Simulation of two-dimensional anisotropic cardiac reentry: Effects of the wavelength on the reentry...

Annals of Biomedical Engineering, Vol. 22, pp. 592-609, 1994 0090-6964/94 $10.50 + .00 Printed in the USA. All rights reserved. Copyright �9 1994 Biomedical Engineering Society

Simulation of Two-Dimensional Anisotropic Cardiac Reentry: Effects of the Wavelength on the Reentry Characteristics

L. J. LEON, F. A. ROBERGE, and A. VINET

Institute of Biomedical Engineering, l~cole Polytechnique and Universit6 de Montrral, Montrral, Canada

Abstrae tmA two-dimensional sheet model was used to study the dynamics of reentry around a zone of functional block. The sheet is a set of parallel, continuous, and uniform cables, transversely interconnected by a brick-wall arrangement of fixed resistors. In accord with experimental observations on cardiac tissue, longitudinal propagation is continuous, whereas transverse propagation exhibits discontinuous features. The width and length of the sheet are 1.5 and 5 cm, respectively, and the anisotropy ratio is fixed at approximately 4:1. The membrane model is a modified Beeler-Reuter formulation incorporating faster sodium current dynamics. We fixed the basic wavelength and action potential duration of the propagating impulse by dividing the time constants of the secondary inward current by an integer K. Reentry was initiated by a standard cross-shock protocol, and the rotating activity appeared as curling patterns around the point of junction (the q-point) of the activation (A) and recovery (R) fronts. The curling R front always precedes the A front and is separated from it by the excitable gap. In addition, the R front is occasionally shifted abruptly through a merging with a slow-moving triggered secondary recovery front that is dissociated from the A front and q-point. Sustained irregular reentry associated with substantial excitable gap variations was simulated with short wavelengths (K = 8 and K = 4). Unsus- tained reentry was obtained with a longer wavelength (K = 2), leading to a breakup of the q-point locus and the triggering of new activation fronts.

Keywordsmlonic model, Propagation, Triggered repolarization, Triggered excitability recovery, Excitable gap, Vortex reentry

INTRODUCTION

It is widely held that unidirectional block is a necessary requirement for the initiation of reentry leading to severe arrhythmias such as tachycardia and fibrillation (1 ,13 ,25- 27). While ultrastructural nonuniformities and inhomoge-

Acknowledgment--This work was supported by the Whitaker Foun- dation, the Medical Research Council, and the Natural Sciences and Engineering Research Council of Canada and by the Fonds FCAR of QuEbec. The contributions of Mr. A. Bleau to the development of soft- ware and Ms K. Lennon to the preparation of the manuscript are grate- fully acknowledged.

Address correspondence to Dr. L. J. Leon, Institute of Biomedical Engineering, Universit6 de MontrEal, P.O. Box 6128, Station Centre- Ville, MontrEal, QuEbec, Canada H3C 3J7.

(Received 29Apr94, Revised 15Jul94, Accepted 16Aug94)

neous tissue properties may play a role in the formation of functional unidirectional block (10), there is increasing evidence from experimental (10,24,26,27) and theoretical (15,28,33) studies that the phenomenon can occur even when the membrane properties are uniform throughout the tissue. It is then believed that the activation sequence and the anisotropic myocardial properties may introduce spatial repolarization differences sufficient to induce functional unidirectional block and initiate reentry (10,27).

Cardiac reentry may be the result of vortex-like activity associated with a meandering center of rotation or pivot point. Several modeling studies have documented these spiral patterns in a two-dimensional setting. The main features include the drift of vortices due to parameter gradients, their anchoring on localized inhomogeneities or de- fects, and the breakup of spiral waves into complicated spa t io - t empora l pat terns (2 ,14 ,18 ,19 ,31 ,32) . Such a mechanism of two-dimensional reentry appears to be a general property of excitable media and thus excludes anatomical or functional peculiarities of myocardium as causal factors of reentry. Supporting evidence was provided by studies in the canine heart in situ (12) and in thin slices of epicardial ventricular muscle (9) in which rotating waves were reported to have a pronounced spiral shape, with an increased curvature near the core or center of rotation. A related two-dimensional simulation study con- cluded that neither the rotation cycle nor the refractory period changed with the anisotropy ratio and that the spiral wave velocity was controlled by the steepness of wave front curvature near the core (20).

As a first approximation, according to the " leading- c i rc le" concept of circus movement of reentry (1), the pathway length of two-dimensional reentry may be taken to correspond roughly to the length of a reentrant loop along which the impulse can continue to circulate. For this reason and because simulations in a ring are faster and less cumbersome than in a sheet model , we first carried out a detailed study of one-dimensional reentry in a ring (29). To examine the applicabil i ty of these observations to the more general problem of two-dimensional reentry, we investigated the dynamics of circus movement in a thin sheet model of myocardium (15,16,21,22). The sheet model has

592

Two-Dimensional Cardiac Reentry 593

uniform membrane properties throughout. Its parallel- cable organization gives rise to continuous propagation in the longitudinal direction and to discontinuous propagation in the transverse direction, analogously to experimental observations in cardiac tissue (26,27). Given a fixed physical organization and constant anisotropy ratio, we describe the effects of different wavelengths on circular propagation. Our goal is to clarify the role of functional block, wave front curvature, and excitable gap fluctuations in determining the dynamics of reentry, using the basic wavelength as parameter.

METHODS

First, a ring model is used to determine the parameters most appropriate to obtain two-dimensional circular propagation. Next, we examine various reentrant patterns using a thin sheet model that can be assimilated to a strip of cardiac muscle in which propagation along the depth of the tissue is neglected.

The Membrane Model

In our previous study of reentry on a ring, we used a modification of the Beeler-Reuter model (called MBR) that incorporated faster sodium current dynamics (3,11). Because of the large computing requirements of two- dimensional reentry simulations, we choose to simplify the MBR model by deleting the slow sodium current inactivation (gating variable j) and keeping Esi constant. In the resulting model (called MBR-1) we have/ion = lna + Isi + IK, where INa = gNa m 3 h ( V - ENa) is the fast sodium current, Isi = gsi df (V - Esi) is the secondary inward current, and I K = IK1 (V) + xllxl (V) is the potassium current. V is the membrane potential, m, d, and xl are gating variables of activation, and h and f a r e gating variables of inactivation. Any gating variable, y, corresponds to a first-order voltage-dependent process described by a steady-state characteristic y~(V) and a time constant "ry(V). g N a = 15 ms/cm 2 and gsi = 0.09 ms/cm 2 are maximum conductances. ENa = 40 mV and Esi = - 74 mV are reversal potentials. IKl and I~1 are empirical functions.

All functions and parameters defining Is~ and I K can be found in the study by Beeler and Reuter (3), and the ex- pressions for m and h can be found in the study by Drou- hard and Roberge (11). The basic qualitative features of reentry on a ring do not appear to be altered when using MBR-1 instead of MBR.

The Ring and Sheet Models

The ring is a closed loop formed by a homogeneous and uniform cable surrounded by a volume conductor of negligible resistivity and infinite extent. The cable (radius a = 5 Ixm) has an intracellular resistivity (p) of 400 ~ cm.

It is described by the well-known cable equation (29) in which V varies with time (t) and distance (x), Cm is the membrane capacitance (1 i~F/cmZ), S is the surface-to- volume ratio (0.4 ~ m - 1), and/ion is the total ionic current provided by MBR-I . Its resting length constant is 598.6 Ixm, and its steady-state conduction velocity is 50 cm/s.

The rectangular sheet model is a single layer of parallel and uniformly spaced identical excitable cables, transversely interconnected by a regular resistive network (16). The regular transverse network of fixed resistors (Fig. 1A) has a constant spacing A = 200 Ixm along each cable. Each resistor, R n , may be viewed as an average representation of a local transverse junction resistance between adjacent myocardial strands. The center-to-center inter- cable spacing is fixed at 100 p~m. Each cable of the sheet is identical to that used in the present ring model.

