An interactive qualitative model in cardiology

COMPUTERS AND BIOMEDICAL RESEARCH 2 8 , 4 4 3 - - 4 7 8

(1995)

An Interactive Qualitative Model in Cardiology P. SIREGAR,* M. CHAHINE,* F. LEMOULEC,t AND P. LE BEUX* *D~partement d'Information M~dicale, University of Rennes 1 School of Medicine, Rue Henri le Guilloux, 35033 Rennes Cedex, France; and ? Laboratoire Traitement du Signal et de l'Image, University of Rennes 1, Cedex, France Received May 16, 1994

Qualitative modeling is a generic term that involves explicit and qualitative representations of the physical world. It can extend the realm of pure mathematical modeling in the sense that qualitative descriptions can, on one hand, simulate complex physical systems and processes and, on the other, produce linguistic descriptions and summaries of simulated system behavior. These summaries should be an essential element of the human/machine interface if truly interactive computational environments are to be developed. In the context of cardiac arrhythmias, a thorough understanding of the underlying processes that lead to the different pathological states is a first step toward optimizing diagnosis and therapy. The CARDIOLAB project is dedicated to cardiology and is aimed at providing a theoretical framework composed of computational models of different grain size and based on different formalisms. One of the intended roles of the framework is to assist researchers, clinicians, and pharmacologists in their quest for a better understanding of rhythmic disorders and ischemic events. In this paper, we present the first element of the framework. It is a cardiac simulator conceptualized in terms of a research field known as qualitative physics. As a simulator, the model's role is to produce fairly detailed descriptions, at different levels of abstraction, of cardiac electrical events when initial tissue-state conditions are given. A crude simulated ECG is also produced as a visual aid. At the end of each simulation session, and upon user request, the system can memorize the initial conditions and the descriptions into an arrhythmia knowledge base. As such, the model can be used as an interactive tool, to grossly delineate the regions in parameter space that correspond to causing or predisposing states leading to specific rhythmic disorders. More refined analysis can thereafter be performed using finer-grained models, the initial conditions of which will have been suggested by the qualitative model. © 1995 Academic Press, Inc.

I. INTRODUCTION In previous papers we have discussed the roles qualitative models could play in the context of an intelligent monitoring system (IMS) (30) and the simplifying assumptions that are introduced as one proceeds from detailed mathematical descriptions to the coarser-grained qualitative models (31). The objective of these preliminary studies was to put forth and support the idea that qualitative models can be the starting point of a multiresolution approach to prediction 443 0010-4809/95 $12.00 Copyright © 1995 by Academic Press, Inc, All rights of reproduction in any form reserved.

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and theory formation. In this paper, a more detailed description of a qualitative model, in its conceptual aspect, is given. Traditionally, the modeling of interacting dynamic processes is carried ofit in terms of the differential and integral calculus. Classical modeling of the heart's electrical activity is a particular instance of this approach. It is generally done via sets of nonlinear differential equations of the Hodgkin-Huxley (HH) type (15, 16, 18, 19) or using what are usually referred to as cellular automata models (4, 15). In both approaches, the heart is modeled as an aggregation of elementary "cells," each of which is governed in the first case, by a particular HH-type equation and in the second by state transition rules. Two kinds of tissues can be modeled: cardiac muscle tissue and nodal tissue like the Purkinje networks with their autorhythmic capabilities. Both HH network formulation (in its discretized form) and cellular automatas can, broadly speaking, be assigned to the class of numerical models. In HH-type networks, the time courses of transmembrane ionic currents are computed as functions of the transmembrane potentials. The cellular automata formulation introduces simplifying assumptions, where the ionic currents are implicitly represented in the form of phase durations. The outputs of both kinds of models are numbers. Analytical and numerical models are today's most precise modeling tools to explain and predict the time dependencies of dynamic systems. However, they can be difficult to manipulate, are computationally very costly, and do not produce qualitative descriptions. Qualitative descriptions convey information that can be used for prediction and control (12, 17), just as images or numbers do. It is a means of communicating between two or more agents (i.e., man-man, man-machine, machine-machine), and a possible role of qualitative models is to produce information of this kind. Fruitful and rich informal thinking about complex systems can be based on nonmathematical language. Dependencies can be expressed in terms of commonsense notions such as causation. In most qualitative models, causal paths can be described explicitly. When the target real-world system is dynamic, causal paths from the system's initial state to subsequent ones can also be produced. They can either be described in extenso in a knowledge base (KB) or dynamically generated during a simulation session. In the context of electrocardiography (ECG), the observation is a time-varying signal whose spatiotemporal characteristics are causally linked to a huge number of distinct underlying time-varying tissue states. That is, it would be virtually impossible to encode in a KB all the possible paths of the activation wavefronts, even if the computational model is grossly detailed. In ECG, the activation paths should be built dynamically, as is done using the numerical approach. Qualitative models can also be used for diagnosis. Diagnosis is, in a sense, opposite to simulation. Second-generation diagnostic systems, also known as "model-based," are anchored around qualitative models (5, 10, 24, 25, 27). Loosely speaking, explanations are provided by going "backward" through the causal paths, from the observation to the cause(s). In what follows, we will mainly deal with simulation (or prediction).

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II. CURRENT RESEARCH

Four problems can be associated with modeling and simulation in ECG (3•): (1) model the electrical conduction system, (2) model the electrical activity of muscle tissues, (3) model the effects of volume conduction, and (4) produce summaries (descriptions) of the salient events or states associated to a particular ECG simulation. To the authors' knowledge, no single computational model addresses these four problems. A number of researchers have developed qualitative models of the heart for diagnostic purposes (5, 27, 29, 30). Some of the models' main characteristics and respective merits have been reported in a previous paper (30). As for numerical models, a thorough review is given in (15). Numerical models can be separated into two broad classes; those that model the heart's conduction system (9, 22) and those that model patches of cardiac muscle tissue (4, 16). The latter type can be further separated into a class that simulates the ECG by computing current densities and the effects of volume conduction (14) and a class that only simulates the isochrones (4). The most complete cardiac muscle models incorporate element-to-element propagation in the three dimensions (15). As noted, however, they are costly in terms of computer time and space, nodal tissues are not included, and they lack qualification capabilities that can limit their full exploitation, as will be developed later. The C A R D I O L A B project is an attempt to address all four aspects related to modeling in ECG, and we are working on integrating declarative knowledge, numerical models, and different levels of descriptions into a single framework (29-31). In fine, the framework (and the artifacts thereof) is expected to fulfill four main roles: basic and pharmacological research, teaching, model-based diagnosis, and monitoring.

