Diagnostic monitoring in anaesthesia using fuzzy trend templates for matching temporal patterns

Artificial Intelligence in Medicine 16 (1999) 183–199

Diagnostic monitoring in anaesthesia using fuzzytrend templates for matching temporal patterns

Andrew Lowe a,*, Michael J. Harrison b, Richard W. Jones c

a Department of Mechanical Engineering, The Uni6ersity of Auckland, Pri6ate Bag 92019,Auckland, New Zealand

b Department of Anaesthesia, Auckland Hospital, Pri6ate Bag 92024, Auckland, New Zealandc Department of Mechanical Engineering, The Uni6ersity of Auckland, Pri6ate Bag 92019,

Auckland, New Zealand

Received 10 February 1998; received in revised form 3 August 1998; accepted 16 October 1998

Abstract

A technique based on the concept of a ‘fuzzy trend template’ has been developed toidentify characteristic patterns in multiple time-series. The method has its foundation infuzzy logic and allows for the intuitive and transparent description of ‘templates’, whichpreserve nuances of vagueness, temporal relationships and quantitative descriptors. Evalua-tion of fuzzy trend templates can provide both belief and plausibility information for use indiagnostic applications. The technique has been applied to the diagnosis of specific problemsin anaesthesia and has demonstrated sensitivity and specificity of 95 and 65%, respectively.Evaluation of fuzzy trend templates is, computationally, relatively efficient and has alloweda real-time implementation. The technique has the potential to be useful in any domain thatrequires temporal pattern recognition based on linguistic rules. © 1999 Elsevier Science B.V.All rights reserved.

Keywords: Anaesthesia; Fuzzy logic; Temporal pattern recognition; Fault detection anddiagnosis; Fuzzy trend templates

* Corresponding author. Tel.: +64-9-3737599 ext. 8668; fax: +64-9-3737479.E-mail address: [email protected] (A. Lowe)

0933-3657/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved.

PII: S0933 -3657 (98 )00072 -4

A. Lowe et al. / Artificial Intelligence in Medicine 16 (1999) 183–199184

1. Monitoring in anaesthesia

It has often been said that the process of anaesthesia, for the anaesthetist atleast, is ninety-five percent boredom and five percent terror. This state of affairsis typical in many monitoring applications and places great demands on thevigilance and objectivity of the anaesthetist [21]. In life-critical situations suchas anaesthesia, lapses in concentration can have dire consequences. In addition,many studies have identified the susceptibility of anaesthetists to ‘data over-load’, whereby excessive amounts of information must be analysed in real-time[7]. Unfortunately, this analysis becomes most difficult at precisely the timewhen it is most critical—that is, when problems occur [31]. For these reasons,it would seem that anaesthesia monitoring is an ideal area of application for‘intelligent’ computer-based systems. It would be hoped that some such systemcould address both the vigilance and objectivity issues mentioned above, andperhaps ease the data processing load by presenting more readily interpretableinformation.

Monitoring in anaesthesia attempts to achieve two main purposes—fault de-tection and diagnosis. Fault detection attempts to identify problems as early aspossible and forms the basis for alarm systems. The commonly used criterionfor alarms—when any one signal exceeds pre-set thresholds—has been foundto be of limited utility due to the complex relationships between measurements[30]. Recent efforts have therefore been directed towards the detection of ab-normal patterns in the available data [18,25]. The determination of what consti-tutes ‘normal’ can, however, be difficult. In the case of anaesthesia, where theprimary objective is to maintain the physiological homeostasis of the patient,the normal state can, for the most part, be considered steady-state [15]. Thisignores the transient processes of induction of, and recovery from anaesthesia,but in any case these result in very much less significant changes than thosecaused by adverse events. Another approach that can be used is to enumerateall (or some) of the abnormal patterns and then detect the presence of suchpatterns in the measured data [32]. The resulting alarms are then specific toparticular types of faults.

Once a fault has been detected it must be identified and diagnosed in orderto instigate the appropriate management procedures. Diagnoses can be formedusing two main methods. The first is diagnosis by first principles [16,19],whereby an attempt is made to recreate the pattern by modelling the effect ofvarious faults. In such a complex system as the human physiology, this processis likely to be highly computationally intensive and not readily implementablein real-time. Alternatively, the detected abnormal patterns can be classified insome manner according to their individual characteristics. If the enumerationapproach to fault detection is also being used then both detection and diagno-sis can be carried out through a single pattern recognition process, resulting inone-step fault detection and diagnosis.

