Simulation of Reentrant Propagation in Cardiac Tissue

Copyright e IFAC Modeling and Cootrol in Biomedical Systems. Galveston. Tcua. USA. 1994

SIMULATION OF REENTRANT PROPAGATION IN CARDIAC TISSUE F.A. Roberge, A. Vinet, L.J. Leon Institute of Biomedical Engineering University of Montreal, Montreal, Canada ABSTRACT

myocardium may occur around a zone of functional UB. In this type of reentry, the absence of a fixed anatomical anchor leaves the zone of UB free to vary in size and to wander through the myocardium at each cycle. The changing characteristics of the reentry pathway are then likely to give rise to cortl>lex tachyarrhythmias which may degenerate into fibrillation (3).

A uniform and continuous ring model of cardiac tissue was used to simulate reentry around a fixed obstacle. Stable one-dimensional reentry was obtained with a sufficiently long ring, and complete block occurred below a minimum ring length. Irregular reentry was observed at intermediate ring lengths, in the fonn of reament or quasi-periodic patterns. Sustained quasi-periodic reentry was associated with a zone of slow conduction which was located on a different part of the ring at each turn. Reentry in two-dimensional myocardium was simulated by means of a redangular sheet of parallel uniform and homogeneous identical cables. Stable two-dimensional reentry occurred around an arc of functional unidirectional block which varied In size and location at each turn. Irregular twodimensional reentry was associated with regions of slow conduction which led to a fragmentation of the arc of unidirectional block and the generation of multiple activation fronts. Overall our simulation results with the ring model closely resemble experimental observations in a canine atrial tissue ring around the tricuspid orifice (1). Similarly, our results on simulated two-dimensional reentry are in accord with experimental observations In isolated strips of myocardium (2).

Unidirectional block may occur in uniform structures with homogeneous membrane properties. Propagation of activation fronts through regions of incompletely recovered excitability would then yield potential gradients which can induce fundional UB and initiate reentry (4). In the present paper the dynamic features of sustained reentry are examined using a ring model and a two-dimensional sheet model of cardiac tissue with spatially uniform membrane and conductivity properties. The ring model is a closed loop formed by a one-dimensional continuous and uniform cable surrounded by an unbounded volume conductor of negligible resistivity. The cable radius is 5 J,1m, the intracellular resistivity is 200 Q-cm, the resting length constant is 837 J,1m, and the steady-state conduction velocity is 71 ;4 crn's. The membrane model (MBR) is a modification of the Bee\er-Reuter formulation to include a faster Na+ current (5,6). The membrane capacitance is 1 J,1F/cm2 and the total ionic current is the sum of a fast Na+ current (ItW, a secondary inward current (I.J carried mainly by calcium ions, and two K+ currents. We have INa .. g Narn3hj(V- EtW where V is the transmembrane potential, g Na - 15 mS/cm2 the maximum condudance, ENa - 40 mV the reversal p0tential, and m,h,j the gating variables. We also have lsi - g sjdf(V- Esi) where g si - 0.09 mS/cm is the maximum conductance, Eai Is a calcium~ependent reversal potential, and d,f the gating variables. Any gating vari-

INTRODUCTION There is substantial evidence that reentry is the most likely underlying mechanism of severe arrhythmias and that unidirectional block (UB) is a necessary requirement for the initiation of reentry. Reentry in rings of cardiac tissue around a fixed obstacle (e.g., around a cardiac orifice) can produce either stable regular rhythms or alternations between long and short intervals, depending on the pathway length. Reentry in two- or three-dimensional

81

able y corresponds to a first-order process descrl>ed by a steady-state characteristic y.(V) and a time constant 'ty(V).

Temporal events at a given site on the ring (Fig.1 B) are described In terms of latency (L), action potential duration (APO), and diastolic Interval (OIA). Reference points are V., Vmin (minimum diastolic potentiaQ, and V-50 (-SO mV repolarization). OIA is a temporal scale which enco~sses ~ changes, and a one-to-one correspondence exists between ~ and OIA in the MBR model. By definition we have OIA - 0 at V-sJ)and, since OIA:.;30 ms at hj - 0.1, ARP -APO + 30 ms. The conduction time (CT), which Is the inverse of the conduction velocity, is measured over a distance of 100 JUll.

