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Indian J Pharmacol. 2010 June; 42(3): 195–196.
PMCID: PMC2937326

A mathematical model to understand the mechanisms of action of class 1 antiarrhythmic drugs

Sir,

The pharmacology of antiarrhythmic drugs is very difficult to understand, specially that of class 1 antiarrhythmic drugs. One drug, quinidine, of group Ia increases the action potential duration (APD) and effective refractory period (ERP) and slows repolarization by blocking Na channels. A second drug, lignocaine, of group Ib, with the same Na-channel blocking action, decreases ERP and APD and accelerates repolarization, while a third drug, flecainide, of group Ic, with the same Na-channel blocking action alters APD and ERP and alters repolarization minimally or does not alter it at all.[1] When students enquire about the mechanism for these variations, it becomes very difficult to explain.

The action potential (AP) in Purkinje fibers can be divided into follwing phases: rapid depolarization phase O, early repolarization phase 1, plateau phase 2, rapid repolarization phase 3 and resting phase, called as slow diastolic depolarization.

Phase 0 (phase of rapid depolarization) is due to the entry of Na ions through fast inward Na+ channels. The cell membrane's electrical charge changes from negative (-90 mv) to zero and finally to positive +30 mv. Transient outward K+ current (Ito) opens briefly immediately after depolarization and contributes to early repolarization (phase 1) and outward (delayed) rectifying K+ current (IK) opens in phase 2 (plateau) and initiates repolarization[2] [Figure 1]

Figure 1
Normal action potential duration of 10 units time (A to C).

Quinidine, procainamide (Ia) and flecainide (IC ) block K+ channels while lignocaine (Ib) does not block K+ channels during repolarization. The class Ia drugs – quinidine and procainamide – block IK - and I to -potassium channels, the class Ib drug – lignocaine – does not block any K + channel while the class Ic drugs – flecainide and encainide – block IK -potassium channels only. Class Ic drugs do not block Ito -potassium channels.[3] “ERP is closely related to the APD.”[4]

In normal AP of Purkinje fibers, the difference of voltage from -90 mv to +30 mv (difference, 120 mv) in 0 phase can be represented by 10 units height from A to B in the diagram. Total time taken from the beginning of 0 phase to the end of phase 3 can be presumed as 10 units time, as represented in the diagram from point A to point C. Ten units height of voltage difference will take 10 units time to return to the resting potential at the end of phase 3. The horizontal axis represents time of completing AP and the vertical axis represents the voltage difference during 0 phase of AP.

Normal voltage difference in the 0 phase of AP in Purkinje fibers is from -90 mv to +30 mv, which is equivalent to 120 mv, and can be represented by 10 units height. We can presume that if a drug blocks the Na+ channels mildly in the 0 phase, the height of AP during the 0 phase will be reduced from 10 units to 9 units. If a drug blocks the Na+ channels moderately in the 0 phase, the height of AP during the 0 phase will be reduced from the normal 10 units to 8 units. If a drug blocks the Na+ channels profoundly, the height of AP during the 0 phase will be reduced from the normal 10 units to 7 units.

We can imagine that 1 unit height during the 0 phase of depolarization will take 1 unit time to complete repolarization or AP. If the height is 7 units, then it will take 7 units time to complete AP. If the height is 9 units, then it will take 9 units time to complete AP.

Quinidine, a class Ia drug, blocks Na+ channels with moderate intensity during the 0 phase of AP. Hence, it reduces the height of the voltage difference from the normal 10 units to 8 units. Quinidine also blocks the Ito-K+ channels during early repolarization of phase 1 and blocks the IK-K+ channels during phase 2 and phase 3 of AP. Blocking of K+ channels during repolarization will prolong the repolarization time. By blocking the Ito channels, quinidine prolongs 1 unit time and by blocking the IK channels, quinidine prolongs 3 units time to complete repolarization. If the K+ channels are not blocked by quinidine, then the time taken to come at the resting potential would have been 8 units, and quinidine could decrease the APD and ERP. But, quinidine, by the blocking Ito and IK channels, prolongs the time of APD equivalent to (8 + 1 + 3) 12 units. The normal time of APD when drug does not block the Na+ and K+ channels is 10 units. Now, it increases to 12 units. Time to complete AP is APD and ERP. In this way, by this mathematical model, we can understand how quinidine increased the APD and ERP. Quinidine also slows repolarization because it increases the repolarization time from the normal 10 units to 12 units [Figure 2 and Table 1].

Figure 2
Increased action potential duration by 2 units time (A to C).
Table 1
Altered action potential of class-1 antiarrhythmic drugs by mathematical model

Lignocaine, a class Ib drug, blocks the Na+ channels mildly and it can be presumed that it will decrease the height of AP during zero 0 from 10 units to 9 units. Lignocaine does not block the Ito and IK channels and that is why it will take 9 units time to come to the resting potential, which is less than 10 units (normal time) to complete APD. Therefore, lignocaine decreases the APD and ERP. Lignocaine slows repolarization because it decreases the repolarization time from the normal 10 units to 9 units [Figure 3 and Table 1].

Figure 3
Decreased action potential duration by 1 unit time.

Flecainide blocks the Na+ channels during 0 phase with profound intensity, which will decrease the height of the 0 phase from 10 units to 7 units. Flecainide does not block Ito (K+ channels) during phase 1 but blocks IK channels during phase 2 and phase 3 of AP. In this way, flecainide increases the time of APD by 3 units. If IK (K+ channels of phase 2 and phase 3) is not blocked by flecainide, then time taken by total AP would have been 7 units only. But, by blocking the IK channels, the time of APD increases from 7 units to 10 units. Ten units is the normal time to complete APD when the drug does not block any channel. In this way, we can explain why flecainide does not alter the APD and ERP and the repolarization time [Figure 4 and Table 1].

Figure 4
Normal action potential duration.

This mathematical model thus explains how quinidine increases the APD and ERP and slows the repolarization. Lignocaine decreases the APD and ERP and fastens repolarization. Flecainide does not alter the APD and ERP and the repolarization time.

Note: All Na+-channel blocker drugs increase the time of 0 phase of AP, but we can ignore it to simplify this model because it is very less in comparison to the total APD.

References

1. Sharma HL, Sharma KK. 1st ed. Hyderabad, New Delhi: Paras Medical Publisher; 2007. Principles of Pharmacology; p. 312.
2. Thomas R. Cardiac drugs. In: Wolff ME, editor. Burger's Medicinal Chemistry and Drug Discovery. 5th ed. Vol. 2. New York: A Wiley International Publication; 1996. p. 164.
3. Thomas R. Cardiac drugs. In: Wolff ME, editor. Burger's Medicinal Chemistry and Drug Discovery. 5th ed. Vol. 2. New York: A Wiley International Publication; 1996. p. 198.
4. Tripathi KD. 5th ed. New Delhi: Jaypee Brothers; 2003. Essentials of Medical Pharmacology; p. 444.

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