SHELL COMPUTER MODEL OF CARDIAC ELECTROPOTENTIAL CHANGES

M. Malik and T. Cochrane’

INTRODUCTION

The cardiac conduction process is one of those biomedical systems, the logic of which and the activities of different parts of which, are comparativelv well known. Consequently it is amenable to *further studv through modelling and simulation. Many properiies of this svtem have been the subject of mathematical stuhies. Action- potential changes of muscle fibres and other parts of the conduction apparatus have been simulated using continuous models based on differential equations1’2, and some complex models dealing with electrocardiogram (ECG) generation have been created3*4.

This paper describes a detailed computer model of the electro-potential changes of the heart. It is oriented more towards a description and discussion of the ideas that were used to develop the model, than to the medical interpretation of results obtained by experimenting with it. A general description of a previous version of the model was published in 1983’. It is based on discrete process sirnulatior?, the cardiac system is decomposed into logically defined elementary parts and the simulation imitates the behaviour of those parts, including their mutual interdependence.

The behaviour of each element is represented by a set of actions (called the simulation events) each of which is supposed to be performed in one moment of the simulation time. Subject to this set of events, an element of the system can be in different states. The time of the next activation of an element does not always have to be given since this can depend on the behaviour of other elenlents; the perfor- mance of one event can change the states and/or the time plans for other elements.

This means that at each moment of the model computation, some elements are planned to be active in the future and some are not_ The structure recording the plans of future events is called the simulation calendar. The program has to search this calend& for the minimal plan corresponding to the event which must be performed next; the reading of the simulation clock is increased to this planned time and the event executed. The perfor- rnance of this event changes the situation of the modelled system and changes the plans for future events (some new plans are added, others may be cancelled). The simulation continues until the simulation clock exceeds a given end value. In this way, all actions of all elements are synchronized and their execution is ordered properly.

The system contains groups of elements which behave according to a common rule (for instance, the actions of all muscle elements in our model correspond to the same principle). Such a general rule is called a simulation process. The CO~J~II~OI~

algorithm describing a simulation process is called its prototvpe. Individual elements whose actions correspond to a prototype are called the process instances of that prototype.

MODEL DESCRIPTION AND ORGANIZATION

The aim of the first stage of the project was to create a model which would simulate the principal known abnormalities of cardiac conduction.

The structure of the model consists of several parts corresponding to the cardiac rmusclc elements and to the more important parts of the conduction system. Altogether, the model has thirteen different process prototypes.

1. Muscle elements of atria

2. Ventricles.

3. Main transmission fibres of the atria1 conduction system.

4.

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1%.

13.

Transmission fibres of His bundle and Tawara branches.

Terminal fibrcs of the atrial conduction svstem.

Ventricular Purkinje fibres.

Pre-excitation sections connecting together the conduction systems of atria and ventricles in a pat hological way.

Sinus node

Atrioventric-uiar node.

Pathological centres radiating their mm

depolarization impulses to the musculature.

Pacemakers.

Special change processes allowing parameters of the model to be changed in the course of a computational experiment.

Another special process modeliing the co- iection of output results and producing the s~rnuiated ECG record.

Figure 1 The thre~dilnensional lorrn of the heart utusc-le is

ulodrlled by a hollow shell with a crntral plane (thr scptunl) and i, crratrd fonl triangular segments piccrd togcthcr nt their rdgts

Figure 2 Each triangular building block is niadr up of 45 basic hrxagonal clrrnrnts, numbered and arranged as shown

The heart musculature is considered as a simpltt

three-dimensional hollow shell structure with a

central section representing the septa. This structure is built up from triangular shaped segments (~~Q-wY 1) each consisting of’ -kj basic hexagonal elements (F@re 2). The triangular form of the basic rnuscie segments enahlcs them 10 t)c pieced together in different \%‘avs and to model different heart shapes.

Elements Lvithin the triangular structure are numbered (identically for each triangif,) and t hc model data contain a table giving the numbers of‘ ncighbours to each element in the triangle. All triangles are numbered and the connections be&veer1 them are described by the numbers of’ both linked structures and by a pointer to a tablr containing the linkage descriptions. Each hexagonal element is addressed by the number of‘ its triangle and bv its position therein. This data organization enables the neighbours of each musculature element to be searched rapidlv.

