x*_{s} is the activation gate variable for *I*_{Ks}, *V* is the membrane potential, and *E*_{K} is the equilibrium potential for K^{+}. The activation of *I*_{Ks} in the original model of Courtemanche et al. is assumed to have one gate represented by squared variables (*x*). It has been experimentally shown, however, that *I*_{Ks} possesses at least two (fast and slow) activation processes (27, 31). The time constant in the original model of Courtemanche et al. represents only the fast process. An additional slow activation process was introduced into the formulation of *I*_{Ks} in our model in a manner similar to that in the Luo-Rudy (LRd) model, the de facto standard guinea pig ventricle cell model (35). We modified this to incorporate two activation processes as follows (2) where *x*_{s1} and *x*_{s2} are the fast and slow activation gate variables, respectively. Although the time constant of *x*_{s1} [τ_{x(s1)}] was set to be the same as the value of the time constant of *x*_{s} in the original model of Courtemanche et al., that of *x*_{s2} [τ_{x(s2)}] was two, four, or six times τ_{x(s1)}. The maximum *I*_{Ks} conductance was adjusted in each case so that the APD in the control at 1,000-ms cycle length (CL) did not change from that of the original model: it was 0.129, 0.187, 0.251, and 0.277 mS/μF for the original and the two-, four-, and sixfold increases, respectively.

Thirty AP simulations were run at each CL. The last AP in each run was used for analysis. The APD was measured at −70 mV, which closely approximates APD at 90% repolarization (APD_{90}).

#### Formulation of kinetic properties of I_{Kr} blockade types.

To incorporate the effects of the three blockade groups, we modified the formulation of *I*_{Kr} given by Courtemanche et al. (2) as follows (3) where *i* represents the blockade type where dofetilide type is the voltage- and time-independent blockade, quinidine type is the voltage-dependent blockade with fast development, and vesnarinone type is the voltage-dependent blockade with slow development, *y*_{i} is the fraction of *I*_{Kr} that is not blocked by drug type *i*, *g*_{Kr} is the maximum conductance of *I*_{Kr}, and *x*_{r} is the activation gate variable,.

Dofetilide blocks *I*_{Kr} in a voltage- and time-independent manner (33). Therefore, dofetilide represents the voltage- and time-independent blockade and *y*_{dofetilide type} was set to 0.1.

On the other hand, the effects of the two types of voltage- and time-dependent blockade were calculated with the following first-order differential equation (4) where *i* represents the blockade type where quinidine or vesnarinone, *y*_{i} is the state variable of the unblocked fraction of *I*_{Kr} in the presence of drug type *i*, *y*_{∞,i}, is the steady-state value of *y*_{i}, and τ_{i} is the time constant for *y*_{i}. Simulations of *y*_{∞,i} and τ_{i} derived from our experimental data for quinidine and dofetilide (33) and vesnarinone (13) are shown in Fig. 1.

*y*_{∞,i} and τ_{i} are expressed as functions of *V* as follows (5) (6) where *i* represents quinidine type or vesnarinone type (7) (8) (9) (10) *y*_{depolarization,i} is the limit set for the minimum of unblocked *I*_{Kr} (*y*_{∞,i}), which was set to 0.1 for both drug types and corresponds to 90% block, α_{i} is the unbinding rate constant for drug type *i*, and β_{i} is the binding rate constant for drug type *i*. The values in the formulas for *y*_{∞,quinidine type} and τ_{quinidine type} were determined from the experimental data of Tsujimae et al. (33) by nonlinear least-squares fitting. In the same way, the values in the formulas for *y*_{∞,vesnarinone type} and τ_{vesnarinone type} were derived from Katayama et al. (13). The experimental data for the temperature sensitivity of *I*_{Kr} blockade by these drugs are not available. However, because it was reported that the temperature dependence of *I*_{Kr} blockade is usually not prominent in various drugs (14), we assumed that the temperature difference among these experiments would not largely affect the present model results. Furthermore, in this study, we adopted the kinetics of dofetilide, quinidine, and vesnarinone from our previous experimental results only to represent the three representative types of *I*_{Kr} blockade by various drugs, and we did not intend to reproduce precisely the experimental results of the effects of these drugs on atrial AP. These formulations are used to simulate the development and recovery from blockade for dofetilide, quinidine, and vesnarinone (Fig. 2). For simulation of development of blockade, voltage steps (1-s duration) from a holding potential of −80 mV to between −20 and +40 mV with a 20-mV increment (Fig. 2, *A–C*) were used. For simulation of recovery from blockade, voltage steps (10-s duration) from a holding potential of +40 mV to between −40 and −100 mV with a 20-mV increment were used (Fig. 2, *D–F*). Although the kinetics of quinidine blockade are complex and consist of at least two exponential components (33), we treated this as a single-exponential process (Fig. 2, *A* and *D*) (33). Block and recovery due to vesnarinone could be represented by single exponentials (Fig. 2, *B* and *E*) (13). Blockade by dofetilide was constant (Fig. 2, *C* and *F*) (33).

