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Mechanisms of atrial fibrillation termination by rapidly unbinding Na⁺ channel blockers: insights from mathematical models and experimental correlates

¹Department of Medicine and Research Center, Montreal Heart Institute and Université de Montréal, Montreal; ²Department of Pharmacology, McGill University, Montreal; and ³Department of Electrical and Computer Engineering, University of Calgary, Calgary, Alberta, Canada

Submitted 11 September 2007 ; accepted in final form 29 July 2008

	ABSTRACT

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Atrial fibrillation (AF) is the most common sustained clinical arrhythmia and is a problem of growing proportions. Recent studies have increased interest in fast-unbinding Na⁺ channel blockers like vernakalant (RSD1235) and ranolazine for AF therapy, but the mechanism of efficacy is poorly understood. To study how fast-unbinding I_Na blockers affect AF, we developed realistic mathematical models of state-dependent Na⁺ channel block, using a lidocaine model as a prototype, and studied the effects on simulated cholinergic AF in two- and three-dimensional atrial substrates. We then compared the results with in vivo effects of lidocaine on vagotonic AF in dogs. Lidocaine action was modeled with the Hondeghem-Katzung modulated-receptor theory and maximum affinity for activated Na⁺ channels. Lidocaine produced frequency-dependent Na⁺ channel blocking and conduction slowing effects and terminated AF in both two- and three-dimensional models with concentration-dependent efficacy (maximum 89% at 60 µM). AF termination was not related to increases in wavelength, which tended to decrease with the drug, but rather to decreased source Na⁺ current in the face of large ACh-sensitive K⁺ current-related sinks, leading to the destabilization of primary generator rotors and a great reduction in wavebreak, which caused primary rotor annihilations in the absence of secondary rotors to resume generator activity. Lidocaine also reduced the variability and maximum values of the dominant frequency distribution during AF. Qualitatively similar results were obtained in vivo for lidocaine effects on vagal AF in dogs, with an efficacy of 86% at 2 mg/kg iv, as well as with simulations using the guarded-receptor model of lidocaine action. These results provide new insights into the mechanisms by which rapidly unbinding class I antiarrhythmic agents, a class including several novel compounds of considerable promise, terminate AF.

state-dependent channel; cardiac action potential; antiarrhythmic drug therapy; computer simulations; heart arrhythmia models

Na⁺ channel-blocking class I antiarrhythmic drugs terminate clinical AF, but their electrophysiological mechanisms of action remain poorly understood (27). One potentially interesting approach to the design of new antiarrhythmic agents is the development of Na⁺ channel-blocking drugs that are AF selective, whether by virtue of a selective action on the atrium or favorable kinetics that provide strong Na⁺ channel blockade during AF and much less effect at the slower rates of sinus rhythm (27). Vernakalant (RSD1235) is an investigational antiarrhythmic agent that shows great efficacy in the termination of recent-onset AF (35) and blocks Na⁺ channels with rapid unbinding kinetics (8). The contribution of Na⁺ channel-blocking actions to vernakalant's clinical efficacy is presently unknown, because the drug also blocks a variety of K⁺ channels (8). Ranolazine also shows atrium-selective Na⁺ channel-blocking actions with rapid kinetics and has promise for AF suppression (3).

AF is considered most commonly to be a reentrant arrhythmia, with a stability that has classically been related to the reentrant wavelength (33). According to the "leading circle" theory, the wavelength, defined as the product of the refractory period and conduction velocity, represents the minimum path length for reentry and therefore determines the size of functional reentry circuits (1). The traditionally accepted explanation for antiarrhythmic drug termination of AF is drug-induced wavelength increases, such that a reduced number of reentrant circuits is possible and AF becomes unstable and stops. Experimental evidence has been presented to suggest that class I antiarrhythmic agents stop AF by increasing the effective refractory period (ERP) and thus the wavelength (33, 42, 44).

Recent data have challenged the established notions of antiarrhythmic drug action by showing that potent Na⁺ channel blockers can terminate AF without increasing the wavelength (45). Indeed, in some instances, a decrease in wavelength precedes termination, and the most characteristic electrophysiological change before AF termination is an increase in the temporal excitable gap (45). It is difficult to determine experimentally whether Na⁺ channel blockade per se can terminate AF because Na⁺ channel blockers used to terminate AF, including class IA agents such as quinidine and class IC agents like flecainide and propafenone, also have important actions on K⁺ channels (18, 34). We (18) previously used a mathematical modeling approach to study the effects of pure Na⁺ channel blockade on AF in a two-dimensional (2-D) canine atrial substrate. In that study, reductions in Na⁺ conductance slowed conduction and terminated AF, in a fashion qualitatively similar to the experimental action of TTX. However, clinically used Na⁺ current (I_Na) blockers do not produce instantaneous and fixed levels of I_Na inhibition; rather, their blocking actions are highly state dependent, and the degree of I_Na inhibition varies dynamically with activation history (13, 36). A realistic analysis of I_Na-inhibiting drug effects on AF, therefore, requires a formulation of state-dependent block.

Extensive recent data have suggested that rapidly unbinding Na⁺ channel blockers (traditionally assigned to Singh and Vaughan-Williams class IB), including vernakalant and ranolazine, are promising AF-terminating drugs (3, 8, 27, 35). To analyze the mechanisms by which such agents act on AF, in the present study, we developed a dynamical model of Na⁺ channel blockade based on the modulated-receptor approach (13) to study state-dependent Na⁺ channel blockade. Cholinergic AF was simulated with the use of a canine atrial tissue model with physiological ionic, coupling, and propagation properties (19). Detailed experiments were performed in a 2-D substrate because of its tractability and simplicity, and the key concepts were then tested in an anatomically realistic three-dimensional (3-D) substrate (41). The drug we chose to model was lidocaine, because, like vernakalant and ranolazine, it has relatively rapid unbinding kinetics and because there are sufficient data in the literature with which to construct a mathematical formulation of lidocaine/Na⁺ channel interactions. Lidocaine was also chosen because of its minimal affinity to K⁺ channels (15, 34), keeping the modeling relatively simple and relevant to drugs acting primarily by Na⁺ channel blockade. We then compared the results of computer simulations with the observed effects of lidocaine on cholinergic AF induced experimentally in dogs by brief (1–10 s) burst atrial pacing during bilateral vagal nerve stimulation.

