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1Department of Biomedical Engineering, Tulane University, New Orleans, LA and 2Medtronic, Inc., Minneapolis, MN ... cal Engineering, The Johns Hopkins University School of Medicine, .... each current can be calculated by Ohm's law. Ii,j.
Annals of Biomedical Engineering, Vol. 29, pp. 35–46, 2001 Printed in the USA. All rights reserved.

0090-6964/2001/29共1兲/35/12/$15.00 Copyright © 2001 Biomedical Engineering Society

Wave Front–Obstacle Interactions in Cardiac Tissue: A Computational Study EZANA M. AZENE,1 NATALIA A. TRAYANOVA,1 and EDDY WARMAN2 1

Department of Biomedical Engineering, Tulane University, New Orleans, LA and 2Medtronic, Inc., Minneapolis, MN (Received 2 June 2000; accepted 19 October 2000)

observations have been made by other researchers.7,12,19 These studies underscore the importance of understanding the interactions between cardiac wave fronts and natural anatomic obstacles such as orifices. A better knowledge of the mechanisms involved in reentry around natural obstacles could aide in the design of new therapies for tachyarrhythmias. A popular therapy for atrial tachyarrhythmias is transthoracic catheter ablation. Over the past 15 yr, the evolution of catheter ablation as a treatment for supraventricular arrhythmias has proceeded at a rapid pace.15,29 Although the tissue properties within ablation lesions have yet to be determined, it is known that the damaged tissue looses its active properties.15 An understanding of the interaction between reentrant waves and obstacles created by ablation will greatly facilitate the enhancement of current therapies involving atrial ablation. Examination of wave front–obstacle interactions has been the focus of both animal4,6,8 and computational models.4,11,16–18,25–27,30,31 The initial work in this field was done by Balakhovsky3 and Krinsky.11 Both studies showed that a temporary obstacle of extended refractoriness was enough to cause rupture of an incident wave, leading to spiral wave development. Pertsov et al. conducted numerical studies of wave front collision with an obstacle in an otherwise homogeneous sheet of tissue.18 The study found that a critical excitability existed below which wave fronts would detach from obstacles after collision. Similar results were obtained by Starobin et al.25,26 in computational models and by Cabo et al.4 in both computational and animal studies. The work of Girouard et al.6 provided experimental evidence of increased loading during wave front pivoting around a fixed obstacle in the guinea pig ventricle. Further, recent research by Street and Plonsey27 focused on transmission of propagation through a region of connective tissue represented as a conductive barrier and demonstrated that the success of transmission is dependent on fiber orientation and direction of propagation.

Abstract—An understanding of wave front–obstacle interactions will greatly enhance our knowledge of the mechanisms involved in cardiac arrhythmias and their therapy. The goal of this computational study is to examine the interactions between wave fronts and various obstacles in a two-dimensional sheet of myocardium. The myocardium is modeled as an isotropic sheet with Luo–Rudy I membrane kinetics. An examination is conducted of wave front interactions with nonconductive and passive-tissue obstacles. Simulations were performed either in environments of reduced myocardial excitability, or with rapid stimulation via a line electrode. The shape of the obstacles and their ability to withdraw current from the active tissue greatly influence wave front–obstacle interactions in each of these environments. The likelihood of wave front detachment from an obstacle corner increases as the curvature of the obstacle corner is increased. A passive-tissue obstacle promotes wave front– obstacle separation in regions of depressed excitability. Under rapid pacing, the presence of the passive obstacle results in wave fragmentation, while the insulator obstacle promotes wave front detachment. The results of this study reveal the importance of obstacle composition and geometry in wave front interactions with cardiac obstacles. © 2001 Biomedical Engineering Society. 关DOI: 10.1114/1.1332083兴 Keywords—Cardiac tissue, Simulation, Obstacles, Wave front propagation, Depressed excitability, Rapid stimulation.

INTRODUCTION Wave front–obstacle interactions have important implications for arrhythmogenesis in cardiac tissue. For instance, there is a great deal of evidence that reentry around natural obstacles in the atria, like the vena cavae and pulmonary veins, may cause atrial fibrillation secondary to atrial flutter.2,9 The concept of reentry around natural obstacles was first presented by Lewis et al.13 who observed a long pathway of excitation extending from the right atrium to the left atrium which they concluded was a reentry around the vena cavae. Similar The current address of Ezana M. Azene is: Department of Biomedical Engineering, The Johns Hopkins University School of Medicine, 720 Rutland Ave., Baltimore, MD 21205. Address corrrespondence to Natalia Trayanova, Department of Biomedical Engineering, 500 Lindy Boggs Center, Tulane University, New Orleans, LA 70118. Electronic mail: [email protected]