III. BASIC FACTORS OF ARRHYTHMIAS Recall that cardiac arrhythmias may be classified as disturbances of impulse formation or disturbances of impulse conduction or as a combination of both (6, 26). Impulse formation is dependent on three factors characterizing an autorhythmic (or pacemaker) cell: (1) the slope of spontaneous diastolic depolarization. (2) the maximum diastolic potential at the end of repolarization, and (3) the magnitude of the threshold. Modifications of these parameters (e.g., by vagal influence), independently or concurrently, may increase or decrease the pacemaker's rate of discharge. Under normal conditions, the sino-atrial (SA) node has the command of the heart's electrical activity. However, other elements of the nodal tissue may exhibit spontaneous, cyclic cell-membrane depolarizations that can propagate through the conduction system to depolarize the entire heart (6). The three major factors of impulse conduction are (1) the rate of rise of the potential upstroke, (2) the amplitude of the action potential, and (3) the length of the refractory periods. A decrease in the slope of the upstroke or in the height of the action potential reduces conduction velocity. Modification of these parameters as a result of decreased tissue states and/or drug uptake may increase, decrease, or altogether stop conduction in the affected regions.

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1V.1. Conceptualization and Model Description Our conceptualization has been carried out within the framework of firstorder logic (FOL). The reason for this choice has to do with the anticipated framework components and complexity, which includes communicating "specialists" (29), symbolic learning paradigms, and natural-language human-machine interface (HMI). In other words, we want to avoid implementing locally adequate representations, say, for simulation, that would be totally unfit to carry out inductive learning or to convey messages to a module (or agent) analyzing an ECG. Therefore, solid grounding of the project in the mathematical apparatus of FOL seemed desirable as a universal language for conceptualization and as a means of avoiding (or reducing) unbridled hacking. This calculus is a representational language with a rigorous syntax, a clear denotational semantics, and sound inference. It is expressive enough so that most of AI systems' declarative knowledge can be represented by its constituent objects, functions, and relations (13). Furthermore, since our final goal is to produce a real-life system (as opposed to pure research laboratory "close-world" programs), the HMI will constitute an essential part. Moreover, computational linguistic representations such as the conceptual graphs developed by Sowa can be rigorously mapped into FOL sentences and vice versa (33). In FOL, conceptualization involves defining the triplet (U,F,R), where U is the universe of discourse, F the functional basis set, and R the relational basis set (13). The actual implementation is based on this conceptualization, but departs from it in terms of form, since implementation constraints such as optimization and programming language specifications had to be introduced. The current model has been implemented in BIM-PROLOG on a Spark 2 workstation. Concerning CARDIOLAB's qualitative model, the universe of discourse is partitioned into four basic sets: ANAT, PHYSIO, PHYSIO-EVENT, and OBS. This partition of U has a direct incidence on the system's task and KB organization which is centered around a blackboard (Fig. 1). ANAT and PHYSIO are the sets of anatomical descriptions and electrophysiological heart-tissue parameters, respectively. These two sets constitute the main corpus of knowledge of the computational model's knowledge base. PHYSIO-EVENT is the set of all dynamic physiological processes. In terms of the model, representatives of this set are dynamically derived in the blackboard by qualitative process simulation (Fig. 1). PHYSIOEVENTS is itself partitioned into three subsets corresponding to three levels of descriptions that we have named the PROCESS, EVENT, and DIAGNOSTIC levels. The objects of the set PROCESS are represented by functional terms whose domain are the sets ANAT and PHYSIO. They designate elementary processes such as "the lower-left ventricle is in an absolute refractory period." OBS is the set of observations, notably the analogical and qualitative representations of the ECG. In the following FOL statements, symbols starting with an upper case letter will represent predicates, functional terms, and constant terms, while those starting with a lower case symbol and written in italic will represent variables. Model elements. Each element is characterized by (1) a name, (2) the type of

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FIG. 1. Qualitative process simulation. The flow of facts and derived information is depicted in this figure. The central working space is a blackboard (BB). Initial conditions are derived from the ARRHYTHMIA, PHYSIOLOGY, and ANATOMY KBs. A sequencer then writes time-stamped PROCESS level event descriptions in the BB. At each BB cycle, the conduction-path topography is reactualized. This task is carried out by a graph-generating and -manipulating task. EVENT and DIAGNOSTIC level concepts are derived from the PROCESS level event descriptions of the BB by abstraction tasks (Abstract1 and Abstract2). Qualitative descriptions are produced by a (presently) crude text generator (Generate text) and a drawing task (Draw-ECG) carries-out the display of the corresponding simulated ECG. Concept generation and ECG simulation are done concurrently with process simulation. At the end of a simulation and upon user request, the system can memorize the case into the arrhythmia KB. This is done by memorizing the initial conditions and the multilevel summaries.

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tissue (muscle or nodal), and (3) the electrical state (or phase). There are 21 elements belonging to muscle or nodal tissue (Fig. 2): • 6 elements for muscle tissue (superior and inferior atria, superior right and

left ventricles, inferior right and left ventricles) • 15 elements for the specialized conduction (or nodal) tissue (SA node,

superior and inferior atria nodal tissue, 4 elements (superior, inferior, left and right) for the A VN, superior and inferior His bundle, two structures for each ventricular branch, superior and inferior Kent bundle). This partition can be justified by the fact that it is the usual level of anatomical detail adopted by clinicians when they explain arrhythmias. Also, the problem of degeneracy resulting from the limited range of values characterizing qualitative simulation put forth by some authors (20) does not concern the present model: one initial condition yields a single solution.

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FxG. 2. The heart model's anatomical decomposition. The model is composed of 21 elements: 6 elements for muscle tissue (superior and inferior atria, superior right and left ventricles, inferior right and left ventricles) and 15 elements for the specialized conduction (or nodal) tissue (SA node, superior

and inferior atria nodal tissue (superior, inferior, left and right), A VN, superior and inferior His bundle, two structures for each ventricular branch, superior and inferior Kent bundle).