	MATERIALS AND METHODS

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Mathematical Model Description and Implementation

2-D model of canine atrial tissue. The Ramirez-Nattel-Courtemanche model of the canine atrial action potential (AP) was implemented (32). Total ionic current (I_ion) is given by the following:

where I_K1 is inward rectifier K⁺ current, I_to is transient outward K⁺ current, I_Kur,d is ultrarapid delayed rectifier K⁺ current, I_Kr is the rapid component of the delayed rectifier K⁺ current, I_Ks is the slow component of the delayed rectifier K⁺ current, I_Ca is Ca²⁺ current, I_ClCa is Ca²⁺-activated Cl^– current, I_K(ACh) is ACh-activated K⁺ current, I_pCa is Ca²⁺ pump current, I_NaCa is Na⁺/Ca²⁺ exchanger current, I_NaK is Na⁺/K⁺ pump current, I_b,Na is background Na⁺ current, I_b,Ca is background Ca²⁺ current, and I_b,Cl is background Cl^– current. An expression for I_K(ACh) was included to simulate vagal effects (19).

The tissue was modeled with a coupled-cable approach introduced by Leon and Roberge (21). In brief, the tissue is composed of cables of cardiac myocytes (radius of 5 µm and resistivity of 75 ·cm). The cables are coupled by resistors (300 k) spaced by 200 µm inserted in a brick wall manner. Fiber resistivity and interfiber resistance values were chosen to match experimental results. Details of the model can be found in Ref. 19.

Modulated-receptor representation of state-dependent drug action. The representation of drug action follows the well-known model proposed by Hondeghem and Katzung (13). We chose this approach because the modulated-receptor model is widely known and can be directly applied to the Hodgkin-Huxley formulation of the I_Na present in the ionic model of the AP that we used. The parameter values for lidocaine were also available, making it possible to easily compare the simulation results with experimental data obtained in the vagal AF dog model. The mathematical model is based on the three states of the Na⁺ channel: resting (R), activated (A), and inactivated (I). I_Na of the canine atrial ionic model is based on the following formulation: I_Na = g_Nam³hj(V – E_Na), where g_Na is Na⁺ conductance, m is an activation variable, h and j are inactivation variables, V is transmembrane potential, and E_Na is Na⁺ equilibrium potential. The j variable reflects slow inactivation. The shift in h gate voltage dependence caused by lidocaine in the model was not applied to the j gate inactivation function. The sum of blocked channels (B) is given by the following: B = R' + A' + I'. After Hondeghem and Katzung, we assume that lidocaine has a high affinity for the activated state and modulates the fast inactivation gate reflected by the h variable of the model. The different states of the model (Fig. 1A, left) are thus given by the following:

We set the basic ion-handling parameters to be identical to the nominal dog atrial ionic model (32) with the lidocaine binding parameters set to k_R = 0.4 ms^–1·M^–1, l_R = 1.0 ms^–1, k_A = 5 x 10⁴ ms^–1·M^–1, l_A = 1.5 ms^–1, k_I = 50 ms^–1·M^–1, l_I 2 x 10^–3 ms^–1, and V = 30 mV (13).

View larger version (39K): [in this window] [in a new window]   Fig. 1. A, left: schematic representation of the Hondeghem-Katzung modulated-receptor model with transition rates from unblocked to blocked channels (k) and from blocked to unblocked (l). Transitions from resting (R) to activated (A) and inactivated states (I) are governed by the Hodgkin-Huxley (HH)-type formulation of the canine atrial ionic model. Right, schematic representation of the guarded-receptor model with affinity to the activated and inactivated states. B: schematic example of phase singularity (PS) identification by the zero-velocity condition. For illustrative purposes, we show a simulation at 1 point in time along with the corresponding wave boundary (thick line) and the wave boundary for a complex 5 ms later (thin line). The PS is defined in the algorithm as the points of intersection (indicated in this case by the red solid dot) between isopotential lines at time t and time t + 1 ms.

Calculations were performed with a time step of 25 µs and up to 16 processors of a 64-processor SGI Origin model 2400 (Westgrid) or of a SGI Altix 3000 high-performance parallel processor (Réseau Québécois de Calcul de Haute Performance). In this way, simulations of 5 s of activity were accomplished in 6 h. The 2-D model was used for most of the theoretical analyses; the 3-D model was then used for the confirmation of key findings.

Potential maps. Propagating wavefronts over the computational substrate were visualized by constructing potential maps. After each millisecond, transmembrane potential at the center of each 5 x 5 cell square (the approximate size of a space constant) on the computational grid was subsampled to a 100 x 200-pixel display grid, providing color maps of the transmembrane potential.

Analysis.
PHASE SINGULARITY ANALYSIS. Wavetips [phase singularities (PSs)] corresponding to points at which all phases intersect were found with the use of the zero-velocity condition as previously described (9). In brief, PSs were defined as corresponding to the intersections between the –60 mV isopotential lines for two frames separated by 1 ms, as shown in Fig. 1B. Each PS was then classified in time using the distance in Cartesian coordinates. Newly created PSs were assigned to new groups. Missing PSs in frame number n were controlled using the position of the PS in the two previous frames (n – 1 and n + 1) and calculating a time trajectory from known groups. PSs in the following frame (n + 1) satisfying the Cartesian criterion (distance with an expected position of <1 mm) were classified as part of the group. Otherwise, the time of frame (n) corresponded to the disappearance of the PS. Time duration and positions of a given wavetip were analyzed from the resulting classification. PSs lasting >90 ms (1 rotation in control AF) were labeled as generators of the activity.

RATIO OF DEPOLARIZED CELLS. The ratio of depolarized cells over time (R_APD) was given by the ratio of cells with a voltage positive to –60 mV to the total number of cells. The time between two activations of a cell (T) were separated into the AP duration (APD) and the diastolic interval. We measured APD as the duration positive to –60 mV (APD_–60), starting when the transmembrane potential became positive to –60 mV during phase 0 of the AP and ending when the cell repolarized to –60 mV. The percentage of depolarized tissue was approximated over time by calculating the number of cells in each time frame n that were positive to –60 mV and normalizing by the total number of cells.

SPATIAL DISTRIBUTION OF APD AND MEAN FREQUENCY DURING AF. Series of APD and T for each cell were collected during AF. Spatial distributions of mean APD and T were calculated by taking the mean of both variables for each cell, yielding a matrix of the mean in space. The spatial distribution of the mean frequency (in Hz) was then calculated as 1,000/T.

The electrophysiological properties of the model in the absence and presence of lidocaine were measured on a homogeneous cable of 10 cm length paced at one end. The ERP was detected by causing a premature activation to the tissue substrate after 20 stimulated activations at basic cycle lengths (BCLs) of 150 or 300 ms. The stimulation amplitude was set to twice the threshold amplitude during pacing to mimic the experimental conditions for ERP measurements. Simulations were also done at thrice threshold in consideration of the possibility that twice threshold might not be on the optimal portion of the strength-intensity curve. The velocity of propagation was measured based on the conduction time between two sites separated by a distance of 5 cm (2.5 cm from the boundaries to minimize boundary effects).

