Math1 Comput. Modeliing, Vol. 14, pp. 576-581, 1990 Printed in Great Britain

0895-7177/90 $3.00 + 0.00 Pergamon Press plc

COMPUTER SIMULATION OF VENTRICULAR FIBRILLATION AND OF DEFIBRILLATING ELECTRIC SHOCKS. EFFECTS OF ANTIARRHY-THMIC DRUGS

Pierre Auger, Alain Bardou **, Alain Coulombs* ** Marie-Claude Govaere**, Luc , Rochettti, Jean-Pierre Schreiber, Dominique Von Eu\N”*, Michel Chesnais***

Laboratoire de Biophysique and * Laboratoite de Pharmacodynamie, Faculte de Pharmacie, 7 Bd Jeanne d’Arc, 21000 Dijon, France. ** INSERM U286, Hopital Broussais, 96 rue Didot, 75674 Paris Cedex 14, France.

*** Laboratoire de Physiologie compatie, Universite d’Orsay, 91405 Orsay France.

Abstract. We present computer simulations of the propagation of the depolarizing wave through the ventricular wall based on the Huygens’ construction method. We simulate different self-sustained conduction troubles related to fibrillation. We calculate the critical size of myocardium needed to induce reentries in either normal or ischemic tissues for isotropic and anisotropic conduction conditions. Then, in order to stop the reentrant activity, simulation of electrical shocks is achieved by depolarizing different percentages of excitable ventricular cells. It is shown that all the excitable cells must be simultaneously depolarized in order to block the reentry. The model is also extended to pharmacological applications on antiarrhythmic drugs. We simulate a decrease in the refractory period leading to spatial and time adjustments of the size of the reentry. We also simulate effects of class III antiarrhythmic drugs increasing the refractory period and blocking the reentry.

Ke_ywords. Computer simulations of ventricular fibrillation; electric defibrillating shocks; antiarrhythmic drugs.

INTRODUCTION

In many cardiac models, a set of coupled differential equations governing the time evolution of electrical potentials and of internal parameters or excitability is chosen for each element of the network. Bidimensional models have been developed and we refer to (F.J.L. Van Capelle and D. Durrer, 1980) and to (M. Delmar et al., 1986). These computer simulations calculate time by time and for each element of the network the electrical potential and they allow to describe the propagation of the depolarizing wave through the tissue cell by cell. Nevertheless, these models involve a lot of variables and their simulations consume a lot of computer time. Very few tridimensional models of this type have been developed such as (A. Winfree, 1988).

In order to limit the consumption of computer time, several authors use simpler models. The heart tissue is represented by a network of elements that can depolarize the nearest excitable elements. For instance, in such a time discrete bidimensional model, any newly depolarized cell can depolarize at the following time interval any excitable cell in a small neighbourhood, either the four nearest cells or the eight nearest cells. In these models, the propagation of the depolarizing wave is described cell by cell and the electrical potential is not calculated continuously with time from chosen differential equations. A simple law of propagation

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is used and computer simulations are realized. We refer to (J.M. Smith and R.J. Cohen, 1984) and (J.M. Smith et al., 1984). In our bidimensional models of the ventricle, we use a particular law of propagation, the Huygens’ construction method.

In the first section, we recall different mechanisms inducing self-sustained conduction troubles mainly triggered by unidirectional blocks, (P. Auger et al., 1987, 1988, 1989). We determine a critical size of a unidirectional block needed in order to induce a reentry. Under this critical size, no reentry appears. This critical size depends upon the values of the conduction velocity and of the refractory period. It also depends on anisotropic conduction rules, i.e. on the orientation of the unidirectional block with respect to the heart fibres direction. The reentry is much favored when the unidirectional block represented by a segment of length I is orthogonal to heart fibres direction.

In a second part, we simulate electrical shocks delivered to the ventricle in order to stop a reentry with different percentages of depolarized cells. It is shown that electrical shocks with smal I percentages (~10%) are totally inefficient and do not modify the initial simulated electrical mapping, that shocks with high percentages (~80%) although they modify the initial electrical mapping induce many secondary self-sustained waves and, that to be sure to stop the conduction trouble it iS necessary to depolarize 100% of excitable cells.

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In a third section, we simulate effects of antiarrhythmic drugs. Here, we only study the influence of either a decrease or an increase of the refractory period on the spatial size and on the time periodicity of self-sustained waves. Increase of refractory period also is able to stop a reentry.

