Vulnerable Window for Conduction Block in a One-Dimensional Cable of Cardiac Cells, 2: Multiple Extrasystoles

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1 Biophysical Journal Volume 91 August Vulnerable Winow for Conuction Block in a One-Dimensional Cable of Cariac Cells, 2: Multiple Extrasystoles Zhilin Qu,* Alan Garfinkel,* z an James N. Weiss* y Departments of *Meicine (Cariology), y Physiology, an z Physiological Science, Davi Geffen School of Meicine, University of California, Los Angeles, California ABSTRACT Uniirectional conuction block of premature extrasystoles can lea to initiation of cariac reentry, causing lethal arrhythmias incluing ventricular fibrillation. Multiple extrasystoles are often more effective at inucing uniirectional conuction block an reentry than a single extrasystole. Since the substrate for conuction block is spatial ispersion of refractoriness, in this stuy we investigate how the first extrasystole moulates this ispersion to influence the vulnerable winow for conuction block by subsequent extrasystoles, particularly in relation to action potential uration restitution an conuction velocity restitution properties. Using a kinematic moel to represent wavefront-waveback interactions an simulations with the Luo-Ruy moel in a one-imensional cable of cariac cells, we show that in homogeneous tissue, a premature extrasystole can create a large ispersion of refractoriness leaing to conuction block of a subsequent extrasystole. In heterogeneous tissue, however, a premature extrasystole can either reuce or enhance the ispersion of refractoriness epening on its propagation irection with respect to the previous beat. With multiple extrasystoles at ranom coupling intervals, vulnerability to conuction block is proportional to their number. In general, steep action potential uration restitution an broa conuction velocity restitution promote ispersion of refractoriness in response to multiple extrasystoles, an thus enhance vulnerability to conuction block. These restitution properties also promote spatially iscorant alternans, a setting which is particularly prone to conuction block. The equivalent ispersion of refractoriness create ynamically in homogeneous tissue by spatially iscorant alternans is more likely to cause conuction block than a comparable egree of preexisting ispersion in heterogeneous tissue. INTRODUCTION In the companion article (1), we analyze the vulnerable winow for conuction block of a single extrasystole in a one-imensional cable of cariac cells with a preexisting graient in refractory perio. In the clinical setting, however, multiple extrasystoles often appear to be involve in the initiation of reentry. For example, uring clinical electrophysiological stuies, programme stimulation with multiple premature extrasystoles is often require to inuce reentry. Relevant to this observation, experimental stuies have shown that ispersion of refractoriness, which correlates with the ventricular fibrillation threshol (2), is moulate by premature extrasystoles (3) an rapi heart rates (4,5). Other experiments (6 11) have emonstrate that ispersion of refractoriness an inucibility of reentry are affecte by the activation sequence of the premature extrasystoles. Previous computer simulation stuies (12 14) have shown that the ability of a premature extrasystole to moulate ispersion of refractoriness epens on action potential uration (APD) restitution an conuction velocity (CV) restitution properties. Using a kinematic moel, Fox et al. (15) showe that the likelihoo of conuction block by multiple premature stimulations epene on APD restitution an CV restitution. In aition, spatially iscorant APD alternans inuce by rapi pacing (16,17) also ramatically increases the ispersion of refractoriness, creating a heterogeneous substrate favoring conuction block an initiation of reentry (12,18 20). However, a etaile analysis of how such factors affect vulnerability to conuction block has yet to be performe. In this stuy, we use theoretical analysis an numerical simulation to exten the analysis for a single extrasystole in the companion article to multiple extrasystoles. First, we stuie how the first extrasystole inuces heterogeneity an moulates the existing ispersion of refractoriness, thereby affecting the vulnerable winow for a subsequent extrasystole. We then characterize the vulnerability to conuction block by multiple extrasystoles. Finally, we analyze how spatially iscorant APD alternans, promote by steep APD restitution an broa CV restitution, affects the vulnerable winow of conuction block. METHODS Mathematical moel We simulate a one-imensional (1D) cable using Phase I of the Luo an Ruy (LR1) ventricular action potential ¼ÿðI ion 1 I sti Þ=C m 1 V Submitte January 6, 2006, an accepte for publication April 24, Aress reprint requests to Zhilin Qu, PhD, Dept. of Meicine (1) (Cariology), Davi Geffen School of Meicine at UCLA, BH-307 CHS, Le Conte Ave., Los Angeles, CA Tel.: ; Fax: ; zqu@menet.ucla.eu. where V is the transmembrane potential, I ion is the total ionic current ensity from the LR1 moel, an D is the iffusion constant, set to cm 2 /ms. For the homogeneous cable, we use Na 1 channel conuctance Ó 2006 by the Biophysical Society /06/08/805/11 $2.00 oi: /biophysj