Because of the parallel-cable arrangement of the sheet, the longitudinal fiat wave conduction velocity (0xF) is equal to that of an isolated cable. Current flow in the transverse direction is governed by R n, p, and A. The transverse fiat wave conduction velocity (0yF) was varied by changing R n, keeping p and A constant. The sheet anisotropy ratio (AR) is defined as AR = 0xF/0y F. The sheet parameters are fixed as follows: length = 5 cm, width = 1.5 cm, Rn = 4 MI), 0xF = 50 crn/s, and 0y F = 12 crn/s. The resulting anisotropy ratio is then about 4:1.

Protocols and Simulations

During propagation in the ring or in the sheet, the action potential characteristics were obtained from measurements based on two fiducial points: the maximum rate of rise of the upstroke (V max) to detect the activation and repolarization to - 50 mV (V_ so) to detect the end of the action potential. The cycle length (CL) is the time interval between two consecutive activations, the action potential duration (APD) is the interval between 12 m a x and V_5o, and the diastolic interval (DIA) is the time elapsed between V_ 50 and the next I? max" The latency of the action potential upstroke, which is normally characterized by the time constant of the foot of the action potential (~foot), is then included in DIA. Since its value is small in the present model (2-8 ms), this approximation is satisfactory for the purpose of the present study. At any node, we then have C L i _ 1 = A P D i - 1 -~- DIAi, where (i - 1) is the beat number of CL i _ 1.

The occurrence of V max was used to define the activation front, to draw the isochrone maps, and to measure the conduction time (inverse of velocity). During repolarization, the h ~ 0.1 crossing level served to detect the position of the excitability recovery front and to calculate its conduction time over a distance of 100 Ixm. This choice of h = 0.1 follows our previous studies of the space-clamped and propagated action potentials (29,30). It is based on the

594 L.J. LEON, F. A. ROBERGE, and A. VINET

A

150 cables x = input junction (1.5 cm) " o = output junction

+

�9 500 patches > (5.0 cm)

B

i iiiNi il

C

20

m . 100 |

Transverse . - " " - . . . . . . . . . .

/I

0 0.3 0.6 0.9

D i s t a n c e ( m m )

FIGURE 1. (A) Diagrammatic representation of the uniform sheet model of parallel cables, 1.5 x 5 cm. Each cable is divided into segments of 100 ixm, and the inter-cable distance is also 100 iLm. Adjacent cables are coupled transversely by a fixed resistance, R,, in a brick-wall pattern. (B) Diagrams illus- trating the stimulation protocols to initiate reentry: line stim- uIus $1 followed by premature area stimulus $2. (C) Spatial spread of the action potential upstroke in the longitudinal and transverse directions.

fact that/Na is much larger than any other current component at threshold, so that excitability recovery is directly proportional to the recovery from INa inactivation (gating variable h). Hence h ----- 0.1 marks the beginning of threshold recovery, or the end of the absolutely refractory period (ARP) when gNa = 15 ms/cm 2. The reasons for choosing distinct markers for the e n d o f the APD (i.e., V_5o) and that of the ARP (i.e., h = 0.1) were explained in relation to reentry on the ring (29).

In addition, there is a one-to-one correspondence between h and DIA, so that DIA is a suitable independent

variable to describe threshold recovery. In fact, h ---- 0.1 corresponds to DIA ~ 30 ms (30). Since APD is measured at V_5o, we haveARP = APD + 30 ms. These results are also valid during propagation (29), When the profile between the front and tail of the propagating wave is continuous, the distance between l) ma x and h = 0.1 (during repolarization) is the wavelength (W) and it coincides with h ~ 0.1.

Our present study with the ring model was based on the methods_ described in (29). After the initiation of stable reentry, the dynamics of propagation were modified by reducing the ring length in a systematic manner. Given the above definition of the wavelength, the excitable gap (Eg) is then the region where h > 0.1. Computations were carried out using a parallel processing method with a spatial increment of 39 p~m and a time step of 2 ixs. Programs were written in FORTRAN and run on Silicon Graphics IRIS-4D/280 (8-parallel processor computer) and IRIS-4D work stations (Silicon Graphicl Mountain View, CA).

In the sheet model, computations were made with a cable segment length (Ax) of 100 txm and a time step (At) of 10 ixs. Longitudinal conduction time was measured as in the ring, In the transverse direction, conduction time measurements were made between corresponding nodes on two adjacent cables (100 Ixm apart). In addition to plotting activation isochrones to describe reentry, we com- puted potential distributions (V-maps) and recovery from sodium inactivation distributions (h-maps) at a given time instant. In V maps; black represents potentials negative to - 60 mV (partial or complete excitability recovery), white represents potentials positive to • 20 mV, and gray represents potentials between - 60 and - 20 mV. In h-maps, black corresponds to h > 0.1 and white to h ~< 0.1. Since h = 0.1 corresponds to the end of the ARP, black areas represent the Eg on h-maps, and white areas represent regions of absolute refractoriness.

As in a previous-study (8), we divided the time constants of activation (%) and inactivation ('re) of Isi by a constant K to obtain a shorter APD and ARP. A reduced ARP corresponds to a shorter wavelength and makes it possible to establish sustained reentry on a shorter ring or in a smaller sheet. In the present study, the sheet size allowed by the available computing capacity was 1.5 x 5 cm. To investigate the possibility of producing sustained reentry in the sheet, we reduced the wavelength using values of K = 2, 4, and 8. Transitions between stable reentry and irregular reentry were found for all K values, but at correspondingly shorter ring lengths with larger K.

Reentry was initiated in the sheet model by a cross- shock protocol, first postulated in theory by Winfree (32) and subsequently used in experimental canine studies (12). First, a transverse plane wave was generated by a conditioning stimulus S 1 (1 ms in duration) applied along the upper longitudinal edge of the rectangular sheet (left


diagram, Fig. 1B). Next, S 1 w a s followed by a premature stimulus S 2 (1 ms in duration) applied over one-quarter of the sheet surface. The time interval between S~ and Sz was chosen such that some excitability recovery existed near the upper longitudinal edge of the sheet. An equivalent result followed when S~ was applied along the left transverse edge of the sheet, as indicated in the right diagram of Fig. lB.

Numerical solutions were obtained using a previously described computing technique (16). Briefly, propagation on any given cable of the sheet is taken to be governed by the standard cable equation. At each time instant, the potential distribution on any cable is first calculated assuming that they are isolated from one another. Then a perturbation is added to account for current flow between adjacent cables. The algorithm can be summarized as follows: (i) calculate Iio, along each cable; (ii) use lio n and the potential distribution at time t i to solve for V along each cable, at time t i+a, assuming that the cables are isolated from one another; (iii) calculate the effects of interactions between cables; (iv) adjust the potential distribution on each cable to account for the transverse connections; (v) update t i and begin the new time step (step 1). Programs were written in FORTRAN and run on a Silicon Graphics INDIGO computer.

On a given cable of the sheet model, the resting length constant of 598.6 ixm and 2tx = 100 I~m allow us to calculate a longitudinal discretization factor of 0.167. The cable is then equivalent to a uniform network in which equal segments (membrane area of 3.14 • 10 -5 cm 2) are joined by resistors of 5.2 MO. This spatial accuracy is sufficient to avoid significant numerical errors (22). For transverse plane wave propagation, the sheet can be assimilated to a discontinuous cable (16) with constant segment length (100 ixm) and periodically varying radius and junction resistance. The spatial period is organized as follows: 4 MI] junction resistance, segment with membrane area of 3.14 • 10 -5 cm 2, 2.6 MI) junction resistance, segment with membrane area of 6.28 • 10 -5 cm 2, 2.6 M ~ junction resistance, and segment with membrane area of 3.14 • 10 -5 cm 2. Thus, although the system is discontinuous, each transverse cable segment is similar to a longitudinal cable segment, and the discretization factor is comparable.

