Rate-dependent activation failure in isolated cardiac cells and tissue due to Na⁺ channel block

We present in silico predictions confirmed in vitro showing the efficacy of class-I antiarrhythmic agents in slowing heart rate at doses well below their IC₅₀ values. The results hinge on differences between isolated cells versus tissue, highlighting the importance of integrative biology in understanding how therapeutic drug doses work.

Abstract

While it is well established that class-I antiarrhythmics block cardiac sodium channels, the mechanism of action of therapeutic levels of these drugs is not well understood. Using a combination of mathematical modeling and in vitro experiments, we studied the failure of activation of action potentials in single ventricular cells and in tissue caused by Na⁺ channel block. Our computations of block and unblock of sodium channels by a theoretical class-Ib antiarrhythmic agent predict differences in the concentrations required to cause activation failure in single cells as opposed to multicellular preparations. We tested and confirmed these in silico predictions with in vitro experiments on isolated guinea-pig ventricular cells and papillary muscles stimulated at various rates (2–6.67 Hz) and exposed to various concentrations (5 × 10⁻⁶ to 500 × 10⁻⁶ mol/l) of lidocaine. The most salient result was that whereas large doses (5 × 10⁻⁴ mol/l or higher) of lidocaine were required to inhibit action potentials temporarily in single cells, much lower doses (5 × 10⁻⁶ mol/l), i.e., therapeutic levels, were sufficient to have the same effect in papillary muscles: a hundredfold difference. Our experimental results and mathematical analysis indicate that the syncytial nature of cardiac tissue explains the effects of clinically relevant doses of Na⁺ channel blockers.

NEW & NOTEWORTHY

We present in silico predictions confirmed in vitro showing the efficacy of class-I antiarrhythmic agents in slowing heart rate at doses well below their IC₅₀ values. The results hinge on differences between isolated cells versus tissue, highlighting the importance of integrative biology in understanding how therapeutic drug doses work.

the fast inward sodium current causes the rapid phase-0 depolarization in the atria and ventricles (8). Class-I antiarrhythmic agents block voltage-gated sodium channels (3, 13, 17, 26) and are clinically effective in terminating some ventricular tachycardias. Even though molecular interactions between drugs like lidocaine and sodium channels have been elucidated (1, 2, 4, 29), the action of therapeutic doses of the drug is not well understood. Doses required to see an effect in single-channel and isolated cell experiments (30 to 400 μmol/l lidocaine) (1, 2, 4) are much higher than those used clinically (5 to 20 μmol/l) (14, 23).

Indeed, Harmer et al. (14) showed that concentrations of drugs well below the IC₅₀ for cell sodium currents cause widening of QRS wave in ECGs, indicating that an integrative physiology approach is required to understand the safety of drug effects in the heart. Lu et al. (21) showed that while both lidocaine and flecainide affect the maximum rate of rise of membrane potential in ventricular tissue and, therefore, conduction velocity, the greater rate-dependent slowing of conduction by flecainide was the likely cause of its lower safety profile. Moreno et al. (25) used computational models and in vitro experiments to show key differences between the effects of lidocaine and flecainide in ventricular cells and in tissue. Fei et al. (11) showed that lidocaine can terminate reentry in canine tricuspid rings, and the authors conjectured that the mechanism behind the successful action of lidocaine was either an increase in the slope of the conduction curve by the fast recovery of sodium channels from block by lidocaine or an induction of functional block in a vulnerable anatomical site. In this article we show the mechanistic reasoning that can be used to understand the results of Fei et al. (11). Our studies attempt to bridge the gap between the understanding of drug action in single cells and in tissue. We used mathematical models to construct testable hypotheses that were confirmed in vitro. Previous work (19) on single cells and tissue with lidocaine did not examine the phenomenon of activation failure that we define as the inability of a suprathreshold pulse to elicit an action potential. In this article we show a large difference in the sensitivities of tissue and single cells to activation failure due to lidocaine at stimulus rates that mimic the onset of ventricular tachycardia.

MATERIALS AND METHODS

Mathematical Models

A mathematical model (28) of the electrical activity in guinea-pig ventricular cells that included the modulated receptor model (16) of use-dependent block of the fast sodium current (I_Na) was used to study Na⁺ channel block by a theoretical class-Ib antiarrhythmic agent. The differential equations for I_Na, based on data from single ventricular cells, were augmented with additional equations for the modulated receptor model describing time- and state-dependent drug binding to Na⁺ channels. Action potentials were elicited by 200-μs duration inward current pulses; the threshold was found to be 21 nA, and a pulse 15% larger than this threshold value was used in all single cell computations. Single cell simulations were performed on a personal computer using a variable step-size Adams-Moulton predictor-corrector numerical integration method as described earlier (28) and using the MatLab package (Mathworks, Natick, MA). Action potential propagation was studied using a chain of 100 such cells with resistive interconnections representing gap junction channels; these simulations were also performed using MatLab. An intercellular gap-junction conductance of 10 μS with a baseline conduction velocity of 70 cm/s was used in all tissue simulations with chains sufficiently long to minimize spatial end effects. All cells in the chain were exposed to a single concentration of the drug in each simulation, i.e., there were no spatial variations in drug concentration. Drug binding rate constants for the modulated receptor model (k_R = 0.4 mM⁻¹·s⁻¹, l_R = 7 s⁻¹, k_A = 47,500 mM⁻¹·s⁻¹, l_A = 200 s⁻¹, k_I = 4,750 mM⁻¹·s⁻¹, l_I = 5 s⁻¹, V = 35 mV) were based on values for lidocaine from Hondeghem et al. (16) and Clarkson et al. (10) and modified to yield computed IC₅₀ values of ∼70 μM for use-dependent block and 200 μM for tonic block at 37°C.