2 806 Qu et al. G Na ¼ 16 ms/cm 2 an the slow-inwar current or the L-type Ca 21 channel conuctance G si ¼ 0:06 ms/cm 2. We also spe up the L-type Ca 21 channel activation an inactivation by ecreasing the time constants t an t f to 75%, i.e., t /0.75 t an t f /0.75 t f. These moifications resulte in a baseline APD of 200 ms an APD restitution steepness close to the experimental measurements in rabbit hearts (22). Slowing of recovery of the Na 1 channel was simulate by increasing the time constant of the j gate in the LR1 moel, e.g., a fivefol slowing of the recovery is achieve by increasing t j fivefol as t j /5 t j. Other parameters are the same as in the original LR1 moel. In homogeneous 1D cable, G K ¼ 0:423 ms/cm 2 was use. In heterogeneous cable, action potential graient was simulate by creating a graient in the maximum conuctance of the time-epenent K 1 channel, i.e., G K ¼ G K ðxþ. For the case of ascening APD graient, it is efine as subsequent premature extrasystole (S3). As shown in Fig. 1 A, S1 is applie at the left en of the cable, an S2 an S3 were applie at the same location (x ¼ l). S2 stimulates two waves propagating in opposite irections (Fig. 1 A), one (the S2 wave) propagates in the same irection, an the other (S2* wave) propagates in the opposite irection as the S1 wave. The DI preceing the S2 wave satisfies (see the companion article for etails) ½ 1 ðxþš x ¼ 1 u 2 ðxþ ÿ 1 Q 1 ðxþ (3) 8 < G Kmax ; if x # x 0 ; G K ðxþ ¼ G Kmax ÿðg Kmax ÿ G Kmin Þðx ÿ x 0 Þ; if x 0, x, x 0 1 h; : G Kmin ; if x $ x 0 1 h (2) where we set x 0 ¼ (L ÿ h)/2. In this stuy, we use a cable length L ¼ 40 mm an h was chosen to be 10 mm. For the escening case, Eq. 2 was use with G Kmax an G Kmin exchange. In this stuy, G Kmax ¼ 0:564 ms/cm 2 an G Kmin ¼ 0:282 ms/cm 2. I sti in Eq. 1 is the stimulation current ensity of the stimuli, which were applie in a 1 mm segment of the cable with strength being ÿ30 ma/cm 2 an uration being 2 ms. S1 was the baseline stimulation an always applie at the left en of the cable with cycle length 1 s, whereas the premature stimuli (the extrasystoles) were applie at either ens of the cable as inicate in each case. Eq. 1 was integrate using the explicit Euler metho with a time step ms an a space step Dx ¼ cm. APD an APD restitution APD was efine as the uration of transmembrane voltage V. ÿ72 mv an the iastolic interval (DI) as the uration of V, ÿ72 mv. APD restitution curves were obtaine by plotting APD as a function of the previous DI. APD was measure from the mile of the cable with a regular S1 pacing train of 500 ms interval, followe by a premature S2 at one en of the cable. with the initial conition 1 ðlþ¼dt S1S2 ÿ a 1 ðlþÿ R l 0 x=u 1ðxÞ, in which DT S1S2 is the coupling interval of the S1 an S2 beats an a 1 (l) is the APD of the S1 beat at location x ¼ l.in Eq. 3, u 2 (x) is the wavefront velocity of the S2 or S2* wave, which is etermine by CV restitution, i.e., u 2 ðxþ ¼g½ 1 ðxþš. Fig. 1 B shows the CV restitution curves obtain form the LR1 moel in the 1D cable for normal an fivefol-slowe Na 1 current recovery, an can be fit by u 2 ¼ u 0 ð1 ÿ e ÿð 1ÿ cþ=t Þ (4) with u 0 ¼ 0.55 m/s an ¼ 0.6. For normal Na 1 channel recovery, t ¼ 10 ms an c ¼ 10 ms. For fivefol-slowe Na 1 channel recovery, t ¼ 50 ms an c ¼ 17 ms. These parameters are use in the kinematic simulations of this CV an CV restitution CV was measure in the same cable by calculating the time DT for the wavefront propagating from x ÿ Dx to x 1 Dx, efining uðxþ ¼2Dx=DT. CV restitution curves were obtaine by plotting CV versus DI measure at the mile of the cable. RESULTS Moulation of ispersion of refractoriness by a premature extrasystole APD an CV restitution strongly influence functional ispersion of refractoriness in homogeneous tissue, an moulate preexisting ispersion of refractoriness in heterogeneous tissue (3,12,14). Here we show how a single premature extrasystole (S2) moulates the ispersion of refractoriness in homogeneous an heterogeneous 1D cables of cariac cells, an, epening on their APD an CV restitution properties, affects the vulnerability to conuction block by a FIGURE 1 (A) Schematic illustration of the S1 an S2 beats in a 1D space. S1 is always applie at x ¼ 0, but S2 is applie in location l, which stimulates two opposite propagating waves (the S2 wave an S2* wave). u is the wavefront velocity an Q is the waveback velocity. (B) CV versus previous DI for normal (h) an 5-fol slowe (s) Na 1 channel recovery, which are fit by Eq. 4 (line). (C) APD versus previous DI. The ata are fit by Eq. 7 (soli line). The ashe line is an APD restitution curve by letting a ¼ 0 an b ¼ 0.4 in Eq. 7, which is a shallower APD restitution curve.