CRITICAL SIZE OF A REENTRY

Isotropic conduction conditions

The propagation law is the Huygens’ construction method in a bidimensional map of the ventricular wall with 2500 elements. In this way, it is assumed that during the time interval Dt, each newly depolarized group of cardiac cells at a point (i,j) is able to depolarize any excitable cardiac cell which can be found in a small neighborhood defined by a circle centered on this point (i,j) with a radius R = c.Dt where c is the conduction velocity. This radius R corresponds to the distance covered by the travelling wave during Dt. One can say that each point of the wave front at time t is the source of a circular wave and that the wave front at time t+Dt is obtained by composition of all these waves.

The computer simulations are presented by sequences of electrical mappings giving the state S(i,j) at each point (iJ) of the surface element at consecutive instants, t=O, Dt, 2Dt, . . . , NDt, where N+l is the number of electrical mappings. We use different grey intensities in order to vizualize the state of cardiac cells. For instance, with a refractory period equal to 4Dt, we use three types of grey for cells in refractory period respectively S=3, 2, 1, and white for excitable cells S=O.

In the case of unidirectional blocks, for a given propagation axis, the wave can only propagate in one direction. In our simulations, a unidirectional block is represented by a segment of length I located at the middle of the surface element. If the wave moves from the left side, it cannot pass through it. Reversely, a wave coming from the right side can pass through it. The conduction is allowed in one direction and is forbidden in the opposite direction. We have already presented simulations of unidirectional blocks and we refer to (Pierre Auger et al., 1988) showing that the wave turns around the unidirectional block and then reenters. These patterns are in good agreement with experimental mappings, see (E. Downar et al., 1988). The reentry occurs when the tissue becomes reexcitable. This is the triggering process of a self-sustained circus motion. Indeed, during the following time intervals, the wave reenters periodically. It is interesting to see that the time periodicity or the frequency of this cyclic process is fixed by the refractory period. Indeed, in order to reenter, the wave must encounter reexcitable tissue, and it follows that the reentry cannot occur before the end of refractoriness. Therefore, the period T of the cyclic circus movement is equal to the refractory period. For a refractory period of about 200 ms, we obtain five reentries per second, i.e. a frequency of 300 reentries per minute.

It can be shown that the reentry does not occur when the unidirectional block is too small, see (P. Auger et al., 1988). The wave divides into two

waves which propagate along the unidirectional block in opposite directions. If the junction between these two waves appears before the end of the refractory period, no reentry can occur due to the small size of the unidirectional block.

As a consequence, there exists a critical size icri which depends on the refractory period and on the conduction velocity so that when I < lcri no reentry occurs and when I > I,d the reentry does occur. This critical size can be easily estimated. Indeed, the distance covered by the two waves propagating in opposite directions along the unidirectional block during a time Dt is equal to 2cDt. As explained previously, in order to trigger a reentry, these two waves must meet before the end of the refractory period t. For constant conduction velocity c and refractory period t, the critical size is thus equal to the distance covered by these two waves during a time precisely equal to the refractory period.

Icrf=2ct=2a. (1)

A is usually called the wavelength. In this way, the critical size is equal to two times the wavelength. Calculation of the critical size for normal and ischemic tissues can be achieved. In the first case, the conduction velocity is about 50 cm/s and the refractory period is about 200 ms. Under these conditions, the critical size calculated from equation (1) is equal to 20 cm. This is a very large value with respect to the real dimension of the heart muscle. For instance, the distance between endocardium and epicardium is about 1 cm in the ventricle. This very large calculated value of the critical size of a unidirectional block shows that such reentries are nearly impossible in normal tissues, with the exception of abnormal conduction pathways. A wide refractory period as well as a fast conduction velocity protects the heart and prevents it from any reentrant activity.

A similar calculation can be achieved for ischemic tissues, where the values of the refractory period and of the conduction velocity are less, t is about 100 ms and c about 10 cm/s, (M.J. Janse and A.G. Kleber, 1981). This leads to a wavelength equal to 1 cm and thus to a critical size equal to about 2 cm. Such a calculation shows that a reentry may occur in ischemic tissues for which the critical size of the unidirectional block becomes comparable to the real dimensions of the heart ventricle.