Isolated Cell Experiments

Ventricular myocytes were isolated as described earlier (6, 30). Female albino Dunkin-Hartley guinea pigs weighing 300–400 g were euthanized by cervical dislocation. The hearts were dissected out and retrogradely perfused via the aorta at 37°C. A Ca²⁺-free solution was used in the first stage followed by a solution containing both collagenase and protease for tissue digestion. Cells were stored in Dulbecco's modified Eagle's medium at room temperature before use. Only rod-shaped ventricular myocytes displaying clear striations were used.

Ringer solution.

The cells were continuously superfused with Ringer solution maintained at 36 ± 0.2°C. The normal Ringer solution consisted of (in mmol/l) 140 NaCl, 5.4 KCl, 1.8 CaCl₂, 1 MgCl₂, 5 glucose, 5 HEPES, and 0.33 NaH₂PO₄, titrated with NaOH to pH 7.4. The intracellular perfusion solution in the pipette electrode consisted of (in mmol/l) 130 KCl, 1 MgCl₂, 5 MgATP, 5 HEPES, 5 creatine phosphate, and 5 EGTA, titrated with KOH to pH 7.4; the resistance of the electrodes, typically 2 to 5 MΩ, were used to measure action potentials in muscle.

The solution in the pipette included EGTA to suppress contractions, which at the higher stimulus rates needed in our experiments could dislodge the cell from the pipette. The cells were used in current-clamp mode in the whole cell-patch configuration using an Axopatch 1D amplifier (Axon Instruments, Foster City, CA). A modified version of the rate ramp protocol (RRP) described below was used to stimulate the cells. A fixed pulse duration of 200 μs was used in all experiments. A Digitimer D4030 (Digitimer, Welwyn Garden City, UK) was used to control the timing of the pulses by triggering a Digitimer DS2A Isolated Stimulator that provided a 200-μs pulse to the step gate input of the Axopatch amplifier. The strength of the stimulus was set to 15% greater than the threshold of each cell. Signals from the amplifier were displayed on a Gould 1604 Digital oscilloscope (Gould Electronics, Ilford, UK), stored for archival purposes on video tape using a Panasonic NV-870 VHS deck (Matsushita Electric, Japan) via a Sony PCM-701ES pulse code modulator (Sony, Japan) and digitized using a CED 1401 (Cambridge Electronic Design, Cambridge, UK) analog-to-digital converter controlled by custom software running on desktop computer.

Tissue Experiments

Eleven male albino Dunkin-Hartley guinea pigs weighing 160–250 g were euthanized by cervical dislocation. The thorax was quickly opened and the heart perfused with Ringer solution before being removed and pinned to a dissection chamber. Papillary muscles 4–5 mm long and less than 1 mm at the widest were carefully dissected from the right ventricle and mounted in a tissue bath (Hugo Sachs Elektronik, March-Hugstetten, Germany). The valvular end of the muscle was secured with a fine silk suture to a Hugo Sachs HSE F30 force transducer; the wall end of the muscle was clipped to the base of the bath. After an initial equilibration period, a preload of 1–5 mN was applied by moving the force transducer assembly with a micrometer.

Ringer solution.

The preparations were continuously superfused with Ringer solution maintained at 36 ± 0.5°C and equilibrated with 5% CO₂-95% O₂ to give a pH of 7.4. The normal Ringer solution consisted of (in mmol/l) 120 NaCl, 4 KCl, 2 CaCl2, 2 MgCl₂, 11 glucose, 23 NaHCO₃, and 0.1 NaH₂PO₄. All chemicals including lidocaine-HCl were obtained from Sigma. Pipettes used to measure action potentials in muscle were pulled using a P-87 Flaming-Brown Micropipette Puller (Sutter Instruments, Novato, CA) contained 3 mol/l KCl and had resistances of 20 to 30 MΩ.

Electrical stimulation and recording.

A bipolar platinum (wire diameter, 250 μm) stimulus electrode was placed at the end of the muscle that was clipped to the base of the bath. A fixed pulse duration of 200 μs was administered using a Digitimer DS2A Isolated Stimulator in all experiments. The strength of the stimulus was always set to 50% greater than the threshold of each preparation and was typically 3 to 5 V. Both action potentials and force measurements were amplified and filtered before being displayed on a Gould 1421 Digital oscilloscope (Gould Electronics) and digitized using a Biopac MP100WS analog-to-digital converter (Biopac Systems, Goleta, CA), which was controlled using the Acqknowledge 3.2 program running on a PowerPC 7600/132 (Apple Computer, Cupertino, CA) computer.

Rate ramp protocol.

Cells and tissue preparations were stimulated at various rates using a RRP shown schematically in Fig. 1A. A basic cycle length (BCL) of 1,000 ms used as each concentration of lidocaine was washed in or washed out for 30 min; this was followed by 18 min comprising three short bursts of stimuli that increased in rate every 30 s. Each burst was followed by a 3-min rest period during which the muscle was stimulated at 1 Hz. At high stimulus rates, the impalement is typically not maintained. In such cases the continuous force measurement was used to measure rate of activation. In all cases where both intracellular potential and force were recorded, every action potential was accompanied by a contraction. The redundant measurements ensured that when impalements could not be maintained, the force measurements indicated whether or not the muscle was excited. Experiments on single cells or muscle preparations in which a one-to-one action potential response to every stimulus could not be maintained under control conditions were terminated; those experiments in which drug effects could not be significantly reversed were terminated and the results discarded.