3 Vulnerable Winow for Conuction Block 807 stuy. Q 1 (x) is the waveback velocity of the S1 wave, which relates to the wavefront velocity as (1) Q 1 ðxþ ¼ u 1 ðxþ 1 1 u 1 ðxþa 1x ðxþ ; (5) where a 1x ¼ a 1 ðxþ=x is the spatial APD graient of the S1 wave an a 1 (x) is etermine by the preexisting heterogeneity. Similarly, the DI preceing the S3 wave satisfies ½ 2 ðxþš x ¼ 1 u 3 ðxþ ÿ 1 Q 2 ðxþ with the initial conition 2 ðlþ ¼DT S2S3 ÿ a 2 ðlþ, in which DT S2S3 is the coupling interval between S2 an S3, an a 2 (l) is the APD of the S2 beat at location x ¼ l. In Eq. 6, u 3 ðxþ ¼g½ 2 ðxþš an Q 2 ðxþ ¼u 2 ðxþ=ð11u 2 ðxþa 2x Þ. a 2x ¼ a 2 ðxþ=x, in which a 2 ðxþ ¼f ½x; 1 ðxþš, i.e., the APD of the S2 beat is etermine by the preexisting heterogeneity an the APD restitution relationship. 1 (x) is etermine by Eq. 3. The APD restitution curve obtaine using the LR1 moel with our control parameters is shown in Fig. 1 C an can be fit by (6) a 2 ¼ a 0 ð1 ÿ ae ÿ 1=25 ÿ be ÿ 1=150 Þ (7) with a 0 ¼ 200 ms, a ¼ 0.8, an b ¼ 0.4, which are use as control parameters in the kinematic simulations. To investigate the effect of APD restitution slope in our kinematic simulation, we change a in Eq. 7 to change the slope of the APD restitution curve, i.e., the APD restitution curve becomes steeper as a increases (see the example in Fig. 1 C). Conuction of the S3 beat fails when 2 (x) reaches c. Due to the nonlinearity, we are not able to analytically obtain the vulnerable winow for the S3 beat even for homogeneous tissue, but one can obtain the vulnerable winow for the S3 beat by numerically solving Eqs. 3 an 6 together. Homogeneous tissue Here we show how the S2 beat creates ispersion of refractoriness leaing to uniirectional conuction block of the subsequent S3 beat in homogeneous tissue. In this case, Q 1 ðxþ ¼u 1 ðxþ ¼u 0,an 1 (x) can be obtaine by solving Eq. 3 for the S2 wave (Fig. 1 A), which satisfies t u 0e ½ 1ðxÞÿ cš=t ÿ u 0 1 ðxþ ¼x 1 C; (8) where C is an integration constant. Since 1 (x) is a function of space, the APD of the S2 beat, etermine by the restitution relation: a 2 ðxþ ¼f ½ 1 ðxþš, is also a function of space. In other wors, ispersion of refractoriness is inuce by the premature S2 beat. The inuce APD graient is etermine as a 2x ¼ f ½ 1ðxÞŠ x ¼ f 1 ðxþ ¼ f e ÿ½ 1ðxÞÿ cš=t 1 u 2 ðxþ ÿ 1 u 0 ¼ f 0 u 0 ð1 ÿ e ÿ½ 1ðxÞÿ cš=t Þ ; (9) in which f 0 =@ is the slope of the APD restitution curve. Therefore, the spatial APD graient inuce by the premature stimulus is proportional to the slope of APD restitution multiplie by the graient in iastolic interval. Using the following relation (base on Eq. 11 in the companion article (1)), a 2x. 1 u c ÿ 1 u 2 ; (10) we can estimate the slope of APD restitution require for conuction block of the S3 wave to occur, i.e., a 2x ¼ f 0 1 ÿ 1 1. ÿ 1 ; (11) u 2 u 0 ð1 ÿ Þu 0 u 2 which leas to f 0. u 2 ÿð1 ÿ Þu 0 ð1 ÿ Þðu 0 ÿ u 2 Þ : (12) For 1 (x) ¼ c, u 2 ¼ u c ¼ð1ÿÞu 0, f 0. 0 base on Eq. 12 or a 2x. 0 base on Eq. 10 is require for conuction block of the S3 extrasystole. For 1 (x). c, u 2. u c ¼ ð1 ÿ Þu 0, a larger slope of APD restitution is require. For example, for 1 (x) ¼ c 1 10 ms ¼ 20 ms, u 2 ¼ 0.43 m/s base on Eq. 4, which results in f 0. 4:3 from Eq. 12, i.e., a minimum APD restitution slope of 4.3 at ¼ 20 ms is neee for S3 block to occur. For 1 (x) c, u 2 u 0, f 0 N is require. Fig. 2 A shows the APD istribution in space for normal an fivefol-slowe Na 1 channel recovery, showing that an APD graient is inuce by the premature extrasystole. Broaer CV restitution increases the APD graient (Fig. 2 A), which increases the vulnerable winow w (open squares versus soli squares in Fig. 2 C). The vulnerable winow increases as a or the slope of APD restitution increases (Fig. 2 C). However, with the control APD an CV restitution shown in Fig.1, conuction block of the S3 beat can only occur for a small range (;5 ms) of S1S2 intervals (Fig. 2 B). For the S2* wave, which propagates in the opposite irection to the S1 wave, 1 (x) can be obtaine from Eq. 3 by substituting u 2 (x) by u 2 ðxþ ¼ÿu 2ðxÞ ¼ÿu 0 ð1ÿ e ÿ½ 1ðxÞÿ c Š=t Þ, which satisfies t 2 u 0lnð2e ½ 1ðxÞÿ cš=t ÿ Þÿu 0 1 ðxþ ¼x 1 C; (13) where C is an integration constant. The inuce APD graient is a ¼ f 0 ÿ 1 2x u 2 ðxþ ÿ 1 ¼ÿf 0 2 ÿ e ÿ½ 1ðxÞÿ cš=t u 0 u 0 ð1 ÿ e ÿ½ 1ðxÞÿ cš=t Þ : (14) Similarly, we obtain the minimum slope of APD restitution require for conuction block of the S3 beat. Using Eqs. 10 an 14, we have (note that S2* propagates towar the negative x axis as illustrate in Fig. 1 A) a 2x ¼ÿf 0 1 u u 0 1, ÿ ÿ 1 ð1 ÿ Þu 0 u 2 ; (15)

4 808 Qu et al. FIGURE 2 Inuction of ispersion of refractoriness by a premature extrasystole in homogeneous tissue from the kinematic moel (Eq. 3). Two cases were stuie: all three stimuli were applie at the same en of the cable (A C) an the S1 was applie at one en, but the S2 an S3 applie at the other en (D F). (A) APD istribution in space (a 2 versus x) of the S2 wave for normal (soli line) an fivefol-slowe (ashe line) Na 1 channel recovery. a 2 was obtaine using a 2 ðxþ ¼200ð1 ÿ 0:8e ÿ1ðxþ=25 ÿ 0:4e ÿ1ðxþ=150 Þ with 1 (x) solve from Eq. 8. (B) Vulnerable winow (shae area) for S3 for ifferent S1S2 coupling interval (DT S1S2 ) obtaine by numerically solving Eqs. 3 7 together for normal Na 1 channel recovery. (C) w versus a (a parameter controls the slope of APD restitution curve in Eq. 7, as shown in Fig. 1 C) for normal (n) an fivefol-slowe (h) Na 1 channel recovery. w was calculate for the minimum S1S2 interval (DT S1S2 ¼ 211 ms) that S2 propagates successfully. (D F) Same as A C but for S1 being applie at one en an S2 an S3 applie at the other en of the cable. w in F was calculate at DT S1S2 ¼ 283 ms for normal Na 1 channel recovery. S2 failure in B an E inicates the S1S2 interval is too short for S2 to stimulate the S2 wave. In numerical simulation, conuction block was consiere to occur if 2 (x), c at any location x. which leas to f 0. u 2 ÿð1 ÿ Þu 0 ð1 ÿ Þðu 0 1 u 2 Þ : (16) For 1 (x) ¼ c, u 2 ¼ u c ¼ð1ÿÞu 0, f 0. 