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Cabo et al. examined the effect of partial Na-channel blockade on wave front–obstacle interactions.4 They elicited elliptical waves beneath a nonconductive filament and identified excitabilities at which the propagating waves detached. Using a ‘‘critical ratio’’ metric, Cabo et al.4 quantified their observations of wave front detachment. Starobin et al. used a similar metric, termed ‘‘charge balance,’’ to help explain wave break in two dimensions.25 In this study, we conduct a further examination of wave front–obstacle interactions in two spatial dimensions. We consider obstacles of two different compositions. Traditionally, obstacles to propagation are modeled as perfect insulators satisfying no-flux boundary conditions.4,17,25,26 Similarly, the first type of obstacle we model here is an insulator. No-flux obstacles have a physiological basis since natural orifices in the atria, for instance, are surrounded by nonconductive connective tissue. However, modeling lesions created via ablation as fully nonconductive is not physiological. These obstacles are able to conduct current and, thus, might interact differently with wave fronts. Keeping that in mind, the other obstacle composition we model, the passive myocardium, allows for current flow between the active tissue and the obstacle. Modeling cardiac obstacles with sharp edges, as is the case in most previous studies,4,16,17,25,30,31 may not always be appropriate since obstacles often have complex geometries. In this study, we model obstacles with both sharp and rounded corners, and examine wave front– obstacle interactions in two different situations. The first situation is an environment of reduced tissue excitability, as created in vivo by class IC antiarrhythmic drugs. The second situation involves rapid pacing of tissue beneath the obstacles via a line electrode. To quantify our observations in the studies involving depressed excitability, we present a method to calculate safety factors in two dimensions that allows us to determine the contribution of individual cells to wave front behavior.

FIGURE 1. Action potential duration „APD… as a function of maximum LRI time-dependent K¿ conductance „ g K,max…. A power function „APDÄ193.65„ g K,max… À0.5621… was fit to the computed data points „triangles… using the method of least squares. The square of the correlation coefficient is 0.992, indicating a good fit to the data points.

where ␴ is the isotropic myocardial conductivity 共1.875 mS/cm兲, C m is the membrane capacitance 共1 ␮ F/cm2 ), and ␤ is the surface to volume ratio of the membrane 共3000 cm⫺1 ). The term I stim ( ␮ A/cm2 ) is the stimulus current density delivered via transmembrane current injection and I ion ( ␮ A/cm2 ) is the Luo–Rudy I 共LRI兲14 ionic current density. We numerically determined the relationship between the maximum LRI time-dependent K⫹ conductance (g K,max) and the LRI action potential duration 共APD兲 共Fig. 1兲, and fitted an analytical function through the resulting data. We used this function to reduce the LRI APD to 112 ms, which corresponds to g K,max⫽3.11 ms/cm2 . The reduction in APD provides a more realistic shape of the action potential 共AP兲 which can be considered as: 共a兲 a typical atrial AP,2 or 共b兲 an AP of tissue in fibrillation. To simulate the presence of an insulator around the tissue domain, we implemented no-flux boundary conditions

MODEL AND METHODS nˆ • 共 ␴ ⵜV m 兲 ⫽0, on tissue boundary,

Tissue Model In our study, we modeled the myocardium as a twodimensional 共2D兲, square 共4 cm⫻ 4 cm兲 isotropic monodomain. In a monodomain representation, cardiac tissue is considered as a continuum instead of as a discrete structure.21,28 As such, the transmembrane potential V m 共mV兲 is defined everywhere in the tissue domain ⍀⫽ 关 0,a 兴 ⫻ 关 0,b 兴 , where a⫽b⫽4 cm. The monodomain equation used to model the tissue is as follows:



ⵜ• 共 ␴ ⵜV m 兲 ⫹ ␤ C m



⳵Vm ⫹I ion⫺I stim ⫽0, ⳵t

共1兲

共2兲

where nˆ is the tissue outward unit normal vector. Obstacles and Stimulation To examine the effect obstacle composition has on wave front–obstacle interactions, we introduced two types of obstacles into the tissue. The first was an insulator; the boundary conditions are those of no current flux across the obstacle border 关Eq. 共2兲兴. The second obstacle consisted of passive tissue, simulated by replacing the active current I ion in Eq. 共1兲 with a leakage

WAVE FRONT –OBSTACLE INTERACTIONS

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cur via other mechanisms. One example is the interaction of wave fronts with refractory tissue during rapid excitation, as occurs during tachyarrhythmia. To simulate this phenomenon, we rapidly paced sheets with obstacles of radius r⫽0. We elicited consecutive waves at pacing intervals between 30 and 70 ms. As each wave approached the obstacle corner, we determined whether it remained attached or detached. Safety Factor Calculation

FIGURE 2. Schematic of myocardial sheet and obstacles used in reduced excitability and rapid excitation simulations. The cross hatched region represents an obstacle with radius of curvature 0 „i.e., r Ä0…. For r Ì0, r is measured vertically from point p . All stimuli were elicited from the line electrode „electrode width not to scale….