Changes In ring length (X) were used to modify the cycle length (CL) and the dynamics of reentry. The potential V. at maxirTIJm dV/dt of the upstroke is used to Identify the time and site of occurrence of the activation fmnt. Excitability or activation recovery (Ar.c> begins at the end of the absolutely refractory period (ARP) and corresponds to the reactivation of the sodium channels. Thus we have ~ - gNahj. The end of the ARP is taken to correspond to hj - 0.1, which is the minimum value to get an action potential in the space-clamped MBR model. The wavelength (WL) corresponds to hj s 0.1 and the excitable gap (EG) to hj>0.1 (Fig.1A).

A

V(mv)

The sheet model can be assimilated to a strip of cardiac muscle in which propagation along the depth of the tissue is neglected. It consists of a rectangular single layer of parallel and unHonnly spaced Identical cables, transversely interconnected by a regular resistive network (7). Each cable of the sheet is identical to that used in the ring model, except for the absence of the Na+ gating variable j, and the Intracellular resistivity which is set at 400 O-cm to yield a conduction velocity of 50 cm's. Except for the conduction velocity, the dynamic properties of the two types of cables are fully equivalent. Nominal values for the longitudinal (9xF) and transverse (9yF) flat wave conduction velocities are 50 and 12 cm/s, respectively. The sheet anisotropy ratio (AR) is defined as AR - 9xP9yF.

hj

'EG·

\+--+'+---,. , WL

,, , --- ,---j----

··

, , ," "

.

hJ·-":

Vmin

,

'"' ,

1

1

I~ X ---+:~ X - - - + 1 1st turn 2nd turn

B

'.

CL;.1

-:L:'-

"

--.:

1·1

In order to obtain reentrant propagation in the sheet model, the pathway length must be easily accommodated by the sheet dimensions at each cycle. As a first approximation, one can take the reentrant pathway as roughly equivalent to that of a one-dimensional ring. Given realistic sheet dimensions of 5 x 1.5 cm (limited by our available computing capacity), a pathway length of 5-10 cm would appear reasonable. Since this value is 2-3 times shorter than the length of our ring model, we reduced the Wl by dividIng 'tct and 'tf by a constant K. Values of K - 2, 4 and 8 were used. It was verified that rings with different K values exhibited the same dynamic regimes, with transitions oCQming at correspondingty shorter ring lengths

1.""1

Figure 1

82

as K increased.

A

-

Reentry was initiated in .the sheet model by a cross-shock protocol similar to that used in experimental studies. Numerical solutions were obtained using a finite element method with a fixed spatial discretization of 50 J.lITl and a temporal step of 2 J.1S. Programs were written In FORTRAN and run on a Silicon Graphics 40-255 computer.

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.!!! CD

"0 >- 200 0

RESULTS AND DISCUSSION

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Xatt

mln

18

14

1. One-dlmenslonal raentry (ring model)

22

B Stable raentry occurs in the ring model for X > Xcm • 19.6 cm, with constant CT and constant CL. The action potential then propagates with unchanging tel11>Oral and spatial characteristics, and the activation and repolarization fronts have the same velocity. Observations at one site are sufficient to describe the solution completely. As the ring length Is shortened to obtain lower CL values (Fig.2A), there Is a concomitant decrease in olA (Fig.2B). The minimum olA of stable reentry is about 100 ms, and CT thus remains essentially constant (Fig.3A). Consequently, the linear relationship CL == X . CT Is valid. Neglecting L we also have CL. APo + olA (Fig.1 B), which indicates that the APD varies in parallel with 0IA(Fig.3B). For a given X, both WL and EG remain constant during successive turns so that WL .. ARP/CT, EG .. (CL-ARP)/ CT, and X '" WL + CT.

I··.. : ...... . . .,.... . ~-; ... s. .•.. . •.

