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Ion Channel Remodeling in Pulmonary Vein Arrhythmogenesis for Atrial Fibrillation

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REVIEW ARTICLE

Ion Channel Remodeling in Pulmonary Vein Arrhythmogenesis for Atrial

Fibrillation

Yung-Kuo Lin

1,2

, Yao-Chang Chen

3

, Shih-Ann Chen

4

, Yi-Jen Chen

1,2 * 1Division of Cardiovascular Medicine, Wan Fang Hospital, Taipei Medical University, Taipei, Taiwan

2Graduate Institute of Clinical Medicine, College of Medicine, Taipei Medical University, Taipei, Taiwan 3Department of Biomedical Engineering, National Defense Medical Center, Taipei, Taiwan

4Section of Cardiology, Taipei Veterans General Hospital, National Yang-Ming University, Taipei, Taiwan

a r t i c l e i n f o

Article history: Received: Oct 27, 2010 Revised: Mar 8, 2011 Accepted: Mar 12, 2011 KEY WORDS: atrialfibrillation; atrium; ion channel; Naþ/Ca2þexchanger; pulmonary vein; ryanodine receptors

Atrialfibrillation (AF) is the most common cardiac arrhythmia seen in clinical practice and can induce cardiac dysfunction and strokes. Mechanisms of AF involve a triggering mechanism from thoracic vein and/ or a substrate mechanism in the atria. Studies showed that elimination of ectopic focus from pulmonary vein (PV) or non-PVs by catheter ablation could cure AF. As noted, PVs are the most important focus in the initiation of paroxysmal AF. The mechanisms of AF are quite complex because of the fact that AF results from a variety of clinical conditions, autonomic tone modulation, and the self-perpetuation phenomenon “AF begets AF.” Likewise, there is a wide range of changes in the function and expression of ion channels (electrical remodeling). Ion channels involved in AF triggers include those mediating calcium homeostasis and nonecalcium ion channels. Abnormal calcium homeostasis and heterogeneous expression of potas-sium channels, pacemaker channels, and stretch- and swelling-activated chloride channels may promote the electrical activity of PV and consequently the occurrence of AF. Shortening of action potential duration in response to decreases in inward currents and/or increases in outward currents can facilitate the genesis and maintenance of AF. Additionally, different underlying diseases may yield different patterns of electrical remodeling. This review summarizes the recentfindings in this area of research.

CopyrightÓ 2011, Taipei Medical University. Published by Elsevier Taiwan LLC. All rights reserved.

1. Introduction

Atrialfibrillation (AF) is the most common cardiac arrhythmia seen in clinical practice and can induce cardiac dysfunction and strokes.1,2However, the mechanism of AF is not fully elucidated. Theories of the mechanism of AF involve two main processes: a triggering mechanism with one or several rapidly firing atrial focus and a substrate mechanism in the atria with the development of multiple reentrant circuits.

Studies showed that ectopic impulses originating in the PVs or non-PV regions could initiate AF, which could be eliminated by catheter ablation of these ectopic focus.3e8The approaches to eliminate PV or non-PV focus for cure of AF become a cornerstone of clinical practice in the current management of AF, in addition to pharmacological treatment. PVs are the most important focus in the initiation of paroxysmal AF. Rapid atrial pacing was found to induce delayed afterdepolarization (DAD) and early after-depolarization (EAD) in PVs.9The influences of autonomic activity on PVs show that acetylcholine can hyperpolarize the membrane potential and inhibit the spontaneous activity in PVs. On the other

hand, isoproterenol accelerates the spontaneous activity and induces EAD or DAD, which can be suppressed by nifedipine.8 Thesefindings confirm the PV arrhythmogenetic potentials.