## RESULTS

#### Incorporation of a slow component into I_{Ks} and its effect on APD prolongation caused by reduced I_{Kr} density.

We began with the original mathematical model for human atrial AP that had been developed by Courtemanche et al. (2). We first examined the frequency dependence of APD under control conditions and in the presence of a voltage- and time-independent *I*_{Kr} blocker represented by dofetilide (Fig. 3*A*). The APs exhibit the spike-and-dome morphology commonly observed in human atrial recordings. Several class III antiarrhythmic agents such as dofetilide cause a voltage- and time-independent blockade of *I*_{Kr} or HERG current (33). The blocking effect of these drugs was termed “dofetilide-type” in the present study and is modeled by reducing the maximum conductance of *I*_{Kr} to 10% of the control (see materials and methods). Dofetilide-type 90% blockade of *I*_{Kr} prolonged the late phase of repolarization at any CL but left AP amplitude and the resting potential unchanged (Fig. 3*A*). This effect is consistent with the experimental data for dofetilide and E-4031 (12, 18). The effect of CL on APD under control conditions and with 90% blockade of *I*_{Kr} is shown in Fig. 3*Ca*. In control conditions, APD slightly increased when CL was increased from 500 to 700 ms, and APD reached a plateau value of ∼300 ms at >800-ms CL. This frequency dependence of the original model correctly captured the feature of the experimental result in the control condition (8). However, when the maximum *I*_{Kr} conductance was reduced to 10% of control to simulate the voltage- and time-independent blockade, APD_{−70 mV} was prolonged at all CLs and the CL-dependent increase of APD reached a plateau value of ∼400 ms at ∼1,000-ms CL. Previous experimental results showed reverse-frequency-dependent prolongation of APD in the atrium at 1,500-ms CL in guinea pig with dofetilide (18) and also in humans with quinidine and flecainide (10). Therefore, the plateau at 1,000-ms CL in the original model is not consistent with the most likely estimation that the reverse-frequency-dependent APD prolongation may also be observed at up to 1,500-ms CL in the human atrium with dofetilide. When APD prolongation was normalized to 500-ms CL, dofetilide-type inhibition of *I*_{Kr} (Fig. 3*Cc*) clearly demonstrated a CL-dependent increase of APD_{−70 mV} only at <1,000-ms CL.

The repolarization current for AP in cardiac myocytes is mainly composed of *I*_{Kr} and *I*_{Ks}. As shown by Jurkiewicz and Sanguinetti (12) in guinea pig ventricular myocytes, although the magnitude of *I*_{Kr} activated during depolarizing pulses remains nearly constant at different CLs because of its quick activation, that of *I*_{Ks} becomes larger as the stimulation rate is increased. As a result, the effect of *I*_{Kr} blockade on AP is relatively small at fast stimulation rate, where the *I*_{Ks} is enhanced. It is thus widely accepted that the different level of activation of *I*_{Ks} at different CLs is the major cause of the reverse-frequency-dependent nature of APD under blockade of *I*_{Kr}. Therefore, we examined the possibility that inappropriate modeling of *I*_{Ks} activation/deactivation may account for the failure of the original model of Courtemanche et al. (2) to reproduce the phenomenon.