current I L ⫽V m /R m , where R m ⫽9100 ⍀⫻cm2 . 1 No boundary conditions were required in this case. Each obstacle was rectangular in shape with its upper left corner coincident with the upper left corner of the sheet. The lower right corner of each obstacle was a quarter circle with radius r varying from 0 to 9 mm 共Fig. 2兲. For r⫽0 共cross-hatched region兲, the obstacle was a true rectangle with its lower right corner at a point 4/3 cm above and 4/3 cm to the right of the lower left corner of the sheet, as indicated by the letter p in Fig. 2. For r⬎0 共entire shaded region兲, the center of the quarter circle was found by traveling vertically a distance r from point p. A reduction of tissue excitability, as caused by class IC antiarrhythmic drugs,10 can promote wave front detachment at obstacle corners.4,25 To simulate reduced tissue excitability, we decreased the LRI maximum sodium conductance (g Na) throughout the active tissue. For each obstacle, g Na was varied from a maximum of 100%g Na to a minimum of 20%g Na . In each simulation, a wave front was elicited by current injection of 250 ␮A/cm2 into a 0.08 cm wide strip of tissue on the left edge of the sheet extending from the base of the obstacle to the base of the sheet 共Fig. 2兲. We then visually inspected the wave front to determine if it remained attached or detached as it traversed the obstacle corner. For obstacles with r⫽0, we also computed safety factors near the obstacle corner to quantify wave behavior in that region 共see ‘‘Safety Factor Calculation’’兲. In addition to separation secondary to depressed excitability, wave front detachment from obstacles can oc-

To quantify wave behavior at obstacle corners, we calculated a 2D safety factor 共SF兲. We extended the 1D cable formulation of SF, first presented by Shaw and Rudy,20 into a 2D formulation. The SF represents the ratio of the amount of charge generated by the cell to excite its downstream neighbors to the least amount of charge required to excite the cell itself.20 A SF⬎1 means that more charge was generated by the cell during excitation than the charge required for excitation. The fraction of SF above 1 represents the actual margin of safety. If SF⬍1, there is not enough charge delivered by the cell to excite its downstream neighbors, so conduction fails. The equation used to calculate SF is20 Q c ⫹Q out 兰 A I c •dt⫹ 兰 A I out•dt ⫽ , Q in 兰 A I in•dt

共3兲

where Q c , the capacitive charge generated by the cell in question, is equal to the time integral of the capacitive current over the integration interval, A. Q out and Q in are the charges generated by current exiting the cell and entering the cell, respectively. The integration interval A spans the duration of the AP upstroke and A ⫽ 兵 t 兩 ( ⳵ V m / ⳵ t)⬎0 其 . However, Shaw and Rudy20 define the integration interval as A⫽ 兵 t 兩 Q m ⫽ 兰 I m •dt⬎0 其 . We were not able to use this definition for A because Q m , the transmembrane charge, was always greater than 0 for cases of highly depressed excitability. Our definition of A is acceptable because it allows us to isolate the AP upstroke, the critical period that determines whether or not a cell will excite its neighbors. We used the computed values of SF, Q c , Q in , and Q out to quantify the behavior of wave fronts as they traversed the obstacle corner. Our SF formulation allows for current flow in two dimensions. Thus, I in and I out , the axial currents resulting from Q out and Q in , can be composed of anywhere from one to four separate currents resulting from charge transfer between adjacent cells 关Fig. 3共b兲兴. Figure 3共a兲 presents the location of several grid points 共in the model they are at a distance of 200 ␮m each兲 in relationship to the corner of the obstacle 共tissue and obstacle not to scale兲. We found that, regardless of whether or not a wave detached, it would always excite

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discretization 共0.02 cm兲 between adjacent cells. Numerical Methods

FIGURE 3. „a…: Corner obstacle of radius r Ä0 in myocardial sheet. The figure shows part of the grid, at a spacing of 0.2 mm, around the obstacle corner „tissue size and obstacle not to scale…. All recordings were made at cell 0 „see the text for explanation…. „b… Resistive network around cell 0 used to compute currents for safety factor calculations.

the cell immediately beneath cell 4 in Fig. 3共a兲. Thus, in order to use SF as a metric to differentiate between attached and detached wave fronts, we made all calculations at cell 0, directly above cell 4. Figure 3共b兲 is a representation of the network of resistors associated with cell 0. The five cells at the ends of the resistors correspond to the cells indicated in Fig. 3共a兲. We defined all currents along the resistors as inward currents. Thus, each current can be calculated by Ohm’s law i 0 ⫺V m I i, j ⫽ 共 V m 兲 •G i ,

共4兲 i 0 (V m ⫺V m ),

where i⫽1,2,3,4 and j is the sign of and i 0 and V m are transmembrane potentials at cell 0 where V m and cell i, respectively. G i is the axial conductance between cell i and cell 0. Thus, the total inward and outward currents are defined as the following sums: I in⫽

兺i I i,⫹ ,

共5兲

兺i 共 I i,⫺ 兲 .