The mechanisms of AF are very complex because AF results from a variety of conditions that cause ion channel remodeling, including aging, congestive heart failure, acute myocardial infarction, ethanol abuse, thyrotoxicosis, obesity, and metabolic syndrome.10e15 Addi-tionally, AF itself causes electrical remodeling, namely, the so-called “AF begets AF,” which plays a significant role in AF pathophysiology. All the factors make the electrophysiological studies for arrhyth-mogenesis of AF more intriguing and sometimes controversial. 1.1. Calcium homeostasis in PVs

Dissociation of PVs yielded single cardiomyocytes with and without pacemaker activity.16Isoproterenol was shown to induce EAD in single PV cardiomyocytes.8,9,16,17 Compared with those without induced EAD, PV cardiomyocytes with isoproterenol-induced EAD have a greater prolongation of action potential duration and a greater increase of L-type Ca2þ currents after isoproterenol, which suggests the potential role of Ca2þ in PV arrhythmogenesis. Moreover, T-type Ca2þcurrents were larger in PV pacemaker cardiomyocytes than in left atrial cardiomyocytes or * Corresponding author. Division of Cardiovascular Medicine, Wan Fang Hospital,

Taipei Medical University, 111 Hsin-Lung Road, Section 3, Taipei 116, Taiwan. E-mail: Y.-J. Chen <a9900112@ms15.hinet.net>

Contents lists available atScienceDirect

Journal of Experimental and Clinical Medicine

j o u r n a l h o m e p a g e : h t t p : // w w w . j e c m - o n l i n e . c o m

1878-3317/$ e see front matter Copyright Ó 2011, Taipei Medical University. Published by Elsevier Taiwan LLC. All rights reserved. doi:10.1016/j.jecm.2011.04.004

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PV non-pacemaker cardiomyocytes.18 Patterson et al19 further showed that an increased Ca2þtransient and NaþeCa2þexchange current (NCX) may enhance EAD formation in canine PVs. Similarly, Wongcharoen et al20also found that PV electrical activity can be reduced by an NCX inhibitor (KB-R7943, Tocris Cookson Inc., St. Louis MO, USA20.”) (Figure 1), which reduced the [Ca2þ]i tran-sient’s amplitude and sarcoplasmic reticulum (SR) Ca2þ stores. Because of Ca2þ influx from inward NCX, L- and T-type Ca2þ currents can trigger a large Ca2þrelease from the SR; thesefindings suggest that calcium homeostasis plays an important role in PV arrhythmogenesis.

Dysfunction of ryanodine receptors (RyRs) induces diastolic Ca2þ leak and activates the transient inward current, leading to membrane depolarization and generating DADs. Honjo et al21

re-ported that a low dose of ryanodine can induce PVfirings, sug-gesting that enhancement of the Ca2þleak also contributes to PV arrhythmogenesis. Similarly, Wongcharoen et al22 showed that FK-506 (Sigma-Aldrich Co., St. Louis, MO, USA) (Figure 2), which dissociates the RyReFKBP12.6 complex and inhibits calcineurin activity, can induce PV burstfirings, which also suggests that RyR dysfunction has an arrhythmogenic potential in the PVs. Moreover, aging was shown to express more RyR in PVs, which takes the potential to enhance Ca2þleak (Figure 3). In contrast, K201 (Aetas

Pharm Co., Tokyo, Japan) may reduce the PVfiring rates, DADs, and transient inward currents through the stabilization of the RyRs, allowing the reduction in the diastolic calcium leak.23

Comparisons of intracellular Ca2þ ([Ca2þ]i) among left atrial cardiomyocytes and PV cardiomyocytes showed larger [Ca2þ]i transients, Ca2þsparks, and SR Ca2þstores in PV cardiomyocytes with pacemaker activity, which suggest that distinctive [Ca2þ]i regulation may contribute to the spontaneous activity of PV car-diomyocytes.24Although Bay K 8644 (Sigma-Aldrich Co., St. Louis, MO, USA), a calcium channel activator, increased [Ca2þ]i, it alone did not induce automaticity in PV non-pacemaker cardiomyocytes. In contrast, the coadministration of Bay K 8644 and BaCl2 (an inhibitor of inward rectifier potassium currents) could induce automaticity in PV non-pacemaker cardiomyocytes. Therefore, resting membrane potential and inward rectifier potassium currents also play a role in the PV spontaneous activity. PV cardiomyocytes with pacemaker activity have been shown to have smaller inward rectifier Kþcurrent and less negative membrane potential. Jones et al25reported that cells from the PV myocardial sleeve showed large elevations in diastolic calcium during activation at physio-logical rates in rabbit model. Cells from the PV share some features with cells from the sinoatrial node but also have distinctly unique features that predispose them to the development of spontaneous activity with a higher stimulated steady-state diastolic calcium.