Activation of *I*_{Ks} contains fast and slow processes (27, 31). *I*_{Ks} in the model of Courtemanche et al. (2) has one fast activation process. Therefore, we incorporated a second variable for slow kinetics (*x*_{s2}) into the activation gate of *I*_{Ks} (*Eq. 2*). The time constant of slow activation of *I*_{Ks} has been reported to be three to five times larger than that of fast activation (31). Therefore, we set the slow time constant [τ_{x(s2)}] to two, four, or six times τ_{x(s1)} (Fig. 4). For each value of τ_{x(s2)}, the maximum conductance of *I*_{Ks} was adjusted to maintain the AP configuration at 1,000-ms CL.

In Fig. 3*B*, the modified model with *I*_{Ks} and τ_{x(s2)} six times τ_{x(s1)} is used to simulate APs at various CLs. The modified model exhibits a spike-and-dome morphology similar to the original model of Courtemanche et al. (2). At 500-ms CL, APD of the modified model was slightly shorter than APD of the original model, but at 1,000- and 1,500-ms CL, APDs of the modified model were nearly the same as APD of the original model. When *I*_{Kr} was reduced to 10% of the control to simulate the voltage- and time-independent blockade, the prolongation of APD with longer CL was more prominent in the modified model.

The effect of varying τ_{x(s2)} on CL-dependent prolongation of APD_{−70 mV} in the presence of the voltage- and time-independent blocker is shown in Fig. 3*Cb*. In control conditions, as τ_{x(s2)} increased, APD_{−70 mV} at 500-ms CL decreased, but τ_{x(s2)} had less effect at ≥1,000-ms CL. When *I*_{Kr} was reduced as τ_{x(s2)} was increased, CL-dependent prolongation of APD_{−70 mV} was more apparent, and the slope of the CL-APD_{−70 mV} relation became steeper. When τ_{x(s2)} was set to four or six times τ_{x(s1)}, APD_{−70 mV} did not reach a plateau until 1,500-ms CL. In addition, normalization of APD_{−70 mV} prolongation caused by reduced *I*_{Kr} density (Fig. 3*Cc*) clearly demonstrated that the prolongation of APD_{−70 mV} caused by *I*_{Kr} reduction depends on τ_{x(s2)}. Clearly, these results more closely resemble experimental results (12, 18) than those obtained with the original model of Courtemanche et al. (2) (Fig. 3*Ca*). Therefore, in the following studies, we used the modified model with τ_{x(s2)} set to six times τ_{x(s1)}.

#### Comparison of delayed rectifier currents in original and modified models.

Figure 5 shows *I*_{Kr} and *I*_{Ks} during APs at various CLs in the original and modified models. Since the formulation of *I*_{Kr} was not modified, the form and amplitude of *I*_{Kr} during each AP at each CL in the original (Fig. 5*A*) and modified (Fig. 5*B*) models were very similar. The peak of *I*_{Kr} during an AP increased as APD was prolonged, because the activation process of *I*_{Kr} progressed during an AP, even in the presence of very rapid C-type inactivation (36). As CL increased, *I*_{Kr} increased by the same amount in both models, although at 500-ms CL, the peak of *I*_{Kr} was slightly smaller during an AP in the modified than in the original model.

In the original model, *I*_{Ks} increased as CL increased, similar to *I*_{Kr} (Fig. 5*A*). On the other hand, in the modified model, *I*_{Ks} decreased as CL increased (Fig. 5*B*), which is consistent with the experimental results shown by Jurkiewicz and Sanguinetti (12). Analysis of the gating variables (gate in Fig. 5) indicates that the difference in the behavior of *I*_{Ks} in the two models may be attributed mainly to the properties of *x*_{s2}. Because its time constant was slow, *x*_{s2} showed only slight variation during an AP: it increased slowly during depolarization and declined slowly during repolarization (Fig. 5*B* gate, thick continuous line). At 500-ms CL, *x*_{s2} at the onset of an AP was high and decreased as CL was prolonged to 1,000 and 1,500 ms. In contrast, the fast gating variables, *x*_{s} in the original model (Fig. 5*A*) and *x*_{s1} in the modified model (Fig. 5*B* gate, thin line), increased rapidly during depolarization and decreased rapidly during repolarization.