共6兲

I out⫽⫺

Because we used an isotropic mesh, we defined the conductances as G i ⫽( ␴␲ a 2 )/L, where ␴ is the conductivity, a⫽2/␤ is the fiber radius, and L is the spatial

For the majority of the simulations, particularly these for which r⫽0, we discretized the domain ⍀ into a 200⫻200 mesh of nodes where each node represented an individual myocyte with LRI membrane kinetics.14 Since a⫽b⫽4 cm, this created a spatial discretization of ⌬x ⫽⌬y⫽0.02 cm. Smaller spatial discretization steps were used to test the convergence of the solution, and also to examine behavior around obstacles with curved corners of small radii. Discretization steps as small as 20 ␮m were used for obstacles with the smallest radii of corner curvature. To estimate the second derivative terms in Eq. 共1兲, we used a second order accurate finite difference scheme. To maintain second order accuracy, we solved the first derivative terms in the boundary conditions with a second order three-point formula.21 To integrate the temporal derivative of V m in Eq. 共1兲, we used a predictor–corrector scheme. The numerical method is described in detail in a previous publication.22 In brief, the scheme was composed of two substeps. The first step used an explicit second order Adams–Bashford formulation and the second step used the implicit third order Adams–Moulton equation. RESULTS Wave Front–Obstacle Interactions in Tissue with Reduced Excitability To examine wave front behavior at obstacle corners during situations of depressed excitability, we varied the maximum Na conductance (%g Na). Additionally, we studied the influence of the corner radius on wave front– obstacle interactions. Figure 4 is a set of plots that describes the relationship between excitability (%g Na), corner radius, and wave behavior. We implemented a least-squares fit to the data gathered to generate the demarcations between the different regions in the plots. In general, for a given excitability, as the radius of the corner was increased, the likelihood of wave front detachment at the corner decreased. This behavior was most profound in the case of the insulator obstacle 关Fig. 4共b兲兴 and much less so for the passive tissue 关Fig. 4共a兲兴 obstacle. In fact, with the passive tissue obstacle, for a

FIGURE 4. Plots of the relationship between excitability „% g Na… and radius of curvature r for: „a… passive and „b… insulator obstacles. The combinations of % g Na and r leading to attachment, detachment, and failed propagation are indicated.

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FIGURE 5. „a…–„c… Wave front detachment without occurrence of reentry before collision with tissue border.

given excitability, a very large increase in corner radius was necessary to switch from detachment to attachment. From Fig. 4 it is evident that, for a given corner radius greater than 0, propagating waves remained attached to the insulator obstacle at lower excitabilities than for the passive obstacle. Figure 5 provides an example of wave front detachment for r⫽0 at 44%g Na in the insulator obstacle case. For the same %g Na , the wave front remained attached to the passive obstacle. When detachment occurred from the insulator obstacle, the wave front made a slight turn at the obstacle corner as if it were going to remain attached 关Fig. 5共a兲兴. However, after making this attempt, the wave front quickly separated from the obstacle 关Fig. 5共b兲兴. In most cases, the wave did not reenter before colliding with the tissue border 关Fig. 5共c兲兴. This figure is consistent with the dynamics of wave front detachment from the edge of a barrier as reported by Cabo et al.4

Figure 6共a兲 compares the SFs calculated at cell 0 for the two obstacles (r⫽0). A SF less than 1 suggests that a wave front cannot supply enough current to overcome the load it experiences as it rounds an obstacle corner. As a result, one would expect detachment to occur for excitabilities with SF⬍1. The insulator and passive obstacles had similar SFs for all %g Na in the range from 50 to 100. In this interval, the largest difference between the two, 1.4%, occurred at 100%g Na . The inset in Fig. 6共a兲 emphasizes the difference in SFs in the interval 40% to 50%g Na . This difference in SFs clearly indicates that in the narrow range of %g Na around 44 the wave front will remain attached to the passive obstacle 共SF⬎1兲, while it will detach from the insulator obstacle 共SF⬍1兲. For %g Na⬎42, the wave front detaches from both obstacles. This is consistent with our observations as shown in Fig. 5. Figures 6共b兲, 6共c兲, and 6共d兲 compare Q in ,Q c and Q out , respectively, for the two types of obstacles we examined.

FIGURE 6. „a… Computed safety factors „SFs… at node 0 of each obstacle „ r Ä0… as a function of excitability „% g Na…. The inset shows the safety factors in the interval 40%–50%g Na . „b…–„d… Plots of Q in , Q c , and Q out , respectively, for each obstacle.