Fig. 1.

In silico predictions of effects of drug block in single cells. A: rate ramp protocol (RRP). A basal stimulation rate of 1 Hz was used during wash-in and wash-out (white in the upper rectangle) of lidocaine. The stimulation rate was increased in steps (shaded areas in top rectangle, shown in more detail in the bottom rectangle) every 30 s from 2 Hz [500-ms basic cycle length (BCL)] to a maximum of 6.67 Hz (150-ms BCL). B: single guinea-pig ventricular cell model sodium current (I_Na) activated by 1-ms voltage-clamp pulses to −40 mV from a holding potential of −90 mV at 20 Hz. Control simulation shows no change in peak I_Na (in nA) with time. Bottom trace shows no channels entering the blocked state. C: model reconstruction of use-dependent block of I_Na by [D] = 100 μmol/l using the same voltage-clamp protocol as in B. Peak inward (negative) I_Na values show the characteristic decay with time indicating use-dependent block. Bottom trace showing the fraction of channels in the drug-bound states illustrates the accumulation of drug block with successive activating pulses. D: single guinea-pig ventricular cell model paced at a BCL of 150 ms. Control simulation shows a 1:1 action potential (AP) response and no channels blocked. E: at [D] = 100 μmol/l of our theoretical drug, the fraction of channels blocked is significant but not enough to cause APs to fail. F: model prediction of activation failure in single cells in the presence of drug paced using the protocol shown in A; at [D] = 200 μmol/l, there is sufficient block of sodium channels to result in a 2:1 response; for every two stimuli, the cell responds with one AP. G: summary of data on model AP response to pacing frequency at indicated drug doses (in μmol/l in inset legend). Response rate is defined as the number of stimuli that result in APs divided by the number of stimuli at a particular stimulus rate. The response rate is dependent on stimulus frequency at sufficiently high drug doses. H: summary of model APs in response to drug dose at stimulus rates (in Hz) in inset legend. Failure of activation of model cells is strongly dependent on lidocaine dose.

Experimental studies conformed to the National Institutes of Health 85-23 Guide for the Care and Use of Laboratory Animals and were carried out under British Home Office Project License PPL 30/1133.

RESULTS

Single Cell Simulations

The use-dependent properties of our theoretical class-Ib antiarrhythmic agent was studied in silico using 1-ms voltage-clamp steps to −40 from a holding potential of −100 mV at 20 Hz (Fig. 1, B and C). Under control conditions, voltage-clamp pulses result in peak I_Na of constant size (∼8 nA) and no block of channels; with [D] = 100 μmol/l (Fig. 1C), the peak inward I_Na decreases with time due to use-dependent block as a result of accumulation of channels in the blocked state. The rate at which the peak current decays is dependent on pulse rate and drug dose (17).

The effect of Na⁺-channel blockers is often understood in terms of voltage-clamp, use-dependence data, and it is implicitly assumed that the same current decay exists in current-clamp experiments and in vivo (14, 33), i.e., that a steady state of sodium channel block will be achieved so that after an initial settling period, the average fraction of channels blocked during the course of an action potential will not vary from beat to beat. Figure 1, D–F, shows the results of single cell current clamp mode simulations using the RRP: Fig. 1D shows cell response under control conditions paced at a BCL of 150 ms (at the end of a burst in the RRP in Fig. 1A); Figure 1E shows the response in the presence of 100 μmol/l of the theoretical drug and agrees with the behavior predicted by Weirich and Antoni (33) for the case of lidocaine. As shown in Fig. 1F, at higher concentrations there is sufficient block of I_Na to temporarily prevent an action potential from being elicited in response to a suprathreshold stimulus; subsequently, as the drug disassociates from sodium channels at resting potentials, enough sodium channels recover from the blocked state to allow an action potential to be initiated successfully at the next pulse. This indicates that pharmacodynamics of drugs under voltage-clamp conditions can be qualitatively different from that under current-clamp or clamp-free conditions: the rapid kinetics of class-Ib drug binding alters cardiac cell excitability from one beat to the next.

This pattern of block is strongly rate dependent as shown in Fig. 1G. In the absence of drug, the response rate of action potentials due to pacing stimuli is 1:1. At high drug doses (0.2–1 mM), simulations show that the response rate can slip to 2:1 or lower rates. Drug dose also plays a large role in block of activation, and Fig. 1H shows the dependence of action potential activation rate on drug dose (0–1 mM). In these graphs the rate of response is plotted against stimulation rate and dose: the rate of response is defined as the number of stimuli that result in an action potential divided by the total number of stimuli at each frequency of the RRP. While the response rate is 1:1 at low drug concentrations and low pacing rates, both rapid pacing and toxic drug concentrations on the order of about 500 μM are required to inhibit action potentials in single cells.

These simulations predict that the high density of voltage-gated sodium channels in cardiac cells does provide a depolarization reserve in isolated cells, and a high degree of block is required to inhibit action potentials transiently.

Simulations of Propagated Action Potentials

The action potential simulations performed with single cells were repeated for a chain of 100 cells. Four cells at one end of the chain were stimulated with current pulses 50% larger than the threshold, and action potentials were allowed to propagate into the rest of the chain. Action potentials in the center of the chain in response to varying stimulus rates and drug concentrations were recorded and analyzed.