0 is require for conuction block of the S3 beat to occur. For 1 (x). c, u 2. u c ¼ð1ÿÞu 0, a larger slope of APD restitution is require. For 1 (x) c, u 2 u 0, f 0. =2ð1 ÿ Þ is require. This is ifferent from the case shown in Eq. 12, where an infinite slope is require. Fig. 2, D F, show the inuce APD graient (Fig. 2 D), the vulnerable winow in the DT S1S2 -DT S2S3 space (Fig. 2 E), an the vulnerable winow w for ifferent APD an CV restitution slopes (Fig. 2 F). Compare to the case shown in Fig. 2, A C, a larger APD graient is inuce an thus the vulnerable winow for conuction block of the S3 wave is larger, but the effect of the slope of CV restitution is smaller (Fig. 2 F). We also numerically simulate a homogeneous 1D cable with the LR1 moel. We first stuy the case in which all extrasystoles occur at the same location. Uner control parameter settings, the maximum APD ifference an graient inuce by the S2 beat is not large enough to cause conuction block of the S3 beat. We faile to etect a vulnerable winow for the S3 beat for either normal or slowe recovery of the Na 1 channel. Base on the kinematic analysis of Eqs , conuction block of the S3 beat can theoretically occur as long as there is an APD graient, yet the vulnerable winow may be very small (e.g., it occurs in ;5 ms range of the S1S2 interval in the kinematic simulation in shown Fig. 2 B). However, the effects of the stimulus strength an uration an the electrotonic coupling may cause a small vulnerable winow to isappear. When premature extrasystoles are applie at a location far away from the S1 stimulus (Fig. 3), we were able to etect a large vulnerable winow for conuction block of the S3 beat. Fig. 3 A shows the APD heterogeneity inuce by the S2 beat for normal an slowe Na 1 channel recovery. Fig. 3 B shows the vulnerable winow for the S3 beat at ifferent S1S2 coupling interval for normal Na 1 channel recovery, which agrees well with that of the kinematic simulation (Fig. 2 E). Fig. 3 C shows the vulnerable winow versus the inuce refractory barrier (Da) for normal an slowe Na 1 channel recovery, showing that the vulnerable winow

5 Vulnerable Winow for Conuction Block 809 FIGURE 3 Inuction of ispersion of refractoriness an conuction block in homogeneous 1D cable of the LR1 moel. The S1 was applie at one en whereas S2 an S3 were applie at the other en. (A) APD graient inuce by S2 for normal Na 1 recovery with DT S1S2 ¼ 280 ms (soli line) an for fivefol-slowe Na 1 channel recovery with DT S1S2 ¼ 287 ms (ashe line). (B) Vulnerable winow (shae area) of conuction block of the S3 beat for ifferent S1S2 coupling intervals with normal Na 1 channel recovery. S2 failure inicates an S1S2 interval is too short for S2 to stimulate the S2 waves. (C) Vulnerable winow size w versus the inuce maximum APD ifference Da for normal Na 1 channel recovery (n) an slowe Na 1 channel recovery (h). Data were obtaine from ifferent S1S2 coupling intervals. is proportional to Da, but the effect of CV restitution is limite, similar to the kinematic simulation (Fig. 2 F). Heterogeneous tissue Here we show how the S2 beat affects ispersion of refractoriness an the vulnerable winow for uniirectional conuction block of the subsequent S3 beat in heterogeneous tissue. In this case, both baseline APD an APD restitution kinetics may vary from location to location (3), an whether the premature S2 enhances or reuces ispersion in refractoriness epens on its location relative to the S1 beat an the preexisting tissue heterogeneity. In Fig. 4 A, we show APD istribution in space for ifferent S1S2 coupling intervals for the case in which S1 an S2 are applie at the same en of the cable an the APD graient is ascening. As the S1S2 FIGURE 4 Moulation of ispersion of refractoriness an inuction of reentry in a heterogeneous 1D cable using the LR1 moel. (A) APD istribution in space of the S2 beat for ifferent S1S2 coupling intervals (from top to bottom, DT S1S2 ¼ 600 ms, 400 ms, 300 ms, an 250 ms) when S1, S2, an S3 were applie at the same en of the cable for an ascening APD graient generate using Eq. 2. (B) Vulnerable winow of conuction block of the S3 beat for ifferent S1S2 coupling interval for the case as in A. The shae area marke S2 wave block is the vulnerable winow of conuction block of the S2 wave. S2 failure inicates the S1S2 interval is too short for S2 to stimulate the S2 waves. (C) APD istribution in space of the S2 beat for ifferent S1S2 coupling interval (from top to bottom, DT S1S2 ¼ 600 ms, 400 ms, 300 ms, an 270 ms) when S1 was applie in one en an S2 an S3 were applie at the other en of the cable for a escening APD graient generate using Eq. 2 by exchanging the G Kmax an G Kmin values. (D) Vulnerable winow of conuction block of the S3 beat for ifferent S1S2 coupling interval for the case as in C. S2 failure inicates the S1S2 interval is too short for S2 to stimulate the S2 waves. (E) APD istribution in space of the S2 beat for ifferent S1S2 coupling intervals (from top to bottom, DT S1S2 ¼ 600 ms, 270 ms, an 250 ms) when S1 an S2 were applie at the same en of the cable for a escening APD graient as in C.(F) APD istribution in space of the S2 beat for ifferent S1S2 coupling interval (from top to bottom, DT S1S2 ¼ 600 ms, 340 ms, an 312 ms) when S1 was applie in one en an S2 was applie at the other en of the cable for an ascening APD graient as in A.