A. Lowe et al. / Artificial Intelligence in Medicine 16 (1999) 183–199 185

2. Pattern recognition and diagnosis

It will be noted from the above discussion that there is likely to be a highdependence on pattern recognition techniques in the monitoring process. Clearly,the effectiveness of whatever pattern recognition technique is employed will greatlyaffect the final performance of the monitoring system. In our particular applicationof anaesthesia monitoring, the system must work in real-time with multiple inputsignals, and be able to reason with temporal relationships both within and betweenthese signals.

It has been shown in similar applications that neural networks[11,20,22,23] andother signal processing methods [6,9] are capable of the required pattern recogni-tion and classification tasks. However, these techniques commonly lack somewhatin their transparency of operation, and often require large amounts of trainingdata. In the domain of anaesthesia, it is beneficial if the diagnoses generated by acomputer-based monitoring system be explainable in order that they are readilyaccepted by anaesthetists. In addition, due to the rare occurrence of many types ofproblems, there does not exist the requisite amount of data to effectively trainsystems based on computer learning.

In contrast, there is abundant expert knowledge (anaesthetists’) on the diagnosisof problems based on the same (or similar) measured signals available to acomputer-based system. Such knowledge is accumulated both from experience andoff-line learning (perhaps derived from first-principles). If this knowledge could beencapsulated in a computer system, then the pattern recognition requirements ofthe intelligent monitor would be met. Fuzzy logic is recognised as a suitable meansof representing expert generated linguistic rules [17,29].

Other researchers have implemented intelligent patient monitoring systems basedon fuzzy logic with varying degrees of success [4,8,14,18,24,34]. Almost all thesystems use fuzzy rules derived linguistically and structured around the presentvalue of measured variables, for example, ‘high blood pressure’. The discriminationof similar events in such systems is dependent on relationships between variables(the rules), and the definitions of fuzzy sets used for classification of quantitativevalues. The fuzzy rules used in these patient-monitoring systems are almostexclusively defined using expert knowledge obtained through interviews. Thus theperformance of these fuzzy logic-based monitoring systems is dependent on theavailable measurements, and their expert fuzzification.

It is not surprising, then, that the most apparently successful applications offuzzy logic techniques are in critical, well-controlled, data-rich environments suchas cardio-anaesthesia [4,8]. Of course, these are also situations that stand to benefitgreatly from intelligent monitoring techniques. In day-to-day general surgery,however, the allowable fluctuations in a patient’s physiological parameters aremuch greater and the static classification techniques normally used in fuzzy logicbecome far less useful. Instead, relative changes and trends become more importantto diagnosis[2,5,15].

One commonly used method of incorporating such information is to creatediagnostic rules that explicitly use past values, for example an antecedent expression


might be ‘current heart rate is high and previous heart rate is medium-high-highand the previous-previous heart rate is medium-high…’. Needless to say, theresulting expressions can become very complex, and the definition of concepts (viamembership functions) such as a ‘medium-high-high’ is not easily done manually.An alternative approach is to define a mathematical relationship based on pastvalues—for example, to find the average slope over a definite time-period. Thisvalue can then be used directly in the fuzzy rule. Human experts, however, do notrecognise patterns this way and it is therefore difficult to interpret linguistic rules inthis manner.

Steimann [26,27] describes a far more satisfactory approach to this problem usingthe concept of a fuzzy course. In essence, fuzzy courses are fuzzy sets on twodimensions—signal level and time—that can be used to represent concepts such as‘rapidly rising and then levelling off’. Their graphical interpretation also provides away for experts to create and verify them. Fuzzy courses have been demonstratedto provide good results in a number of medical domains requiring temporalreasoning. Their use in domains where separate trends must be identified as a singlepattern is more limited, however. This has been recognised by Steimann whosuggests the possibility of segmented fuzzy courses.

Haimowitz [12,13] describes a non-fuzzy system that is similar in concept to fuzzycourses but allows segmentation. The TrenDx system uses more traditional meth-ods (interval analysis and regression) for identifying individual trends. However itcan represent temporal relationships such as ‘following’ by grouping trends intotrend templates. TrenDx allows the representation of uncertainty in some templateparameters by the ability to specify locations in time as temporal intervals.Uncertainty associated with the trends themselves must be accounted for by eitherconstraints (which do not produce smooth output), or careful design of regressionformulas.

This paper extends the concept of template systems to fuzzy domains. The result,which we term ‘fuzzy trend templates’, provides a formal mechanism for thesegmentation of fuzzy courses. They thus make it possible to represent the vagueanalogues of Allen’s [1] thirteen crisp, fundamental, temporal relationships. Inaddition, both signal levels and temporal relations can be defined fuzzily, allowingthe intuitive specification of fuzzy rules from linguistic expressions. As a furtherextension to the concept of fuzzy trend templates, we have also utilised fuzzymeasure theory from the mathematical theory of evidence [17] to provide additionalinformation specific to diagnostic—as opposed to pattern matching—tasks.