As depicted in Fig. 1C, the spread of the action potential upstroke is about 0.7 mm in the longitudinal direction and, with the inter-cable distance of 100 ixm, roughly 0.35 mm in the transverse direction. The ratio of the longitudinal to the transverse spread is then about 2, compared with a ratio of about 4 for the corresponding flat conduction velocities (AR of 4:1). Hence the brick-wall pattern of transverse connections must not be expected to yield a simple scaling of propagated events along the major axes, in the usual sense of continuous two-dimensional media in

which the ratio of conduction velocities is equal to that of the resting length constants. In principle, however, it should be possible to choose the sheet parameters to obtain approximately equal ratios of flat conduction velocities and spatial spread of the upstroke along the two major axes. This particular case was not investigated. Note also that the inter-cable distance of 100 ixm does enter into the calculations and was chosen at this value solely to establish the desired Oyv of 12 cm/s (16). Consequently, it specifies the structural organization of the sheet and must not be used to change the anisotropy ratio.

The location of the q-point at each time instant was determined automatically from a pair of membrane potential distributions taken 20 Ixs apart. Edge detection techniques were used to find the border between excitable and refractory regions, which is the demarcation between white and black regions on V-maps. The direction of edge movement was calculated at each point. When the comparison of the two maps indicated a displacement in the direction of the activation potential gradient, the point was taken to belong to the activation front. All points that did not meet the criterion were discarded. Then the endpoint of the activation front contour toward the center of the sheet was taken as the q-point, following verification that it was stationery in the 20 Ixs time interval between the two V-maps.

RESULTS

As a first step, we use the ring model to examine the effects of shortening the time constants of lsi on the APD and its range of variations. Next we describe the effects of wavelength changes on the dynamics of two-dimensional reentry.

Reduced APD in the Ring

For the nominal case of K = 1, reentry in a continuous and uniform ring model has been found to have the following main characteristics (29):

1. For stable regular reentry, CL, DIA, and A P D are the same at all nodes. They satisfy the relation CL = DIA + APD, and they define a constant potential profile propagating at constant velocity. This occurs if the ring length (X) is large enough to allow a sufficient Eg. As X is decreased, the conduction velocity of stable reentry remains nearly constant so that CL, DIA, and APD are reduced. However, since dAPD/dDIA < 1 in this range, variations in APD are always less than DIA variations. Stable reentry is transformed into sustained irregular reentry when X falls below a critical value (Xc~it), such that DIA for stable propagation would fall in the region where dAPD/dDIA > 1. For nominal parameter values in the MBR model, this occurs when

596 L J. LEON, F. A. ROBERGE, and A. VINET

DIA is still far from the beginning of the ARP, and a large Eg then exists ahead of the activation front.

2. Irregular reentry terminates spontaneously if X is reduced below a minimum value (Xmin).

3. Sustained irregular reentry (Xmi n < X < Xcrit ) is quasi- periodic, except for a narrow range of ring lengths near Xcrit where the nature of the reentry could not be clearly established.

4. For an observer moving with the activation front, quasi-periodic circulation is characterized by a periodic temporal oscillation of D1A and APD values, with increasing amplitude at shorter ring lengths. Observa- tions at a fixed point on the ring produced sequences of alternating D1A, APD, and CL values. The amplitude of the alternans is continually changing, and the alternating patterns are temporarily broken by short sequences of decreasing or increasing values. This basic pattern of alternations is even more marked with MBR-1 than with MBR.

5. The complexity of quasi-periodic reentry arises from a combination of factors, namely, the slope of the APD i

versus DIA i relation, which is then larger than 1, the transient slow conduction over a portion of the ring, and the occurrence of triggered secondary excitability recovery that creates abrupt Eg changes. Close to Xc,~t, however, quasi-periodic reentry occurs in the absence of triggered secondary recovery and the Eg fluctuates in a more regular manner.

The nominal conduction velocity of an MBR-1 cable is 50 cm/s. In the ring, the value of Xcrit is then 13.7 cm for K = 1 and 5.9, 3.44, and 2.73 cm for K = 2, 4, and 8, respectively (Fig. 2; Table 1). During stable reentry, the conduction velocity is constant (about 50 crn/s) and the

O3 E v t -

t -

__1

o

O

250

200

150

K = 2

~+ + +

-I- +

+ +

+ - t - _1 . +

4

100 ' ' ' 4 6 8 10

Ring Length (cm) FIGURE 2. Reentry in a uniform ring with K = 2. Cycle length (CL) is proportional to ring length 00 for X > 6 cm. At X = 6 cm, reentry becomes irregular.

TABLE 1. Sustained reentry on the ring

K = 2 K = 4 K = 8

Stable reentry X (cm) 6 4 3 W (cm) 4.2 3.0 2.25 Eg (cm) 1.8 1.0 0.75 APD (ms) 84 60 45 DIA (ms) 36 20 15 CL (ms) 120 80 60

Irregular reentry X (cm) 5.90 3.44 2.73 APD (ms) 16-128 24-69 26-51 DIA (ms) 26-108 11-55 8-29 CL (ms) 117-135 70-94 52-67 0 (cm/s) 43.6-50 44-50 44-50

following relations are applicable: CL ~ X/O, X = W +

Eg, and CL i_ 1 = APDi-1 + DIAi, where i is the beat number. These relations can be verified using the data of Table 1. For example, prior to the bifurcation at X = 6 cm with K = 2, a velocity of 50 cm/s yields W = 4.2 cm and Eg = 1.8 cm, and the corresponding C L i _ 1, A P D i - 1, and DIA i values are 120, 84, and 36 ms, respectively.

Another major feature of one-dimensional reentry with increased K is a drastic shrinkage of the zone of sustained irregular reentry (Xmi n < X < Xcrit ) . It is about 0.05 cm with K = 2, 4, or 8 (Fig. 3). The main desired effect of a larger K is to decrease the APD (or wavelength) of the reentrant impulse, as well as its range of variations. This is successfully accomplished, as indicated in Table 1, and the reduction in the range of APD variations is paralleled by similar effects on DIA and CL.

A D

B E

1ool K=, I 0

C F

t K=8 t K=8 ~ 100 100

. . . . . . . . . . . . . . . . . . . . . . . . . . 1.o o r' so ........................... 1oo ,so Ring length (cm) Diastolic interval (cm)

FIGURE 3. Reentry in a uniform ring with K = 2, 4, and 8. Action potential duration (APD) versus ring length (X) (A-C), and APD versus diastolic interval (DIA) (D-F). APD values above and below the horizontal dotted line correspond to regular and irregular reentry, respectively.


Despite the above changes in reentry characteristics with larger K values, the basic qualitative features of quasi-periodic reentry continue to be present. At the bifurcation (Xcrit), the slope of the APD i versus DIAi relation becomes larger than 1 (Figs. 3D and 3E); there is slowing of conduction caused by short DIAs, as attested by a decrease of about 12% in conduction velocity with K = 2, 4, and 8 (Table 1); and triggered waves of secondary repolarization and excitability recovery are manifested (29).

In summary, accelerating the kinetics oflsi allows us to obtain reentry on shorter rings as K is increased and to generate sustained irregular reentry at lower critical ring lengths. The region of sustained irregular reentry is reduced to a narrow range of ring lengths just below Xc~it. The range of CL, APD, and DIA variations is decreased with larger K values, and the minimum DIA is always small enough to induce slow conduction over a portion of the ring. Hence, whereas the basic qualitative features of one-dimensional reentry are not changed with larger K values, the APD and wavelength are shortened, and the variability of irregular reentry is progressively reduced.

Two-Dimensional Reentry

Given a parallel-cable sheet model of 1.5 • 5 cm and a fixed anisotropy ratio (AR = OxF/Oy v ~ 4:1), consider a flat transverse wave front moving from top to bottom in Fig. 1A. Action potentials are then triggered at input junctions, and propagation on the first cable takes place in both directions, toward neighboring output junctions. Since a given output junction is located midway between two input junctions with the present transverse resistive network, two action potentials collide at a given output junction. From there on, each output junction supplies current to an input junction on the next cable, and the same process is repeated. This behavior creates a spatial diversity of potential levels and action potential characteristics over the surface of the sheet. In particular, the potential at an output junction is slightly lower than the potential at an input junction, and the collisions at output junctions make I) max larger at these points.