6 810 Qu et al. coupling interval ecreases, the APD graient eceases. This ecrease in APD graient causes the vulnerable winow of conuction block of the S3 beat to ecrease (Fig. 4 B). In Fig. 4 C, we show APD istribution in space for ifferent S1S2 coupling intervals for the case in which S1 an S2 are applie at the two ifferent ens of the cable an the APD graient is escening. The APD graient increases as the S1S2 coupling interval ecreases. As a consequence, the vulnerable winow of conuction block for the S3 beat increases as the S1S2 interval ecreases, reaching a maximum before S2 fails to inuce a propagating wave in the cable (Fig. 4 D). Conuction block of S2 wave occurs before S2 fails to inuce a propagating wave in the former case (Fig. 4 B), but not in the later case (Fig. 4 D) since the baseline APD graient is not large enough for S2 to be blocke. In Fig. 4 E, we show APD istribution in space for ifferent S1S2 coupling intervals, for the case in which S1 an S2 are applie at the same en of the cable, but the APD graient is escening. As the S1S2 coupling interval ecreases, the APD graient first eceases but then increases. In Fig. 4 F, we show APD istribution in space for ifferent S1S2 coupling intervals for the case in which S1 an S2 are applie at the two ifferent ens of the cable an the APD graient is ascening. The APD graient ecreases first but then increases as the S1S2 coupling interval ecreases. Effects of multiple extrasystoles in a homogeneous cable In general, the iastolic interval preceing the Sn 1 1 wave for multiple extrasystoles can be moele by the ifferential equation ½ n ðxþš x ¼ 1 u n11 ðxþ ÿ 1 Q n ðxþ (17) with the initial conition n ðlþ ¼DT SnSn11 ÿ a n ðlþ, in which DT SnSn11 is the coupling interval between Sn an Sn11 an l is the location at which the extrasystoles occur. Q n ðxþ ¼u n ðxþ=ð11u n ðxþa nx Þ an u n11 ðxþ ¼g½ n ðxþš. a nx ¼ f ½ nÿ1 ðxþš=x is the spatial graient in APD of the Sn beat, in which f[ n-1 (x)] is the APD restitution function. Using the APD restitution function (Eq. 7) an CV restitution function (Eq. 4), one can stuy numerically the DI an APD istribution an conuction block in a 1D cable for multiple extrasystoles. Multiple programme extrasystoles We first stuy the case of multiple extrasystoles whose intervals are pre-selecte. Fig. 5 A shows APD istributions in space inuce by multiple extrasystoles from the same location. The APD graient inuce by the S2 beat is ascening, which gave rise to a small vulnerable winow for conuction block of the S3 beat, as shown in Fig. 2 B. The APD graient inuce by the S3 beat after an almost minimum S2S3 interval (DT S2S3 ¼ 65 ms) is escening, which prevents conuction block of the S4 beat. The APD graient inuce by the S4 beat after an almost minimum S3S4 interval (DT S2S3 ¼ 125 ms) is ascening an much larger than that of the S2 beat. Due to this increase in graient, the vulnerable winow for conuction block of the S5 beat after the S4 beat is much larger (Fig. 5 B) than that of the S3 beat after the S2 beat (Fig. 2 B). The vulnerable winow of the S5 beat ecreases as S2S3 interval increases (inset in Fig. 5 B), an finally isappears. Fig. 6 shows APD istributions in space inuce by multiple extrasystoles in a 1D cable of the LR1 moel, which are very similar to those from the kinematics moel shown in Fig. 5 A. Although conuction block oes not occur for the S3 beat, since the APD graient inuce by S2 is not large enough, conuction block of the S5 beat occurs ue to the substantial increase in APD graient after the S4 beat. Slowing of the Na 1 channel recovery causes larger graients (compare Fig. 6, A an B). Multiple ranom extrasystoles We stuie multiple ranom extrasystoles to aress the issue of how vulnerability is affecte by multiple extrasystoles in homogeneous tissue in general. We first stuie the inuction of conuction block by multiple ranom extrasystoles in the kinematic moel (Eq. 17). The etaile simulation protocols are state in the legen of Fig. 7. Fig. 7 A FIGURE 5 Inuction of ispersion of refractoriness by multiple extrasystoles in homogeneous 1D cable from the kinematic moel (Eq. 3). (A) APD istribution in space for the S1, S2, S3, an S4 beats, which were applie at the same en of the cable. DT S1S2 ¼ 211 ms, DT S2S3 ¼ 65 ms, an DT S3S4 ¼ 125 ms. (B) The vulnerable winow w (shae area) for the S5 beat versus the S3S4 interval for DT S1S2 ¼ 211 ms an DT S2S3 ¼ 65 ms. Inset shows w for DT S1S2 ¼ 211 ms an DT S2S3 ¼ 200 ms.