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18 22 Ring length (cm) Figure 2

irregular alternation between short and long intervals, and the minimum CL, APo, and olA values become smaller as X is decreased. Hone were to take recordings at two distant sites on the ring, the two patterns would appear completely irregular and unrelated. Assuming the latency to be negligible, the relationship C4-1 .. APo~1 + olAj (Fig.1B) Is applicable at any given site. The APo and olA vary in parallel (Fig.3B), and the wavefront detection criteria Va and hj > 0.1 continue to be applicable.

Irregular reentry begins at X < Xcm .. 19.6 cm, and complete block occurs at X < Xmin - 12.8 cm (Fig.2). The action potential waveform changes contiooouslyas it travels around the ring, the conduction velocity of the activation and repolarization fronts may be different, and both may vary along the pathway. Observations at multiple sites are then necessary for afull description of irregular sustained reentry. Activity recorded at a single site shows that irregular reentry is characterized by a range of CL and olA values which increases as X Is shorter (Fig.2). The pattern Is an

The occurrence of short olAs during irregular reentry reflects the existence of a state of incomplete excitability recovery at Vmin. When olA is quite short (e.g. < 60 ms), ~ .. g Na hj is very low and the maxi-

83

mum dV/dt of the upstroke is quite small also. Then

turns to the same location on the ring during several

the eT of the activation front is increased (Fig.3A) and

successive cycles, all DIA values reQJr periodically

conduction block occurs If X is short enough (viz., X <

over a distance of X(1 +8). In other words, there is a

12.8 cm, Fig. 2). On this basis we may distinguish

well-defined DIA waveform as a function of time dur-

between recurrent reentry (19.0 < X < 19.6 cm), when

ing successive turns around the ring. A further char-

eT

along the pathway is essentially constant, and

acteristic of sustained quasi-periodic reentry is a trig-

quasi-periodic reertry (12.8 < X< 19.0 cm), when sIow-

gered secondary repolarization front which tends to

ing of conduction occurs over a limited portion of the

maximize the length of the excitable gap at each (1 +8)

ring.

turns.

A

2. Two-dImensional reentry (sheet model) Stable two-dImensional reentry was obtained

_

with K - 4 and AR - 4:1. A line stimulus (S1) extend-

60

ing over the upper longitudinal edge of the rectangu~

E

~

tar sheet initiates a flat transverse wavefront. When a

.-... E

second stirrulus (S2) is applied over the full upper left quadrant after a delay of 165 ms, conduction is blocked

()

in the transverse direction but the recovered excitabil-

20

ity In the upper right CJ.Iadrant allows longitudinal propagation to take place. The situation at 200 ms after the

o

100

200

onset of S1 is illustrated in the top panel of Fig.4, where potentials positive to -20 mV are represented by the

B

white region, potentials negative to -60 mV by the black region, and potentia Is between ~O and -20 mV by the grey region. As the activation front progresses longitudinally in the upper right quadrant, the lower half of

en 150

-er

the sheet Iooses its refrac:toriness and transverse con-

c(

also into the left lower quadrant. The resulting curved

E

duction becomes possible. Then the lower right quadrant is progressively invaded, and propagation moves

a..

wavefront defines the initial phase of rotating propa-

50

gation.

o

100 DIAi (ms)

200

At 240 ms (Fig.4). the elliptic tip begins to enter the upper left quadrant and it "turns the corner'" between 240 and 280 rns. In each panel the black arrow

Figure 3

points to the arc of UB around which the wavefront is

Quasi-periodic reentry involves periodic DIA

rotating. The first full turn is nearly completed at 300

changes extending beyond one full tum around the

ms. The horizontal thin white line on the panels at

ring. It is characterized by a rotation number (1+8),

280 and 300 ms indicates the length of the longitudi-

where 8 is an irrational number smaller than unity.

nal excitable gap; its large magnitude reflects the sta-

Therefore, while the same activation front never re-

bility of the reentrant pattern. At 340 ms the second

84

A

200_m.s--------------

B -...-...... -...........

... ,_.... "-'" .......

260

}400

Figure 5 smaller arc of UB. This exarT1'le illustrates the variations in size and location of the arc of UB during stable reentry.