3-D simulation. A previously described morphologically realistic model of the atria was used for 3-D simulations (40, 41). The model is composed of cables, similar to the 2-D substrate in this study, but the cables follow 3-D paths, with cables arranged in layers for each atrium to provide thickness. The atria are electrically connected only via Bachmann's bundle, the sheath of the coronary sinus, and the rim of the fossa ovalis. The same ionic model was used as in the 2-D sheet.

Reentry was induced with a S1–S2 protocol as previously described (41). Based on this previous work, ACh-release islands were chosen, which resulted in sustained fibrillatory activity. Islands with [ACh] = 3.78 nM and a radius of 0.3 cm were uniformly distributed over the atria. The window of vulnerability was determined for the timing of S2 to induce reentry.

Experimental Methods

Animal preparation. All animal handling procedures were approved by the Animal Research Ethics Committee of the Montreal Heart Institute and followed guidelines of the Canadian Council for Animal Care. Seven mongrel dogs (mean weight: 26.9 kg, range: 19.4–32.0 kg) were anesthetized with morphine (2 mg/kg sc) and -chloralose (120 mg/kg iv followed by 29.25 mg·kg^–1·h^–1) and ventilated mechanically. Body temperature was maintained at 37°C, and a femoral artery and both femoral veins were cannulated for pressure monitoring and drug administration. A median sternotomy was performed, and bipolar electrodes were hooked into the right and left atrial appendage for recording. Five silicon sheets containing 240 bipolar electrodes (Research Centre, Sacre-Coeur Hospital, Montreal, QC, Canada) were sutured onto the atrial surfaces. A programmable stimulator was used to deliver 2-ms square-wave current pulses. The current intensity was adjusted to twice threshold under both control and drug conditions. The ERP was determined as the longest S1–S2 interval that failed to capture the atria. The vagus nerves were isolated and divided in the neck. Bipolar hook electrodes (stainless steel insulated with Teflon except for the distal 1–2 cm) were inserted within and parallel to each vagus nerve. Bilateral bipolar vagal stimulation (VS; 0.1 ms square-wave pulses, 10 Hz, 60% of voltage for asystole) was applied continuously during AF induction by a brief train of atrial burst pacing (10–15 Hz, 2-ms pulses, 4 times the atrial capture threshold) and maintained to allow for sustained AF. A ventricular demand pacemaker was used to stimulate the ventricle at 80 beats/min when the ventricular rate became excessively slow during VS.

Open-chest electrophysiological study. The atrial ERP was measured at the right atrial appendage at BCLs of 150, 200, 250, 300, and 350 ms, with 10 basic stimuli (S1s) followed by a premature stimulus (S2) with 5-ms decrements. The longest S1–S2 interval failing to capture was defined as the ERP. Conduction velocity was measured in the right atrial free wall at BCLs of 150, 200, 250, 300, and 350 ms. AF induction was performed by atrial burst pacing with 10 Hz, four times the threshold current, for 1–10 s.

Experimental protocol. Baseline AF duration and electrophysiological parameters were obtained in the absence and presence of VS under control condition. After sustained AF of >20 min during VS had been confirmed, lidocaine (loading dose of 2 mg/kg iv followed by a maintenance infusion at 2 mg·kg^–1·h^–1) was started. If the drug terminated AF, electrophysiological measurements were performed in both the presence and absence of VS. If AF was not terminated, VS was discontinued to allow for AF termination, and measurements were then repeated as described above.

Activation mapping. Five silicon plaques containing 240 bipolar electrodes were attached to the epicardial surfaces of both atria as previously described (22). Electrophysiological mapping was conducted with the Cardiomap system (Research Center, Sacré-Coeur Hospital and Institute of Biomedical Engineering, École Polytechnique and Université de Montréal).

Frequency analysis. The frequency content of fibrillatory activity was calculated by fast Fourier transformation of filtered (fifth-order Butterworth) rectified epicardial bipolar extracellular potentials during AF (29). The dominant frequency (DF) for the 240 electrodes was then determined based on the peak in the power spectrum.

Statistics. Normality of distributions was verified using a Kolmogorov-Smirnov hypothesis test. One-way ANOVA was used to compare experimental measurements corresponding to a normal distribution; otherwise, the nonparametric Kruskal-Wallis test was applied. Multiple-comparison procedures with the Bonferroni correction served to compare multiple group means if ANOVA showed significant overall differences. P 0.05 was considered to be statistically significant.

	RESULTS

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Results of Model Simulations: 2-D Model

AF termination as a function of lidocaine concentrations. We first defined three different [ACh] distribution conditions that resulted in sustained AF with different patterns of activity. Various lidocaine concentrations were then applied after 1, 2, and 3 s of simulated sustained AF for each condition (3 different distributions of ACh, 6 lidocaine concentrations applied at 3 standard time points for a total of 54 simulations). Figure 2 shows the concentration dependence of lidocaine termination in the 2-D model. Overall, lidocaine concentrations of 30 µM were required to terminate AF, and 89% termination occurred at 60 µM lidocaine (Fig. 2A). Lower concentrations of lidocaine were needed to terminate AF with larger [ACh] spatial periods (e.g., threshold concentration of 30 µM for 5-cm distribution vs. 50 µM for 1.25- and 2.5-cm distributions; Fig. 2B).

View larger version (15K): [in this window] [in a new window]   Fig. 2. Percentage of atrial fibrillation (AF) termination as a function of lidocaine concentration ([lidocaine]) in the mathematical model. A: overall termination rates. B: termination rates for each ACh concentration ([ACh]) distribution studied. For comparison, lidocaine terminated experimental AF with 86% efficacy (6 of 7 experiments).

View larger version (28K): [in this window] [in a new window]   Fig. 3. Simulated tissue electrical properties with increasing lidocaine concentration ([lidocaine]). A: effective refractory period (ERP) at basic cycle lengths (BCL) = 150 (red) and 300 ms (blue) with corresponding action potential (AP) duration (APD) at 90% repolarization (APD90; continuous curves). Simulations with two (2x) and three times (3x) threshold are shown with the line types indicated. B: velocity of propagation at BCLs = 150, 300, and 600 ms. C: corresponding wavelength of propagating activity showing first a decrease followed by an increase at [lidocaine] > 45 µM for both BCLs = 150 and 300 ms. D: minimum and maximum steady-state block of Na+ current for BCLs = 150 and 300 ms.