7 Vulnerable Winow for Conuction Block 811 both normal an slowe Na 1 channel recovery, at least five extrasystoles are neee to cause conuction block in the cable an the vulnerability to conuction block correlates linearly with the number of ranom extrasystoles. The vulnerability is much higher in the case of slowe Na 1 channel recovery. These results agree well with the kinematic theory shown in Fig. 7 A. The reason that five extrasystoles are neee in this case is that the graient inuce by the S2 beat is too small for conuction block of the S3 beat, an the graient inuce by S3 beat is always escening, which prevents conuction block of the S4 beat (see Fig. 6). Only after S4 beat is the graient large enough for conuction block of the S5 beat. FIGURE 6 Inuction of ispersion of refractoriness by multiple extrasystoles in homogeneous 1D cable of the LR1 moel. Shown are APD istributions for the S1, S2, S3, an S4 beats, which were applie at the same en of the cable. (A) Normal Na 1 channel recovery. DT S1S2 ¼ 220 ms, DT S2S3 ¼ 100 ms, an DT S3S4 ¼ 150 ms. (B) Fivefol slowe Na 1 channel recovery. DT S1S2 ¼ 230 ms, DT S2S3 ¼ 110 ms, an DT S3S4 ¼ 170 ms. shows the percentage of the simulations in which conuction block occurs versus the number of extrasystoles. The total number of simulations for each ranom extrasystole sequence is The vulnerability linearly correlates with the number of ranom extrasystoles applie, an broaening the CV restitution curve substantially increases vulnerability. In both cases, the minimum number of stimuli is three extrasystoles. We then stuie how APD restitution slope affects vulnerability with the ranom extrasystoles protocol. Fig. 7 B shows the percentage of simulations in which conuction block occurs versus a, a parameter controls the slope of APD restitution in Eq. 7. The case of nine ranom extrasystoles is shown. The vulnerability increases with the slope of APD restitution, in a sigmoial manner. Note that the maximum slope of the APD restitution excees one when a becomes greater than 0.15, whereas the vulnerable winow begins to increase at a being aroun 0.25, showing the importance of the slope of APD restitution curve. We also stuie the inuction of conuction block by multiple ranom extrasystoles in a homogeneous 1D cable of cells using the LR1 moel. The extrasystoles all occurre at the same location (x ¼ 0) of the cable. Fig. 7 C shows the percentage of simulations in which conuction block occurs. The total simulations for each extrasystoles case are 500. For Spatially iscorant APD alternans an vulnerability in a homogeneous cable Spatially APD iscorant alternans can be inuce in a 1D homogeneous cable by rapi pacing at fixe pacing cycle length (PCL). Fig. 8, A an B, show the APD istributions for two alternating beats using control parameter settings, with normal or slowe Na 1 channel recovery, respectively. With normal Na 1 channel recovery, iscorant alternans occurs when 170 ms,pcl,190 ms, whereas with slowe Na 1 channel recovery, it occurs when 190 ms, PCL, 240 ms. Fig. 8 C shows the refractory barrier (Da) inuce by spatially iscorant alternans an the vulnerable winow (w) for conuction block by an aitional premature extrasystole for ifferent PCLs. The premature extrasystole was applie at the same location as the rapi pacing site after the 30th beat. Fig. 8 D plots w against Da for the case of slowe Na 1 channel recovery, showing that once a threshol graient is reache, w increases linearly with Da, as in all other conitions shown in this an the companion article. Note that in the same homogeneous cable, the ispersion of refractoriness inuce by a single premature extrasystole is not large enough to cause conuction block of a subsequent extrasystole. However, iscorant alternans inuce by rapi pacing resulte in a large vulnerable winow. The APD graients for the S1 beat shown in Fig. 3 were obtaine for the minimum S1S2 coupling intervals at which S2 propagates successfully; in other wors, they are the largest graients that can be inuce by a premature extrasystole in the homogeneous cable for the two cases. However, much larger APD graients can be inuce uring iscorant APD alternans ue to phase reversal of the iscorant alternans in space. Besies inucing large spatial APD graients, iscorant alternans creates aitional conitions favoring conuction block. In the results shown in Fig. 8, the vulnerable winow for slowe Na 1 channel recovery is larger than that for normal Na 1 channel recovery for the same refractory barrier (compare PCL ¼ 175 ms in normal Na 1 channel recovery with PCL ¼ 200 ms in slowe Na 1 channel recovery in Fig. 8 C). In aition, w is as large as Da in the case of

8 812 Qu et al. FIGURE 7 Vulnerability to conuction block cause by multiple ranom extrasystoles. (A) The percentage of the simulations that conuction block occurre versus the number of ranom extrasystole applie for normal (n) an fivefol-slowe (h) Na 1 channel recovery in a homogeneous 1D cable by numerically solving Eq. 17. The stimulus was given when n (0). 0 with 0 ¼ c 1 60j. j is a ranom number uniformly istribute between [0,1]. Conuction block is consiere to occur when n (x), c at any location x an for any beat number n. Eqs. 4 an 7 were use. The total number of simulations for each ata point is 2000 an the percentage is calculate as the portion of the 2000 simulations in which conuction block was observe. (B) The percentage of simulations in which conuction block occurs versus a (Eq. 7) by numerically solving Eq. 17. The number of ranom stimuli is nine. (C) The percentage of the simulations that conuction block occurre versus number of ranom extrasystoles applie for normal (n) an fivefol-slowe (h) Na 1 channel recovery in a homogeneous 1D cable of the LR1 moel. All extrasystoles were applie at the same en of the homogeneous cable. Control parameters were use. The extrasystole was given when n (0). 