Figure 2, A–D, shows action potentials in the center cell when paced at a 150-ms cycle length (6.67 Hz) with increasing drug concentration. Figure 2A shows that there is no sodium channel block under control conditions, and action potentials travel down the chain from the stimulus site at a 1:1 response rate—each pacing stimulus results in an action potential. Increasing the drug concentration to 5 μM (Fig. 2B) does result in a significant and varying fraction of channels being blocked but not enough to cause failure of conduction. At 7 μM (Fig. 2C) we start to see action potential failures resulting in a 2:1 response, only one out of every two stimuli result in an action potential. As in the case of the single cell simulations, each activation failure provides enough time for channels to recover from block so that the subsequent stimulus is successful. At 10 μM (Fig. 2D), we see that a higher fraction of channels are blocked, but we still see the same 2:1 or 50% response rate we saw with 7 μM and pacing at a BCL of 150 ms. At 10 μM, conduction block, a 2:1 response, occurs starting at a pacing cycle length of 200 ms (5 Hz) while the response is still 1:1 at 7 μM and 5 Hz. At 20 μM, a 2:1 response is seen starting at an even lower pacing rate i.e., 4 Hz. At 50 μM, a 1:1 response is seen at 1 Hz and 4:1 response is seen at 2 Hz followed by complete inexcitability at pacing frequencies above 2 Hz. These data are summarized in Fig. 2, E and F, which show that the response rates were strongly rate dependent as well as dose dependent, respectively. Unlike the single cell results, response rates drop off at much lower doses of the theoretical class-Ib antiarrhythmic agent.

Fig. 2.

In silico prediction of propagating APs in a chain of cells. Model prediction of APs propagating in a chain of 100 guinea-pig ventricular cells. One end of the chain is stimulated at increasing rates. APs are from the cell at the center of the chain at the highest pacing rate (BCL = 150 ms). A: control simulation shows a 1:1 response of stimuli to APs (top trace) and the fraction of sodium channels in the blocked state (bottom trace) in the middle cell of the chain. B: simulated propagating APs (top trace) and sodium channel block (bottom trace) with all cells in the chain exposed to 5 μmol/l drug. The amount of block is time varying depending on the state of the channel, increases during channel activation and inactivation and decreases in the resting state. The amount of block at 5 μmol/l drug is insufficient to cause conduction block. C: simulated propagating APs (top trace) with all cells exposed to 7 μmol/l drug. Activation failure, a 2:1 response, is evident at this low dose. Response rate is computed as the number of APs divided by the number of stimuli. The bottom trace shows the fraction of sodium channels blocked; AP failure results in larger effective diastolic interval during which the drug can disassociate from the channel, allowing a successful AP at a subsequent stimulus. D: simulated propagating APs (top trace) with all cells exposed to 10 μmol/l drug. The higher amounts of block (bottom trace) are seen at this drug concentration compared with A–C. As in C, the 2:1 response results in a lower effective AP rate and a consequent rate-dependent lengthening of AP duration. E: summary of model predictions: rate of response of propagating APs shows strong dependence on pacing rate (abscissa). Control conditions (○, no drug, top line) result in a 1:1 response regardless of pacing. 7 μmol/l drug (▲) results in a 2:1 or 50% response rate when the chain is paced at a BCL of 150 ms. At 20 μmol/l drug (■), the onset of conduction failure occurs at a pacing frequency of 4 Hz, whereas at 50 μmol/l drug (⧫ bottom line), the chain is inexcitable at pacing rates above 2 Hz. F: summary of model predictions: rate of response of propagating APs shows strong dependence on drug dose. The data in F are the same as that used for E but with the abscissa showing drug dose and different lines for different pacing frequencies. Pacing at 1 Hz results in a 1:1 response (○, top line) regardless of drug dose. When paced at 2 Hz, a 4:1 response is seen at 50 μmol/l drug (▲, long dashes). The short-dash line (●) shows the 4-Hz responses with a 2:1 response starting a 20 μmol/l drug. At the highest pacing rate of 6.7 Hz (□), 2:1 responses occur at 7 μmol/l drug and continue till at 50 μmol/l drug, the chain is inexcitable.

Our in silico data show a clear difference between the responses of isolated cells compared with tissue: a cell is susceptible to activation failure only at high drug doses (200 μM), whereas multicellular preparations showed activation failure at much lower doses (7 μM). The depolarization reserve we see with isolated cells is depleted by therapeutic drug doses in tissue. The greater sensitivity of multicellular preparations to Na⁺ channel block compared with isolated cells explains the mechanism by which conduction block by drugs might terminate a reciprocating ventricular tachycardia.

We tested this hypothesis in vitro using isolated cardiac cells and papillary muscle preparations. The class-Ib antiarrhythmic lidocaine was used to test the above predictions as it has one main target in cardiac cells: voltage-gated sodium channels, whereas other antiarrhythmic drugs have significant effects on potassium and calcium channels as well. Harmer et al. (14) found that the free plasma concentrations of lidocaine in patients is in the range 7.1–12.8 μM. They obtained an IC₅₀ of 30.9 μM for use dependent block of hNav1.5 currents by lidocaine at room temperature. This pattern of high IC₅₀ values and significantly lower therapeutic doses is seen in all class-I antiarrhythmic agents.