Global effects of lidocaine on AF activity. We studied the spatial-temporal organization of AF by examining the spatial distribution of local DFs obtained from fast Fourier transformations of simulated activity. Examples of vagal effects with and without lidocaine application on the DF distribution are shown in Fig. 4, A–C. Mean DF values based on the frequencies of activation at each cell in the substrate showed the changes shown in Fig. 4D. Simulated VS increased DFs more importantly with smaller [ACh] spatial distributions. Lidocaine decreased the frequency of AF in a concentration-dependent fashion and progressively decreased the DF differences among the three spatial periods. Similarly, the spatial standard deviation in frequency (Fig. 4E) increased with VS and then decreased with lidocaine, although large effects are seen with a low drug concentration and no further increase follows. The mean number of PSs varied with the spatial distribution of [ACh]. Figure 4F shows that the number of PSs was small in the absence of VS but increased with the addition of ACh ([ACh]_max = 15 nM) through the creation of PSs in low-[ACh] regions. Decreasing the spatial period of [ACh] distribution diminished the number of PSs. Lidocaine reduced the mean number of PSs but had a larger effect when there were more short-lived PSs in control, i.e., for the largest spatial period of [ACh] (5 cm).

View larger version (58K): [in this window] [in a new window]   Fig. 4. Spatial distribution of dominant frequencies (DFs) during AF simulations. A: control (CTL) condition without [ACh] (activity is not sustained). B: similar to experimental vagal stimulation (VS) with [ACh] spatial heterogeneity having a spatial period of 5 cm. C: same as B but with lidocaine at 40 µM. D: mean of frequency during AF for each group. CTL corresponds to the activity without [ACh], whereas VS corresponds to the substrate with heterogeneous [ACh]. Subsequent data show changes with increasing [LIDO] in the heterogeneous [ACh] substrate. E: SD of the spatial mean frequency shown for each group as a measure of frequency heterogeneity. F: mean numbers of PSs shown for each group.

View larger version (28K): [in this window] [in a new window]   Fig. 5. Reentry characteristics in homogeneous canine atrial tissue simulations ([ACh] = 7.5 nM) for increasing lidocaine concentration. A: frequency of the reentrant activity. B: three examples of generator core-tip trajectory showing increased meandering with increasing lidocaine concentration. C: minimum rectangular surface covering the position of the PS for a 2-s period.

View larger version (62K): [in this window] [in a new window]   Fig. 6. Example of a simulation in which AF was sustained by numerous generators. A: [ACh] distribution with a spatial period of 5 cm and the concentration indicated by degree of shading. B: positions of generators are shown by colored lines in the second panel and duration by the color scale. C: spatial distribution of APD positive to –60 mV (APD–60) with a minimum of 25.9 ms and maximum of 66.8 ms. D: snapshot of the transmembrane voltage (V; color scale at left) with the position of the PSs indicated by solid circles following S2 at 200 ms. Subsequent panels show snapshots of transmembrane voltages at the simulation time points indicated.

View larger version (62K): [in this window] [in a new window]   Fig. 7. Termination of AF sustained by multiple generators (same conditions as in Fig. 6) with lidocaine concentration of 40 µM. A, top left: positions of generators from the time of lidocaine application (tLIDO = 1,280 ms) showing increased meander, decreased number of generators, and termination of the longest-lived generator in the lower boundary of the tissue (compare with AF under the same conditions in the absence of lidocaine; Fig. 6A). Bottom left, distribution of APD–60. Subsequent panels show snapshots of the transmembrane potential at the time points indicated. Note the thinning of waves and the decrease in their number. B: time variation in the number of PSs in control (black curve) and lidocaine (red curve) showing a marked decrease in the number and reduced variation after tLIDO (vertical dotted gray line). C: fraction of Na+ channels blocked at two sites (positions are indicated in A, bottom left) with a larger fraction in the region having a longer APD. D: variation of the ratio of depolarized cells over time (RAPD) with lidocaine (red curve) compared with control (black curve), exhibiting a mean decrease in RAPD over time with lidocaine. E: transmembrane potential variation before and after lidocaine application.

View larger version (73K): [in this window] [in a new window]   Fig. 8. Failure to terminate simulated AF with lidocaine concentration of 40 µM when applied at a later time (tissue characteristics are identical to those in Figs. 6 and Fig. 7). A, top left: positions of generators from tLIDO = 2,000 ms showing a delocalization from low-[ACh] areas. Failure to terminate occurs, despite increased meandering, because the generator that sustains AF fails to annihilate on boundaries or other collision conditions. Bottom left, distribution of mean APD under control conditions. Subsequent panels show snapshots of transmembrane potential. B: time variation in the number of PSs in control (black curve) and lidocaine (red curve) showing a long-term decrease in the number and lesser variation over time. C: fraction of blocked Na+ channels at two sites (positions indicated in A, bottom left) with greater block in the region with longer APD. D: variation of RAPD with lidocaine (red dashed curve) compared with control (black curve), exhibiting a mean decrease in RAPD with lidocaine. E: sequence of APs before and after lidocaine (administration time indicated by vertical dotted gray lines) shows slower activity, similar but less marked compared with the case shown in Fig. 7.

View larger version (61K): [in this window] [in a new window]   Fig. 9. Termination of simulated AF with lidocaine concentration of 60 µM when applied at the same time as in Fig. 8 (tissue characteristics are identical to those in Figs. 6–8). A, top left: positions of generators from tLIDO = 2,000 ms showing a delocalization from low-[ACh] areas. Bottom left, distribution of mean APD under control conditions. Subsequent panels show snapshots of transmembrane potential. B: time variation in the number of PSs in control (black curve) and lidocaine (red curve) showing a rapid decrease in the number. C: fraction of blocked Na+ channels at two sites (positions indicated in A, bottom left) with greater block in the region with longer APD. D: variation of RAPD with lidocaine (red dashed curve) compared with control (black curve), exhibiting a rapid decrease in RAPD with lidocaine. E: sequence of APs before and after lidocaine (vertical dotted gray lines) shows a decrease in frequency and diminished AP amplitude.

View larger version (74K): [in this window] [in a new window]   Fig. 10. Sustained activity in a simulation with lidocaine concentration of 60 µM and a spatial period of 2.5 cm. The layout is the same as in Fig. 7. The activity after tLIDO is sustained by a single PS that is renewed over time through a local transient block near the tip that induces the creation of a new single PS that then acts as the primary generator. A, top left: positions of generators from tLIDO = 1,280 ms. Bottom left, distribution of APD–60. Subsequent panels show snapshots of the transmembrane potential at the time points indicated. B: time variation in the number of PSs in control (black curve) and lidocaine (red curve) showing a marked decrease in the number and reduced variation after tLIDO (vertical dotted gray line). C: fraction of Na+ channels blocked at two sites (positions are indicated in A, bottom left) with a larger fraction in the region having a longer APD. D: variation of RAPD with lidocaine concentration = 40 µM (red curve) compared with control (black curve). E: transmembrane potential variation before and after lidocaine application.