300 ms Irregular twcHilmenslonal reentry was simulated with K -2 (instead of K ., 4) and the same AR -

~-u-

4:1. The resulting larger ARP. greater range of CL. APD. and DIA variations. and increased WL made it impossible for the sheet to accommodate the path-

340 ...._________....

way length for more than a few cycles. As depicted by the isochrone maps for three conserutive turns in Fig.S. there is first a distortion of the isochrones (410-490 ms. left lower quadrant) which is followed at the next

Figure 4

turn (490-570 ms) by a split of the arc of UB and the appearance of an additional activation front. Exten-

turn of reentry is well under way and the

sive fragmentation of the arc of UB occurs at the next

transmembrane potential distribution reveals a stable

turn (570-650 ms). giving rise to multiple activation

pattern despite the short longitudinal excitable gap.

fronts.

The displacement of the activation front can be

The minimum DIA of two-dimensional irregular

followed on the isochrone map of Fig.5. During the

reentry is short enough to induce conduction velocity

second turn (300-380 ms. as indicated beside each isochrone). the front rotates around an elongated and

changes. as indicated by the distorted isochrones of Fig.S. Thus slow propagation takes place at some

slightly oblique arc of UB defined by the tip of the successive isochrones (corresponding to the arrows in

points of the sheet. particularly in the vicinity of the arc of UB. It was not possible to fully examine the fea-

Fig.4). Similar1y. during the third turn (400-460 ms).

tures of two-dimensional irregular reentry in the present

the extreme limit of each isochrone maps a somewhat

study because the simulation runs were too short.

85

being limited the

to

less than 10 cycles. Nevertheless,

occurrence

of

the

triggered

cessive turns of reentry. The sum of these similarities strongly suggests that recurrent and quasi-periodic

secondary

reentry were present In the atrial tissue preparation.

repolarization wave was easily visible on the transmembrane potential distrbJtions.

Simllar1y, the siroolated regular and Irregular pat-

A

tems of two-dimensional reentry appear to be representative of experimental observations in isolated strips of myocardium (2) and during ventricular tachycardia

(8).

For example, in isolated thin strips of rabbit

myocardium, the induction of reentry was greatly facilitated by lowering the extracellular potasskJm concentration, thereby decreasing both the refractory period and the conduction velocity. Since the resulting

B

reentry was unstable and self-terminating, It would appear that the average pathway length was too long to fit the size of the preparation. Splitting of the arc of UB into two parts in this case resembles the simulation results depicted In the second panel of Fig.6. Using carbamylcholine to shorten the refractory period, with-

out substantially affecting the average conduction ve-

locity, these authors observed a dramatic accelera-

c

tion of the reentrant rhythm, a shortening of the APD, and a reduction In the size of the arc of UB. As

a

result the reentry is more stable and all these effects are consistent with the behavior of the present model.

REFERENCES 1. Frame LH, Simson MB, Circulation, 1988; 78:12n-

Figure 6

1287 2. Smeets JLRM, Allessie MA, et al., Clrc Res, 1986;

CONCLUSIONS

58:96-108 3. Han J, Moe GK, Circ Res, 1964;14:44-60

OveraH, the irregular patterns of reentry otiained

4. Quan W, Rudy Y, Circ Res, 1990;66:367-382

in the ring model are similar to the spontaneous oscil-

5. Beeler GW, Reuter H, J Physiol (London), 19n;

lations seen during unstable tachycardias in the ca-

286:1n-210

nine atrial tissue ring around the tricuspid orifice (1) .

6. Drouhard JP, ROberge FA, Comp Biomed Res,

There is good qualitative agreement on the following

1987; 20:333-350

points: alternations between long and short CL, APD,

7. Leon W, Roberge FA, Circ Res, 1991; 69:379-395

and DIA values; parallel variations in APD and DIA; a

8. Cardinal R, Verrneulen M, et al., Circulation, 1988;

range of CL variations substantially smaller than cor-

n :1162-1176.

responding ranges of APD and DIA variations; slow conduction at different pathway locations during suc-

86