This brick-wall structure of anisotropic myocardium was chosen because of its satisfactory correspondence with experimental observations (26,27) and the capacity of the model to reproduce typical elliptic activation wave fronts following point source stimulation (15). Because transverse propagation cannot be scaled in a uniform manner by the transverse resistance, the sheet can be consid- ered inhomogeneously anisotropic, despite uniform membrane properties throughout. Since Oxv (50 crn/s) is larger than 0y F (12 cm/s), the long axis of the elliptic contour is aligned with the longitudinal direction of the sheet, and the axial current is also largest in that direction. The axial

current is minimum along the transverse axis where the conduction velocity is lowest. The curvature of the activation front is maximum longitudinally, close to the tip of the elliptic contour, and minimum transversely. Along an intermediate direction between the two major axes, the axial current has both a longitudinal and a transverse component. The presence of a transverse axial current creates a drag effect on the wave front and slows longitudinal propagation, such that the velocity of the elliptic front is lower than that of a flat front (15). This model behavior is also in good agreement with experimental observations (17). One can then expect the magnitude of the drag effect to increase with the curvature, thereby leading to a more pronounced reduction in longitudinal conduction velocity (33). If the curvature is sufficiently large, the quantity of current deviated away from the longitudinal axis may be such that the remaining current is too small to sustain propagation so that conduction block occurs. The excess longitudinal axial current over the minimum just sufficient to maintain propagation is called the safety factor of longitudinal propagation (15).

Moving Center of Rotation. The initiation of two- dimensional reentry is illustrated in Fig. 4 using potential distributions (V-maps) and distributions of recovery from sodium inactivation (h-maps). On V-maps, potentials negative to - 60 mV are represented in black, potentials positive to - 2 0 mV in white, and potentials between - 6 0 and - 20 mV in gray. There is no visible gray zone on the potential distribution during the action potential upstroke because of the very narrow spread of potential. On h-maps, black corresponds to h > 0.1 and white to h ~< 0.1. Because h drops abruptly to zero during the upstroke, the h = 0 crossing is an accurate marker of activation (29). Thus, because of the abrupt changes in V and h during the activation, the representation of this event (white-black interface) is nearly identical on both types of maps.

Conversely, a gray zone is visible on the potential distribution during repolarization because of the rather wide spread of potential, and the black-gray interface (at - 60 mV repolarization) defines the repolarization front. On the other hand, since the h = 0.1 crossing is in very good correspondence with the end of the ARP (29,30), it is an accurate marker of excitability recovery. Hence the h-map provides a good representation of the activation and recovery fronts at a given instant, and the distance between the two defines the Eg.

In a sheet with K = 8 and AR ~ 4:1, stimulus S 2 was applied at a coupling interval of 110 ms to initiate an activation front at the interface between the upper two quadrants. The V-map and h-map at 130 ms after S 1 in Fig. 4 provide an instantaneous picture of the initiation of circular propagation. They depict a single curved activa-


FIGURE 4. Initiation of two-dimensional reentry with AR = 4:1 and K = 8. Potential distribution (V-map, left column) and distributions of recovery from inactivation (h-maps, right column) at 130-190 ms after stimulus $1, with $2 occurring 110 ms after $1. A = activation front due to $2; R = recovery front due to $2; R' = recovery front due to $1; q = point of junction between the A and R fronts. Points 1 and 2 are recording sites used in Fig. 13. See text for map interpretation.

tion (A) front that progresses both longitudinally toward the right edge of the sheet and transversely toward the lower right quadrant. The rotating recovery (R) front at that moment is propagating transversely and curving slightly. Rotation at that particular time instant is centered at the point where the A and R fronts join together (Fig. 4, arrow). This singular point (called the q-point) has been defined for spiral waves (33). It is the only point that the A and R fronts have in common, and it is the pivot around which the wave front is developing its rotating pattern.

Thus the A and R fronts curl around a virtual axis located at the q-point. But the q-point is not fixed, and it drifts within a small area as the wave front pursues its rotation. Events at 150 ms (Fig. 4) indicate that the curling A front is invading the lower half of the sheet and moving longitudinally leftward. There are two recovery fronts, a flat one (R') belonging to S 1 and a curling one (R) belonging to $2. R' extends horizontally over the full length of the sheet and moves downward, whereas R propagates upward within the upper left quadrant. The q-point, defined by the junction of the A and R fronts at 150 ms, has moved slightly leftward compared with its position at 130 ms. As rotation continues at 170 and 190 ms, the flat R' front approaches the bottom edge of the sheet and eventually disappears. When the A front crosses the interface between the upper and lower quadrants (slightly before 190 ms), the first turn of rotating propagation is just being completed. Successive turns follow basically the same pattern.

At each point on the curling A and R fronts, the con-

duction velocity depends on the anisotropic conductivi- ties, the state of excitability recovery, and the local curvature. Occasional contacts of the wave fronts with the sheet boundaries may also affect the velocity. Since the q-point is common to both the A and the R fronts, the two must have the same velocity at that particular location.

In excitable media, the dynamic process of activation is quite different from that of excitability recovery. The A front propagates because the diffusion current brings the local potential above threshold and triggers a local depolarization process controlled by the sodium conductance. For propagation to be maintained, there must be a sufficient area of membrane that can be activated ahead of the front. This area is defined by the Eg. If the Eg is too small, the A front enters a zone of incomplete excitability recovery and higher threshold, the diffusion current is partly ineffective, and the conduction velocity decreases. Simi- larly, the local A front velocity decreases with curvature because a greater area of membrane ahead of the front must be brought above threshold by the same quantity of diffusion current.

Contrary to the A front, the R front moves away from a repolarized region and propagates into a depolarized one. Its localization is rather arbitrary because it does not correspond to a threshold crossing event. Even in the absence of any diffusion current, the R front can still be detected, since a spatially distributed recovery necessarily follows in the wake of the A front. In some instances, a source of diffusion current may develop behind the A front (such as triggered repolarization, as described later) and


accelerate the recovery process for a while. Hence, the position and movement of the R front, as defined here, reflect the delay after the passage of the A front, as well as the integrated effects of diffusion currents affecting the plateau phase of the action potential.

When the A front develops a high curvature, it is slowed down by the drag current. This occurs in the vicinity of the q-point (e.g., at 150 and 190 ms in Fig. 4), such that the q-point has a tendency to move slowly also. Correspondingly, the R front tends to follow the A front and to reflect any perturbation that may have accelerated or delayed the repolarization process in the interim. Since the R front is detected at h = 0.1, the repolarization is then close to the diastolic potential and the repolarization gradient is quite low. Thus the diffusion currents on the R front are small and the drag effects negligible. Therefore, contrary to the A front, the R front velocity is not signif- icantly affected by its curvature in this case.

On the other hand, the R front is attached to the q-point and, as depicted in Fig. 4, the rotating patterns yield a narrow Eg near the center of rotation. There is then an additional slowing effect imposed on the A front, which is also exerted on the q-point. In this way, since the R front tends to pursue its trajectory, it is pulled by the nearly immobile q-point and forced to rotate. By pivoting around the q-point, the R front moves away from the A front and produces an increased Eg. In a similar way, since the A front's head continues to move forward, the A front also pivots around the q-point. Hence the curling R front always precedes the curling A front around the center of rotation.