A critical aspect of lidocaine's action was its ability to reduce primary rotor core-tip PS anchoring and increase generator meandering. To better understand these effects, we analyzed the dynamics of reentry in the presence of a single simplified [ACh] heterogeneity. The spatial dispersion of [ACh] is shown in Fig. 11A, with the smallest amount of ACh at the center of the sheet. The corresponding mean APD dispersion during reentry, shown by the grayscale in Fig. 11B, showed a maximum APD in the center surrounded by a circular region of lower APDs. The trajectory of the PS during 10 s of activity showed anchoring in the high-APD region in control (lidocaine concentration = 0 µM, blue line in Fig. 11B), whereas the core-tip PS showed much freer movement when the lidocaine concentration = 50 µM (red line in Fig. 11B). The enhanced meandering with increasing lidocaine concentration corresponded to changes in the velocity of PS meander. The distribution of PS velocity is shown in Fig. 11C for reentry in control and with three different lidocaine concentrations (20, 40, and 50 µM). An increase in lidocaine concentration from 0 to 20 µM did not change the velocity distribution, but greater concentrations increased high-velocity PS meandering. When minimum and maximum [ACh] were fixed, increasing the [ACh] distribution distance (AChD; in cm) from the minimum to maximum concentration increased the amplitude of APD variation (APD_amp1, where APD_ampl = maximum – minimum APD) because the smoothing effect of electrotonic coupling during the repolarization phase of the AP was limited to a smaller distance. Figure 11D shows the conditions under which the core-tip PS remained anchored in the low-[ACh] zone (the dark gray region corresponding to conditions for PS anchoring). Greater APD_amp1 was needed to disconnect a PS from its anchored region when lidocaine concentration was greater. Without lidocaine, the generator remained anchored for PSs with APD_ampl values of >15 ms, whereas in the presence of 50 µM lidocaine, anchoring required APD_ampl values of >32.5 ms. Figure 11E shows the relation between the angular velocity of spiral wave rotation and PS (generator) meander velocity for different AChDs, with angular velocity decreased by increasing lidocaine concentrations (from 0 to 10, 40, and 50 µM). In all cases, greater angular velocity was associated with reduced PS meander velocity and increased anchoring. Thus, by reducing excitability, lidocaine decreased spiral wave angular velocity and decreased anchoring.

View larger version (36K): [in this window] [in a new window]   Fig. 11. Determinants of spiral wave meander in the absence and presence of lidocaine. A: the grayscale represents [ACh]. A low-[ACh] region ([ACh] = 0 nM) surrounded by a high-[ACh] region ([ACh] = 15 nM) is centered in the tissue (AChD corresponds to the distance between the minimum and maximum [ACh]). The remainder of the tissue is set to the midlevel of ACh to minimize wave break ups and the formation of other generators outside the heterogeneity. B: mean APD–60 during the simulation (10 s) is shown as a grayscale, with a maximum of 50 ms in the center and minimum 25 ms in the high-[ACh] region [resulting in an APD variation amplitude (APDampl) of 25 ms]. The generator (induced by a S1–S2 protocol) is anchored in the low-[ACh] region in control (blue line) while the same generator with lidocaine concentration of 50 µM (red line) meanders in and out of the region before terminating in the upper boundary. C: effects of lidocaine on the meander velocity distribution of the generator core tip (calculated by taking the distance traveled by the PS in 1 ms divided by the time interval). Each curve shows the distribution of velocity for different lidocaine concentration ([LIDO]) values. Increasing [LIDO] widens the distribution and increases the velocity of the generator. D: anchoring effect of [ACh] heterogeneity as a function of APDampl and [LIDO] shows that a generator can still be anchored (dark gray region) at greater [LIDO] but that such anchoring requires a progressively greater APDampl. E: the meander velocity of the spiral generator's core tip is inversely proportional to the angular velocity of the generator, indicating that meander is increased when there is slowed generator rotation with lidocaine.

To confirm our findings in a more anatomically realistic model, we performed simulations in a 3-D atrial substrate. Simulations were performed with S2s covering the window of vulnerability with and without the addition of 60 µM lidocaine. In the absence of lidocaine, the window of vulnerability was 95 ms. Reentry generated within this window was of two types, depending on the timing of S2: ectopic beats in the first 75 ms caused short-lasting reentry with a duration of 814 ± 93 ms, whereas S2s in the last 20 ms resulted in sustained reentry with a quasistable pattern lasting >5,000 ms (fibrillatory activity). The initial type of reentry was anatomical, with activity travelling between the left atrium and right atrium via the Bachmann's bundle, coronary sinus, and/or fossa ovalis. The reentry then established rotors in either atrium and quickly degenerated into sustained quasistable behavior. Activity was maintained by a large rotor in one of the atria or sometimes by irregular, mobile and small rotors. Four episodes of AF were chosen, for which reentrant activity was irregular and wavebreaks were associated with primary rotor movement from the left atrium to the right atrium and back over time. At 2,000 ms, the state of the simulation was saved, and the simulation was continued with no lidocaine (control condition) or with the addition of 60 µM lidocaine.

The control cases (no lidocaine applied) maintained quasistable behavior for at least another 3,000 ms. Upon exposure to lidocaine, reentry was extinguished for all cases at an average of 358 ± 9 ms after t_LIDO. Termination invariably involved increased wave size and reduced curvature associated with slowing of the primary rotor wavefront. Interactions between wavefronts were reduced, eliminating wavebreaks. Propagating wavefronts were extinguished when they encountered geometrical boundaries or refractory tissue. These effects were attributable to lidocaine-induced decreases in Na⁺ source current, which, particularly in the face of spatially heterogeneous increases in current sinks related to I_K(ACh) enhancement, likely altered liminal-like requirements for sustained propagation. The effects of lidocaine on wavefront propagation and mechanisms of termination were qualitatively very similar to those of the 2-D simulations. To analyze the effect of lidocaine in preventing sustained AF development, additional simulations were run for 2,000 ms with 60 µM lidocaine present from the beginning (before the ectopic pulse was applied). In the presence of lidocaine, only nonsustained anatomic reentry could be induced, with an average duration of 555 ± 61 ms and a 75-ms window of vulnerability (vs. 95 ms before lidocaine). Thus, lidocaine both terminated fibrillatory activity and prevented its establishment. Examples of control AF and lidocaine-induced termination are shown in Supplemental Fig. 1. Movies of control AF and termination by 60 µM lidocaine are also available online as supplemental material: the control case (movie 1: AF_Lido0uM.avi) and termination by lidocaine application at 600 ms (movie 2: AF_Lido60uM_at_600ms.avi) or 2,000 ms (movie 3: AF_Lido60uM_at_2000ms.avi).¹

Experimental Observations

Vagal AF was terminated in dogs with 85.7% efficacy (6 of 7 dogs). Termination occurred 4 ± 2 min following the onset of the 5-min lidocaine loading injection. Lidocaine prevented the reinduction of sustained AF in two-thirds of the dogs.