As a result of these differences, deactivation of the product of *x*_{s1} and *x*_{s2} in *Eq. 2* for the modified model is slower than deactivation of *x* in *Eq. 1* in the original model. Consequently, at short CL in the modified model, *I*_{Ks} showed little deactivation at the end of repolarization, and its initial jump at the onset of the next AP was large; with increasing CL, deactivation could develop further, and the amplitude of the initial jump decreased markedly. After the initial jump, *I*_{Ks} increased during an AP, but the combination of a slower activation and the decline of the initial current meant that the peak of *I*_{Ks} during an AP decreased as CL was prolonged. On the other hand, in the original model, deactivation was more rapid, and the initial jump of *I*_{Ks} was less at 500-ms CL and was influenced less by increasing CL. *I*_{Ks} increased rapidly during an AP because of its fast activation kinetics, and peak *I*_{Ks} increased with CL. Finally, in the original model, the relative proportions of *I*_{Kr} and *I*_{Ks} showed little change with CL [Fig. 5*A*; *I*_{Kr}/(*I*_{Kr} + *I*_{Ks})]. In the modified model, the contribution of *I*_{Kr} to the total repolarizing current increased with CL (Fig. 5*B*). Thus it is the modified model that can reproduce the experimental data for the frequency-dependent behavior of *I*_{Ks} and APD.

#### Effects of I_{Kr} blockers on APD prolongation.

Experimentally, it is also recognized that excessive AP prolongation is quite divergent among dofetilide, quinidine, and vesnarinone (10, 12, 32). We next examined how voltage- and time-dependent kinetics in *I*_{Kr} blockade affect the reverse frequency dependence of APD prolongation in the model. Although dofetilide and E-4031 block *I*_{Kr} in a voltage-independent manner, quinidine and vesnarinone do so in a voltage- and time-dependent manner (13, 33). Both drugs inhibit the current more potently and rapidly as the membrane potential is more depolarized. However, at the same depolarized potential, the development of block is much faster with quinidine than with vesnarinone. To isolate the effects of these three typical kinetic properties of *I*_{Kr} blockade, we fixed their maximal effect to correspond to 90% block of *I*_{Kr}. Thus we calibrated the blockades with different kinetics by assuming the same possible maximal block effect, so that the effects of the kinetics of *I*_{Kr} blockade on APD can be compared selectively. This is represented in the calculations as a steady-state unblocked fraction of 0.1 (see materials and methods).

Figure 6 illustrates the effects of these drug types on APD at different CLs in absolute and relative terms. Dofetilide-type blockade exhibited the strongest and the most prominent reverse-frequency-dependent increase in APD_{−70 mV} (Fig. 6*A*). Quinidine-type blockade was less effective than dofetilide-type blockade but showed a similar reverse frequency dependence. Vesnarinone-type blockade was weaker than quinidine- or dofetilide-type blockade, and it showed no frequency dependence at >800-ms CL. The result in relative terms also more clearly demonstrates no frequency dependence of vesnarinone-type blockade at >800-ms CL; thus frequency dependence is similar for vesnarinone-type blockade and control (Fig. 6*B*). These results are consistent with experimental findings (10, 12, 32). We conclude that the modified model can reproduce the characteristic frequency-dependent prolongation of atrial APD by the *I*_{Kr} blockers with the three representative kinetics.

Figure 7 shows the effects of the three drugs on AP configuration and the two delayed rectifier currents *I*_{Kr} and *I*_{Ks}. At 500-, 1,000-, and 1,500-ms CL, all three drug types significantly prolonged AP without affecting the AP amplitude and the resting potential (Fig. 7). With increasing CL, quinidine- and dofetilide-type blockade progressively prolongs APD to a similar extent, whereas vesnarinone-type blockade became less effective (see also Fig. 6*B*).