Each hexagon represents one independent instances of the relevant simulation process. Both muscu- lature process prototvpes (atria1 and ventricular) arr similar and consist of three events. These events describe the depolarization and repoiarization of’ one element in a simplified wav. If a depolarization signal has been accepted, the element sends it (with a delay) to all neighbouring elements and other neighbouring structures (conduction fibres, utl- protected pathological centres, etc.) which are able to accept it. It then waits a longer interval which corresponds to the repolarization period and for which no depolarization signal can be transmitted to this element.

Main conduction system

The main transmission system (again, treated separately for atria and ventricles) consists of independent linear sections which can be COII-

netted together to create an image of’ the conduction tree. The beginning and end of each section are separated to permit fibre orientation to be defined. III a manner similar to the data organization of the musculature blocks, the conduction sections are numbered, a data table containing the numbers of front and back neigh- bours for each section. Each main section can also contact a Purkinje fibre, the sinus nodf, (atria1 sections only) or the AV node. Ail contacts of Purkinje fibres (or fine terminal fibres in the atria) and of the nodes are saved in another data table.

The simulation process prototype is different for the transmission fibres of atria and of ventricles. The main ventricular fibres are able to radiate their oww impulses corresponding to their own sponta- neous frequencies. The state diagranl of the process prototype of a main ventricular transmission section is shown in Figure ~3. Signal transmission consists of three separate events: SE,, after a signal has been accepted or created the section sends it

f.‘ardlac elertropotential model: M. Malik and 7: Cochranr

Figure 3 State diagram of the simulation process of one ventricular main conduction section. Signal elaboration consists of three events (SE,,,,,) which are described in the text. WS is the initial state in which the process waits before radiating its own impulses. WP is the state in which the process waits for the natural period of the section. The ‘sig’ state changes are explicit changes when a signal has been received, the other changes are implicit ones made according to time plans

anterogradely with a certain delay; SE,, the signal is sent retrogradely with a longer delay; SE,, the section waits for its repolarization period when no signal can be transmitted to it. The organization of the atria1 main conduction section process is similar, though simplified by having no sponta- neous activity.

Prukinje apparatus and atria1 terminal fibres

The Purkinje apparatus and the atria1 terminal fibres have been modelled as in the main transmission system. The linear fibres represent the final branching of the whole transmission system: those which have no front neighbours contact the central elements of musculature triangles and may initiate depolarization of the contacted hexagonal element. As for the main sections, the terminal branching fibres are oriented.

Nodes Special simulation processes are used to describe the sinus and atrioventricular nodes. Both structures are unique, therefore each of their simulation processes exists in only one instance. The sinus node process is quite simple. It works in two states: it creates the depolarization signal and then periodically transmits it to the atria1 main transmission sections; subsequently the node waits for its repolarization period. Transmission sections contacting the node may activate and depolarize it retrogradely.

In contrast, the realization of the AV node is the most complicated process in the model. As with the ventricular main transmission sections, the node is able to radiate its own impulses when not activated for a certain time. Such a nodal signal is radiated to the ventricles and the atria. In addition, the node sums the signals accepted from the atria1 conduction system and a ventricular signal is

Figure 4 State diagram of the simulation process realizing the AV node. Signal elaboration states (SE,,,,,) and waiting states

(WP, WS) are similar to those of the process of the ventricular conduction section. The atrial signal is transmitted (with a delay) to the ventricles only if the sum of contributions received exceeds a radiation threshold: sl = received signal contributions, rt = radiation threshold. The ‘sig a’ changes are explicitly caused by an atrial signal. The ‘sig v’ changes are caused by a ventricular signal and the other changes are implicit. The explicit atria1 and/or ventricular changes can be switched off by protecting the node

produced only when the sum exceeds a certain threshold. The state diagram of the node is shown in Figure 4.

The model permits either or both nodes to be ‘protected’, which means that no signal can be transmitted to them. Such a protected node is completely independent.

Pathological structures

Input data structures also allow pathological pre- excitation fibres and/or pathological centres to be introduced into the model. A pre-excitation fibre is modelled by special transmission sections which, at one end, contact the atrial and, at the other, the ventricular conduction apparatus. The section may transmit atria1 signals directly to the ventricles and similarly, ventricular signals may be trans- mitted to the atria. Several pre-excitation sections may be introduced and each of them may be uni- directional or bi-directional.