In Vitro-Isolated Cell Experiments

The threshold current pulses for initiating action potentials in isolated guinea-pig ventricular myocytes showed cell-to-cell variations but were found to be roughly of the same magnitude as those in the simulations of single cells. Cells were stimulated with the same RRP (Fig. 1A) as used in the simulations. An example of the response of a cell to lidocaine is shown in Fig. 3A: under control conditions, a one-to-one response is always observed; in the presence of 100 μmol/l, a 2:1 response was consistently seen (Fig. 3B); the lower response rates were readily reversed when the cell was reexposed to normal Tyrode solution without lidocaine (Fig. 3C). Drug effects were reversible and consistent. As in the case of the simulations, the response rate of single cells was strongly rate as well as dose dependent (Fig. 3, D–F). These experiments showed that, qualitatively, single cells responded as predicted by the simulations. High lidocaine concentrations and high pacing rates both inhibit action potentials transiently.

Fig. 3.

In vitro myocyte and papillary muscle response to lidocaine. A–C: AP responses of a representative isolated guinea-pig ventricular myocyte paced using RRP shown in Fig. 1A. Shown are 1-s durations at the 5-Hz segment of the RRP. A: control conditions. B: after wash-in of 500 μmol/l lidocaine. C: after wash-out; cf., Fig. 1, D–F, for in silico results. Stimulus artifact is evident in all recordings. D and E: summary of isolated-myocyte AP data vs. pacing frequency and lidocaine dose. D: AP response rate vs. pacing frequency at 50 (n = 6), 100 (n = 8), and 500 (n = 6) μmol/l. E: AP response vs. lidocaine concentration at various pacing frequencies (inset legend). All data points are means ± SE; cf., Fig. 1, G and H, for in silico results. F–H: AP and force responses in a representative papillary muscle paced using RRP. Shown are 1-s durations at the 6.7-Hz segment of the RRP. F: control conditions. G: after wash-in of 5 μmol/l lidocaine. H: after wash-out. Papillary muscle was stimulated at one end, and the recording electrode was placed two-thirds of its length down from stimulus site. Stimulus artifact is evident, and AP response in the presence of the drug is in between a 2:1 and a 3:2 response. I and J: summary of papillary muscle AP response data vs. pacing frequency and lidocaine dose. I: AP response rate vs. pacing frequency at 5, 10, and 50 μmol/l lidocaine (means ± SE, n = 5–11) J: AP response rate vs. lidocaine concentration at various pacing frequencies (inset legend); data are means ± SE (n = 5–11). Note order of magnitude difference in lidocaine needed to block APs in single cells (Fig. 3F) vs. papillary muscle; cf., Fig. 2, E and F, for in silico results.

Papillary Muscle Experiments

Action potentials and force measurements from a guinea-pig papillary muscle under control conditions (Fig. 3G), when exposed to 5 μmol/l lidocaine (Fig. 3H) and subsequent washout of lidocaine (Fig. 3I), qualitatively confirm simulation predictions of the effects of 7 μmol/l of the theoretical class-I drug in chains of cells. Some variability in action potential activation was observed: only one muscle preparation did not respond to this low dose of 5 μmol/l lidocaine; all muscle preparations responded to the higher doses. As in the case of the single cells, the response rate was both rate dependent as well as dose dependent, and the results of 11 experiments are summarized in Fig. 3, J–L. Figure 3, J and K, shows the dependence of action potential response to stimulation data for 5 μmol/l and 50 μmol/l lidocaine, respectively. Figure 3L shows the summary of dose-dependence data at a stimulus rate of 150 ms with the connected lines representing data from the same animal.

These experiments directly confirmed the predictions made by the simulations on chains of cells. Action potential failures were observed much more readily in tissue preparations than in single cells when exposed to lidocaine. Our simulation results (Figs. 1 and 2) are on the conservative side: predicted action potentials failure occurs at higher stimulus rates and higher doses in the simulations. The in silico and in vitro results indicate that use-dependent sodium channel block results in failure of activation of action potentials in cardiac tissue at high stimulus rates with therapeutic doses of class-I antiarrhythmics.

DISCUSSION

We showed that the rapid pharmacodynamics of lidocaine result in beat-to-beat changes in cardiac cell excitability. Unlike voltage-clamped cells where a sodium current can always be elicited by an externally applied voltage change, the cumulative effect of use-dependent block of sodium channels causes cells to become transiently unresponsive to stimuli under current-clamp or clamp-free conditions. We showed that when a cell fails to activate because of Na⁺ channel block, it can rapidly recover from block at resting potentials, thereby allowing it to respond successfully to a subsequent stimulus. Furthermore, we showed that both isolated cells and tissue preparations show similar activation failure but at very different doses of lidocaine. This difference in dose explains the therapeutic effect of lidocaine in conversion of ventricular tachycardia to stable sinus rhythm.

A Thought Experiment to Simplify and Understand Cell Activation

Figures 1 and 2 show data obtained by employing the complete set of 22 differential equations per cell. To understand the mechanism behind activation failure, we use a thought experiment in this discussion to examine and understand the principal components of the upstroke of the action potential. The main current active at resting membrane potentials is the inward rectifying potassium current I_K1 (31). I_K1 is inward for potentials below the reversal potential for potassium (E_K) and outward but with strong rectification at more positive potentials. This can be seen in the current-voltage (I–V) plot for I_K1 in Fig. 4A. While a static I–V diagram is sufficient to describe I_K1, the I_Na responsible for the upstroke of the action potential has complex kinetics. The description of Hodgkin-Huxley (15) description of this current is as follows:

$$I_{\text{Na}}={\text{G}}_{\text{Na}}{\text{m}}^3\text{h}\left(V-{\text{E}}_{\text{Na}}\right)$$ (1)

Fig. 4.