To determine the mechanisms underlying these different effects, the unblocked fractions of *I*_{Kr} (*y*_{i}), the total amount of repolarization current (*I*_{Kr} + *I*_{Ks}), and the contribution of *I*_{Kr} to the repolarization current [*I*_{Kr}/(*I*_{Kr} + *I*_{Ks})] were examined (Fig. 7). The dofetilide-type blocking effect (*y*_{dofetilide type}) was constant and independent of CL (Fig. 4, unblocked fraction). Vesnarinone- and quinidine-type drugs cause voltage- and time-dependent blockade of *I*_{Kr} (Figs. 1 and 2), with blockade during depolarization and recovery during repolarization. The corresponding values for *y*_{quinidine type} and *y*_{vesnarinone type} decrease with depolarization and increase with recovery (Fig. 7, unblocked fraction). The values for *y*_{quinidine type} and *y*_{vesnarinone type} are maximal just before each AP. As CL increases, these maximum values for *y*_{quinidine type} and *y*_{vesnarinone type} increase as the time allowed for unblocking increased. At the onset of an AP, *y*_{quinidine type} decreases, and during the AP it reaches a minimum value of ∼0.2 irrespective of CL. This shows that, in practical terms, quinidine blocks *I*_{Kr} in a frequency-independent manner because of its fast blocking kinetics. Therefore, quinidine- and dofetilide-type blockades have functionally similar effects on *I*_{Kr} during AP, although they have very different kinetics. In contrast, the minimum value of *y*_{vesnarinone type} was dependent on CL; that is, maximum blockade of *I*_{Kr} by vesnarinone was attenuated as CL was prolonged.

The total delayed rectifier current increases with increasing CL under control conditions (Fig. 7, *I*_{Kr} + *I*_{Ks}) because of the increase in *I*_{Kr} [Fig. 7, *I*_{Kr}/(*I*_{Kr} + *I*_{Ks}); see also Fig. 5*B*, *I*_{Kr}]. In the presence of a vesnarinone-type drug, total *I*_{K} also increased with increasing CL, because the relative proportion of *I*_{Kr} also increased. As a result, vesnarinone-type blockade did not induce reverse-frequency-dependent prolongation of APD. In contrast, in the presence of dofetilide- and quinidine-type drugs, total *I*_{K} decreased as CL was prolonged. This is due to their constant maximum blockade of *I*_{Kr} over a wide range of CL and the reduction of *I*_{Ks} with longer CL. As a result, APD prolongation with CL was enhanced by dofetilide- and quinidine-type blockade.

## DISCUSSION

The major findings in this study are as follows. *1*) Incorporation of a slow process of activation into *I*_{Ks} enabled a human atrial AP model to reproduce the reverse-frequency-dependent elongation of APD that is associated with the reduction of *I*_{Kr}. *2*) This model could account for the mechanisms underlying the difference between the frequency-dependent effects of the three kinetically distinctive types (dofetilide, quinidine, and vesnarinone) of *I*_{Kr} blockade on APD. *3*) The analysis with the models predicted that the drugs with the voltage- and time-dependent *I*_{Kr} blockade with slow kinetics may be good candidates without the adverse effect of arrhythmias.

There are many mathematical models for cardiac APs, e.g., models for Purkinje fibers (19, 20), rabbit sinoatrial node (4), and ventricular cells of rat (22, 23), guinea pig (35, 40), dog (9, 38), and human (24) and models for atrial cells of dog (15, 25) and human (2, 21). No model is perfect, but each has its use for examination of different properties of cardiac AP behavior, with their respective limitations taken into account. In practical terms, it is important to adjust models according to the purposes for which they will be used.

A problem in pharmacology is the proarrhythmic action of many drugs. A number of compounds inhibit *I*_{Kr} in cardiac myocytes and, therefore, have the potential to prolong the AP in a reverse-frequency-dependent manner, which could induce torsades de pointes in the ventricle. Because of this side effect, drugs such as terfenadine, astemizole, grepafloxacin, droperidol, and cisapride have been withdrawn or restricted in their clinical application (26). Also, the development of a number of new pharmacological agents was stopped when they were shown to block *I*_{Kr}. A mathematical model of the cardiac AP that can reproduce reverse-frequency-dependent APD prolongation on *I*_{Kr} blockade could be used to assess the risk of such side effects.