3. Fuzzy trend templates

The basic pattern-matching unit in fuzzy trend templates is the fuzzy course. Afuzzy course C0 is a fuzzy relation between time t and another (real-valued) variablex, which we will term the ‘signal’. In other words, it is a mapping between crisptime and fuzzy sets x on the same universe as that of x. We will express thisrelationship by


x=C0 (t) (1)

Physically, a fuzzy course can be considered the allowable spread in the course ofa signal still considered following a trend [27]. In effect it represents the course ofa signal and some associated vagueness that may change with time.

Eq. (1) allows for all possible mappings between time and signal level as it allowsfor non-convex fuzzy sets x. For example, a fuzzy course could be created toexpress a signal being ‘either high or low’. In most applications, however, limitingx to convex fuzzy sets still preserves sufficient expressive power as they can becombined using fuzzy set operators such as the fuzzy union and intersection.

In the case that fuzzy courses are defined as mappings to normalised, convexfuzzy sets (that is, they are mappings to fuzzy numbers [10]), they can be definedusing fuzzy number theory. This results in some conveniences. For example, itbecomes possible to define specific fuzzy courses such as ‘rising rapidly’ using aparametric approach. In this instance, a suitable function might be

C0 (t)=m� t�k0 (2)

where m is a positive fuzzy number, k0 is some initial fuzziness and � and � arefuzzy multiplication and addition operators (their behaviour being governed by theextension principle [10]). The membership function of an example of this type offuzzy course is shown in Fig. 1. We stress again that although defining fuzzycourses using fuzzy numbers is convenient, it is not required in the derivations thatfollow.

In addition to the fuzzy course, we also assume that we have access to themeasured (sampled) signal. Using angular brackets �� to denote a sequence, andsquare brackets [�] to denote a sampled value, we may represent the sequence ofmeasurements of x at times ti, i=1 to n by

�x�n=�x [t1], ... , x [ti ], ... , x [tn ]�

Fig. 1. Membership function for a linear fuzzy course.


The membership of the ith sampled point to the fuzzy course is given by themembership of signal level to the fuzzy set found by evaluating C0 at ti. Usingstandard nomenclature this can be written as

mC0 (ti)(x [ti ])

The membership of the series of discrete points to the fuzzy course is found byaggregation of the individual memberships.

mC0 (�x�n)=h(mC0 (t 1)(x [t1]), ... , mC0 (tn)(x [tn ]))

The aggregation function h is defined as some mapping

h : [0, 1]n� [0, 1]

In the strictest sense, h represents the ‘and’ aggregation of the memberships (that isall points must lie within the fuzzy course) and the minimum operator is appropri-ate. That is, the membership of the series to the fuzzy course is given by

mC0 (�x�n)= min15 i5n

mC0 (ti)(x [ti ]) (3)

Eq. (3) corresponds to the definition given in Steimann [27]. Depending on thecertainty of the trend definition and the degree of noise or artifact on the measuredsignal it may become necessary to move away from t-norm type aggregationoperators. The generalised mean, of the form

ha(a1,a2, ... , an)=�1

n%n

i=1

aia�1

a

is often more suitable in such applications. The parameter a determines the extentto which the mean resembles the minimum (a= −�), arithmetic mean (a=1) ormaximum (a=�) function.

As it has been described so far, a fuzzy course is defined over all time. In mostpractical applications this is not useful—trends generally occur over a finiteduration, relative to a specific point in time. This can be considered a ‘segment’ ofthe time course of the signal.

A fuzzy segment represents the allowable courses of a signal o6er a possibly fuzzychange in time such that the measured signal can be considered to be following atrend. A fuzzy segment S0 is defined as a triple

S0 =S0 (t0)= (t0, C0 , d0 ) (4)

where t0 is a crisp time origin, C0 is a fuzzy course and d0 is a duration. Thefunction-like notation is used in this paper where t0 must be defined parametrically.The duration d0 is a fuzzy number defined on the domain of the fuzzy course andwill usually have the additional constraint that its membership function is greaterthan zero over the entire duration of the fuzzy segment. Mathematically, theconstraint

md0 (0+ )+md0 (0− )\0


Fig. 2. Parametric definition of a trapezoidal membership function.

(that is, membership to d0 at times slightly greater or less than zero are not bothzero) is sufficient as d0 is convex. In practice, this requirement reflects the idea thata signal following a trend for less than the full expected duration must still beconsidered to have followed the trend to a certain degree. The membership functionshown in Fig. 2 would be a suitable fuzzy number if c=a\0 or b=dB0.