AnisotroDic .conduction.c.onditions

It has been shown experimentally that the conduction velocity was connected to the relative orientation of the direction of propagation with respect to the heart fibres directions. It has been proved that the conduction velocity was about three times larger for a direction of propagation parallel to the heart fibres than for a perpendicular direction and we refer to (D.E. Roberts et al., 1979). Let Cpar be the conduction velocity parallel to heart fibres and c er the conduction velocity perpendicular to heart gbres. We have:

cpar = 3+,,. (2)

578 Proc. 7th Int. Conf. on Mathematical and Computer Modelling

In order to take into account the anisotropy of conduction, we will now consider a surface element of the ventricle with horizontal cardiac fibres. Consequently, we have modified the Huygens’ construction principle so that the wave front positions at time t+Dt is now given by the envelopes of half ellipses centered on the points of the wave front at time t. The long axis of these ellipses is three times larger than the short one and corresponds to the heart fibres direction.

along this block is three times smaller than in isotropic conditions. In the case of a unidirectional block orthogonal to the direction of heart fibres, the critical size is now given by equation (3) :

lwi = 2/3.C+.t = 2.*.t = 2l3.h. (3)

For normal tissue. the same calculation of lcri can be achieved (spur = 50 cm/s, t = 200 ms) leading to a critical size of about 7 cm. This is still a high value and it shows that although anisotropy can favor a reentry, it remains quite impossible for normal tissues. For ischemic tissues, where cpar = 10 cm/s, t = 100 ms, the critical size is about 7 mm. This last result demonstrates that small unidirectional blocks may induce reentries and that anisotropism yet increases the risks for reentrant activities.

FIG. 1, Anisotropic conduction conditions.

Figure 1 shows the triggering of a reentry induced by a unidirectional block. As previously described, the wave divides into two waves propagating along the unidirectional block, but contrary to the case of isotropic conditions, their junction is delayed because the transverse propagation iS slowed down. Indeed, as the unidirectional block is orthogonal to the direction of heart fibres, the conduction velocity of the two waves propagating

ELECTRICAL DEFIBRILLATING SHOCKS

This section presents simulations of electrical defibrillating shocks. First of all, a self-sustained conduction trouble is initialized by ectopic cells in asymmetric conditions, see (P. Auger et al., 1988). In a second step, we simulate an electric shock delivered to the surface element of the ventricle in order to try to stop it. In our simulations, the percentage P of cardiac cells which are depolarized by the electric shock can be chosen. It is assumed that only excitable cardiac cells can be depolarized by the electric shock.The electric shock is applied at the sixth time interval. Effect is seen on the next time interval 7Dt. We have realized three simulations with a small percentage P = 2%, a large one P = 90% and finally P = 100%.

Lowpercentaqe of deDolarized Cells

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FI_G 2. Defibrillatina electric shock withP.?2”/~

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At time 6Dt, an electrical shock is delivered to the ventricle surface element with a percentage P = 2% of excitable cardiac ceils depolarized by it. The following time interval 7Dt displays the effect of the shock. Figure 2 shows that the percentage of depolarized cells is too low and that the electric shock is inefficient to stop the reentry. Moreover, one can see that the electrical mapping is similar after the shock than before. After time interval 12Dt, things happen as if no shock had been delivered.

FIG. 3. Defibrillatinqelectric shockwith P = 90% .

In figure 3, we present a simulation of an electric shock with a high percentage of excitable depolarized cardiac cells, P = 90%. One can see that at time interval 7Dt, almost all the excitable cardiac cells have been depolarized. The next time intervals show that the very small number of cells that has not been depolarized is able to initialize again multiple reentries. These secondary reentries are sources of several self-sustained rotating waves. Consequently, one sees that this electric shock is also inefficient. But contrary to the previous example, the electrical mapping is changed. The initial pattern is destroyed and the secondary reentries generate a new electrical periodic pattern.

100% Depolarized cells

FIG. 4. 100°/a deflbrillatina electric shock.

The two previous figures show that at time of shock delivery, the non depolarized cardiac cells by the shock excitable can still initialize self-sustained conduction troubles. Consequently, to be sure to stop the reentry, it is necessary to deliver an electric shock depolarizing 100% of the excitable cells. Figure 4 shows that the wave front of the depolarization wave meets everywhere cardiac cells in refractory periods and thus cannot propagate anymore. Such a shock is efficient and stops definitively the conduction trouble.