0 with 0 ¼ c 1 60 j. The percentage was calculate as the number of simulations in which conuction block occurs against the total simulations for each ranom extrasystoles case. The total number of simulations for each case is 500. slowe Na 1 channel recovery. This seems to be opposite to the result for a single extrasystole in which slowing the recovery of Na 1 channel tens to reuce the vulnerable winow (see companion article (1)). To reconcile this, we use Eq. 3 to calculate w of the premature S2 extrasystole uner two conitions: 1), the wavefront velocity of the S1 beat is constant u 0, an 2), the wavefront velocity of the S1 wave varies as in iscorant alternans, following the CV restitution relation. In both cases, the same spatial APD istribution was use. Fig. 9 shows the results an simulation etails. For normal Na 1 channel recovery, w ¼ ms if the S1 wavefront velocity is constant but w ¼ 112 ms if the S1 wavefront velocity varies as in iscorant alternans, similar to the simulation in the LR1 moel. For slowe Na 1 channel recovery, w ¼ 110 ms if the S1 wavefront velocity is constant but w ¼ 128 ms if the S1 wavefront velocity varies as in iscorant alternans, similar to the simulation in the LR1 moel. The reason is that the wavefront velocity uring iscorant alternans is slower, so that the waveback velocity Q 1 (x) is also slower. The slowing of the waveback velocity FIGURE 8 Inuction of ispersion of refractoriness an conuction block ue to spatially iscorant alternans in homogenous 1D cable. Control parameters were use except for the j gate of the Na 1 channel. (A) APD istribution in space for two consecutive beats for PCL ¼ 175 ms in the case of normal Na 1 channel recovery. (B) APD istribution in space for two consecutive beats for PCL ¼ 200 ms in the case of fivefol-slowe Na 1 channel recovery. (C) The inuce refractory barrier Da (soli symbols) by iscorant APD alternans an vulnerable winow w (open symbols) for the S2 wave versus PCL for normal (triangles) an slowe (squares) Na 1 channel recovery. Da was efine as the APD ifference between a minimum an its neighboring maximum in one beat (the beat marke by the open circles was use). (D) w versus Da for the case of slowe Na 1 channel recovery. The vulnerability to conuction block was examine by a S2 stimulus after a 30 beats APD alternans. The APD istribution in A an B was for 29th beat () an 30th beat (s).

9 Vulnerable Winow for Conuction Block 813 FIGURE 9 Vulnerable winows for ifferent S1 wavefront velocities, but the same spatial APD graient, with normal an fivefol-slowe Na 1 channel recovery. (Soli bar) Obtaine by solving Eq. 3 with constant S1 wavefront velocity, i.e., u 1 ðxþ ¼u 0.(Shae bar) Obtaine by solving Eq. 3 with S1 wavefront velocity varies as in the iscorant alternans, i.e., u 1 ðxþ ¼u 0 ð1 ÿ e ð1ðxþÿcþ=t Þ.(Open bar) From LR1 moel in homogeneous 1D cable. Normal recovery: the spatial APD istribution of the S1 beat is a 1 ðxþ ¼167 ÿ 123=ð1 1 e ðxÿ11:5þ=3 Þ, which was fit from the 30th beat (for x ¼ 0 24 mm) of the iscorant alternans shown in Fig. 8 A. 1 ðxþ ¼PCL ÿ a 0 ðxþ in which a 0 ðxþ ¼ =ð11e ðxÿ13:5þ=3 Þ was fit from the 29th beat (for x ¼ 0 24 mm) in Fig. 8 A. PCL ¼ 175 ms. Fivefol slowe recovery: the spatial APD istribution of the S1 beat is a 1 ðxþ ¼176 ÿ 128=ð11e ðxÿ10:5þ=2 Þ, which was fit from the 30th beat (for x ¼ 0 22 mm) of the iscorant alternans shown in Fig. 8 B. 1 ðxþ ¼PCL ÿ a 0 ðxþ in which a 0 ðxþ ¼ =ð11e ðxÿ11:5þ=2 Þ was fit from the 29th beat (for x ¼ 0 22 mm) in Fig. 8 B. PCL ¼ 200 ms. Note that uring iscorant alternans, the cycle length CL(x) is not constant in space but varies, however, in a magnitue much smaller than APD; therefore, it is reasonable for us to calculate 1 (x) using PCL instea of CL(x). of the S1 wave increases the likelihoo of conuction block of the S2 wave. Therefore, iscorant alternans creates a substrate favoring conuction block by inucing a steeper an larger refractory barrier an by causing the slowing of the S1 wavefront velocity. DISCUSSION In this stuy, we use kinematic analysis an numerical simulations to stuy vulnerability to uniirectional conuction block cause by multiple extrasystoles. Our major finings are: 1. In homogeneous tissue, a large ispersion of refractoriness, which requires steep APD restitution an broa CV restitution, can be inuce by premature extrasystoles, thereby creating a substrate for conuction block of subsequent extrasystoles. 2. In heterogeneous tissue, however, a premature extrasystole can either reuce or enhance the ispersion of refractoriness epening on its location an propagation irection with respect to the S1 beat, which in turn may reuce or enhance the vulnerability to conuction block of a subsequent extrasystole. For instance, in the real heart, the sinus beat propagates from enocarium to epicarium so that an extrasystole arising from the enocarium will reuce the inherent transmural ispersion of refractoriness, whereas an epicarial extrasystole will enhance the inherent ispersion. 3. The vulnerable winow increases linearly with the number of ranom extrasystoles arising from the same spatial location in homogeneous tissue. 