Pathological centres represent parasystolic activity. Such a centre is connected to the central element of one musculature triangle and periodically activates the musculature in this area.

Pacemakers Routines realizing different pacemaker modes can be linked to other parts of the model. The interface between pacemaker routines and the main part of the model consists of universal algorithms corres- ponding to atria1 and ventricular sensing and pacing (definitions: sensing area, the set of muscle blocks in which the depolarization wave is detec- ted; sensing key, a key which can switch sensing off

268 J. Rmmed Eng. 1985, Vol. 7, October

or on; pacing area, the set of muscle blocks which the pacemaker can stimulate). Realizing these algorithms in this way enables the different pacemaker types to be described quite simply.

In addition, the combination of two or more pacemakers operating at the same time can be evaluated by the model. Each separate pacemaker creates one Instance of the pacemaker process and these instances are synchronized by a simple searching structure.

The current version of the model contains a libraq of pacemaker modes which include those corres- ponding to the ICHD pacemaker classification’.

Results collection The model produces a simulated ECG record in three orthogonal XYZ projections. The orthogonal projections have been chosen for their simplicity. The full 12-channel records (with other special possibilities) are expected to be introduced at the next stage. A simulation run produces output in raw digital form; the raw data are filtered, smoothed and plotted.

To produce the output signal, the polarization changes from the whole heart muscle are summed. Depolarization and repolarization of each musculature element are described by a common sampled function which is the same for all hexagons (see section on model electrophysiology). These sampled values are read as input data, allowing the function to be changed.

The way in which the depolarization and re- polarization of each hexagon influences the changes of the main polarization vector of the whole heart muscle depends on the topological position of this element. The data for each muscu- lature segment contain three projection constants (one for each output channel) which describe the topological position of the triangle, it is supposed that the positions of all hexagons are the same for each segment. If a muscle element is depolarized, the common output signal is multiplied by the projection constant of the corresponding muscu- lature segment and by the level on which the depolarization signal has been transmitted to that element; the products obtained for all muscle elements are summed.

Global synchronization In the model, a large number of elements is used to construct the atria1 and ventricular musculature. This implies a great deal of searching to find the minimal planned values. Therefore, a specially structured calendar is used to allow these values to be found quickly. Searching is performed in a three level hierarchical way.

At the topmost level, the simulation process proto- types e.g. sinus node, atrial conduction section, atrial muscle etc. are ordered in a global binary tree. Each element of this tree has associated with it

b .-------- 4’

t muscle 0 elements

Figure 5 The upper, global tree contains pointers to the lower, local trees. Each lower tree belongs to one musculature block and contains lists which represent the time plans of muscle elements which are in the same state. All elements in the same list are treated togvthrr

a lower level local tree, or local list, which is ordered according to the planned times of indivi- dual instances of the selected process prototype. All elements in the same muscle triangle and in the same state (representing the same simulation event) are ordered in a linear list and are treated together. The lists belonging to one triangular block are ordered in a binary tree and the triangular blocks themselves are ordered in another global binary tree according to the values of their local minimal plans. This tree of triangular blocks represents the lower level tree associated with the musculature process prototype. (Figure 5).

With this searching scheme, it can be shown that, for the least complex case approximately 5 x lo3 comparison operations are required for one modelled cardiac cycle (at worst, approximately lo5 comparisons are needed). This is to be compared with lo6 - lo8 comparison operations if searching is done using balanced binary trees.

MODEL ELECTROPHYSIOLOGY

The organization of the model mirrors the electrophysiological properties of the real heart.

Pacemaker activity

Electrical activity is initiated, in the normal situation, by a pacemaker located in the sinus node region which periodically transmits a depolarizing signal to the atrial transmission sections. After transmitting this signal the node waits for its repolarization period. Transmission sections contacting the node may sometimes activate and depolarize it retrogradely.

In the event of failure of the sinus node or, perhaps in addition to firing of the sinus node, the atrioventricular node and the ventricular con- duction branches may produce pacemaker activity. These subsidiary pacemakers reach depolarization threshold more slowly than the sinus node, they do not therefore normally reach threshold on their own but are depolarized by the impulse generated in the sinus node. Modelling a disease or the action of drugs, the sinus node pacemaker may be usurped by one or other of these pacemakers and arrhythmias may occur. In the model, automaticity, conduction velocity and refractory periods are altered simply by changing the input data.