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Physiological parameters. The adopted electrophysiological parameters are identical to those formulated for the cellular automata models (31). Their values and range have been derived from the literature (3, 6, 28). Transmembrane potentials and current flows are not accounted for. Only factor (1) of impulse formation (see section III) is being addressed. Factors (1) and (2) of impulse conduction are "chunked" into a single macroscopic quantity: the depolarization-phase duration while the refractory period is made explicit. We thus have the following element attribute durations: • For muscle tissue elements, the resting state, the upstroke activation time, the relative and absolute refractory periods. • For nodal tissue elements, the slow diastolic depolarization time, the upstroke

activation time, the relative and absolute refractory periods. Hence, each nodal tissue element (to the exception of the Kent bundle, where we assume that no ectopic pacing can occur) is capable of generating its own impulses, and in the normal sinus rhythms, all distal elements are discharged by the sinus node (SN) since a frequency gradient--from proximal to distal elements--has been incorporated in the model. Because of this possibility, events implying AV nodal pacing, such as may occur in total AV blocks (Fig. 11), can be simulated. The tissue characteristics are defined by relations, For instance, the sinus node slow-diastolic depolarization is given by the relation Phase-duration(SN,slow-diastolic-depolarization,500) and both kinds of tissues are partitioned into the 21 basic building blocks or elements.

IV.2. The Physiological Characteristics Introduced into the Model Each phase is qualified as excitable or not excitable. A model element's state transition follows rules describing excitability, frequency adaptation, and conduc-

tion velocity adaptation. Excitability. An element of the cardiac muscle type in a "resting state" or a "relative refractory period" will "depolarize" in the presence of an incoming (or afferent) impulse. Likewise, an element designating nodal tissue in a "slow diastolic depolarization" or "relative refractory" will be "discharged" by an incoming impulse. In the absence of an incoming impulse, when the end of the "slow diastolic period" is reached, then a nodal element will depolarize. Finally, any model element whose state-description denotes an "absolute refractory period" will remain unchanged in the presence of incoming impulses. Frequency adaptation. This property is modeled by a function particularized for each structure. The refractory period of a patch of cardiac tissue adapts to the rate of its successive depolarizations. When the frequency increases, the refractory period decreases and inversely. In the model, the duration associated with the refractory period of the nth beat is given by a function of the form

T,, = g(F,_,, D),

[1]

where Fn-1 is the current depolarization rate of the structure (calculated with the last two depolarizations) and D is the duration of the refractory period for

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a normal 70 bpm rate. The function g has been defined empirically and is a piecewise linear function (21) (Fig. 3). Conduction velocity adaptation. In accordance with known physiological data (5), conduction delays are modeled by adapting each element's depolarization duration to the afferent impulses' prematurity. An element in a partial-refractory phase will be depolarized by an incoming impulse but the rise time will be increased by a factor proportional to the partial refractoriness. Each model structure is thus characterized by a function that allows to calculate the duration of the structure's depolarization in the case where an "afferent premature impulse" message is present. This model parameter adaptative property is described by the relation (21) (Fig. 4)

D ÷ = f ( P , D) At P=I---

[2]

R'

where P is the prematurity factor, At is the time lag between the beginning of the refractory state and the instant the impulse reaches the structure, R is the relative refractory period, and D is the normal depolarization duration (i.e., corresponding to a frequency of 70 bpm). To the author's knowledge, there is no exact expression for the function f. However, the function depicted in Fig. 4 has been extrapolated from data obtained in the guinea pig (11). To summarize, an element's absolute refractory period is reduced when the cycle length is reduced, and conduction velocity in a given element is also reduced if an afferent impulse reactivates the element in its relative refractory period.

T n (ms)

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D,

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FIG. 3. Frequency adaptation of the refractory phase. Nodal tissue, notably the AVN, adapts to the frequency of incoming impulses. Qualitatively speaking, as frequency increases, the refractory phase duration decreases up to a limiting value. This relationship is concreted by the above empirical function, where T, is the refractory phase duration at the nth cycle, and F,_~ is the frequency corresponding to the preceeding cycle. A particular value, A, corresponds to the refractory phase duration of "normal" nodal tissue in a 70-bpm rhythm context.

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P F~G. 4. Conduction velocity adaptation. Each model element is characterized by a function that depicts the depolarization duration as a function of the prematurity (see text). At this space scale, the depolarization duration is the time taken by a wavefront to propagate across the patch of myocardium represented by the model element. When prematurity occurs early, at the beginning of the relative refractory period (AT near 0), P = 1 - At/R is close to 1 and the impulse is slowed almost to a stop. On the other hand, if At is close to the refractory duration, R, then quasinormal conduction occurs.

IV.3. The Explicit Representation of Time Since one of the system's requirements is to describe in linguistic terms the events that occur during a simulation session, the temporal support of all e v e n t s - from the finest grained to the most g l o b a l w n e e d s to be explicitly stated. Informally speaking, the system should be able to make assertions such as "event A occurred between to and tl" or "event A overlapped even B." Most importantly, it should infer the proper temporal supports of global, more abstract events derived from the P R O C E S S level event. A t many space scales, any event can be described as a finite set of explicit temporal relations between one or more elementary events (although at the atomic and subatomic scales, quantum mechanics show otherwise, but this point is not addressed here). A t the space scales considered here we have assumed that intentional definitions of global events can be derived in the form of temporal orderings of more elementary events whose definitions are also intentional. To rigorously differentiate time-invariant facts (or by convention considered as such) with changing ontologies, logicians have extended F O L into the so-called temporal logics (1, 2, 23) where explicit representations of times characterize the c o m m o n understanding and usages of such concepts as "occurrences" or events. Hence, one of the objectives of this research area is to provide a commonsense approach to time. Allen and Hayes' (1, 2) and M c D e r m o t t ' s (23) models of time are both extensions of first-order logic and belong to the family of the so-called "reified temporal logics" in which propositions are made into objects: The temporal support of all events is explicitly

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represented by reifying these concepts into functional terms, the temporal aspects of which are then indicated by asserting proposition types to be true (or to hold at particular times). In the present context, the propositions are events such as "the upper atrium is depolarizing" or "an afferent impulse coming from the lower bundle branches to the ventricles occurred." These propositions are represented as terms of a particular predicate, the "Holds" predicate which explicitly represents the temporal occurrence of the events. Those of Allen and Hayes are based on intervals and those of McDermott on points or instants, but both formalisms are formally equivalent (34). Our conceptualization explicitly uses both formalisms since we consider events that have durations, events that are "instantaneous," and states. An elementary, temporally marked event Pi is defined by

Pi = Holds(Di, Ii),

[3]

where D,- E PHYSIO-EVENT and Ii is the time interval over which Di holds. For instance, the fact that a structure S is in a partial refractory state during an interval [to, tl[ will be represented by Holds(Partial-refractory-state(S), [to, tl[). Similarly, an "instantaneous" temporally marked event can be defined by

Pi = Holds( Di, [t, t]) -= Holds( Di, at(t)).