The SR Ca2þ-ATPase (SERCA2a) is critical in uptaking [Ca2þ]iin cardiomyocytes to maintain SR Ca2þcontent. Lee et al26found that the tumor necrosis factor-alphaetreated PV cardiomyocytes had a significant decreased SERCA2a with a compensatory increased NCX, which may induce larger DAD, large transient inward currents, and an enhanced PV arrhythmogenesis. These findings

Figure 1 KB-R7943 decreased thefiring rate of a pulmonary vein pacemaker car-diomyocyte in a concentration-dependent manner. Reproduced with permission from Oxford University Press.

Figure 2 FK-506 induced burst firing in a pulmonary vein (PV) pacemaker car-diomyocyte from an aged animal. Reproduced with permission from Elsevier.

Figure 3 Quantification of the expression level of ryanodine receptor (RyR) in pulmonary veins from young and aged animals. (A) Summary of densitometry quan-tification. (B) Representative Western blot images (a-actin as an internal control to confirm even loading). Reproduced with permission from Elsevier.

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may contribute to inflammation or heart failureerelated AF. Simi-larly, aging also reduces SERCA2a, which may underlie the higher AF during aging.22

1.2. Non-Ca2þionic currents in PVs

Potassium currents have been suggested to contribute to the determination of pacemaker activity in cardiomyocytes. PV car-diomyocytes differ from atrial carcar-diomyocytes in manifesting more depolarized resting membrane potentials because of a lower level of inward rectifier potassium currents (IK1). Therefore, PV car-diomyocytes are prone to trigger arrhythmia because of a reduced threshold for excitation.27 Moreover, the hyperpolarization-activated time-dependent current (IKH) has been observed in a subset of PV cardiomyocytes and has been suggested to play a role in PV electrical activity. Chen et al27demonstrated that the fast-firing (2.5 Hz) PV pacemaker cardiomyocytes had less negative maximum diastolic potential and steeper slope of diastolic depo-larization than those of slow-firing (<2.5 Hz) PV pacemaker car-diomyocytes. Moreover, the PV beating rates are linearly correlated with the slope of diastolic depolarization and maximum diastolic potential. The fast-firing PV pacemaker cardiomyocytes had smaller transient outward current (Ito) and IK1but similar sustained delayed rectifier Kþ current (IKsus), and rapid delayed rectifier currents (IKr), as compared with slow-firing PV pacemaker cardiomyocytes (Figure 4). After suppression of IKHand IK1with barium, fast-firing PV pacemaker cardiomyocytes manifest a higher incidence and current density of pacemaker currents (If) than slow-firing PV cardiomyocytes. However, the percentage of If-positive PV car-diomyocytes was significantly less than that of sinoatrial and atrioventricular nodal cardiomyocytes.

Cell swelling secondary to myocardial ischemia might be the underlying mechanism of ischemia-related AF genesis. Lee et al28

found that hypotonic solution induced larger swelling-activated chloride current (ICl,swell) in PV pacemaker cardiomyocytes than in PV non-pacemaker cardiomyocytes or atrial cardiomyocytes. Compared with atrial cardiomyocytes, hypotonic solution shortened the action potential duration and increased the cell width to a greater extent in the PV cardiomyocytes. Moreover, hypotonic solution decreased the PVfiring with a decrease in the transient inward currents and DADs. Thesefindings suggest that the ICl,swellplays an important role in the electrical activity of the PV cardiomyocytes. On the other hand, hypertonicity can increase the spontaneous beating rates.29In addition, hypertonicity increased the transient inward currents (Iti) and NCX to a greater extent in PV cardiomyocytes than in atrial cardiomyocytes. These findings suggest that cell morphology was associated with PV electrical activity. In support of this, external stretch of PV tissue preparations could increase PV spontaneous activity and triggered activity, which was suppressed by stretch channel inhibitors of gadolinium or streptomycin.30

Figure 4 IeV relationship of Ito, IKsus, IKr-tail, and IK1in fast and slow pacemaking cardiomyocytes from PV. *p< 0.05 and **p < 0.01 versus the slow PV pacemaking cardiomyocytes. Reproduced with permission from John Wiley and Sons. PV¼ pulmonary vein.