This behavior is depicted in Figs. 4 and 5. The flat transverse component of the R front (upper left quadrant of the h-map at 170 ms in Fig. 4) has disappeared at 190 ms, and the R front is now in a position to develop its full curling pattern around the q-point. In this particular case, the reentry is maintained over several consecutive turns, and the end of the eighth turn (at 680 ms) is shown in Fig. 5A. At this instant, the curling A front is well delineated

FIGURE 5. Reentry with AR = 4:1 and K = 8 at 680 ms after $1. (A) V-map and h-map as in Fig. 4 (B). V(left) and h (right) vary along the horizontal dotted line of (A), Symbols are as defined in legend of Fig, 4.

and its propagation is ensured by large depolarization gradients, as indicated at the intersections between the A front and the horizontal reference broken line (upstrokes A in Fig. 5B). On the other hand, the repolarization is spread over a large area around the A front, and its spatial distribution has a much lower longitudinal potential gradient (Fig. 5B). The right and left gradients have about the same value in this case, despite the contact with the border of the sheet on the left-hand side, which prevents full repolarization (maximum of about - 3 0 mV). Although this incomplete repolarization does not affect the R front velocity appreciably, it does prevent full excitability recovery at the A front, as indicated by the slightly lower h amplitude on the h-graph of Fig. 5B. On the whole, boundary effects do not have a significant direct influence on the R front, but they may decrease the upstroke (INa) and thus reduce the A front velocity. Because of the large magnitude of INa, however, a noticeable reduction in conduction velocity can occur only as a result of a major decrease in h, as illustrated during slow propagation in the ring model of reentry (see Fig. 10A of the accompanying paper [29]).

In the vicinity of the q-point, the high curvature at the A front's head (h-map; Fig. 5A) slows the rightward propagation despite the substantial longitudinal Eg of about 0.5 cm. Hence the q-point is also moving very slowly and is in fact almost immobile under these conditions. Repolar- ization begins to occur over a small area behind the q-point, as shown on the V-map of Fig. 5A. When examined along the horizontal reference broken line, this early repolarization appears as a dip in potential between the two A fronts in Fig. 5B. As explained below, this behavior is indicative of the phenomena of triggered secondary repolarization and excitability recovery.

Regular Circus Movement. When the wavelength is relatively short, the reentrant wave can be easily accommo- dated by the dimensions of the sheet, and the minimum Eg is such that regular circus movement is possible. This occurs with K = 8, which defines a wavelength of 2.25 cm just before the onset of irregular reentry in the ring (Table 1). The corresponding CL and APD are 60 and 45 ms, respectively. A given turn of reentry is illustrated by the isochrone map of Fig. 6A, which depicts the A fronts at every 10 ms. This particular turn at 430--470 ms occurs between the last set of maps of Fig. 4 (190 ms) and the set of Fig. 5A (680 ms). The full dot at the tip of each isochrone is the q-point (arrow), which drifts slowly over a small area.

The h map of Fig. 5A depicts the typical pattern of sustained circus movement in the anisotropic sheet; that is, a curling R front followed closely by a curling A front and separated from it by the Eg. The most important Eg variations are expected to occur in the longitudinal direction,


ms in Fig. 6B result from the peculiar shape of the R front, which tends to display a wide longitudinal expansion. This behavior of the R front cannot be explained in terms of velocity changes. Instead, it appears that the phenomenon is due to triggered secondary repolarization and excitability recovery associated with very slow propagation of the A front when the Eg becomes very small.

FIGURE 6. Reentry with AR = 4:1 and K = 8. (A) Isochrones corresponding to A fronts of Figs. 4 and 5A, plotted at 10 ms interval. The full dot at the tip of each isochrone is the q-point. (B) Activation-recovery distribution (A.R-map) representing the superposition of three h-maps at 340, 360, and 380 ms after stimulus $1. The area between the A and R fronts is the excitable gap (Eg). The full dot is the q-point. (C) Locus of q-points for ten successive cycles of reentry.

where differences in A and R front velocities are amplified by the intrinsically larger longitudinal conduction velocity (0xF ~ 4 �9 0yF). This occurs primarily in the neighborhood of the q-point, where the A front velocity is drastically reduced by the small Eg, whereas the R front velocity is not subjected to such a restriction. Hence, despite its tendency to rotate around the q-point, the R front can move away from the A front and increase the longitudinal Eg. This behavior is further illustrated by the activation- recovery map of Fig. 6B, which displays three patterns similar to that shown on the h-map of Fig. 5A. The large longitudinal Eg ahead of the A front's head ensures the unimpeded A front propagation toward the right at 340 and 360 ms, and toward the left at 380 ms. In other words, because of the slow displacement of the q-point, the rotating R front can distance itself from the A front and facilitate propagation by increasing the Eg. The locus of q-points during several turns of reentry with K = 8 (Fig. 6C) shows the rather limited drift associated with these conditions. Because of the short wavelength, the sheet dimensions and AR provide ample room for circus movement, which then exhibits a stable sustained pattern.

The Eg deformations near the q-point at 340 and 380

Triggered Secondary Repolarization and Excitability Re- covery. Given the above results, it may be expected that a longer basic wavelength will make it more difficult to achieve sustained circus movement reentry under similar conditions. To examine this hypothesis, we simulated two-dimensional reentry with K = 4, corresponding to a basic wavelength of 3 cm and an APD of 60 ms near the bifurcation point in the ring (Table 1).

Consider the set of V- and h-maps at 310 ms in Fig. 7 (obtained with K = 4), which display patterns resembling those of Fig. 5A (obtained with K = 8). Potential changes along the two orthogonal reference broken lines show that the A front's head is driven by a large upstroke (A in Fig. 8A), as was also the case in Fig. 5B. The tongue of repolarization behind the q-point, centered roughly at the intersection of the orthogonal reference lines, corresponds to a dip in plateau potential (asterisk in Fig. 8A) located at a short distance behind the A front's head. Potential changes along the transverse reference line show that the upward moving A front (upper left quadrant) is driven by a considerably attenuated upstroke (A in Fig. 8B), owing to the small transverse Eg (see h map at 310 ms; Fig. 7). Hence the A front velocity is temporarily reduced until the R front moves away. This perturbation causes a transfer of charges from an upstream region of membrane to the site of slow conduction in order to sustain propagation, thus giving rise to the observed dip in plateau potential (29).

At later times, the A front's upstroke has resumed its normal amplitude (A in Figs. 8D and 8F), since the transverse Eg is now sufficiently large (h-maps at 340-350 ms; Fig. 7). The dip in plateau potential behind the A front reaches the diastolic level at 340-350 ms (asterisk in Fig. 8) and reflects the enlargement of the tongue of secondary repolarization in the V-maps of Fig. 7. As the A front's head moves rapidly away, an elongated gray-black link persists between the nearly stationary tongue of "secondary" repolarization and the moving region of "primary" repolarization ahead of the A front.

The progressive growth of secondary repolarization on the V-maps is accompanied by the emergence of a small black island of secondary excitability recovery on h-maps, visible in this case at 340-350 ms in Fig. 7. In the meantime, the q-point remains attached to the A front, near the A front's head, and moves rapidly in the rightward direction. Thus the onset and development of secondary repolarization and excitability recovery are not governed by


FIGURE 7. Reentry with AR = 4:1 and K = 4. V-maps (left column) and h-maps (right column) as in Fig. 4, with same symbol definitions.

the properties of the q-point. An important additional feature of these events is their relative immobility, such that they both eventually merge completely with the primary repolarization zone and primary Eg at 380 ms. As a result, there is a jump increase in the Eg, which then becomes quite large in the longitudinal direction.

Similar events take place near the A front's head in the longitudinal direction (just beyond 350 ms in Fig. 7; not illustrated) when the longitudinal Eg becomes small enough to slow the A front. The net results appear at 380-390 ms, where the tongue of secondary repolarization is well developed on the V-maps, and a new island of secondary excitability recovery is visible on the h-map at 390 ms.

The pattern shown at 300 ms in Fig. 7 is also the result of a similar process that was triggered at some earlier time

(not illustrated). In this case, the merging of the secondary and primary repolarization areas is already completed at 300 ms, as is the merging of the secondary and primary zones of recovered excitability. Then, since the R front (defined by secondary recovery) is nearly immobile, the fast moving A front's head catches up and the Eg decreases rapidly.

These phenomena are related to the triggered waves of secondary repolarization and excitability recovery described during one-dimensional reentry in the ring (29) when the wave front catches up with the wave tail and produces a large decrease in conduction velocity. The same underlying mechanisms apply in both cases, with proper allowance for the differences between one- dimensional reentry and two-dimensional anisotropic reentry. The main features are as follows:


Longitudinal

A

>

Transverse

B ~" 2o t 31o ms "~A

-6o

C 2 0 | , ~ a40ms ~..~ -20 . . . . . .