Properties of lidocaine on tissue characteristics. VS reduced ERP (Fig. 12A). Lidocaine tended to increase the ERP, but the changes were small and not statistically significant. No significant conduction velocity changes occurred between baseline and VS (Fig. 12B). Lidocaine significantly decreased conduction velocity in a frequency-dependent fashion, with a mean 32.4% decrease versus control at a BCL of 150 ms and a 22.4% decrease at a BCL of 300 ms.

View larger version (8K): [in this window] [in a new window]   Fig. 12. Experimental data. A and B: mean ± SE data for ERP (A) and conduction velocity (CV; B) in vivo at BCLs = 300 and 150 ms. *P < 0.05 and **P < 0.01 vs. CTL; P < 0.05 and P < 0.01 vs. VS.

View larger version (41K): [in this window] [in a new window]   Fig. 13. Experimental data. A: electrode area used to record activity during AF. LA, left atrium; RA, right atrium; BB, Bachmann's bundle; LAA, LA appendage; RAA, RA appendage; FW, free wall; PW, posterior wall; IW, inferior wall; AVR, atrioventricular ring; IVC, inferior vena cava; SVC, superior vena cava. B–D: mean spatial distribution of DFs in AF (missing values are shown as open circles and correspond to signals with ventricular artifacts for >5 dogs) without VS (CTL; B), with vagal stimulation (C) and with VS and lidocaine (D). Black edges on colored circles show the electrodes having a change in amplitude compared with the minimum DF, which is >2 times the SD in control DF in B.

View larger version (10K): [in this window] [in a new window]   Fig. 14. Experimental data. A: DF for AF without VS (CTL), with VS, and after lidocaine with VS (LIDO). **P < 0.005. B: SD of spatial DF. *P < 0.05.

	DISCUSSION

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Relation to Previous Observations Regarding I_Na Blocker Effects in AF

Classical theories of AF and antiarrhythmic drug actions, centered on the multiple wavelet reentry and leading circle concepts, suggest that Na⁺ channel blockers should be ineffective in terminating AF because they reduce the wavelength by slowing conduction (26). However, clinical experience has consistently demonstrated the ability of Na⁺ channel blockers to terminate and prevent AF, and an experimental study (45) has demonstrated clear efficacy despite a lack of wavelength increase. We (18) have previously showed that reductions in I_Na conductance lead to slowing and enlargement of primary AF generators, terminating AF in a fashion consistent with experimental findings. In the present study, we extended these observations in several ways by 1) modeling state-dependent I_Na blockade as occurs with clinically used antiarrhythmic drugs; 2) testing applicability in an anatomically realistic 3-D model of AF; 3) confirming in vivo relevance with the same clinically used antiarrhythmic drug that was modeled in the computational analysis; and 4) assessing the mechanisms underlying primary rotor destabilization due to state-dependent I_Na block. Our modeling results suggest an increased organization of atrial activation before class I antiarrhythmic drug-induced AF termination as well as greater variations in AF cycle length due to primary rotor hypermeander. Both of these phenomena have been documented in previous experimental studies (2, 17, 45).

Theoretical Aspects

In the present study, we found that the termination of AF by a dynamic representation of lidocaine action in the mathematical model follows the principles of spiral wave reentry in excitable media. Mandapati et al. (24) showed that decreased excitability expands the core area while slowing ventricular fibrillation. A key determinant of all types of persistent reentrant activity is the source-sink relationship related to the propagating activation wavefront. A wavefront can propagate as long as unexcited but excitable cells (the "sink") have their Na⁺ channels activated by the diffusion current moving forward from depolarized cells at the edge of the front (the "source"). The diffusion current acts as a drain on the source, so that if a relatively small source is attached to a larger sink, the loss of source current caused by the sink may reduce the current available for excitation to the point that propagation fails. There is thus a critical relationship between the source current for excitation and the mass of tissue being excited, which electrotonically sinks the source current (4). Because lidocaine blocks I_Na and decreases the source current, the activation of sink cells becomes more difficult, particularly close to spiral core tips, where wavefront curvature is more pronounced (4, 47). Such effects are complicated by the varying background of enhanced I_K(ACh), which opposes inward I_Na and functionally reduces source current, in our heterogeneous [ACh] substrate. On the other hand, the APD abbreviation caused by I_K(ACh) reduces the current sink, accelerating and stabilizing reentrant rotors (19). Thus, rotor dynamics in the presence of lidocaine will be a complex function of the I_Na-opposing effects of I_K(ACh), the rotor-accelerating and -stabilizing effects of I_K(ACh), and the source current-reducing effects of lidocaine. Of these phenomena, the I_K(ACh) effects are also present under control conditions, so it is primarily the I_Na-related phenomena that change with lidocaine application. A relatively large source enables the spiral to rotate rapidly, creating a smaller core (24). When the source is reduced by drug-induced I_Na blockade, the path followed by the spiral wave tip increases because more time is needed to activate cells downstream from the propagating wavefront, thus increasing core size and reentry period similar to reentry in weakly excitable media (11). Slowed rotation reduces anchoring of the rotor core, increasing meandering. Increased meandering and enlarged spiral wave core dimensions lead to AF termination, a result that is not predicted by the wavelength concept, highlighting its limitations. In mathematical models of ventricular tissue, constant and direct I_Na blockade also promotes spiral wave meander but stabilizes spiral waves without terminating reentrant activity (31), indicating important differences between atrial and ventricular tissue responses.