Table 1 Effects of AF-related interventions on action potential and ionic currents in a pulmonary vein arrhythmogenesis model

AF-related intervention APD EAD DAD ICa-L Iti Ito IK1 If Beating rates

Rapid atrial pacing Y [ [ Y [ Y e [ [

Thyroxine Y [ [ [ [ [ [ [ [

High temperature Y [ [ [ [ ? [ [ [

Isoproterenol Y[ [ [ [ [ e e ? [

Angiotensin II [ e [ [ [ Y Y [ [

Tumor necrosis factor-alpha Y e [ Y [ [ e ? e AF¼ atrial fibrillation; APD ¼ action potential duration; EAD ¼ early after-depolarization; DAD¼ delayed afterdepolarization; ICa-L¼ L-type Ca2þ current; Iti¼ transient inward currents; Ito¼ transient outward currents; IK1¼ inward rectifier potassium currents; If¼ pacemaker currents; [ ¼ increase; Y ¼ decrease; e ¼ unchange; ? ¼ unknown.

Y.-K. Lin et al. 110

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1.3. Electrophysiological characteristics of PV in AF models

AF is well known to be enhanced in the presence of precipitating factors.9,10,16,26,31,32 Table 1 summarizes the electrophysiological characteristics of PVs in AF models. The most consistentfinding in these AF-related situations was an increase in DAD and transient inward currents in PV myocytes. Genesis of EAD and increase in PV beating rates were also observed frequently. However, changes in the AP duration, L-type Ca2þcurrents, Ito, and IK1were more vari-able under these conditions. Therefore, enhanced triggered activity seems most likely an important contributor to PV arrhythmo-genesis. In support of this hypothesis, NCX inhibitor (KB-R7943), RyR stabilizer (K201), angiotensin II receptor blocker (losartan), and endothelin 1 (Table 2) consistently decrease transient inward current and DAD in PV cardiomyocytes. Therefore, these interven-tions may potentially decrease the occurrence of AF.20,23,32,33 2. Conclusions

Alterations in electrical activity, ionic currents, and calcium handling play an important role in the arrhythmogenesis of PV cardiomyocytes. Using agents selectively regulating the ionic currents underlying the electrical activity of AF triggers is the potential target therapy for atrial tachyarrhythmias.

Acknowledgments

The present work was supported by the following grants: DOH100-TD-B-111-003 from the Center of Excellence for Clinical Trial and Research in Neuroscience; NSC97-2314-B-038-030-MY3, NSC98-2314-B-010-031-MY3, and NSC99-2628-B-038-011-MY3 from the National Science Council of Taiwan; 99wf-eva-02, 100swf01, and 100swf06 from Taipei Medical University-Wan Fang Hospital; and V99C1-120 and V98C1-037 from Taipei Veterans General Hospital. References

1. Kannel WB, Abbott RD, Savage DD, McNamare PM. Epidemiologic features of chronic atrialfibrillation. N Engl J Med 1982;306:1018e22.

2. Wolf PA, Abbott RD, Kannel WB. Atrialfibrillation as an independent risk factor for stroke. The Framingham Study. Stroke 1991;22:983e8.

3. Haissaguerre M, Jais P, Shah DC, Takahashi A, Hocini M, Quiniou G, Garrigue S, et al. Spontaneous initiation of atrialfibrillation by ectopic beats originating in the pulmonary veins. N Engl J Med 1998;339:659e66.

4. Chen SA, Hsieh MH, Tai CT, Tsai CF, Prakash VS, Yu WC, Hsu TL, et al. Initiation of atrialfibrillation by ectopic beats originating from the pulmonary veins: electrophysiological characteristics, pharmacologic responses, and effects of radiofrequency ablation. Circulation 1999;100:1879e86.

5. Pappone C, Rosanio S, Oreto G, Tocchi M, Gugliotta F, Vicedomini G, Salvati A, et al. Circumferential radiofrequency ablation of pulmonary vein ostia: a new anatomic approach for curing atrialfibrillation. Circulation 2000;102:2619e28. 6. Chen SA, Chen YJ, Yeh HI, Tai CT, Chen YC, Lin CI. Pathophysiology of the pulmonary vein as an atrial fibrillation initiator. Pacing Clin Electrophysiol 2003;26:1576e82.