Muscle cell activity

The only electrical activitywhich contributes to the collected signal is that arising from the depo- larization and repolarization of muscle elements; the conduction segments merely convey pacemaker impulses to different part of the myocardium. Each muscle element is supposed to be either depola- rized or in the resting state. The changes between these two states are instantaneous.

The way in which depolarization and repolarization of muscle elements influences the polarization vector of the heart as a whole, depends on the direction in which the polarization wave radiates through the musculature. This dependence on the direction of propagation has been simplified and only two directions are introduced: the normal and the abnormal one. To enable the model to dis- tinguish normal signal radiation from abnormal signal radiation (for instance, when a pathological fibre bypassing the AV node is present) positive and negative depolarization signal levels have been introduced. The normal direction of signal radia- tion is defined by the anatomical orientation of the main sections of the conduction apparatus.

A common output signal describes the action of each hexagonal element. These signals, multiplied by the projection constants and by the signal level, are summed only when depolarizing the elements; however, the form of the output signal (Figure 6.) reflects also the repolarization changes. The choice of this output function is not a trivial matter since the exact relationship between individual cell action potential and the final output form is not known. We have chosen an approximation to the first derivative of the myocardial action potential. This gives the correct form of the P - T, wave (depolarization and repolarization of atria) on simulated ECG records. The situation for the

ventricles is not so simple, because of a variation in action potential both from endocardium to epi- cardium and from base to apex!. Since the repolarization wave radiates through the ventric.les in the opposite (base-apex) direction to the depolarization wave, the part of the first derivative of the myocardial action potential which corres- ponds to repolarization, is multiplied by the negative signal radiation level (i.e. it is positive) in the ventricular output signal. This approximation enables us to use our model for rhythm studies (see below) but it is a less adequate representation where exact morphology of the electrocardiogram is important. Enhancement of the model to permit more detailed morphological studies is discussed below.

DIFFERENCES FROM EARLIER MODELS

To our knowledge, the first attempts to model the electropotential changes of the heart muscle on a cotnputer were made in the 1960s. Those models may be divided into two groups according to their methods of operation. Models of the type described by Wartak9 divide the ventricular musculature into several parts, about twenty or thirty, and describe the electropotential changes of each part as an equivalent dipole; a computer was used to integrate the changes of all dipoles together. Models of the other type”, deal with the proliferation of a depolarization wave through ventricular muscle. The musculature is approxi- mately homogeneous and consists of a great number of independently acting elements; the polarization wave radiates through this structure.

Both approaches are combined in the model of Geselowitz et uL.~*~, in which the ventricular musculature is modelled by a cubic network of several thousand elements. The action of each element is treated as a dipole and global syn- chronization is made using Huyghens’ Principle. This model is probably the most developed of the so-called multidipole models and we shall comparr our model with it.

Figure 6 The approximarc form of the cmnn~on output signal. According to the direction of thr depolarization wavr through the atrial and vmrricular nlusrulaturc, the r~polarization part of the signal has hccn usrd on thr po>itivc kvel for the atria (a) and on the negative level for thr vrntriclcs (v)

The model of Geselowitz et al. is restricted to the ventricles, a restriction which prohibits its use for modelling most rhythm changes; for this, the abnormal action of nodes, conduction svstem and atria1 musculature are very important. Similarly, to model a case in which the depolarization wave is radiated through the musculature in an abnormal way, the model of Geselowitz et al. would need other experimental data defining the activation isochrones, just for this special case. This model is therefore used for studying pathologies of individual heart contractions, especially those which depend on changes of the action-potential in pathologically altered regions of the myocardium.

The main application of our model is in heart rhythm studies (we present some results in the next section of the paper). Pathologies of the conduction system (bundle branch block, for instance) can be modelled, too, but for such studies, abnormal potential radiation across the heart wall is more important. Rhythm studies were also the reason for introducing simulated pacemakers into the model. The idea of testing complex pacemaker modes using a computer model is very promising.

Our model has been developed independently and is different in several respects. These differences are both in the principal project implementation ideas and in the application areas addressed. We start with the conceptual differences.

Conceptual differences Geselowitz el al. model the musculature of ventricles by a three-dimensional image built from several cross-sections taken along planes perpen- dicular to the base-to-apex axis. We consider both the musculature of atria and of ventricles, which are modelled by arrays of points ordered in tangential pieces. This means that the model of Geselowitz et al. can accurately represent radiation of depolarization and repolartzation waves across the heart wall, while our model offers only the possibility to model changes along the heart wall (which is less important for mophological studies).