Interactions Among ACh Concentrations/Distributions and Tissue Excitability in Determining AF Stability

ACh activates I_K(ACh) and thereby decreases the refractory period and hyperpolarizes the cell membrane (25). Such effects accelerate reentrant rotors maintaining AF and increase their angular velocity (30). The increase in angular velocity is accompanied by a decrease in the displacement and meander velocity of PSs corresponding to rotor core tips (Fig. 11E). Our simulations indicate that larger spatial gradients in [ACh] produce larger APD gradients because of reduced electrotonic smoothing and thereby functionally anchor rotor core tips in longer-APD zones with a relative paucity of ACh. Na⁺ channel blockade terminates AF by removing this anchoring effect and causing AF-maintaining rotors to meander more freely around the tissue substrate until they annihilate against a boundary or obstacle. These observations suggest that factors that reduce APD, thereby increasing the angular velocity and functional anchoring of rotors, should reduce the ability of Na⁺ channel-blocking drugs to terminate AF. This has, in fact, been the clinical observation with AF-related electrical remodeling, which both reduces APD and induces resistance to Na⁺ channel blocker-induced AF termination (7, 39).

Novel Aspects of the Present Study and Potential Importance

This study is the first to our knowledge to use realistic mathematical models to study the effects of a state-dependent Na⁺ channel-blocking drug on AF. It is also the first to study class I antiarrhythmic drug-induced termination of AF in an anatomically realistic 3-D computational substrate. We chose to analyze a class IB antiarrhythmic drug with rapid unbinding kinetics from the Na⁺ channel and were fortunate to find enough data from the literature to model the state-dependent I_Na-blocking effects of a drug (lidocaine) that we could also administer in an experimental model of AF. We were thus able to show qualitative agreement between model simulations and experimental observations.

There is presently great interest in developing novel therapeutic agents for AF (27). With the results of the Cardiac Arrhythmia Suppression Trial (5), clinical interest in class I agents greatly decreased. Recently, however, there has been increasing evidence that rapidly unbinding Na⁺ channel blockers may constitute safe, effective, and selective agents for AF suppression (3, 8, 35). In the present study, we provide a mechanistic rationale for the efficacy of a rapidly unbinding Na⁺ channel blocker and show that AF selectivity is conferred by strongly rate-dependent actions that produce little conduction slowing at sinus rhythm rates but important I_Na blockade at the rapidly discharging rates of AF. Additional mechanisms, including differences in molecular Na⁺ channel composition of atrial versus ventricular cells and discrepancies in state-dependent blockade owing to quite different AP conformations in the atrium versus ventricle, may also provide bases for atrium-selective Na⁺ channel blockade. Nonetheless, the mechanisms of drug efficacy in AF would likely be similar to those that we have uncovered in the present work.

Vagal tone plays an important role in clinical AF (46), and spatially heterogeneous refractoriness shortening is a key component of the underlying mechanisms (23). The present results are therefore relevant to physiologically and clinically relevant phenomena.

Potential Limitations

Any mathematical model is, by definition, an oversimplification of the biological reality, and that is certainly true of the models used in the present study. There are well-established limitations to the modulated-receptor model that we used (16, 36). We chose this model because it can reproduce use-dependent Na⁺ channel block and is a model that has been widely used in the past. The simulations were done with parameter values originally published in 1977. The use of an equilibrium voltage shift is biophysically undesirable, and it would be better to specify dynamic transition rates between the various blocked states that produce the apparent voltage shift in the model. However, the latter were never clearly defined by Hondeghem and change to Hondeghem and Katzung and cannot be defined by us with the data available. We used the formulation described here because it represents a working formulation as defined in the modulated-receptor model and provides a useful description of state-dependent lidocaine block. Consideration of additional experimental data published since the 1977 modulated-receptor model article might help to refine the model. However, our goal here was not to optimize this model but to use it to simulate state-dependent drug action, and the model as applied here provided state-dependent actions that resulted in predictions in qualitative agreement with experimental observations. Alternative models include the allosteric effector approach (16), which necessitates a Markov model for the Na⁺ channel, or Starmer's guarded-receptor model (36). The allosteric effector model would have required complex modifications of the canine atrial ionic model to incorporate Markovian rather than Hodgkin-Huxley formulations for I_Na, beyond the scope of this article. We could also have chosen the guarded-receptor model as the basis for simulating lidocaine-I_Na interactions, since parameter values for lidocaine are also available (10). A study by Hanck et al. (12) has suggested that lidocaine alters channel kinetics. The guarded-receptor model assumes that fundamental channel kinetics are unaffected by Na⁺ channel blockers, although "guarding" of drug access to the channel can produce apparent alterations in channel kinetics.

To determine the extent to which our findings depend on the specific model used to simulate the lidocaine-Na⁺ channel interactions, we simulated the guarded-receptor formulation (APPENDIX) and directly compared our results using the modulated-receptor model with simulations using the guarded-receptor model for similar conditions. To do this, we simulated the lidocaine effects on AF (ACh spatial period of 5 cm, 3 different application times) with the guarded-receptor model for lidocaine (37). We then compared the results with those obtained with the Hondeghem-Katzung model under the same substrate conditions. Termination rates with either model are shown in Table 1. Both models showed low termination rates at 40 µM lidocaine (1 of 3 simulations, respectively), and both showed uniform efficacy (3 of 3 simulations) at 60 µM. Details of blocking kinetics and spiral wave dynamics before termination in one simulation are shown for the guarded-receptor model (60 µM lidocaine) in Fig. 15. The rate of I_Na block increase was slower with the guarded-receptor model than under comparable conditions with the modulated-receptor model, resulting in slower overall block onset and less fluctuation in block during each cycle. Likely because of the slower block onset, termination occurred somewhat later with the guarded-receptor model (2.22 ± 0.21 s after application) than with the modulated-receptor model (0.52 ± 0.29 s, P < 0.001) in the five simulations with the same conditions for which termination occurred with both. More importantly for our purposes, the termination mechanisms observed with the guarded-receptor model are quite similar to those observed for the modulated-receptor model. Specifically, wavebreak was reduced, spiral wave rotation was slowed, the generator spiral core was enlarged, and enhanced meandering led to termination by collision with boundary conditions or refractory tissue zones.