A: primary mechanisms involved in myocyte depolarization and activation failure. Current-voltage (I–V) relation for the inward rectifier current I_K1 model used in the simulations. B: model current-voltage relations for I_K1 and I_Na. Bottom curve (dotted) shows I_Na from Eq. 2; middle trace (dashed) shows I_Na obtained using voltage-clamp simulations; top trace (solid) shows computed I_Na in the center cell of a chain during a propagating AP: current magnitudes as well as the voltage-dependence vary significantly in these three conditions. I_Na values from Eq. 2 and from voltage-clamp pulses are much larger than the currents seen in a cell during a propagating AP. In addition, I_Na during a computed AP reaches its peak at a cell membrane potential of +4 mV, whereas the peak current in response to voltage-clamp pulses occur at −20 mV. C: schematic current-voltage relations for the simplified system comprising I_K1 and I_Na. This I–V relation results in a phase space diagram as shown at bottom. In diastole, the system rests at the stable equilibrium E_K; if a stimulus is strong enough to take the system to the right of the unstable equilibrium (threshold) E_th, inward I_Na will push the system towards E_Na, which is a stable equilibrium of the reduced system. Thus, in single cells, there is a well-defined threshold stimulus size beyond which all stimuli will take the system toward the sodium reversal potential. D: schematic view of effect of sodium channel blockade in a one-dimensional model of AP propagation with simplified dynamics. The I–V relation can be used to predict whether a stimulus will propagate or not: in order for AP to be able to propagate successfully, not only does there have to be a region of inward current, the area, A₂, of the I–V region between E_th and E_Na where the current is inward has to be greater than the area, A₁, under the curve between E_K and E_th. The relative areas do not affect single cell AP responses, but the more stringent conditions for tissue AP propagation results in much greater sensitivity to lidocaine.

where G_Na is the maximum total cell sodium current conductance (0.5 μS), m and h represent the activation and inactivation gates of the sodium channels, V is the cell membrane potential, and E_Na is the Nernst potential for I_Na (49 mV). A static I–V diagram for I_Na can be constructed for the upstroke of the action potential by making two assumptions as part of this thought experiment (these simplifications were not used to construct Figs. 1 or 2): 1) m is fast: it reaches its equilibrium value, m_∞, at each voltage instantaneously; and 2) h is slow: it does not change from its resting value, h₀, close to 1. The above approximations yield a simpler expression for the activation phase:

$$I_{\text{Na}}\simeq {\text{G}}_{\text{Na}}{\text{m}}_{\infty }^{\text{3}}{\text{h}}_{\text{0}}\left(V-{\text{E}}_{\text{Na}}\right)$$ (2)

This simplification is solely for the purpose of understanding the dynamics of the upstroke of the action potential and was not used in any of the results shown earlier. A plot (Fig. 4B) of the simplified current in Eq. 2 has a peak sodium current of around −25 nA. Voltage-clamp analysis of the single cell model (Fig. 4B) results in a peak inward current that is half as large (−12.5 nA), indicating that the above assumptions, as a first-order approximation, overestimates the actual inward current.

The ventricular cell model used in the above simulations has the same maximal conductance for both I_Na and I_K1 in line with recent findings of reciprocal Nav and Kir2.1 channel expression in cardiac cells (24). Despite the matching maximal conductances, the rectification properties of I_K1 results in a relatively small (1 nA) positive current at −65 mV, suggesting that isolated cells have a large depolarization reserve. During the upstroke of the action potential, the rate of membrane potential change is slower than in voltage-clamp conditions, and this is even more pronounced during a propagating action potential. A plot of the computed I_Na in the center cell of the chain of cells at cell membrane potentials (Fig. 4B) during a propagating action potential shows an I–V relationship with significantly smaller currents (about 2 nA) than in voltage-clamp simulations indicating significant inactivation even before activation is completed. This N-shaped I–V profile in Fig. 4B of the combined effect of the available I_K1 and I_Na during an action potential is key to understanding when action potentials are generated in isolated cells versus muscle. Although voltage-clamp data show large voltage-gated sodium currents in isolated cardiac cells, the actual Na⁺ currents during action potentials in tissue is much closer magnitude to I_K1, indicating a smaller depolarization reserve in tissue.

Lidocaine blocks sodium channels in the activated (open) and inactivated states (4, 9) with rapid association and disassociation kinetics. In whole cell voltage-clamp experiments, block by lidocaine occurs at approximately the same rate as the channel inactivation process, making it difficult to distinguish between the open and inactivated state block. The net effect is that the peak I_Na developed during an action potential will be substantially decreased by drug blockade in a manner similar to channel inactivation. During the plateau of the action potential, sodium channels are inactivated and lidocaine remains bound to these channels. Repolarization to resting membrane potentials returns sodium channels to the “resting” state, allowing lidocaine to disassociate from channels. At high stimulus rates mimicking ventricular tachycardia, there is insufficient time for lidocaine to disassociate, resulting in accumulation of lidocaine block. Accumulation of sodium channel block due to previous action potentials will further reduce the effective maximum conductance of I_Na and thus decrease the inward sodium current at all cell membrane potentials. Thus the effect of lidocaine can be approximated by a temporarily reduced peak inward current in the I–V diagram (lower curve in Fig. 4D).