D ~ ~ 340ms

/

E F L 3ooms

-20 . . . . . . . .

> -60 > -oo L/ _ ~ . J

0 2 4 0 i Position (cm) Posit ion (cm)

350 ms

FIGURE 8. Longitudinal (left column) and transverse (right column) potential variations along the orthogonal broken lines at 310, 340, and 350 ms in Fig. 7. A = activation front. Asterisk marks the site of early local repolarization (A and B), or full triggered repolarization (C-F).

1. Triggered repolarization is due to a dip in plateau potential arising from a current sink created by the slow propagation of the preceding A front. The sequence of events was described in detail for the ring (29).

2. Since the trigger is a dip in plateau potential occurring at a specific location relative to the causal A front, secondary repolarization and recovery are independent of the subsequent behavior of the A front, q-point, and primary R front.

3. In two dimensions, the site is bounded by slow moving secondary repolarization and recovery fronts. Because of the sheet anisotropy, the longitudinal dimension develops more rapidly and gives rise to an elongated pattern. When the secondary fronts merge with the primary fronts, there is a jump increase in the area of full repolarization and Eg.

4. Slow propagation of the secondary fronts in any direction results from the low potential gradient and the small transmembrane current. Full secondary repolarization is a regenerative process involving mainly the delayed activation of Isi, since at this early stage I K is not appreciably involved because of its slow time constant of activation (29). In consequence, triggered recovery fronts in the vicinity of the q-point are nearly stationary, and as soon as the activation front acceler- ates, the Eg decreases rapidly.

Some of these features are summarized on the activation-recovery maps of Fig. 9. The large Eg reduction at

320-360 ms is accompanied by a slowing of the q-point, which reaches its rightmost position around 360 ms (Fig. 9B). The island of recovered excitability appears between 320 and 340 ms, it grows larger at 360 ms, and the merging with the primary recovery front is accomplished at 380 ms.

The appearance of triggered waves of repolarization and excitability recovery corresponds to a new dynamic

FIGURE 9. A-R-maps of reentry as in Fig. 6B, with AR = 4:1 and K = 4. Timing corresponds to that of Fig. 7. Expanding island of triggered secondary recovery at 340 and 360 ms, which merges with the primary repolarization front at 380 ms.


regime in which the recovery is not directly linked to the A front and q-point. Its manifestation requires a propagating impulse of minimum complexity, as provided here by the different time scales of the upstroke, plateau, and repolarization phase of the action potential and related excitability fluctuations. In a similar vein, it has been shown that the presence of a piecewise time constant for the recovery variable of the FitzHugh-Nagumo model intro- duced peculiarities in the spatial distribution of the vul- nerable window to induce reentry in a two-dimensional medium (14). Here, however, because of the presence of the triggered recovery waves, this phenomenon is not equivalent to the compound quasi-periodic rotation of a single vortex with unique R and A fronts (2,34). The global dynamics now depends on the respective behavior of the primary recovery front and triggered recovery wave and eventually on their collision.

Circus Movement with Triggered Repolarization and Ex- citabilityRecovery. The events depicted in Fig. 10 provide an illustration of the contribution of triggered secondary repolarization and excitability recovery to sustained circus movement with K = 4. Starting at 280 ms on the isochrone map of Fig. 10A, we see that the A front's head travels rapidly in the longitudinal direction. The nearly

FIGURE 10. Reentry wi th AR = 4:1 and K = 4. (A) Isochrones plotted at 10 ms interval and q-points as in Fig. 6A (B) A ,R - map as in Fig. 6B at 410 -480 ms after stimulus $1. (C) Locus of q-points for ten successive turns of reentry.

elliptic curve expands progressively at 300 and 320 ms as it approaches the right edge of the sheet. The A front at 280 ms can be compared with that at 410 ms in Fig. 10B, which occurs about one turn later. As with K = 8 in Fig. 6, the more rapid longitudinal propagation of the R front's head brings about a progressive increase in the magnitude of the longitudinal Eg. There is a large distortion of the R front near the q-point at 410 ms in Fig. 10B, owing to the collision with a triggered front of secondary recovery.

In the meantime, the q-point travels more slowly in a slightly oblique downward direction and reaches its rightmost position at 300 ms in Fig. 10A. Comparing Figs. 6C and 10C, we see that the shape of the Eg region near the q-point amplifies the drift of the q-point with K = 4. It thus appears that the triggered waves of secondary repolarization and excitability recovery have an important influence on the behavior of the q-point.

Overall, all other things being equal, an increased wavelength amplifies the differences in A and R front velocities and thus makes the Eg more variable. Because of the merging of the triggered recovery waves with the primary R fronts, the q-point has a greater tendency to move about, and it drifts over a broader locus during successive turns.

Unsustained ReenO~y. Since sustained reentry was obtained with K = 8 and K = 4, we examined the situation involving a larger basic wavelength. With K = 2, the minimum wavelength of stable reentry in the ring is 4.2 cm and the corresponding APD is 84 ms (Table 1). The 40% increase in wavelength, compared with the previous case with K = 4, may then be expected to lead to unsustained two-dimensional reentry.

The isochrone maps of Fig. 11 display two apparently regular reentry cycles (Figs. l l A and l lB) , followed by progressive changes leading to termination (Figs. 11C and 11E). In comparison with Fig. 10A (K = 4), we see that the longer wavelength yields activation fronts of generally greater curvature in the longitudinal direction and generally lower curvature in the transverse direction. The locus of q-points also extends over a larger area in Figs. 11A and 1 lB.

The first sign of potentially unstable behavior is the appearance of visible distortions of the isochrone curves at 410-440 ms in the lower left quadrant of Fig. 11C. As illustrated by the h-map at 420 ms in Fig. 12, these distortions are linked to the occurrence of a very small Eg in this area of the sheet. The locus of q-points (Fig. 11C) also exhibits a peculiar shape. The behavior of the q-point is then governed by a more complex pattern of repolarization and excitability recovery, as shown at 420 ms in Fig. 12. The main feature in this case is a residual white spot on the V-map, just ahead of the q-point, surrounded by a gray

604 L.J . LEON, F. A. ROBERGE, and A. VINET

A

B

C

E

FIGURE 11. Reentry with AR = 4:1 and K = 2. Isochrones plotted at 10 ms interval and q-points as in Figs. 6A and 10A. Time (in milliseconds) is measured from the onset of S 1.

contour. This situation brings about a decrease in the Eg and a slowing of the q-point. The result is a modified q-point locus that is indicative of forthcoming repolarization and recovery disturbances. For example, the h-map at 420 ms shows a wide separation between the primary R front and the large black island of recovered excitability on the right, such that the absence of a large Eg makes it impossible for the A front to accelerate longitudinally in the usual manner (e.g., as in Fig. 7).

In this particular example, the consequence at the next turn is a split of the q-point locus, as shown in Fig. 11D and at 500-560 ms in Fig. 12. A triggered recovery wave becomes visible on the h-map at 500 ms, and it grows substantially at 520-530 ms. Whereas the region between the primary R front and the island is partly repolarized at

520-530 ms, as shown on the V-map, the Eg remains very small and no connection is made between the two areas of recovered excitability on the h-map. As a result, the A front 's head and q-point move away from the island and a pair of new activation fronts are initiated on the edge of the island (arrows on the h-map at 540 ms). A similar process is repeated by the appearance of two triggered waves at 560 ms in Fig. 12 (black islands).

The creation of an additional pair of q-points at 540 ms corresponds to the appearance of a new isochrone segment on the map of Fig. 11D. Transverse and longitudinal propagation of the q-points accompany the expansion of this new activation front. If the pattern were stable, one would expect to see the typical figure-of-eight reentry observed in experimental preparations (7,10). In the present case, however, the pattem is unstable, and at the next turn, additional activation fronts are created (Fig. l lE). This extensive splitting of the q-point locus indicates that the reentry progresses toward its termination.