View larger version (67K): [in this window] [in a new window]   Fig. 15. Simulation of lidocaine action with the guarded-receptor model. Shown is an example of the termination of AF sustained by multiple generators (the control case is depicted in Fig. 6) by lidocaine (60 µmol/l) under the conditions of an [ACh] distribution with a spatial period of 5 cm and a maximum [ACh] = 15 nmol/l. A, top left: positions of generators from tLIDO = 1,280 ms showing increased meander, decreased number of generators, and termination of the longest-lived generator in the lower boundary of the tissue, similar to the result obtained with the Hondeghem-Katsung modulated-receptor model with [LIDO] = 40 µmol/l (Fig. 7). Comparison of the drug models with the same [LIDO] (40 µmol/l) shows a longer time before termination with the guarded-receptor model (2.9 s after tLIDO) compared with the modulated-receptor model (0.9 s after tLIDO). Bottom left, distribution of APD–60. Subsequent panels show snapshots of transmembrane potential at the time points indicated. Note the thinning of waves and the decrease in their number following simulated lidocaine application, similar to the change in dynamics seen in simulations with the modulated-receptor model. B: time variation in the number of PSs in control (black curve) and lidocaine (red curve), showing a marked decrease in the absolute number and the variation in numbers after tLIDO (vertical dotted gray line). C: fraction of Na+ channels blocked at two sites (positions are indicated in A, bottom left), with a larger fraction of channels blocked in the region having a longer APD. The increase in fractional block is slower with the guarded-receptor model, and fractional block shows less oscillation during each rotation of the remaining generator. D: time-dependent variation of RAPD with [LIDO] = 60 µM (red curve) compared with control (black curve), exhibiting a mean decrease in RAPD with lidocaine. E: transmembrane action potentials before and after lidocaine application (indicated by the vertical dotted gray line).

In addition, we used a very simple (albeit clinically relevant) paradigm of AF: that of increased vagal tone. The important effects of cardiac disease and atrial remodeling, which have profound influences on the AF substrate (28), have not been considered and would be appropriate subjects for future research. Lidocaine is not generally considered a clinically useful drug for AF termination, and yet we found it to be successful in both our mathematical model and in the corresponding canine in vivo paradigm. A previous study (6) has similarly demonstrated the efficacy of lidocaine in terminating vagotonic AF in the dog. Interestingly, that study also found striking lidocaine-induced changes in activity during AF as reflected by fast Fourier transforms, with a mean 59% decrease in DF that is quantitatively quite similar to the 54% DF decrease we saw in the smallest spatial [ACh] gradient substrate with 50 µM lidocaine. A recent in vitro study (3) in a canine ACh-induced AF model similarly observed the efficacy of lidocaine and the rapidly unblocking Na⁺ channel blocker ranolazine. The discrepancy between lidocaine's efficacy for AF termination in the present and previous canine studies (6) and its reputation for being ineffective for clinical AF may be related to interspecies differences or to a need for larger lidocaine doses for atrial versus ventricular effects (20). In any case, our objective in the present study was not to revisit the use of lidocaine as a drug for clinical AF therapy but rather to exploit lidocaine as a prototype class IB antiarrhythmic agent for which we have sufficient information to create a model of state-dependent I_Na blockade and that is readily available for intravenous testing in experimental animals. It would be very interesting to obtain detailed information about the state-dependent I_Na-blocking properties of drugs like vernakalant or ranolazine and then correlate their predicted actions in mathematical models with observed in vivo effects; however, such a study is outside the scope of the present work.

Lidocaine is administered in vivo as a loading dose over several minutes, which imposes a significant period of time before steady-state effects are achieved and making temporal correlations with the mathematical model very difficult. The time required for computer simulation of realistic electrical activity in a tissue sheet makes it practically impossible at the moment to simulate continuous electrical activity over several minutes of lidocaine administration. Instead, we simulated the effects of steady-state lidocaine at a specific time after AF onset. Nevertheless, there was good quantitative agreement in electrophysiological changes and AF termination efficacy between the mathematical model and experiments. Further improvements in computational speed may make it possible in the future to more closely simulate experimental and clinical drug administration conditions. We used twice threshold stimuli to define the ERP in experiments because this is the standard experimental and clinical approach and compared the results with simulations using the same criterion. There are theoretical reasons why stronger stimuli might provide more valid ERP assessments of drug action. We therefore repeated our ERP simulations with thrice threshold stimuli and obtained qualitatively similar results (Fig. 3A). Further consideration of optimized stimulus intensity for the assessment of drug actions may be appropriate for future work.

In this study, we specifically addressed mechanisms of rapidly unbinding Na⁺ channel blocker action on simulated and experimental AF. We did not address the issue of the effects of varying intrinsic AP parameters and differences in drug kinetics on I_Na-blocking drug action in AF. It would be interesting in future work to examine the role of Na⁺ channel-blocking kinetics over a wide range in determining the efficacy for AF as well as the rate and tissue selectivity of action.

	APPENDIX: IMPLEMENTATION AND SIMULATION OF THE GUARDED-RECEPTOR MODEL

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The guarded-receptor model (37) was implemented in the 2-D sheet with affinity to the activated and inactivated states (the schematic representation is shown in Fig. 1A). The model is described by the following:

where B_A and B_I are the sums of blocked channels from activated-state and inactivated-state affinity, respectively. [D] is the concentration of the drug, and B_O is the sum of blocked channels in the open state. Simulations of lidocaine action were obtained with a binding rate (k_A) = 1,370.0 ms^–1·M^–1 and an unbinding rate (l_A) = 1.3 x 10^–5 ms^–1 for the open state and a binding rate (k_I) = 60.0 ms^–1·M^–1 and an unbinding rate (l_I) = 2.3 x 10^–4 ms^–1 for the inactivated state. For direct comparison between the drug models, we simulated lidocaine action, modeled by the guarded-receptor model, on AF obtained with an ACh spatial period of 5 cm and [ACh]_max = 15 nM (corresponding to results shown in Fig. 6) and compared the results directly between the two models under identical conditions. The stimulation protocol for AF initiation was the same as used for simulations of the Hondeghem-Katzung modulated-receptor model, and the same three different application times of lidocaine were simulated. The effect of four different lidocaine concentrations (30, 40, 50, and 60 µM) was studied.

	GRANTS

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This work was supported by grants from the Canadian Institutes of Health Research, the Quebec Heart and Stroke Foundation, the Fondation Leducq (European-North American Atrial Fibrillation Research Alliance award), and the Mathematics of Information Technology and Complex Systems Network of Centers of Excellence (to S. Nattel), a Heart and Stroke Foundation of Canada Fellowship (to P. Comtois), and an Alberta Heart and Stroke Foundation award (to E. Vigmond).

	ACKNOWLEDGMENTS

 
The authors thank Nathalie L'Heureux for technical assistance and France Thériault and Luce Bégin for secretarial help with the manuscript.

	FOOTNOTES

 

Address for reprint requests and other correspondence: S. Nattel, Montreal Heart Institute, 5000 Belanger St., Montreal, QC, Canada H1T 1C8 (e-mail: stanley.nattel{at}icm-mhi.org)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

¹ Supplemental material for this article is available online at the American Journal of Physiology-Heart and Circulatory Physiology website.

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