Assuming that membrane currents other than I_K1 and I_Na are minimally activated for the few milliseconds immediately after a stimulus, the subsequent cell depolarization can be described using the following:

$$C_{\text{m}}\frac{\text{d}V}{\text{d}t}=-I_{\text{K1}}-I_{\text{Na}}$$ (3)

where C_m is the total cell membrane capacitance. The above equation can be written in the following form:

$$\frac{\text{d}V}{\text{d}t}=-f\left(V\right)$$ (4)

where f is a function of V alone. We see from the I–V relation for the sum of I_K1 and I_Na that Eq. 4 has three equilibria: one at E_K, where I_K1 is at its reversal potential and I_Na is yet to be activated; one at a threshold potential, E_th, where the outward I_K1 is balanced by a partially activated inward I_Na; and a third equilibrium at E_Na, where I_K1 is zero and I_Na is at its reversal potential. During the depolarization phase, E_K and E_Na are “attracting”: if V is close to these potentials, the system will tend toward these values. E_th is considered unstable or “repelling” since values of V on either side of E_th will tend toward one of the other two equilibria. This is shown as a phase-space diagram in Fig. 4C, which only explains the upstroke and not the later phases of the cell action potential. In a single cell, getting V positive to E_th is sufficient to trigger an action potential. As long as the I–V curve has a negative segment, a suprathreshold stimulus will push the cell towards E_Na. The simplified model indicates that even a substantial block of I_Na cannot prevent a single cell from reaching threshold and explains our observation that single cells are not susceptible to activation failure.

Difference Between Activation in Isolated Cells Vesus Tissue

The propagating action potential was modeled in silico using a chain of coupled cells. In the continuum limit, as the number of discrete points in a segment of finite length goes to infinity, the one-dimensional strand of cells can be modeled as a partial differential equation as follows:

$$C_{\text{m}}\frac{\partial V}{\partial t}=\frac{{\partial }^{2}V}{\partial x^2}-f\left(V\right)$$ (5)

In such a system it has been established (12, 27) that a propagating impulse can only occur if the area (A₂) under the curve of f(V) in Eq. 5 from E_th to E_Na is greater than the area (A₁) under the curve f(V) from E_K to E_th as shown in Fig. 4D. In terms of ionic currents, this means that the excitability due to I_Na has to overcome the stabilizing effect of I_K1. This implies that the conditions for action potential propagation in tissue are more stringent than in the case of single cells: not only does there have to be a section of the I–V curve where the total membrane current is negative, the curve should be sufficiently negative to allow propagation in the case of tissue preparations.

In theoretical models of discrete coupled cells, it was shown (20) that the strength of gap junction coupling plays a large role in the ability of the system to support propagating waves. The lowest value of gap junction coupling conductance needed in the one-dimensional chain of cells to allow action potential propagation is 0.05 μS, at which point the threshold stimulus needed to initiate an action potential is the same as that in a single cell and the conduction velocity is a very slow 3 cm/s. As the coupling conductance is increased, both the size of the threshold stimulus needed and the conduction velocity both increase. One would expect that there could be a continuum in the amount of depolarization reserve and, therefore, in threshold drug concentrations needed to inhibit action potentials as the coupling conductance is varied. Surprisingly, such a drug concentration continuum was not seen: whereas 5–7 μM of the drug is required to inhibit propagating action potentials for gap junction coupling conductances between 5 and 15 μS, only slightly higher values are needed with progressive uncoupling: 7–10 μM drug for 2 μS and 10–15 μM drug for lower conductance values. In vitro experiments using gap junction uncouplers could be used to test this prediction.

Limitations of present study.

The original modulated-receptor model was used to reconstruct use-dependent block of I_Na in simulations in this study. This model is based on sodium currents in tissue preparations (10, 17) and does not satisfy microscopic reversibility. The modulated receptor model and the parameters we used in our simulations predicted in vitro results at lower doses but cause more persistent block and abolished action potentials at high doses (Figs. 1 and 2), whereas this latter effect was not seen in vitro (Fig. 3). However, despite these limitations, the predictive power of the model is confirmed by the in vitro experiments and is qualitatively consistent with the findings of Moreno et al. (25), who found conduction block with lidocaine only under pathological conditions, e.g., heart failure, at higher doses.

The experimental design in this study was optimized to simplify the analysis of data and the interpretation of results. Unlike Moreno et al. (25), we did not study the effects of drugs on conduction velocity since our focus was solely on the effects of lidocaine on action potential generation and the differences between isolated cells and tissue. A chain of cells is sufficient to model plane wave action potential propagation, but since the myocardium has a complex three-dimensional anatomical structure with anisotropic gap junction conductance distribution (32), this study is, therefore, conservative in that propagation in actual tissue will involve a greater three-dimensional load on each cell, increasing the likelihood of propagation failure.

A limitation in the thought experiment is that it ignores the effects of I_Na activation time. The analysis of Hunter et al. (18) showed that activation delay has a large effect on the relation between sodium conductance and conduction velocity. Conduction velocity shows a fourth root dependence on conductance when no activation delay is involved. This changes to an eighth root dependence when activation delay is included. Near the critical point at which enough block of sodium channels has occurred to approach conduction failure, this effect can be very significant. Conduction failure may then occur even when the condition A₁ < A₂ is satisfied since the safety factor (18) may be insufficient to guarantee conduction through anatomical heterogeneities. These factors further enhance the differences in drug concentrations required to cause activation failure in single cells versus whole tissue. Thus the above analysis is a conservative prediction of activation failure.

The parameters of block and unblock of sodium channels by our theoretical drug were chosen for a generic class-Ib antiarrhythmic agent based on limited existing data that did not include action potential block. While there are differences between the predictions of the model and experimental observations, the basic phenomena of lower action potential response rates at single cell and tissue due to sodium channel block predicted in silico were confirmed in vitro. A lidocaine-specific model that reconstructs effects in isolated cells and conduction block in tissue will require additional fine-tuning or a more sophisticated model (25).