To simulate the distribution of potentials within their model ventricle, Geselowitz et al. assigned activation times and action potential shapes which were derived from laboratory experiments, and from values reported in the literature. Our model contains an image of the conduction system and traces the proliferation of the impulse from the sinus node along its conducting fibres and throughout the cardiac musculature; i.e. the activation sequence (which in the model of Geselowitz et al. is taken from measured activation isochrones) is included as an integral part of our model. In addition the other main components of the cardiac conduction system are included to provide a complete description. In this sense, the model reported here is an improvement, but it is more difficult to validate by direct laboratory experiments (we shall revert to this point).

The last difference between the two approaches is the way in which the ECG signal itself is generated. Geselowitz et al. consider a dipole at each musculature point and describe its changes using a discrete approximation to the action-potential curve at that point. Our model perceives each musculature point as an independently changing potential and describes its action by a common signal form, which corresponds to the first derivative of the action-potential curve. The model of Geselowitz et al. can easily describe such pathological situations in which the action-potential curve is abnormal for some musculature regions; this is also possible in our model but needs more complex preparation of the output signal. In addition, the three-dimensional form, based on anatomical data, is a more realistic representation than our heart-shaped shell; our approach has been chosen for its lower computational complexity.

Application differences The different applications of the models follow from their conceptual differences and are almost complementary.

It might appear that the best way to develop an even more realistic model would be to combine above approaches. This would certain,ly be pos- sible. However, the matter is not so smlple, we shall discuss this further below.

the

MODEL REALIZATION AND EXPERIMENTS

The current version of the model has been developed on an ICL-4/72 computer. The source text is written in standard Fortran-IV and has approximately 12000 lines. Result calculation requires approximately 160 kbytes. The calculation speed depends on the complexity of the modelled situation. Simulation of one complete heart beat needs approximately 50 s of CPU time.

Input data and configuration Atria1 musculature, including the atria1 septum, has been modelled by 54 triangular blocks, i.e. 2430 hexagons; the musculature of ventricles, including the septum, by 88 blocks, 3960 hexagons.

The radiation speed between two neighbouring musculature elements (the time necessary to transmit the signal from one hexagon to the other) has been chosen as 8 ms for atria and 4 ms for ventricles. These constants have been determined by comparing simulated curves and natural ECG records.

The projection constants for the triangular muscle segments correspond not only to their topological position, but also to the thickness of the musculature area which the triangle represents. The proportion of output contributions of one triangle in different parts of the musculature has been approximated as 1:2:4:8 for atria, right ventricle, left ventricle and apex respectively. These values have been determined empirically.

Cardiac electropotentzal model M. Malik and ?: Cochmne

Figure 7 Clobal organization of the conduction system. lines represent the main conduction sections, other lines represent the Purkinje fibres. Black dots are connections

Bold

between neighbouring fibres, the triangles correspond to the places where the Purkinje system contacts the musculature blocks

The detailed map of the conduction system which was prepared for the previous version of the model5 has been modified and the main sections and Purkinje fibres have been separated, (Figure 7). In atria, 22 main sections and 43 terminal fibres and in ventricles, 37 main sections and 143 Purkinje fibres have been considered. The topology in which the conduction tree contacts the musculature corresponds approximately to the real situation.

Experiments

We have mentioned that our model is especially suited for rhythm studies. It enables different rhythm pathologies to be described, individually and when present, simultaneously. In addition, the special system change processes allow the simula- ted situation to be transfigured in the course of computational experiment (These routines may contain a random number generator and intermit- tent situations can be modelled in this way.)

Figure 8 shows simulation of a normal ECG and simulation of a sinus tachycardia combined with a pathological process changing the conduction velocity along all fibres of the left Tawara branch. (Z?pre 9) contains the simulated ECG of a fast sinus and fast nodal tachycardia The result shown in Figure 10 has been obtained when three different pathological centres with different frequencies are considered. Figure I I shows a ‘bigeminal P-wave’ rhythm when complete AV block with retrograde VA conduction is present Finally, the results in

Figure 8 Simulation result of: a, a normal situation and, b, a sinus tachycardia combined with a pathological process altering

the electrical axis by changing the conduction velocity along all fibres of the left Tawara branch

r: --‘-7 T-,--,-,-l-T1-‘_ & ,

X

I

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Figure 9 Simulated ECG of a, a fast sinus tachycardia in which P waves fuse with T waves and b, a fast nodal tachycardia in which each QRS complex is followed by a

negative P wave

272 J. Btonted Eng. 1985, Vol. 7, October

Cardiac electropotential model. M. Malih and 7: Cochrane

“. I -’ _- .^~-_l __“__ --.C_r-.