EFFECTS OF ANTIARRHYTHMIC DRUGS

Drugs belonging to different classes have antiarrhythmic effects. These drugs act at the cellular levels by modifying the properties of the cardiac cells. Drugs can have multiple effects by modifying the refractory period, the conduction velocity, the excitability threshold of cardiac cells and the ratio between parallel and transverse conduction velocities. For instance, class Ill ones act by increasing the refractory period of cardiac cells, see (A. Bril and L. Rochette, 1987) and (C. Lambert et al., 1987).

In the two next sequences, we are only considering a variation in the refractory period while all the other parameters are assumed to remain constant. In this way, we simulate a variation in a single parameter, the refractory period while the conduction velocity and all other parameters are unchanged. In general, antiarrhythmic drugs have complex effects and can make vary many parameters in the same time. Thus, our simulations do not correspond to specific drugs, but it is rather a computer experiment in order to determine what is the effect on a reentry of the variation of a single parameter, the refractory period. First of all, we study a decrease in the refractory period and in a second step an increase in the refractory period.

Decrease in refractory period __.

Figure 5 presents a simulation where a periodic conduction trouble is initialized by a unidirectional block. At the beginning of the sequence, the population is homogeneous with a refractory period equal to 6Dt, corresponding to five different colours. At the end of the sequence, it can be seen that the refractory period is now equal to 4Dt. This decrease in the refractoy period can result of ischemia. It has been proved experimentally that in ischemic tissues, the refractory period was smaller than in normal tissues.

Figure 5 shows that two effects occur, concerning respectively the spatial size of the area where the reentry is located and the time periodicity of the cyclic reentry. Firstly, one sees that the spatial size of the reentrant area decreases with the refractory period. Indeed, the spatial size is approximately given by the critical size calculated in first section, which is equal to two times the wavelength A. When the refractory period decreases, it makes also decrease the wavelength x and as a consequence

the critical size lcri = 2.X. For instance, in the example of figure 5, the refractory period decreases of about l/3 and it can be seen that the spatial size

550 Proc. 7th Int. Cor$ on Mathematical and Computer Mode&g

of the reentrant area also decreases of about 113. Secondly, figure 5 also shows that the time Spontaneously, the spatial size of the reentry gets frequency of the reentry increases when the adjusted on the new value of the refractory period. refractory period decreases. Indeed. in order to

reenter the’depolarizing wave must wait for the end of refractoriness. When the refractory period decreases, the depolarizing wave can reenter more frequently. In figure 5, the refractory period decreases of about II3 and the time frequency increases of about l/3. The frequency of the reentry varies from about 3 per second, i.e. 160 per minute to about 5 per second, i.e. 300 per minute. The characteristic time scale of the reentry is also adjusted on the refractory period.

Increase in refractonf ceriod

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EIG_6, !n_crea.se_ Iln the_&actoy period.

FIG. 5. Decrease in the refractorv period, In the simulation of figure 6, we assume the inverse process of figure 5, i.e. an increase in the refractory

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period. Like previously, a reentry is initialized by a unidirectional block. At the beginning of the sequence, the refractory period is equal to 4Dt and at the end, it is equal to SM. Figure 6 shows that the wave front of the periodic wave which is always travelling at the same constant conduction velocity is going to meet cells with a longer refractory period. While before, there was a good synchronization, i.e. that the wave front always appeared when the tissue was becoming newly excitable. Now, on the contrary, the wave front meets cardiac cells in the middle of their refractory periods and thus cannot propagate anymore. We can speak of an effect of desynchronization between the travelling wave periodic motion and the periodicity of the depolarization-repolarization process. The effect is to stop immediately the reentry.

To conclude this section, we can say that any drug which makes increase the wavelength A has an antiarrhythmic effect. On the contrary, any drug which makes decrease the wavelength favors the reentry, i.e. it is going to accelerate the time frequency and to shorten the critical size.

CONCLUSION

Our computer simulations allow us to study several mechanisms inducing self-sustained conduction troubles leading to an uncoordinated contraction of cardiac fibres related to fibrillation. These simulations show that an important parameter is the critical size or the wave length. Any process which makes increase this critical size has an antiarrhythmic effect. On the contrary, any process which makes decrease the critical size favors reentries and arrhythmias.

In the future, we would like to extend this model to a three dimensional version taking into account the real shape of the heart. The Huygens’ construction method can be easily extended to three dimensions by considering spheres of radius c.Dt in isotropic conduction conditions and ellipsoids with a large axis cPar.Dt and small axis cper.Dt in anisotropic conductlon conditions corresponding to heart fibres.

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This work was supported by a grant from Region de Bourgogne and also by a grant from CNAMTS.