High pacing rates and insufficient perfusion can result in accumulation of extracellular potassium ([K⁺]_o), which shifts the potassium reversal potential and, through I_K1, the resting membrane potential of the cell to a more depolarized level. Sufficient depolarization may result in inactivation of sodium channels, making activation failure more likely. While this may occur in our in vitro muscle experiments, we did not address [K⁺]_o changes in our in silico experiments. Any change in [K⁺]_o will occur at the rate of a few micromoles per liter per second and would not be fast enough to explain the beat-to-beat changes in cell excitability seen in our in silico and in vitro experiments. Our results indicate that [K⁺]_o accumulation does not have a primary role in rate-dependent activation failure due to lidocaine.

Implications of Present Study

In the wake of the Cardiac Arrhythmia Suppression Trial report (7), it became increasingly clear that a deeper understanding of the action of antiarrhythmic drugs is needed. Recent studies (14) have shown that IC₅₀ values of drug block in isolated cells under voltage-clamp conditions are an order of magnitude larger than therapeutic doses and plasma levels of antiarrhythmic drugs. With much of the work on drug effects being performed in single cells where there is very little spatial effect or in whole animal or whole heart where the spatial effect is much too complex to analyze closely, there is a need for preparations of intermediate complexity. While it has been shown that action potential propagation failure can occur in tissue (11, 25, 32), a mechanistic explanation of this effect has been lacking. Isolated cell experiments use toxic concentrations of sodium channel blockers to examine their pharmacodynamics. Our study attempts to remedy this situation: we show a dramatic 20- to 100-fold difference in the sensitivity of isolated cells versus tissue to lidocaine due to electrotonic effects. Our in silico and in vitro results explain the mechanism of action of therapeutic doses of lidocaine in terminating tachycardias.

GRANTS

This work was supported by grants from the Burroughs-Wellcome Fund, the British Heart Foundation, and the Wellcome Trust.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

A.V., D.J.P., and D.N. conception and design of research; A.V., A.J.S., D.J.P., and D.N. performed experiments; A.V. and A.J.S. analyzed data; A.V. interpreted results of experiments; A.V. prepared figures; A.V. drafted manuscript; A.V. and D.J.P. edited and revised manuscript; A.V., A.J.S., D.J.P., and D.N. approved final version of manuscript.

Appendix A: Mathematical Modeling And Sensitivity Analysis

Models were used in a predictive setting, and the hypotheses generated in silico were tested using in vitro experiments. Model equations and all parameter values including the modulated receptor model (10, 16) were previously published (28) and are not reproduced here.

Local and global sensitivity analysis of the model showed that the observed phenomena were robust. In addition to examining changes in behavior of solutions, sensitivity analysis can also be used to show stability of solutions.

Local Sensitivity Analysis

Local sensitivity was computed using the “internal numerical differentiation” method of Bock (5) and the more accurate “iterative approximation based on directional derivatives” method of Maly and Petzold (22). The system of 22 differential equations were integrated for a single action potential with the drug concentration set to 0. The 22 states were augmented with a system of 1,408 sensitivity factors (the sensitivities of the 22-state variables to each of the 64 parameters listed above) to yield a total of 1,430 values at each point in time during the action potential.

The two methods produced slightly different quantitative data, but the most sensitive parameters were identical. Small variations in parameters such as drug (lidocaine) concentration set to 0 can have significant changes in a number of state variables, but this is expected for any parameter with a 0 value. The voltage-dependent kinetic terms were considered constant and not varied; if these were to be considered as parameters, they would certainly be the most sensitive ones for the gating variables. All computations were performed in MatLab using the ode15s stiff system solver through the sens_ind and sens_sys packages.

Global Sensitivity Analysis

Local sensitivity analysis tells us much about how the model behaves in response to small changes in parameters. In the case of linear parameter dependence as in the case of sensitivity of V_m to membrane conductances like the sodium channel conductance G_Na, local sensitivity is also valid for large parameter changes. However, parameters with nonlinear dependence often require global sensitivity analysis. We took the approach of selecting critical parameters and modifying each parameter in isolation. The parameters selected were the ones that were critical to drug binding, the on and off rates for the kinetics of drug binding to sodium channels. At lower doses of lidocaine, changes in the membrane potential in response to variations in drug binding rate parameters were minimal, suggesting that the model is relatively insensitive to parameter changes.

We used high doses (500 μM) to see significant variation in behavior with parameter changes; furthermore, each of the six rate parameters were increased or decreased by relatively large amount (20%) while keeping all others the same. Significant changes are only seen in the case of increasing the rate of binding (k_a) or decreasing the rate of unbinding (l_a) to the activated (open) state and in these cases the result of a second stimulus in a 2:1 response is even smaller. This is to be expected since these are the largest rate parameters in the modulated receptor model in the case of lidocaine block and thus fractional changes in these rates will have the largest effect. Altering the conductance between cells did not significantly alter the lidocaine concentration at which action potential failure occurs. The gap junction conductance used in the above simulations was 10 μS and neither reducing nor increasing it by 50% changed the above results: between 5 and 15 μS, there was no conduction block for 5 μM of the drug, whereas 7 μM was sufficient to cause failure of conduction and a 2:1 response.

Despite the high dose of lidocaine and the large parameter variations in these global sensitivity simulations, the end result is that the response of the model is not significantly altered, suggesting that the model is robust and that simulation results are not overly sensitive to parameter variations.