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Propagation of the planar wave failed at 26%g Na and 27%g Na for the insulator and passive cases, respectively. These critical excitabilities are depicted in Figs. 4共a兲– 4共b兲 by horizontal lines. The behavior of a failing wave took on one of two forms. The first type of behavior, type I, was characterized by a wave that never detached from the bottom of the obstacle, but failed to propagate after traveling a short distance. This represents the wellknown decremental propagation at low excitability. The insulator obstacle was associated with this type of propagation failure. Type II failure was characterized by a wave that remained attached to the bottom of the obstacle only briefly before detaching. After detaching, the weakened wave front propagated down and to the right before dying at the bottom border of the tissue. We observed this behavior after wave detachment from the passive tissue obstacle. Wave Front–Obstacle Interactions Under Rapid Excitation To examine wave front–obstacle interactions in refractory tissue, we rapidly paced tissue beneath obstacles of radius r⫽0. However, in order to explain behaviors we observed as a result of rapid excitation, we first examined the properties of waves near the obstacles created by a single pacing stimulus. Figure 7共b兲 is a plot of APD versus the distance from the base of each obstacle to nodes a–e in Fig. 7共a兲 for a wave created by a single pacing stimulus. 关To generate a smooth curve, we also determined APDs at two nodes between nodes a and b and three nodes between nodes d and e. These are shown in Fig. 7共b兲.兴 Figures 8共a兲 and 8共b兲 are plots of the actual APs at nodes a–e. Note that the APDs beneath the passive tissue obstacle first decrease then increase as one moves down and away from the obstacle. We also calculated the maximum upstroke velocities beneath each obstacle 共at node a). We found that beneath the insulator obstacle, the maximum upstroke velocity 关 (dV m /dt) max兴 was 47.13 mV/ms, much larger than 36.66 mV/ms for the passive tissue obstacle. The differences in plateau magnitude, APD, and (dV m /dt) max are most likely a product of the varying abilities of the two obstacles to withdraw current from the active tissue. The low maximum upstroke velocity at the wave front pivot point is consistent with the findings of Girouard and co-workers.6 During rapid stimulation, there was a marked difference in the appearance of the first propagating wave beneath the insulator obstacle as compared to the passive tissue obstacle. In particular, the first wave front beneath the passive obstacle had a visible dispersion of refractoriness 关Fig. 9共a兲兴 while the wave below the insulator obstacle was perfectly planar 关Fig. 9共d兲兴. We can see from both Figs. 7 and 8 that the APD beneath the insu-

FIGURE 7. „a…: Schematic of the locations beneath the obstacle at which APDs were calculated. APDs were also determined at two nodes between a and b and three nodes between d and e . These five nodes are not shown in the schematic. „b… Plots of APD vs distance from bottom of obstacle to nodes a – e in „a….

lator was constant 共112 ms兲, whereas the APD for the passive tissue obstacle was less than 112 ms as far away from the obstacle as 6.2 mm. During pacing at 60 ms intervals, the dispersion of APD near the passive obstacle resulted in the formation of a wave fragment after the third pacing stimulus 关Figs. 9共b兲–9共c兲兴. However, wave fragmentation did not occur at 55 or 63 ms coupling intervals. Because the APD of each node beneath the insulator was approximately identical, no wave fragment formed beneath that obstacle 关Figs. 9共d兲–9共f兲兴. The existence of a dispersion of refractoriness immediately below the passive obstacle had an important and interesting consequence when pacing at 60 ms. Figure 10 illustrates this. The wave fragment described above, and labeled 1 in Fig. 10共a兲, propagated with its top end attached to the passive obstacle while its bottom end ‘‘dangled’’ in the active tissue. 关Note that Figs. 10共a兲 and 9共c兲 represent the same moment in time and differ only in their scale gradients.兴 Since wave fragment 1 was not attached to the tissue boundary at its lower end,

WAVE FRONT –OBSTACLE INTERACTIONS

FIGURE 8. „a…–„b… Plots of action potentials recorded at nodes a – e .

it quickly reentered after rounding the obstacle corner 关Figs. 10共b兲–10共c兲兴. As wave 1 continued to reenter, it collided with wave 2, the next wave to reach the obstacle corner 关Figs. 10共c兲–10共d兲兴. In response to this collision wave 2 fragmented and lost its anchor to the tissue boundary 关Fig. 10共d兲兴. As a result, wave 2 collided with the right border of the tissue and reentered due to current reflection 关Figs. 10共e兲–10共f兲兴. After reflecting from the right border, wave 2 disrupted wave 3 关Figs. 10共f兲– 10共g兲兴, and the sequence of fragmentation and reentry continued 关Figs. 10共h兲–10共i兲兴. Even at coupling intervals as low as 30 ms, wave fronts did not detach from the passive tissue obstacle. However, we did observe wave front detachment from