Temporal Features of Reentry. With a short wavelength (K = 8), the reentry is rapid and shows only small cycle length fluctuations: CL = 66 --- 4 ms (Table 2). APD and D1A measurements taken over the whole surface of the sheet during several successive tums indicate a low degree of variability: APD = 48 --- 3 ms andDIA = 18 +- 3 ms.

The largest fluctuations in action potential characteristics occur in the neighborhood of the locus of q-points, as illustrated in Fig. 13A from a tracing taken at point 1 in Fig. 4. Velocity and Eg variations in this region (Fig. 6) lead to wide fluctuations in maximum and minimum membrane potential levels. On the whole, there is a sustained shift of diastolic potentials toward depolarized values. On the other hand, observations at a site away from the q-point locus (point 2 in Fig. 4) show relatively small variations in action potential amplitude and diastolic potential (Fig. 13B). The conduction velocity remains essentially constant in this region, since the minimum Eg is fairly large.

A larger wavelength (K = 4) yields a longer and more variable cycle length of 88 -+ 14 ms (Table 2). The corresponding APD and DIA values show a slightly lower degree of variability: APD = 63 --- 9 ms and DIA = 25 -+ 10ms .

Increasing the basic wavelength to about 4.2 cm (K = 2) reduces the cycle length further and dramatically increases the range of fluctuations: CL = 117 -+ 36 ms, APD = 81 -+ 31 ms, and DIA = 36 - 18 ms. Typical tracings taken at points 1 and 2 in Fig. 12 are displayed in Fig. 13C and 13D, respectively. The neighborhood of the triggered wave (Fig. 13C) is the site of very short APDs and DIAs, and the range of values is large. On the other hand, there are much more limited variations in action


FIGURE 12. Reentry with AR = 4:1 and K = 2. V-maps (left column) and h-maps (right column) as in Figs. 4 and 7. Arrows on h-maps point toward q-points. Timing corresponds to that of Fig. 11.

potential characteristics away from the q-point locus and triggered wave region (Fig. 13D).

DISCUSSION

Circus movement in a parallel-cable sheet model of myocardium was initiated via a conditioning planar wave of excitation (stimulus S~) intersected by a voltage gradient applied perpendicularly (premature stimulus $2). If the timing of S 2 is chosen properly, the result is a premature activation front that propagates toward the tail of the conditioning wave (32). The presence of spatially different recovery states results in curling of the activation front around a moving center of rotation and generation of reentrant activity. The initial location of the center (first turn of reentry) is largely determined by the timing of S z. In the present study, the initial center of rotation was chosen near the middle of the sheet in order to minimize boundary

TABLE 2. Reentry in the sheet model with AR = 4:1

K = 8 K = 4 K = 2

APD(ms) 4 8 - + 3 63-+ 9 81 -+31 DIA (ms) 18 + 3 25 +- 10 36 -+ 18 CL (ms) 66 -+ 4 88 -+ 14 117 +- 36

effects. During subsequent turns, its location depended on the dynamics of reentry. It meanders in the vicinity of the middle of the sheet during sustained reentry (Figs. 5-10), and multiple centers of rotation appeared during unsustained reentry (Figs. 11 and 12).

In the present study, the AR was kept unchanged at a value of about 4:1, the sheet dimensions were fixed at 1.5 • 5 cm, and the impulse wavelength was modified by changing the duration of the action potential plateau. This modification was achieved by dividing the activation and inactivation time constants of Isi by an integer K, thereby modifying the action potential plateau without affecting too drastically its repolarization phase, which depends primarily on the outward potassium currents. The implica- tions of different K values were assessed in a preliminary way from reentry simulations on a uniform ring, using ring lengths just above the limit for stable reentry (just above the horizontal dotted line in Fig. 3), thus providing the data of Table 1. These reference values indicate that the basic wavelength and APD increase as K is decreased. Although the wavelength of two-dimensional reentry changes continuously over the sheet during reentry, it is nevertheless related to a specific K value. On this basis, we can infer that a shorter wavelength of reentry is associated with a larger K value in the sheet, in correspondence with the observed fairly regular sustained reentry


A 5O

0

E >

-50

B 50~

0

> -50

K = 8

i , i , i , i ,

4O0 800 Time (ms)

' i , i , i i

' ' , L i , ~ , 7 i ~i !

, I ' 4 ( ~ 0 ' I 8 0 0 '

Time (ms)

C

50

-SO

D

5O

~o

-SO

K = 2

i i

i r

400 800 Time (ms)

I I t

4 0 0 8 0 0

Time (ms)

FIGURE 13. Strips of activity corresponding to recording sites near (site 1) and far from (site 2) the q-point. (A and B) Sites I and 2, respectively, for K = 8 (Fig. 4). (C and DI Sites 1 and 2, respectively, for K = 2 (Fig. 12).

with K = 8, increasingly meandering sustained reentry with K = 4, and unsustained reentry with K = 2.

The present structural organization of the sheet model was chosen on the basis of experimental observations on cardiac tissue (23,27). In the longitudinal direction, the high density of intercellular connections, the rapid conduction of impulses, and the smooth action potential waveforms justify the choice of a continuous-cable representation. In the transverse direction, the relative sparsity of intercellular junctions, the lower conduction velocity, and the more abrupt membrane potential fluctuations are in agreement with a structural brick-wall pattern. Al- though this arrangement gives rise to microscopic fluctuations in action potential characteristics (16), the macro- scopic features of propagation, as described by flat or elliptic wave fronts (15), are quite similar to those observed in sheets of symmetrically connected cells (22). On the other hand, since transverse propagation does not scale uniformly with the transverse resistance in the present model (see Methods and Fig. 1C), it can be expected that some features of reentry will change with the anisotropy ratio. These questions are currently under investigation, and preliminary results indicate that simple scaling of vortex-like reentry by the AR (20) does not occur in the present model.

The results of the present study are related to those described in a companion paper on one-dimensional reentry in the homogeneous ring (29). In the ring, the transition between stable and irregular reentry, or its spontane-

ous termination, was established in terms of a critical ratio between the basic wavelength of the reentrant impulse and the ring length. Similarly, in the sheet, the ratio of the basic wavelength to the sheet size determines whether the reentry is stable, irregular, or terminates spontaneously. As in studies using the FitzHugh-Nagumo model with a short action potential duration (19,20), simulations with a short basic wavelength tend to yield spiral-like reentrant patterns. With a fixed sheet size and fixed AR, a longer basic wavelength increases the importance of boundary effects. Such effects are likely to be more important in the longitudinal direction because of the larger conduction velocity.

These basic character is t ics of s imulated two- dimensional reentry are in accord with experimental observations in isolated strips of rabbit atrium in which the initiation of reentry was greatly facilitated by lowering the extracellular potassium concentration, thereby decreasing the basic wavelength and the conduction velocity of the reentrant impulse (25). Comparable results were obtained in the same study using pharmacological agents to either shorten the wavelength in order to obtain sustained reentry, or lengthen the wavelength in order to have sponta- neous termination.

Reentry Around a Fixed Obstacle

As a first step, the main features of two-dimensional circus movement can be discussed in relation to reentry


around a fixed obstacle, using a sheet with an inert central core. Consider a hypothetical concentric rectangular hole with dimensions proportional to those of the sheet, and assume that the conduction velocity along its perimeter is constant longitudinally (50 crn/s) and transversely (12 cm/ s). If the hole is large enough to leave only a narrow conducting strip around the perimeter of the sheet, the pathway is then equivalent to a ring and we can calculate a cycle length of 440 ms for a pathway length of 13 cm. Assuming an ARP of about 225 ms with K = 1 (29), the equivalent excitable gap (Eg = 0 (CL - ARP) = 6.34 cm) is nearly half the ring length and thus amply sufficient to ensure stable reentry.