Figure 10 Simulated result of three different protected pathological centres with different frequencies. The centres are located in thp left ventricle, in the right ventricle and in the ventricular septum

/ -T--_- I-- - , ” Y-r I -: ._ . .--. _ .^_

I .& ;p . __.r _.... __T.-__.._T___..T.” ..“...r_ .^._. -1

Figure 11 Model of a rare situation in which complete AV

block with retrograde VA conduction and concealed nodal rhythm are present. Each QRS complex is followed by a negative P wave which starts a new period of the sinus node, the normal P wave is earlier than the nodal beat (the sinus frequency is higher than the nodal one) but has no effect on

the ventricles

@+ure 12 compare the effects of WI, DVI, VDD and DDD pacemakers’ under the same conditions of sinus bradycardia and complete AV block

PERSPECTIVES AND DISCUSSION

Model validation The version of the model described here is obviously not the final one. Further versions are in preparation. This was one reason why the current version was validated only on the final results (the results of the model were compared with natural ECG records in those cases where the principal aetiology is known). It would certainly be possible to trace the model computation and to compare it with laboratory data; however, some serious limitations of the model are known and, therefore, external validation is sufficient at this stage.

Figure 12 A comparison of the effects of: a, WI, b, DVI.

c, VDD and d, DDD, pacemaker under the same underlying conditions of sinus bradycardia with complete AV block

J. Biomd Eng. 1985, Vol. 7, Octoh 273

Cardiac elertropotential model: M. Malik and 7: Cochrane

Further development

The most important simplifications which were included in the current model have already been mentioned. We have suggested also, that a combina- tion of the ideas of our model with those of Geselowitz et al. is probably not the best course for future development_ Some simplifications are common to both these models. The most impor- tant is the point-like description of the musculature which supposes that the potential-change prolifera- tion through the musculature is homogeneous, which it is not. It would be more exact to consider muscle fibres and to describe the musculature, not by a point network, but by a fibre graph network, which would take into account the histological orientation of fibres in different parts of the heart wall. Signal proliferation along a fibre and across a bundle of fibres” may be described by differential equations. Algorithms solving these equations can be realized in the form of a simulation process; such a process can be synchronized with the manner in which the depolarization and repolari- zation signal is radiated through the heart muscle. Such synchronization must however, depend on signal proliferation along the conduction system fibres and not on externally obtained activation isochrones.

For rhythm studies, the cycle rate dependence of signal transmission velocities and duration of repolarizatlon has to be included in the model. This means that the history (several past events) needs to be saved for each node, as well as for conduction element or conduction elements area. The process algorithm would have to handle these past records in a systematic way.

It is evident that in realizing these ideas (and others not mentioned here) the time and storage complex- ity of the model will increase many times. There- fore, it is suggested that two different versions of the model be developed, a simpler and a more rea- listic one. The simpler version could be used in the initial stages of a research project and, perhaps, for educational purposes. Currently, it seems that it would not be expedient to develop a fully realistic version because of its immoderate storage and time demands. It would be better to develop different versions oriented towards specific problems e.g.

one for morphological studies, rhythm studies, etc. Each new version would improve the simpler model in a specific direction. Such a package would be more powerful than a fully universal model. For morphological studies (for instance, if the influence of ischaemia on the ST segment were modelled) it would be sufficient in most cases to model one or two cardiac cycles. Model accuracy could be traded for a slower computation. If a special rhythm study examining the influence of one or more pathological centres is to be modelled over several hundred cardiac beats, high compu- tation speed would be an important criterion.

ACKNOWLEDGEMENTS

The authors thank Mr P. Bures from the University Computing Centre, Prague, for his helpful sugges- tions at final program installation and Miss K. Hnatkova from the Department of Computer Science, Charles University, Prague, for her help with graphical output.

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