[4]

For example, the fact that an afferent impulse to structure $1 came from an adjacent structure $2 at a given time instant t can be expressed by Holds(Afferentimpulse(Sb $2), at(t)). Finally, a state description (following McDermott) such as "the atrium was depolarizing at t" can be given by Hols(Depolarizing (Atrium), at(t)).

1V.4. Other Model Features Rapidity and ECG simulation. As mentioned, E C G generation can be essential to correlate the timing of electrophysiological events with E C G pattern variations and morphology. This model is a crude propagation-type heart model. E C G synthesis is carried out concurrently with qualitiative process simulation. The latter is carried out at faster-than-real-time speed. The major time-consuming simulation task is the E C G display. An equivalent dipole is assigned to each element and oriented perpendicular to it. The dipole amplitudes' time course is dependent on the size of the element and follows a sum of sinusoidal function whose coefficients and phases have been defined empirically for retrograde and anterograde wavefronts. Although the model representation on the screen is planar, the model is assumed to be on an ellipsoidal surface that can be rotated parallel to the frontal plane in order to account for the heart's different positions in the chest. That is, there are two coordinate systems associated with the E C G simulation (Fig. 5): one linked to the heart and the other to the thorax. The mathematical expressions describing the time courses of the elementary vectors are associated with the heart coordinate system, while the standard leads are associated with the thorax coordinate system. By this means the user can define the electrical axis of the heart. At each time increment, a vectocardiogram (VCG)

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Yc

FIG. 5. Coordinate system associated to the simulation of the ECG. The coordinate system linked to the heart is given by (Xc, Yc, Zc), and the coordinate system solid to the body is given by (Xb, Yb, Zh). Zc is traditionally directed toward the back, and c~ is the heart's electrical axis in the frontal plane, The user can define the heart's electrical axis.

is generated by summing each elementary vector. It is then projected on two derivations chosen by the user. More specifically, ECG synthesis is carried out in five steps: 1. Calculation of the local vector's time course produced by each structure in a coordinate system linked to the heart. The result is a local action potential. 2. Calculation of the total vector by addition of the local vectors. 3. Calculation of the VCG by projecting of the total vector in a coordinate system linked to the thorax. 4. Projection of the VCG on selected leads. 5. Screen display of the resulting simulated ECG. The empirical constructions notwithstanding, the local action potentials satisfy the following constraints: (i) their temporal supports vary according to the activestate durations of the corresponding structure; (ii) their amplitude is adequate to the size of the generating element; and (iii) their sum projected on any given lead yields acceptable waveforms (see Figs. 9 and 10). For instance, local hypertrophy can be represented by increasing the amplitude and duration of the activation wavefronts corresponding to the elements associated with hypertrophy, and infarct by reducing the amplitude of the activation wavefronts corresponding to elements associated to the region of infarct. Shortcomings. Since potential fields or current densities are not treated by the model, the magnitude of the threshold and the maximum diastolic potential (factors (2) and (3) of impulse formation) and the amplitude of the action potential (factor (2) of impulse conduction) are not explicit. As a result, (i) a "cell" will always be activated by an incoming impulse if it is not in an absolute refractory period, and (ii) pacemaker discharge of distal nodal cells due to the

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spread of electrotonus over ischemic tissue (6, 19) cannot be accounted for. Indeed, at the space scale of tissue patches, their response to electrotonus will depend on the average transmembrane resting-state potential of the constitutive cells and on the extracellular electric field intensity. The other obvious insuffiency has to do with the model's gross anatomical decomposition. Atrial flutter and fibrillations and ventricular fibrillations cannot be adequately simulated. As a corollary to the above-mentioned simplifications, the simulated ECG is too crude to allow detailed (in terms of medical requirements) morphological analysis. Hence, the simulated signal should mainly be regarded as a visual aid excluding any quantitative analysis. The manner by which each elementary dipole evolves is not based, as it should be, on potentials computed from transmembrane currentsource densities (14) and by computing volume conduction effects (7), but on somewhat arbitrary mathematical functions which, when summed, yield an acceptable ECG. Computational tractability (near real-time speed), process intelligibility, and interactivity have been gained at the expense of a reduction of a certain number, and kind, of interactions that can be modeled. However, as already suggested, this model is intended to be at the forefront of the HMI, between the user and the detailed numerical models (Fig. 12 and 13).

V. PROCESS SIMULATION AND THE HIERARCHY OF CONCEPTS

Browsing through the literature and auditioning physicians show that certain key concepts are invariably invoked, including in daily clinical practice. In ECG, these concepts are generally physiological events and constitute landmarks in problem-solving activities such as diagnosis. A possible representational scheme then is to derive explicit representations of these landmarks that constitute sets of "mental constructs" at different levels of detail. Many events may be described in terms of temporal orderings of finer-grained events. Thus, the explicit representation of time and concept hierarchies are key elements of the representation problem. A fundamental question that arised during conceptualization was whether a generating set of concepts could be identified from which all the landmark, higher level concepts could be derived. Simulation could then proceed in terms of the generating set only, i.e., the generating set would be deductively closed at the simulation level. The answer proved to be affirmative. Referring to section III, they are the factors of impulse formation and conduction or the chunked quantities thereof. They belong to what we have labeled the PROCESS level, because process simulation is carried out at this level. Other two, more global levels are derived from the basic event sequences of the PROCESS level: they are the EVENT level and the DIAGNOSTIC level (Fig. 1). All three sets form the set PHYSIO-EVENT of the universe of discourse. Their role is to produce simulation descriptions that cover the three fundamental aspects of cardiac rhythm (26): (1) The rhythm has an anatomical origin, (2) the rhythm has a discharge sequence, and (3) the rhythm has a conduction sequence. Any complete description of a cardiac rhythm should refer to all three aspects. As

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an example, consider the following diagnostic: "Normal sinus rhythm of 70 beats per minute with a 2:1 A V block." The anatomical origin is the SA node, the discharge sequence is normal at 70 bpm, and the conduction sequence is a 2:1 AV block. This is a D I A G N O S T I C level cardiac rhythm description. The role of the abstraction tasks (Fig. 1) is to derive such sentences from the elementary PROCESS level events by composing landmark concepts of one level into those of higher levels. V.1. The P R O C E S S Level As introduced, the canonical, landmark concepts of the PROCESS level are the tissue-phase transitions. Process simulation is performed at this level. Simulation proceeds as follows: when an element is "activated," it "generates" an impulse afferent to contiguous structures. This state of affairs is expressed by the creation of an object, an afferent impulse, which can be viewed as a message with distinct effects on the recipients of the message. Here the recipients are the contiguous elements which will respond positively if they are in an excitable state (and therefore undergo a forced state transition) and negatively otherwise. In other words, the landmark concepts associated with impulse formation are slow-diastolic depolarization, depolarization, and pacemaker discharge while the landmark concepts associated with impulse conduction are the refractory states, the resting state, and afferent impulse. An instance of the generation of an afferent impulse description is Vx3y Holds(depolarizing(x), [to, tl [)/% Adjacent(x, y) D Holds(Afferent - impulse(x, y, at(h))),