7. Chen YJ, Chen SA. Electrophysiology of pulmonary veins. J Cardiovasc Electro-physiol 2006;17:220e4.

8. Chen YJ, Chen SA, Chang MS, Lin CI. Arrhythmogenic activity of cardiac muscle in pulmonary veins of the dog: implication for the genesis of atrialfibrillation. Cardiovasc Res 2000;48:265e73.

9. Chen YJ, Chen SA, Chen YC, Yeh HI, Chan P, Chang MS, Lin CI. Effects of rapid atrial pacing on the arrhythmogenic activity of single cardiomyocytes from pulmonary veins: implication in initiation of atrial fibrillation. Circulation 2001;104:2849e54.

10. Chen YC, Chen SA, Chen YJ, Chang MS, Chan P, Lin CI. Effects of thyroid hormone on the arrhythmogenic activity of pulmonary vein cardiomyocytes. J Am Coll Cardiol 2002;39:366e72.

11. Wang TJ, Parise H, Levy D, D’Agostino RB, Wolf PA, Vasan RS, Benjamin EJ. Obesity and the risk of new-onset atrialfibrillation. JAMA 2004;292:2471e7. 12. Watanabe H, Tanabe N, Watanabe T, Darbar D, Roden DM, Sasaki S, Aizawa Y.

Metabolic syndrome and risk of development of atrialfibrillation: the Niigata Preventive Medicine Studydmultivariable models. Circulation 2008;117: 1255e60.

13. Lin YK, Chen YJ, Chen SA. Potential atrial arrhythmogenicity of adipocytes: implications for the genesis of atrialfibrillation. Med Hypotheses 2010;74: 1026e9.

14. Allessie MA, Boyden PA, Camm AJ, Kléber AG, Lab MJ, Legato MJ, Rosen MR, et al. Pathophysiology and prevention of atrialfibrillation. Circulation 2001; 103:769e77.

15. Chen YC, Chen SA, Chen YJ, Tai CT, Chan P, Lin CI. Effect of ethanol on the electrophysiological characteristics of pulmonary vein cardiomyocytes. Eur J Pharmacol 2004;483:215e22.

16. Chen YJ, Chen SA, Chen YC, Yeh HI, Chang MS, Lin CI. Electrophysiology of single cardiomyocytes isolated from rabbit pulmonary veins: implication in initiation of focal atrialfibrillation. Basic Res Cardiol 2002;97:26e34. 17. Lo LW, Chen YC, Chen YJ, Wongcharoen W, Lin CI, Chen SA. Calmodulin kinase

II inhibition prevents arrhythmic activity induced by alpha and beta adrenergic agonists in rabbit pulmonary veins. Eur J Pharmacol 2007;571:197e208. 18. Chen YC, Chen SA, Chen YJ, Tai CT, Chan P, Lin CI. T-type calcium current in

electrical activity of cardiomyocytes isolated from rabbit pulmonary vein. J Cardiovasc Electrophysiol 2004;15:567e71.

19. Patterson E, Lazzara R, Szabo B, Liu H, Tang D, Li YH, Scherlag BJ, et al. Sodium-calcium exchange initiated by the Ca2þ transient: an arrhythmia trigger within pulmonary veins. J Am Coll Cardiol 2006;47:1196e206.

20. Wongcharoen W, Chen YC, Chen YJ, Chang CM, Yeh HI, Lin CI, Chen SA. Effects of a Naþ/Ca2þexchanger inhibitor on pulmonary vein electrical activity and ouabain-induced arrhythmogenicity. Cardiovasc Res 2006;70:497e508. 21. Honjo H, Boyett MR, Niwa R, Inada S, Yamamoto M, Mitsui K, Horiuchi T, et al.

Pacing-induced spontaneous activity in myocardial sleeves of pulmonary veins after treatment with ryanodine. Circulation 2003;107:1937e43.

22. Wongcharoen W, Chen YC, Chen YJ, Chen SY, Yeh HI, Lin CI, Chen SA. Aging increases pulmonary veins arrhythmogenesis and susceptibility to calcium regulation agents. Heart Rhythm 2007;4:1338e49.