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the insulator obstacle for coupling intervals between 30 and 55 ms. Figure 11 illustrates the behavior at a pacing interval of 50 ms. Note in Figs. 11共g兲–11共h兲 that wave front number five detaches from the insulator obstacle and subsequently reenters. In contrast, wave front number five does not detach from the passive obstacle 关Figs. 11共c兲–11共d兲兴. Figure 12 depicts recordings from cell 0 for each obstacle. From Fig. 12共a兲, it is evident that no waves separated from the passive tissue obstacle since there is an uninterrupted train of APs, each separated by approximately 50 ms. However, in the insulator case, there is an alternating pattern of attachment 共successful AP兲 and detachment 共small electrotonic response兲. There are slight irregularities in the magnitude and timing of the APs in Figs. 12共a兲–12共b兲. In the passive case, the gradual decrease in AP magnitude with time probably occurs because it took a few excitations for the cell to fully adapt to the rapid stimulation. In the insulator case, action potential irregularities are due primarily to the reentry of detached waves 关Fig. 11共h兲兴. These detached wave fronts could reenter and collide with the obstacle corner at the same time as a new wave front generated by the pacing electrode. The results were APs with exaggerated magnitudes, indicated by the symbol ** in Fig. 12共b兲. Also, a reentering wave front could reattach at the obstacle corner before the arrival of the next wave, creating a premature AP. This early AP then caused the next generated wave to detach. Early APs generated by detached wave fronts are indicated by the symbol * in Fig. 12共b兲. DISCUSSION Wave Front–Obstacle Interactions in Tissue with Reduced Excitability Starmer et al.23,24 have shown that the Na-channel blockade depresses tissue excitability and increases the

FIGURE 9. Transmembrane potential distributions in the active tissue during rapid pacing at 60 ms intervals from a vertical line electrode beneath each obstacle. „a…– „c… illustrate the development of a wave fragment beneath the passive obstacle. No wave fragment forms beneath the insulator obstacle „d…–„f…. Note: The scale is not continuous but discrete to better illustrate the dispersion of APD.

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FIGURE 10. Transmembrane potential distributions in the active tissue during rapid pacing at 60 ms intervals from a vertical line electrode beneath the passive obstacle. Each wave is assigned a unique number for the purposes of identification and retains that number from plot to plot.

vulnerable period, thereby creating a substrate ripe for reentry. In the first part of this study, we depressed tissue excitability by reducing the maximum Na conductance g Na and then examined the interactions of planar wave fronts with obstacle corners of varying curvature. In general, we found that as the curvature of the obstacle decreased 共i.e., an increase in the radius of curvature兲, so did the excitability at which detachment occurred 共Fig. 4兲. Panfilov and Keener16 suggested that in vivo an infarct may act as a suitable substrate for reentry provided it has a high enough curvature. Our results concur on this point since we observed detachment at higher excit-

abilities from obstacles with higher curvatures. In essence, wave fronts are more likely to detach from obstacles with sharper corners. And once waves are shed from obstacles, they can then reenter, as suggested by Panfilov and Keener,16 and cause arrhythmia. From Fig. 4 it is clear that for a given radius of curvature, wave fronts remain attached to the insulator obstacle at lower excitabilities than to the passive obstacle. This is probably due to the comparatively large AP plateau magnitude 共Fig. 9兲 and (dV m /dt) max near the insulator obstacle. Both of these serve to increase the likelihood of wave front attachment by supplying more

FIGURE 11. Transmembrane potential distributions in the active tissue during rapid pacing at 50 ms intervals from a vertical line electrode beneath the passive „a…–„d…, and insulator „e…–„h… obstacles. Each wave is assigned a unique number for the purposes of identification and retains that number from plot to plot.

WAVE FRONT –OBSTACLE INTERACTIONS

FIGURE 12. Recordings at cell 0 near „a… passive, „b… insulator obstacles for the stimulation shown. APs created by reattachment of detached wave fronts are indicated by **. APs of elevated magnitude created by simultaneous collision of reentering waves and new waves are indicated by *.