[s]

which simply states that "an impulse afferent to an element y and coming from an element x is produced at some time point t~ if x is adjacent to y and if x was depolarizing during some interval [to, hi. Note that while the Adjacent relation designates a predefined, static concept in the conceptualization (and encoded as a constant in the physiological KB), events are time-stamped by intervals or time points and are dynamically generated in the blackboard. An example of a forced state transition is the discharging of a pacemaker. It occurs when an afferent impulse depolarizes the pacemaker before or when it reaches the end of its natural slow-diastolic depolarization period. This fact can be written as VxVy Holds(In - slow - diastolic - depolarization(x), at(t)) /% Holds(Afferent - impulse(y, x), at(t)) /% Phase - duration(Slow - diastolic - depolarization, x, D)/% 12 is t + D D Holds(Depolarizing(x), 12), [6] where the relation Phase-duration(slow-diastolic-depolarization,x,D) denotes the (physiological KB) fact that pacemaker x has a slow-diastolic depolarization of duration D. Now, given a simulation interval, say intsim, the PROCESS events

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generated in the blackboard can be abstracted into more global ones by an abstraction task as we see now. V.2. The E V E N T Level At this level, we introduce a special kind of object in the conceptualization: that of directed graphs. They are derived by an abstraction task (Fig. 1), where graph construction, manipulation and abstraction rules constitute the essentials of the task. The starting point is the successive depolarization of contiguous structures as obtained at the P R O C E S S level and from which chronicles are built in the form of directed graphs. More specifically, we define a chronicle as a directed tree structure, constructed over some time interval I: Gchron,int = (X~h .... int, E~hro.,~,,t), where Xch .... int is the set vertices and E~h.... ~nt the set of edges. The elements of Xch .... im are each designated by a doublet (x, t), where x is a structure ~ A N A T and t denotes the depolarizarion onset of x . Schron,in t and Echron,in t are built as follows: If, during the interval intsim, the following P R O C E S S level facts have been derived Holds(Depolarizing(x), [to, q D Holds(Afferent - impulse(x, y), at(t1)) Holds(Depolarizing(y), [tl, t2[) then add (x, to) and (y, q) to Xch .... in, and add ((x, to), (y, tl)) to Ech .... in,. Hence, a chronicle is a way of representing a "percolating" conduction path of a propagating wavefront (Fig. 6). given a chronicle, C A R D I O L A B ' s graphmainpulating task generates an object which we accept as an argument of a F O L predicate: that of embedded lists. This is just another way of representing the path of the given propagating wavefront. We have the following definition: A conduction path occurring over a given time interval, int, is a list representation of the corresponding partial or total chronicle. That is, an object, p, whose status is given by the (reified) predicate Conduction-path(p) is an ordered list (with respect to the temporal dimension) constructed from a chronicle and defined as follows. Let Lchr.in t = (((X0, to), (X1, tl)), ((Xl, tl), (X2, t2)) . . . , ((Xn-1 t,-1, xn, tn))) be the ordered list representation of C~h.... int, where xi designates a depolarizing structure and ti the onset time; then the conduction path can be defined as a list of elements p - (eo, el . . . . .

em),

where int = [to, tn [ and the element ej is either a node of the form (xi, tj) or a list. For example, from Fig. 6, we have

457

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FIG. 6. Associating a chronical with an instance of reentry. (a) A traveling impulse arriving at A is blocked on F but reenters via B-C-D-E-F. (b) The corresponding chronicle is a branching-time graph that can be represented by an ordered list (see text). Note the elementary circuit (B, tl) ... (B,/8)-

Lch . . . .

[t0,t8[ = (((A, to), (B, t,)), ((B, t,), (C, t2)), ((B, tl), (H, t2)), ((C, t2), (D, t3), ((D, t3) , (G, t4)), ((D, t3), (E, ts), ((E, ts), (F, t6)),

((F, t6), (H, t7)), ((n, tT), (a, t8))) and the corresponding list representation, p, is given by p = ( ( h , to), (B, tx), ( ( H , t2)), ((C, t2), ( D , t3), ( ( G , t4)),

((E, ts), (F, t6), (H, tT), (B, ta)))).

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The object, p, can be built from the Lch ..... int graph and list operations. Most importantly, a new E V E N T level fact is written in the blackboard, namely,

Holds(Conduction-path(p), int). Conduction path characteristics. From the foregoing discussion, the central concept of the E V E N T level is that of branching path represented by the object p. Most landmark concepts of the level designate particular attributes to these paths. For instance, conduction paths may be linear, branching trees, elementary circuits, or complex circuits. Associated with these attributes we may have a normal traveling impulse, a simple reentry, or self-sustained circus movements. These attributes are derived from the structures of the embedded lists by the list and graph-manipulating task. To resume, the E V E N T level includes the following families of landmark concepts (see Fig. 7): • Conduction path attributes (anterograde; retrograde; subpath; circuit; reentry; elementary-circuit; self-sustained-circuit; branch-to-branch; normal, abnormal, or aberrant; sinus traveling inwulse; traveling-impulse). • Conduction modality attributes (slow, slowing, normal). • Blocks (directional (anterograde, retrograde), exit block, entrance block, protection). • General conduction attributes (path and modality) (homogeneous, inhomogeneous, simultaneous, concealed). All concepts are defined intentionally. Here are some examples: Attributes associated to the conduction path topography. A commonly used concept in clinical practice is that of traveling impulse. What is usually conveyed when this concept is evocated is the impulse origin of the propagating wavefront as well as the conduction sequence. We thus have the following definitions: V I, Vpcmk V lwr-lfs Vp Holds(Traveling - impulse(p, pcmk, lwr - lfs), I) ¢:* Holds(Conduction - path(p), I) A Origin (p, p c m k )

[7]

A Pacemaker(pcmk) A Lower - leafs(p, lwr - lfs), where the predicate Origin is defined by Origin(p) = y ¢:* y = First(p), the function symbol First designates the first element of the list p, and lwr-lfs is a list of the leafs of the branching wavefront. Using the hierarchy of structures, this list can be reduced by change-of-scale rules. Of particular interest is the sinus traveling impulse. It is a traveling impulse whose origin is the SA node that verifies the functional term equivalence:

Sinus - traveling - impulse(p, lwr - lfs) =- Traveling - impulse(p, SA - node, lwr - lfs). As another example, consider a particular instance of a branch-to-branch path in which the impulse may propagate linearly from the lower LBB to the

AN INTERACTIVE QUALITATIVE MODEL IN CARDIOLOGY

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lower RBB. This state of affairs can be expressed in terms of chronicles and paths:

Vto, Vh Vp'Vp nolds(Lin - path(p',p), [to, hD A First(p') = ( L L B B , to) A Last(p') = ( L R B B , h)

[8]

D Holds(Branch - to - branch - path(p'), [to, h[), where the predicate Holds(Lin - path(p', p), [to, tt[) designates the fact that a traveling impulse followed a conduction path p containing a linear path p' (a path without circuits) that started in the lower left bundle branch and ended in the lower right bundle branch and that the time interval during which this linear path propagation occurred lasted between to and h. It is common usage for clinicians to qualify conduction paths as being normal or abnormal Normality is considered when nothing abnormal occurred. For a conduction path we have:

VpVI Holds(Conduction - path(p), I) A Normal(p) ¢~ -1 (Holds(Conduction - path(p), I) A Abnormal(p)).

[9]

An instance of an abnormal path is when an accessory pathway has been visited. In the present model, only the Kent pathway is considered, but the definition can be extended to the James or Mahaim bundles (6). Directional blocks. A first-order temporal logic definition of an exit block is determined with respect to a discharging pacemaker and the state of its surrounding tissue:

Vx Pacemaker(x) A Holds(Slow- diastolic-depolarization(x), I•) A 11 = [to, tt[ A Holds(Absolute - refractory - phase(surround(x),/2)

[10]

A (Overlaps(It, 12) V In(ll, 12) V Meets(ll, 12)) D Holds(Exit - block(x), at(h)), where In([to, h[, [t2 t3[) ¢=> t2 < to < tt < t3 and surround(x) designates the tissue that surrounds the region denoted by the variable x and where the temporal relations Overlaps, In, and Meets are Allen's (1). Attributes associated to the conduction modality. Conduction may be normal, abnormal, or inhomogeneous. It is dependent on the conduction speed that, given a region, can be slow, slowing, increasing, anterograde, or retrograde. The conduction modality over a time interval will be considered abnormal if there are structures in the conduction path where the propagation is slow, slowing, or retrograde or if a block occurred during the time interval. Directional attributes anterograde and retrograde are based on topographical knowledge and the velocity attributes slowing or increasing on temporal differences of the instantaneous velocity. On the other hand, the attribute slow can only be

460

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WENCKEBACH3:2 - The f i r s t cycle is a Wenckebach4:3 which thereafter stabilizes into a Wenckebach 3:2 . The sinus rhythm is 79 bpm and a]ong AVN refractory period (980 ms) characterizes the AV Junction. - I n t e r e s t i n g leads : OI, V3 .

....,......a

FIG. 7. T h e main and the simulation windows. A t the beginning of each simulation session, the main window is displayed. Prestored cases in the form of the initial conditions can be selected from the arrhythmia KB via the arrhythmia button. If a new case is to be explored, the user can choose one of the prestored cases (that include normal sinus rhythms) and thereafter modify the physiological parameters corresponding to the selected case (see Fig. 8). Once the initial conditions have been determined, the duration of the simulation is then given by entering the desired value at the level of the duration field of the simulation window. T h e current simulation time is given by the time

AN INTERACTIVE QUALITATIVE MODEL IN CARDIOLOGY

461

established on the basis of prior statistical information. To date, this statistical information has not been introduced in the KB as we are still eliciting this information from the literature. Putting aside this question, an abnormal conduction will be recognized for some conduction path p if for some subpath p ' the conduction speed is slow or retrograde or if a block occurred:

Holds(Abnormal - conduction - modality(p, I) ¢=~ 3t9', 311 In(It, 1)/k Subpath(p',p) /k Holds(Conduction - speed(p', slow), 11) V Holds (Conduction - direction (p', retrograde), It) V 3 r 3 s 3 t Holds(Block(r, s), at(t))/~ Within(t, I). [11] From the previous definitions, any conduction path that contains a circuit or a branch-to-branch conduction exhibits an abnormal conduction. In a manner similar to conduction path topography, normality for the conduction modality is established if no abnormality is detected. In this case, the conduction is said to be homogeneous. From the foregoing discussion, normal conduction occurs during some time interval if the conduction path is normal and h o m o g e n e o u s conduction occurs:

V l V p Holds(Normal - conduction (p), I) ¢:~ Holds(Normal - Conduction - path(p), I) /k Holds (Homogeneous - conduction (p), I).

[12]

Finally, derived E V E N T concepts include normal conduction in the ventricles,

aberrant conduction in the ventricles, conduction failure of the sinus impulse, and uentricular fusion beat. V.3. The D I A G N O S T I C Level While the E V E N T level is concerned with paths and path attributes, the I ~ I A G N O S T I C level corresponds to the description and labeling of normal rhythms and rhythmic disorders: the level is that of a cardiologist diagnosing an E C G (Fig. 7). It is based on the two previous levels. The concept set and the corresponding international definitions include • Impulse formation: premature beat, capture beat, escape beat, escape rhythms,

sinus bradycardia, sinus tachycardia, atrial extrasystole, A Vnodal extrasystole,

field. Two leads among the 12 standard derivations can be chosen. As process simulation proceeds, the ECG, the current state of each structure, and the corresponding qualitative descriptions can be displayed. Here we have EVENT and DIAGNOSTIC level descriptions. At the end of a simulation epoch, the process can be further pursued by modifying the duration field value. If not, then the user can increase the arrhythmia KB by clicking the save button. The ensuing system actions are (i) memorize the initial conditions and (ii) memorize the multilevel descriptions.