Inasmuch as dofetilide has been approved by the US Food and Drug Administration for treatment of AF (6), the importance of elucidating the action of *I*_{Kr} blockade on atrial APD prolongation is increasing. The model of the human atrial AP of Courtemanche et al. (2) could not reproduce reverse-frequency-dependent elongation of APD with blockade of *I*_{Kr}. To reproduce this property, we introduced a slow process of activation and deactivation of *I*_{Ks}, which has been identified experimentally (27, 31). The modified human atrial AP model reproduced reverse-frequency-dependent prolongation of atrial APD, which results from the reduction of maximum conductance of *I*_{Kr} by voltage- and time-independent *I*_{Kr} blockers such as dofetilide and E-4031. It also differentiates between the effects of the voltage- and time-dependent blockade caused by quinidine- and vesnarinone-type *I*_{Kr} blockers. Therefore, the model can be used to assess the frequency-dependent action of different types of pure *I*_{Kr} blockade on the human atrial AP. Because the simulation results with pure *I*_{Kr} blockade are very close to the experimental results of the effects of dofetilide, quinidine, and vesnarinone on atrial APD (10, 18, 32), it was also suggested that the effects of these drugs on *I*_{Kr} are the major determinants for atrial APD, although quinidine is known to affect a number of ion currents, including *I*_{Na} and *I*_{to}, and vesnarinone is known to increase *I*_{Ca,L} (16, 39). Indeed, our preliminary model study showed that additional blockade of *I*_{Kur} and *I*_{to} only marginally enhanced the reverse frequency dependence of APD prolongation induced purely by the quinidine-type blockade of *I*_{Kr} (not shown).

This study showed that the slow activation/deactivation of *I*_{Ks} plays an important role in reproducing the reverse-frequency-dependent APD prolongation by *I*_{Kr} blockers in an atrial model. The Luo-Rudy model (35, 40), which is the de facto standard for the guinea pig ventricular cell, also has fast and slow components of *I*_{Ks} activation; in this example, the slow time constant is four times the fast (35) (Fig. 4). Reconstitution of *I*_{Ks} with KvLQT1/minK exhibited three activation processes: fast, slow, and very slow (27). The time constants for fast and slow were 0.68 and 1.48 s, respectively, at +40 mV, which approximate the fast and slow activation processes of *I*_{Ks} that we use. We have not incorporated the very slow time constant (8.0 s at +40 mV) into *I*_{Ks}. If this slower activation process were incorporated into our model, the reverse-frequency-dependent APD prolongation associated with the dofetilide-type of blockade of *I*_{Kr} would become more prominent. Further experimental studies are needed to identify the gating processes of *I*_{Ks} for further improvement of the model.

APD is shorter in atrial cells from AF patients than from normal subjects; this is due to electrical remodeling of atrial myocytes, where expression of ion channel currents, including *I*_{Ca,L}, *I*_{to}, *I*_{Kur}, and *I*_{K1}, is altered (5). Courtemanche and colleagues (3) incorporated such changes into their original model and examined the effects of modification of the different outward currents on atrial AP configuration. The modified human atrial AP model that we present here could also be extended to an AF-atrial AP model that will be suitable for assessing the effects of drugs with the potential to treat AF.

In conclusion, this simulation showed that slow activation and deactivation of *I*_{Ks} underlie the reverse-frequency-dependent atrial APD prolongation induced by *I*_{Kr} blockers. It also demonstrated that the effects of these compound types on APD prolongation depended on the kinetics of their interactions with *I*_{Kr}. Further studies may provide useful information for understanding the underlying mechanisms of cardiac arrhythmias and estimating the balance between antiarrhythmic effect and proarrhythmic risk of *I*_{Kr} blocking agents. This approach may also be useful for the in silico design of antiarrhythmic agents.

## GRANTS

This work was supported by the Leading Project for Biosimulation “Development of Heart and Lung Models for In Silico Prediction of Drug Action and Clinical Treatment” from the Ministry of Education, Culture, Sports, Science, and Technology (Japan) to Y. Kurachi.

## Acknowledgments

We thank Dr. I. Findlay and A. O'Meara for comments about the manuscript.

## Footnotes

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- Copyright © 2007 by the American Physiological Society