current 共large AP magnitude兲 at a faster rate 关large (dV m /dt) max] to neighboring cells. This suggests that a nonconductive obstacle is less likely than a conductive obstacle to initiate reentry secondary to wave front detachment in situations of depressed excitability. In addition to being higher than those for the insulator, the detachment excitabilities on the passive obstacle phase diagram 共Fig. 4兲 change very little as the radius increases. This suggests that the ability of the obstacle to withdraw current is just as important, if not more so, than curvature when it comes to wave front detachment from obstacle corners. According to Ikeda et al.,8 an increased source-to-sink ratio can cause successful depolarization of cells near the obstacle border, resulting in successful attachment of a reentrant wave. Conversely, a decreased source-to-sink ratio can lead to wave shedding. Obstacles that can remove charge from the active tissue, like the passive obstacle, decrease the source-to-sink ratio, thereby decreasing the likelihood of maintained attachment. In this study, we used a quantitative formulation of SF to measure the source-to-sink ratio during simulations with depressed excitability and r⫽0. To calculate the SF, we extended the 1D approach of Shaw and Rudy20 into 2D. For the passive and insulator cases, our SF formulation yielded detachment excitabilities in perfect agreement with those we observed visually. Cabo et al. used the critical ratio R cr /W as a metric to relate the excitability of the myocardium to the propensity for wave front detachment.4 R cr represents the criti-

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cal radius of excitation and W represents the width of the wave front. Cabo et al. found that wave front detachment occurred from a thin nonconductive obstacle for critical ratios greater than 1. In that respect, the critical ratio is similar to our SF formulation since we observed detachment for SF⬍1. However, the critical ratio used by Cabo et al. does not take into consideration the ability of the obstacle to act as a current source or sink and, as a result, to influence wave front behavior. Other formulations of a 2D SF also do not take into account the effect of the obstacle on wave front behavior.25 However, any method for quantifying wave front detachment must include the source/sink behavior of the obstacle itself, particularly if the obstacle is conductive. As far as we know, the formulation for a 2D SF presented here is the only one that takes into consideration the ability of obstacles to act as sources and sinks. In addition, our formulation explicitly includes the currents that contribute to the SF, allowing us to determine the contribution of individual cells to the wave behavior. We observed two types of propagation failure beneath the obstacles. Type I, which occurred only beneath the insulator, did not involve detachment of the weakened wave from the obstacle 共i.e., decremental conduction兲. However, type II failure, associated with the other obstacle, did involve detachment, coupled with a progressive shortening of the extent of the wave. Efimov et al. have shown that when a wave front propagates in a medium of depressed excitability, the extent of the wave 共in the direction perpendicular to propagation兲 decreases as propagation progresses.5 Based on this observation, one may predict that the excitability at the time of type II failure would be lower than the excitability at the time of type I failure. However, the opposite is observed here since type I failure occurs at 26%g Na and type II failure occurs at 27%g Na . Thus, the reason for the distinction between type I and type II failure cannot be the intrinsic tissue excitability. Instead, we believe that during type II propagation failure, the passive obstacle acts as a current sink, withdrawing charge from the weakened wave front. The net result of this charge removal is similar to a reduction in excitability near the obstacle, causing the wave to detach. Detachment of a weakened wave can have potentially proarrhythmic side effects if the wave invades an area of nominal excitability and reenters. Wave Front–Obstacle Interactions Under Rapid Excitation At a pacing interval of 55 ms or less, we observed wave front detachment only from the insulator obstacle. At first, this may seem to contradict our simulations involving reduced excitability 共see ‘‘Wave Front– Obstacle Interactions in Tissue with Reduced Excitability’’ in the ‘‘Results’’ section兲 in which the insulator

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obstacle was the least likely to promote wave front detachment. However, this argument assumes that the chief mechanism of detachment in both cases involves depressed excitability. As mentioned in ‘‘Wave Front– Obstacle Interaction in Tissue with Reduced Excitability’’ in the ‘‘Discussion’’ section, detachment under conditions of depressed excitability 共in the absence of rapid stimulation兲 is likely due to a lessening of the maximum AP plateau potential and (dV m /dt) max . However, another factor, refractoriness, also plays a role during rapid excitation. The APDs near the insulator obstacle are longer than those near the passive obstacle 共Fig. 8兲. Therefore, under conditions of rapid stimulation we would expect the tissue near the insulator obstacle to remain refractorily longer than the tissue near the passive obstacle. Following that logic, we would also expect waves to detach from the insulator obstacle at longer pacing intervals. This is indeed the behavior we observed. In the case of the passive obstacle, reentry occurs secondary to wave fragmentation 关Figs. 10共a兲–10共i兲兴. However, fragmentation occurs only over a small range of pacing intervals 共roughly between 54 and 62 ms兲. You will recall from Fig. 7共b兲 that close to the passive obstacle, the APD is near 100 ms. However, as one moves further from the obstacle, the APD increases to 112 ms. This 12 ms dispersion of refractoriness is at the root of wave fragmentation beneath the passive obstacle. The reason for this is twofold. First, in order for fragmentation to occur, enough time must elapse between successive wave fronts for the tissue near the obstacle corner to recover. In other words, before the next wave arrives at the corner, enough time must elapse for an AP of 100 ms to exit its refractory period. Second, the time between successive wave fronts must be short enough to prevent recovery of tissue far from the obstacle 共APD⫽112 ms兲. Otherwise, instead of fragmentation, a planar or nearplanar wave is generated. Pertsov et al.17 used a FitzHugh–Nagumo type model to examine the interaction of autowaves with thin nonconductive filaments under conditions of rapid stimulation. They modeled a filament extending from the right border of the tissue into the center and paced from a line electrode across the tissue’s bottom border. They found that, after collision at a critical pacing interval, waves would separate from the filament. In some cases, wave detachment would produce a chaotic pattern of reentry near the edge of the filament. Similarly, our findings suggest that there exists such a critical pacing interval above which wave detachment from obstacle corners occurs, possibly causing reentry. In the case of the insulator obstacle, this interval is somewhere between 55 and 60 ms.