A decrease in the hole size yields a minimum conducting pathway along the internal perimeter of the perforated sheet and forces the reentry to adopt a shorter cycle length. For example, suppose that the internal perimeter is 3.25 cm (hole of 0.375 x 1.25 cm). Under the hypothesis of constant longitudinal and transverse conduction velocities along the internal perimeter, the cycle length is reduced by a factor of 4 to 110 ms. Given the ARP of 225 ms with K = 1, it is clear that reentry would be impossible under these conditions. It is then necessary to shorten the ARP (or the wavelength, since the two quantities are equivalent when the conduction velocity is constant). With K = 2 (ARP = APD + 30 = 114 ms) the ARP would still be larger than the cycle length of 110 ms and reentry would be impossible. Using K = 4 (ARP = 90 ms) yields an Eg

of 0.53 cm, which is about 16% of the pathway length of 3.25 cm. Given our simulation results on one-dimensional reentry in a homogeneous ring (29), reentry at constant velocity is unlikely with such a short Eg. It is therefore necessary to reduce the ARP further, and an impulse defined by K = 8 (ARP = 75 ms) is likely to be appropriate, since the calculated Eg of 0.97 cm would represent about 30% of the pathway length. These observations provide a kind of upper bound for stable circus movement, since they do not include the decrease in velocity associated with both premature activation and wave front curvature.

In the homogeneous ring model, conduction velocity fluctuations correspond to quasi-periodic reentry (29). The principal underlying mechanism is one of transient slow conduction at each cycle, accompanied by a triggered wave of secondary excitability recovery. Since an analogous phenomenon was observed during reentry in the present sheet model (Figs. 7-9), a similar source of cycle length and conduction velocity fluctuations could be pre- sumed to occur in the perforated sheet. Thus circus movement around the hole would occur in the form of curling activation and repolarization fronts consistent with quite different propagating conditions at the inner and outer pe- rimeters of the sheet. In particular, a high A front curvature would prevail near the internal perimeter when the velocity decreases, and a lower curvature would be seen

along the outside perimeter, where the velocity would remain essentially constant. Differences in curvature, leading to different A front velocities, would then constitute an additional source of Eg variations, which would reinforce the effects of slow conduction and triggered secondary excitability recovery.

With the present sheet model, the above considerations indicate that two-dimensional reentry around a fixed obstacle may be expected to be roughly analogous to one- dimensional reentry in a ring. When the basic wavelength is short relative to the minimum pathway length, conduction velocity fluctuations along the internal perimeter are likely to be small and no triggered waves secondary repolarization and excitability recovery would be expected. On the other hand, if the basic wavelength is long enough relative to the minimum pathway length, the reentry could be quasi-periodic (29). If it is the case, the main pattern would then be one of alternations between long and short CL and action potential characteristics and cyclic wavelength and Eg changes. This behavior would result from the following circle of events: small Eg ~ transient slow conduction ~ triggered waves of secondary repolarization and excitability recovery ~ large Eg ~ transient rapid conduction ~ small Eg . . . Differential conduction velocities of the A front due to curvature differences could also be a factor causing Eg variations. This analogy is incomplete, however, because the A front of two- dimensional propagation has much greater directional flexibility than in the ring, particularly in the presence of a wide Eg created by the collision between the primary recovery front and the triggered wave of secondary excitability recovery.

Experimental studies of two-dimensional anisotropic reentry around a fixed obstacle have produced results of limited interest so far, mainly because of the large dimensions of the central inexcitable core. In some studies of isolated rabbit myocardium preparations (4,5), the ratio of the impulse wavelength to the pathway length gave rise to a large Eg, which made it possible to have stable reentry since the inexcitable core dimensions were comparatively large also. The description of echo reentry (6) in this preparation is of some interest, but since it arises through a process of micro-reentry away from the central core, it is only marginally related to the principal phenomenon of functional block.

Reentry Around a Core of Functional Block

In the uniform sheet, the center of rotation is defined by a unique point of junction (the q-point) between the activation (A) and recovery (R) fronts at each instant. During reentry, the meandering q-point defines a locus of partly depolarized membrane that is the site of conduction block. This locus is shaped by the AR, the sheet dimensions, and the basic wavelength of the reentrant impulse.


As pointed out in a companion study on one- dimensional reentry in a ring model (29), the occurrence of triggered secondary excitability recovery and complex reentry patterns requires a membrane model of minimum complexity. In the case of the MBR and MBR-1 models (see Methods), there are three different time scales gov- erning the development of membrane potential and excitability fluctuations: short time constants for the action potential upstroke, medium time constants for the plateau, and long time constants for the repolarization phase. If similar results are to be obtained with simplified models of membrane kinetics, it would be necessary to meet these minimum requirements, as seems to be the case with a particular version of a FitzHugh-Nagumo model used in a recent study (18).

The success of computer simulations, however, requires that proper attention be paid to the accuracy of numerical solutions, particularly in light of the important role played by excitability properties at the microscopic level. On the temporal scale, the required time step is determined by the fastest time constant of the system, and the conditions for accurate solutions are chosen in a straightforward manner. Difficulties arise with respect to spatial discretization. Clearly, if the spatial grid is too sparse, important discretization errors will occur and basic dynamic phenomena will be filtered out. As pointed out in an earlier study (22), a discretization factor (the ratio of the space step over the resting length constant) of 0.1 gives a good approximation of the continuous case and a value of 0.2 may represent an upper limit. These requirements are satisfied in the present study. In some published works on two-dimensional reentry (8,20), however, there is a possibility of large discretization errors, which may have an impact on the qualitative features of the results. In brief, given the importance attributed to the continuous reaction-diffusion equation as an underlying model of reentry (32), simulation studies must provide reasonably accurate solutions if the results are to be of interest.

The locus of q-points is somewhat similar to a moving zone of unexcited membrane around which circus movement takes place at each cycle. It is a very small area that, in the present anisotropic sheet, tends to be thin and elongated. We have not yet studied in detail the dynamics of the q-point, and only some qualitative insight derived from a comparison with one-dimensional reentry on a homogeneous ring (29) is presently available to us. Slowing of conduction around the q-point locus and the generation of triggered waves of secondary repolarization and excitability recovery are similar to the events observed in the ring during quasi-periodic reentry. The precise features of sustained irregular reentry in the sheet remain to be established, and we cannot conclude at this stage that it is quasi-periodic. There are wide variations in APD and DIA values over the surface of the sheet during successive cy-

cles, and short action potentials are necessarily associated with local slow conduction. The alternation patterns seen during one-dimensional reentry might well be present during irregular reentry in the sheet, but a reliable spatio- temporal study of several consecutive cycles is needed to draw firm conclusions.

The locus of q-points during a reentry cycle in the model is analogous to the arc of unidirectional block described in experimental studies (4,5,24,25). The analogies include the changing shape of the area of unidirectional block at each cycle and its displacement over the surface of the sheet. So far, because of the absence of direct information on the dynamics of the recovery waves in the experimental preparation, it has not been PoSsible to describe the arc of unidirectional block in terms of a well- defined q-point and to detect the occurrence of triggered secondary repolarization and excitability recovery. Im- proved experimental techniques are needed to assess the importance of the predictions of the present study.

Similarly, the process of reentry termination in the sheet is very complex and involves a fragmentation of the locus of q-points, associated with the generation of additional activation fronts (Fig. 11). We have not studied the corresponding behavior of the recovery front in this case, and there are technical difficulties in describing the phenomenon in terms of Eg variations in the model. This is a desirable goal, however, and this problem is the subject of ongoing studies. At a minimum, we can suggest that the splitting of the q-point locus into two segments, giving rise to the well-known figure-of-eight pattern of two- dimensional reentry (7,10), may be a higher level of ir- regularity. It might perhaps be reasonable to suggest that further fragmentation of the q-point locus into several segments could be a prelude to the onset of fibrillation.

In conclusion, we can say that a relatively short wavelength (K = 8 or K = 4) affects the meandering of the q-point, but we do not know the precise role played by the triggered waves in this respect. With a longer wavelength (K = 2), which leads to the breakup of the q-point locus, the triggering of activation fronts at a distance constitutes a novel and intriguing feature of the dynamics of two- dimensional reentry.

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Simulation of two-dimensional anisotropic cardiac reentry: Effects of the wavelength on the reentry...

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