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A Vnodal tachycardia, ventricular escape rhythm, ventricular extrasystole, ventricular parasystole. • Impulse conduction: accessory conduction-path, fusion, normally and abnormally conducted sinus impulse, A V-blocks (first, second, and third degree). Here are some examples: Conduction disturbance. An example is the Wenckebach phenomenon (Fig. 13a) characterized by cycles of increasing PR intervals followed by a dropped QRS complex revealing a total AV block. A new cycle is then initiated. The increase in the PR interval results from a monotonically slowing of the conduction velocity in the AV junction until propagation is totally blocked. This state of affairs can be summarized by the following rule: Holds (Slo wing - conduction (A VN), Ii) A Holds(Traveling - impulse(p, SN, (LAN), 12) A Meets(Ii,/2) A Holds(Block(SN, UA VN), at(I~))

[13]

A I3 is I1 U 12 D Holds(Wenkebach, I3), where I2 is the starting point of the interval 12, L A N designates the lower atrial nodal tissue, and UAVN the upper AVN. Note that the statement Holds(Slowing - conduction(A VN), Iz) is an E V E N T level description built from the lower PROCESS level descriptions showing the hierarchies that are built during qualitative process simulation. Impulse formation disturbance. An example is that of an AVnodal extrasystole. This event is characterized by a ventricular contraction whose impulse origin is the AVN. Using the traveling-impulse description of the E V E N T level, we have

VpVIHolds(A VN - extrasystole(p), I) ¢:* Holds(Traveling - impulse(p, A VN, (R V, Lie)), I),

[141

where change-of-scale rules have been applied to obtain the list (RV, LV). Likewise, consider an AVnodal tachycardia. This is often described as a ventricular pacing at a rate between 150 and 200 bpms and whose impulse origin is in the AVN. In our conceptualization, this state of affairs can be written as

VrVI Holds(Ventricular - pacing(A VN, (RV, L V), I) Holds(pacemaker - rate(A VN) = r), I) A Between(r, 150,200) D Holds(AVnodal - tachycardia, I),

[15]

where Holds(Ventricular - pacing(A VN, (RV, L VV), I) designates the fact that both left and right ventricles are placed b3~the A VN during some interval I. Finally, sinus tachycardia, ventricular tachycardias, and all bradyacardias

AN INTERACTIVE QUALITATIVE MODEL IN CARDIOLOGY

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can be defined in a similar way. Complex arrhythmias involving different causes may be difficult to define a priori. In a manner similar to Bratko et al. (5), we are working on inductive schemes to derive these definitions from the E V E N T and the predefined D I A G N O S T I C landmark concepts. VI. USER INTERFACE AND CASE EXAMPLES

The main window is the starting point of a simulation session (Fig. 7). The arrhythmia button allows the selection of prestored cases partitioned into the impulse, conduction, and combined disturbance tricothomy. The load and run button triggers the display of the station window. It allows the user to control the simulation and to visualize the outcome. The main features of the simulation windows are the 2D graphical model, the simulated E C G , and the (semi-) naturallanguage text descriptions at different levels of description (PROCESS, E V E N T , or D I A G N O S T I C ) . U p o n starting a simulation, the user can choose one of the previously stored cases by clicking on the pathological case button or by defining himself the initial system state by clicking the "physiology" (see below). By clicking the "action" button, simulation begins, the unfolding of the simulation, i.e., step by step (the action button) or by defining beforehand the duration of the simulation using the " d u r a t i o n " field of the simulation window. Ongoing E C G then appears on the screen at the pace of process simulation. The window size corresponds to 4 sec of ECG. The vertical bars on the frame correspond to 200 msec as it is classically done in electrocardiography. The command board also allows the user to specify if the Kent accessory pathway should be introduced (i.e., the Kent button). The physiological window (Fig. 8) is of fundamental interest for the user. It is via this window that he/she can interactively specify the physiological parameters. The left-hand side is a menu with 21 structures (more if on chooses to add ectopic pacemakers). When a structure is selected, a window exhibiting its corresponding parameter set and current parameter values appears on the screen. The user can then either save those value by c~cking the "save durations" button on the command board or modify one or all of the said parameters. This interactive process can be pursued for any of the other structures. Once the modifications are done, simulation can proceed by clicking the action button. The lower right side of the window allows the introduction of an ectopic pacemaker in the selected structure and the determination of its physiological parameters (frequency, starting time of ectopic activity, if it is protected, etc.). At the end of a simulation session, if the sequence of events exhibits interesting features, the initial conditions and semi-natural-language descriptions can be saved and stored in one of the predefined classes of the rhythmic disorders or form a new subclass (Fig. 1). Hence, in all the ensuing simulation sessions, this new case can be chosen via the command board of the main window (Fig. 7). Figure 9 illustrates impulse formation disorders, while Fig. 10 illustrates combined impulse formation and impulse conduction disorders. Finally, Fig. 11 exhibits a simulation of a therapeutic intervention.

464

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F]6. 8. The physiology window. This window allows the user to add anatomical attributes (e.g., a Kent accessory pathway, one or more pacemakers) to the "normal" heart model. Most importantly, physiological parameter values of any given structure can be interactively modified via this window. A structure is selected, and the desired modifications may be performed. This interactive process is operative at the beginning of a simulation session and at the end of a simulation epoch. By this means, drug action at a specified time can be simulated (see Fig. 10).

To summarize, the visulaization capability, the possibility of interactively determining structural and physiological attributes, and the possibility of increasing a l m o s t w i t h o u t limits the r h y t h m i c d i s o r d e r K B u n d e r l i n e t h e p o t e n t i a l i t y o f t h e m o d e l as a c o n s t i t u e n t o f a c o m p u t a t i o n a l r e s e a r c h a n d t e a c h i n g e n v i r o n m e n t . F u r t h e r m o r e , b y o n l y m e m o r i z i n g t h e initial p a r a m e t e r v e c t o r a n d t h e c o r r e s p o n d i n g s y n t h e t i c e v e n t d e s c r i p t i o n ( i n s t e a d of, say, a signal) i m p o r t a n t d a t a c o m p r e s s i o n is a c h i e v e d .