Limitations and Impact of Study The most significant limitation of this study is the use of a 2D model. The 3D structure of the heart is important in reentry 共see, for instance, Jalife and Gray9兲. In particular, subendocardial pathways and pectinate muscle bundles act as alternate routes for current flow through which reentrant paths can form. We also use a simplified representation of the obstacles. Due to the fact the electrical properties of ablation lesions are unknown, we chose to retain, throughout the passive-tissue obstacle, the conductivity values of the myocardium. Further, we assume that each obstacle is a homogeneous medium. However, it is well known that during ablation there is a gradient of tissue damage because of thermal diffusion.15 This produces a gradient of electrophysiological properties in and around the damaged tissue. A gradient of electrical properties will produce a gradient of electrical responses to stimuli, possibly altering wave front– obstacle interactions as compared to a homogeneous case. Thus, taking into account heterogeneities in conductivity within and around the obstacle could be necessary to accurately model tissue damage. Despite these limitations, the results presented in this study have strong implications for the modeling of natural obstacles and obstacles created by cardiac ablation, and for our understanding of wave front–obstacle interactions in general. We have shown that the choice of obstacle corner geometry is an important consideration. In general, we showed that as the curvature of an obstacle corner is increased 共i.e., the radius, as shown in Fig. 2, is decreased兲, so increases the likelihood of wave front detachment from the corner. This agrees with results presented by Ikeda et al.8 who found that as the radius of a circular hole in the tissue increases, the likelihood of spiral wave attachment also increases. It is also consistent with the findings of Girouard et al.,6 which demonstrated an increased load during wave front pivoting. We found that the behavior at the pivot point is, however, modulated by the current withdrawing ability of the obstacle since the importance of obstacle curvature to wave front detachment decreases as current withdrawing ability increases. We have demonstrated that a change in the current withdrawing ability of a homogeneous obstacle can have marked effects on how waves interact with it. First, a conductive obstacle is more likely to promote wave front–obstacle separation than a nonconductive obstacle in regions of depressed excitability. As such, when excitability is decreased, conductive obstacles are more likely to induce arrhythmia secondary to reentry. Thus, when the excitability of the myocardium is depressed, as occurs during the use of class IC antiarrhythmic drugs, we would expect weakly conductive tissue to be more proarrhythmic than natural nonconductive obstacles.

WAVE FRONT –OBSTACLE INTERACTIONS

Second, conductive obstacles significantly alter wave behavior near the tissue–obstacle interface. In our studies with depressed excitability, we found that failed propagation beneath the conductive obstacle resulted in detachment of the failing wave. This is a potentially proarrhythmic consequence if the weak wave invades a region of nominal excitability and reenters. Also, a dispersion of APD forms beneath conductive obstacles. The extent of the dispersion appears to be dependent on the conductivity of the obstacle, with higher conductivities leading to greater dispersions. These APD gradients can lead to wave fragmentation and reentry in situations of rapid stimulation. Third, we found that completely nonconductive obstacles behave differently under varying conditions known to promote wave front detachment. For example, we found that the insulator obstacle is less likely to promote wave detachment under conditions of depressed excitability and more likely to do so under conditions of rapid pacing. Overall, our results suggest that before an accurate physiological model of an obstacle can be made, it is important to first establish its actual electrical and geometric properties. It may not be acceptable to model all obstacles with sharp edges just as it may not be acceptable to model all obstacles as completely nonconductive barriers. Further research is needed to determine the exact composition of tissue subsequent to ablation and other types of myocardial damage as well as to determine how to best represent natural obstacles, such as orifices, in computational models. Once this has been achieved, more accurate models of wave front interactions with cardiac obstacles can be created and used to more fully understand the mechanisms of arrhythmogenesis and its therapy. ACKNOWLEDGMENTS This work was supported in part by a grant from Medtronic, Inc., National Science Foundation Award No. BES-9809132 and GIG Award No. DMF-9709754, the Louisiana State Board of Regents Support Fund under Grant No. LEQSF共1998-01兲-RD-A-30, and a Louisiana State Board of Regents fellowship. Part of this work was used to satisfy the MS degree requirements of E.M.A. at Tulane University.

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