23. Chen YJ, Chen YC, Wongcharoen W, Lin CI, Chen SA. K201, a novel antiar-rhythmic drug on calcium handling and arrhythmogenic activity of pulmonary vein cardiomyocytes. Br J Pharmacol 2008;153:915e25.

24. Chang SH, Chen YC, Chiang SJ, Higa S, Cheng CC, Chen YJ, Chen SA. Increased Ca2þsparks and sarcoplasmic reticulum Ca2þstores potentially determine the spontaneous activity of pulmonary vein cardiomyocytes. Life Sci 2008;83: 284e92.

25. Jones SA, Yamamoto M, Tellez JO, Billeter R, Boyett MR, Honjo H, Lancaster MK. Distinguishing properties of cells from the myocardial sleeves of the pulmo-nary veins. A comparison of normal and abnormal pacemakers. Circ Arrhythm Electrophysiol 2008;1:39e48.

26. Lee SH, Chen YC, Chen YJ, Chang SL, Tai CT, Wongcharoen W, Yeh HI, et al. Tumor necrosis factor-alpha alters calcium handling and increases arrhyth-mogenesis of pulmonary vein cardiomyocytes. Life Sci 2007;80:1806e15. 27. Chen YC, Pan NH, Cheng CC, Higa S, Chen YJ, Chen SA. Heterogeneous

expression of potassium currents and pacemaker currents potentially regulates arrhythmogenesis of pulmonary vein cardiomyocytes. J Cardiovasc Electro-physiol 2009;20:1039e45.

28. Lee SH, Chen YC, Chen SY, Lin CI, Chen YJ, Chen SA. Swelling activated chloride currents in the electrical activity of pulmonary vein cardiomyocytes. Eur J Clin Invest 2008;38:17e23.

29. Lee SH, Chen YC, Cheng CC, Higa S, Chen YJ, Chen SA. Hypertonicity increases rabbit atrium and pulmonary vein arrhythmogenesis: a potential contributor to the genesis of atrialfibrillation. Clin Exp Pharmacol Physiol 2009;36:419e24. 30. Chang SL, Chen YC, Chen YJ, Wangcharoen W, Lee SH, Lin CI, Chen SA. Mechanoelectrical feedback regulates the arrhythmogenic activity of pulmo-nary veins. Heart 2007;93:82e8.

31. Chen YJ, Chen YC, Chan P, Lin CI, Chen SA. Temperature regulates the arrhythmogenic activity of pulmonary vein cardiomyocytes. J Biomed Sci 2003;10:535e43.

32. Chen YJ, Chen YC, Tai CT, Yeh HI, Lin CI, Chen SA. Angiotensin II and angiotensin II receptor blocker modulate the arrhythmogenicity activity of pulmonary veins. Br J Pharmacol 2006;147:12e22.

33. Udyavar AR, Chen YC, Chen YJ, Cheng CC, Lin CI, Chen SA. Endothelin-1 modulates the arrhythmogenic activity of pulmonary veins. J Cardiovasc Elec-trophysiol 2008;19:285e92.

Table 2 Effects of pharmacological agents on action potential and ionic currents in pulmonary vein cardiomyocytes

Pharmacological agents APD DAD ICa-L Iti Ito IK1 If Beating rates

KB-R7943 [ Y Y Y Y Y ? Y

K201 [ Y Y Y ? ? ? Y

Losartan [ Y e Y Y Y e Y

Endothelin-1 Y Y Y Y Y [ [ Y

APD¼ action potential duration; DAD ¼ delayed afterdepolarization; ICa-L¼ L-type Ca2þ current; Iti¼ transient inward currents; Ito¼ transient outward currents; IK1¼ inward rectifier potassium currents; If¼ pacemaker currents; KB-R7943 ¼ a Naþ/Ca2þ exchanger inhibitor; K201¼ a ryanodine receptor stabilizer; Los-artan¼ an angiotesin II receptor blocker; [ ¼ increase; Y ¼ decrease; e ¼ unchange; ?¼ unknown.

數據

Figure 2 FK-506 induced burst firing in a pulmonary vein (PV) pacemaker car- car-diomyocyte from an aged animal
Table 1 Effects of AF-related interventions on action potential and ionic currents in a pulmonary vein arrhythmogenesis model
Table 2 Effects of pharmacological agents on action potential and ionic currents in pulmonary vein cardiomyocytes

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