AN INTERACTIVE QUALITATIVE MODEL IN CARDIOLOGY

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VII. PERSPECTIVES

Tutoring and research. A number of applications are planned for the model. Computer-assisted tutoring is one of the most immediate applications since both signal and multilevel summaries are provided by the model. During "testing" sessions, linguistic descriptions can be occulted (Show/Hide texts button of the simulation window, Fig. 7), while the signal is generated. The student can carry out his/her own diagnostic of the ECG. He/she can thereafter compare the diagnostic with the process/event/diagnostic descriptions provided by the simulator. Most importantly, their are not limits to the number of cases that the user can generate. Following simulation session, new cases can be memorized into the arrhythmia KB (Fig. 1). Pharmacological research is another area of investigation where models can be used to test the theoretical influence of antiarrhythmic agents. It is indeed impossible to mentally predict all the effects of such agents since they can act singly or in combination on the refractory periods, global conduction speed, local conduction tensors, and the excitability thresholds. The complexity of the effects of antiarrhythmic agents is corroborated by the fact that in many instances, the cure has proven to be worse than the desease, as recent studies show. Introducing models to ellucidate when and how a given agent (or a combination of agents) should be preferred to another is a step toward the rationalization of a basically empirical approach to therapy and to a better assessment of the risk/benefit tradeoff. A minimal condition is that the model includes physiological parameters that are known to be directly affected by drug actions. The model and the framework's quantitative counterparts constitute a step in this direction since drugs modify the phase durations of the target tissues. Diagnosis and monitoring. One of the most challenging applications using computational models resides in their introduction within diagnostic and/or monitoring systems (5, 27, 29, 30). Three aspects characterize model-based diagnosis: (1) hypothesis generation, (2) hypothesis testing, and (3) hypothesis discrimination. When a computational model is used to derive observations in the hypothesis gefieration phase, the inference process is known as abduction (8, 24). In our framework the inference pattern is the following: 1. Generate a pattern 2. Compare the generated pattern with the real pattern 3. If the patterns match, then provide an explanation (the hypothesis) by infering the sequence of events (PROCESS, EVENT, DIAGNOSTIC) and/or the initial conditions, else go to i. Although conceptually simple to understand, an unconstrained approach to abductive inference as described here is computationally intractable, even using a model as grossly decomposed as CARDIOLAB's qualitative model. Indeed, suppose some ECG needs to be explained, and suppose further that we focus the search to nodal tissue only. Then, if we impose that each parameter takes on only five values, then there are 5.252 or approximately 1016 different instances of the model's nodal tissue initial parameter vector (32). Searching for one, or a subset of instances, such that the ensuing simulation(s) reproduce(s) an ECG

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that comes near to the observed data is a task whose complexity is exponentially difficult. That is, any blind, exhaustive search is d o o m e d from the start. A priori knowledge of the physiology of diseased tissues and intelligent search procedures and inductive learning schemes to discover pathognomonic conditions need to be introduced, as will be discussed in following papers. Using this added information and control structure, the use of interactive qualitative models with highlevel message-passing capability could be the gold-standard means of initiating the scientific endeavor of associating p a r a m e t e r space regions with the full range of significant behavioral modes. This is a condition sine qua n o n of performing abductive inference as it is understood here. Barring unusual luck and physiological knowledge notwithstanding, this task is virtually impossible to perform by directly using numerical models and particularly the H H - t y p e nonlinear differential equations. To overcome this computational pit, one can assume (or at least hope) that realistic behavioral modes can be put forth using simple models and that m o r e detailed models can be initialized on this basis to produce m o r e accurate predictions. This is a research p r o g r a m that we are actually pursuing by including cellular a u t o m a t a models within the f r a m e w o r k (Figs. 12 and 13).

VIII. DISCUSSION AND CONCLUSION In this paper, we described the qualitative model of the C A R D I O L A B framework. The system addresses four aspects of E C G simulation, namely simulating the heart's conduction system, simulating cardiac muscle activity, simulating the time course of the vectocardiogram, and producing multilevel qualitative descriptions of the ongoing simulations. A central concern in simulation studies is the adequacy of a designed model with respect to its intended goal. Models of cardiac electrical activity may differ in complexity, level of description, and representation. Depending on the desired level of detail, analytical, cellular automatas, and qualitative models can be used. Their advantages and shortcomings can be summarized in terms space and time complexities, ease of interpretation, and clinical relevance. In this paper, the most important feature was qualification in its descriptive role. Two other major roles of qualification are explanation and diagnosis. Explaining an E C G presupposes that explicit descriptions of simulated processes can be derived from the appropriate representations and the corresponding inference engines. H e a r t

FIG. 12. Cellular automata modeling. Simulating the onset of ventricular fibrillation with a 2D, 25,000-element cellular automata model. The predisposing factor is 10 patches in the ventricles exhibiting a dispersion of repolarization (range = 120 msec) and a slowed (more than twofold) and dispersed conduction (range = 90 mrn/sec) in the ventricles. The trigger is a lower-right bundle branch extrasystole (T = 5 msec) initiating three reentran't circuits (two in the right ventricle and one in the left; T = 275-800 msec). The resulting condition is a desynchronization of the ventricles as can be seen in the depolarization wavefronts and the simulated ECG.

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functions, components, and structural hierarchies are made explicit in the framework's qualitative model. Diagnostic-level concepts such as the wenckebach phenomenon are dynamically produced during qualitative process simulation (Fig. 13a). They are established from the lower EVENT level descriptions such as a slowing of conduction oelocity and blocked impulse in the A V junction. The latter are themselves built in terms of concepts of the still lower PROCESS level. It is these hierarchies of concepts that form the core of the system's multilevel descriptive and explanatory power. In addition to descriptive power, multiscales modeling in the spatial and temporal domains appears to be a necessity. Quantitative analysis of rhythmic disorders and QRS-complex morphology requires much more detailed models than the qualitative model presented. For instance, micro-reentry in ischemic tissue or PVB-induced ventricular fibrillation (Fig. 12) can only be reproduced by models composed of thousands of individual elements. Reentry can be obtained by imposing a statistical dispersion of the elements' repolarization phase durations. The system includes a 25,000-element celhdar automata model. Hence, the framework allows a spatiotemporal multiscale approach to studying cardiac electrical activity. Since the clinical context is of prime importance, factors relevant for studying arrhythmias and ischemias are taken into accout. They include the durations of cell phases that characterize impulse formation and conduction (e.g., slowdiastolic depolarization) and adaptative properties such as the rate-dependent repolarization durations or the recovery-state dependency of conduction speed. An important question that we are currently addressing is the delimitation, in parameter space, of the regions of converging behavior between the different grain-sized models. This point, which is essential to correctly establish the interface between the models, will be discussed in a following paper. Within the regions of convergence, qualitative models may play an essential role in model-based diagnosis. Indeed, explaining an observation by tuning real-valued model parameters so that the model output matches the observation is computationally unsolvable if brute force search procedures are applied. Grossly defined parameter values may be derived from qualitative models. Although computationally extremely costly, it is conceivable that by imposing a small, discrete set of initial values to each of the model parameters, critical and noncritical parameter space regions may be grossly delimited. Verifications could thereafter be performed using the more detailed models. A validation phase should follow in the appropriate experimental and clinical settings. In this demanding context, interactive qualitative models can be used in two ways: As an explanatory module, lying at the interface between detailed qualitative models and the user, and as the starting point of a multiresolution process, paving the way to more complex quantitative analysis.

ACKNOWLEDGMENT This work was suupported in part by the Conseil Regional de Bretagne.

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