4. Spatially iscorant alternans, which is promote by steep APD restitution an broa CV restitution, creates a particularly volatile substrate for conuction block. The equivalent ispersion of refractoriness in homogeneous tissue create ynamically by spatially iscorant alternans is more likely to cause conuction block than a comparable egree of preexisting ispersion in heterogeneous tissue. This is ue to the ynamic variation in CV uring spatially iscorant alternans, which causes slowing of the S1 wavefront velocity, promoting collision with the trailing wavefronts. In general, steep APD restitution an broa CV restitution promote ispersion of refractoriness in response to extrasystoles an thus increase vulnerability to conuction block. Implications for inuction of reentry in cariac tissue The ifferences that we observe between homogeneous an heterogeneous tissue are interesting. In homogeneous tissue or milly heterogeneous tissue, reentry can occur if a critical tissue area is epolarize in the refractory phase of a previous excitation (23,24), either by cross-fiel stimulation of a large area with barely suprathreshol current strength, or by a point electroe with the very high current strength to epolarize a sufficiently large area (25 28). Uner physiological conitions, however, a spontaneous barely suprathreshol single extrasystole meets neither of these criteria. Base on our analysis, a premature extrasystole after an S1 beat (e.g., the sinus rhythm) generates two propagating waves (Fig. 1 A) in opposite irections, which inuce ifferent egrees of ispersion of refractoriness (Fig. 2). This allows a critically time secon extrasystole (S3) from the same location to block in one irection, but not in the other irection. Thus, in homogeneous two-imensional or threeimensional tissue, inuction of reentry requires at least two extrasystoles, in which the first generates an asymmetrical istribution of refractoriness resulting in uniirectional conuction block of the secon extrasystole. Preexisting tissue heterogeneity is not a requirement for initiation of reentry for a point stimulus, as long as multiple extrasystoles with proper timing an electrical restitution properties are present. This is also emonstrate by simulations using multiple ranom extrasystoles (Fig. 7). Extrasystoles arising from the same location ue to automaticity or triggere activity (29 32) are common in real hearts. In heterogeneous tissue, on the other han, the first extrasystole may either enhance or reuce the ispersion of

10 814 Qu et al. refractoriness, epening on its coupling interval an location relative to the previous beat. Our analysis shows that when the S2 extrasystole occurs in the short APD region an propagates in the same irection as the S1 beat, it attenuates the ispersion an thus reuces the vulnerable winow for a subsequent S3 beat. If it propagates in the opposite irection to the S1 beat, it increases the ispersion an thus the vulnerable winow for the subsequent S3 beat. When the S2 extrasystole occurs in the long APD region, it first attenuates an then increases the ispersion of refractoriness as the coupling interval ecreases, irrespective of its irection of propagation. Since sinus beats conuct from enocarium to epicarium, this suggests that an extrasystole arising from the enocarium will ecrease transmural ispersion, whereas an extrasystole arising from the epicarium will increase the transmural ispersion. Thus the subsequent S3 extrasystole is more likely to inuce reentry if it arises from the epicarium than enocarium. This is opposite to the case of a single extrasystole in heterogeneous tissue analyze in the companion article. In this case, the graient of refractoriness require to cause conuction block was less for a single extrasystole arising from the enocarium than the epicarium. Note that conuction block may not occur with an extrasystole arising from the mimyocarium. However, this extrasystole can attenuate ispersion at long coupling intervals but increase ispersion at shorter coupling intervals, which may create a substrate for conuction block for a subsequent extrasystole from the same location. The importance of electrical restitution on the initiation of cariac arrhythmias The slope of APD restitution has been shown to be an important control parameter in the genesis of alternans (33) an stability of reentry (34 36). Specifically, when the slope is steep (.1), APD alternans often occurs uring rapi pacing (33,37) or reentry aroun an obstacle (34,38), an spiral wave breakup occurs in tissue (35,36). APD an CV restitution are also critical parameters that regulate the vulnerability to uniirectional conuction block an thus the initiation of arrhythmias. First, steep APD restitution an CV restitution generate spatially iscorant alternans (12,13), which creates a substrate that is very vulnerable to conuction block. Secon, steep APD restitution an CV restitution are also require for a single or multiple premature extrasystoles to cause a large ispersion of refractoriness, sufficient to cause conuction block. Experimental stuies (39 42) have emonstrate the importance of APD restitution slope on the stability of reentry, an it will be informative an clinically important to investigate experimentally the importance of APD an CV restitution on the vulnerability to reentrant arrhythmias in real cariac tissue. It will also be interesting to stuy how APD an CV restitution interact with anatomical structures to moulate ispersion of refractoriness, given that both experiments (43) an computer simulations (44) show that obstacles enhance the ispersion of refractoriness ue to ionic heterogeneities. Limitations In aition to the limitations outline in the companion article (1), the kinematic equation (which oes not inclue electrotonic coupling) becomes quantitatively inaccurate in comparison with the ionic moel (which has electrotonic coupling) for the case of multiple stimulations, since electrotonic coupling has important effects on the ynamics of cariac conuction an ispersion of refractoriness (45 48). Nevertheless, the kinematic moel is still qualitatively correct. This work was supporte by National Institutes of Health/National Heart, Lung, an Bloo Institute grants P50 HL53219 an P01 HL078931, an the Laubisch an Kawata enowments. REFERENCES 1. Qu, Z., A. Garfinkel, an J. N. Weiss Vulnerable winow for conuction block in a one-imensional cable of cariac cells: 1. Single extrasystoles. Biophys. J. 91: Laurita, K. R., S. D. Girouar, F. G. Akar, an D. S. Rosenbaum Moulate ispersion explains changes in arrhythmia vulnerability uring premature stimulation of the heart. 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