The membership of a series �x�n to a fuzzy segment that has t0=0 is given by

mS0 (�x�n)= max15 i5n

min(mC0 (�x�i), md0 (ti)) (5)

That is, it is the best fit of the data to the fuzzy course conditional on the data alsospanning an adequate length of time (the fuzzy duration). As before, otheraggregation operators may be used in the place of the maximum and minimumoperators shown here if it is warranted by the application. For instance, in Eq. (5)the membership of the sequence to the fuzzy segment is at least as large as themaximum of all sequences ending before the current time increment n :

mS0 (�x�i)]mS0 (�x�i−1)Ö2] i]n

If the fact that a signal closely follows a trend for a long time is significant, then ageneralised mean may be more appropriate than the maximum.

A fuzzy segment with t0"0 can be represented by an equivalent triple

S0 (t0)= (0, C0 (t− t0), d0 + t0)

however in this form the membership of ‘duration’ represented by d0 + t0 is nolonger necessarily greater than zero at t=0+ or t=0− (as t0 is crisp).

The membership mS0 (�x�n) given in Eq. (5) can be considered the ‘belief’ Bel(�)that the trend represented by S0 has occurred in the signal x. We will use thenotation

Bel(S0 )=Bel(S0 ��x�n)=mS0 (�x�n) (6)

In fact, the true belief would be dependent on the sampling process that generates�x�n from the true signal x(t). For a constant sampling rate, only trends that havemost of their frequency components at less than half the sampling rate will beobservable. For the purposes of this paper we will assume that all informationuseful for detection and diagnosis is available in the sampled sequence.


The ‘plausibility’ Pl(�) of the trend occurring (or having occurred) is related tothe belief that the trend has not occurred. This belief is influenced by the degree towhich the sampled sequence lies outside of the fuzzy segment. The plausibility maybe defined by

Pl(S0 )=Pl(S0 ��x�n)=1−Bel(S0( )=1−mS0( (�x�n) (7)

where S0( is the fuzzy segment representing the ‘complement’ of the trend S0 . It isoften convenient to define it by its membership function

mS0( (�x�n)=n(mS0 %(�x�n)) (8)

where n(�) is a (monotonic, decreasing) fuzzy complement function1 such that

n(m)+m51 (9)

and S0 % is defined as the fuzzy segment equivalent to the original segment S0 exceptthat its belief is not dependent on duration. That is,

mS0 %(�x�n)= max15 i5n

mC0 (�x�i) (10)

This independence is necessary because a short time span of the data does notprovide evidence against a diagnosis, although it does reduce the amount ofevidence supporting the diagnosis.

S0 could also be defined using an entirely separate fuzzy course, however Eqs.(7)–(10) guarantee that the requirement Pl(S0 )]Bel(S0 ) is met. The proof of this isgiven in the appendix.

The plausibility in Eq. (7) then can be rewritten as

P1(S0 )=1−n�

max15 i5n

mC0 (�x�i)�

So far we have defined a fuzzy segment as a means of calculating whether asequence of sampled data is following a vague trend at a particular point in time.Both belief and plausibility measures can be calculated. In practical applications itis often useful to be able to recognise trends in multiple signals where the startingtimes of the trends are also vague. These collections of trends can be related to‘events’ in time. For this purpose we introduce the concept of a fuzzy trendtemplate.

A fuzzy trend template T0 is defined by a 5-tuple consisting of a crisp time datumt, a relative fuzzy starting time t0, a sequence of data �x�, a fuzzy segment S0 anda list of up to m sub-templates, which themselves are fuzzy trend templates.

T0 (t)= (t, t0, �x�, S0 , {T0 1, ... , T0 m})

The starting time of a sub-template is relative to the starting time of its parent. Thetwo degrees of freedom represented in t0 and the duration of the fuzzy segment d0together allow the representation of all temporal relationships between templates

1 Klir and Folger [17] discuss the other properties required of this function.


(that is, events). In the case of a template not having a parent (that is, it is the roottemplate) then the fuzzy starting time is relative to zero. Note that although t0 hasbeen defined to be a fuzzy number it could, in fact, be crisp.

The membership of a fuzzy trend template with no sub-templates evaluated usinga set �X� of available data made up of N individual sampled sequences

�X�={�x1�, … , �xN�} (11)

is related to the membership of the fuzzy segment applied at every point in thefuzzy offset. That is,

mT0 (�X�n)= supmt 0

(t 0)\0min(mS0 (t 0+ t)(�x�i), mt 0

(t0))

For a template that has sub-templates, the membership is defined as a template-spe-cific function F of all the memberships, also aggregated over the fuzzy startingtime:

mT0 (�X�n)

= supmt 0

(t 0)\0min(F(mS0 (t 0+ t)(�x�i), mT0 1(t 0+ t)(�X�), ... , mT0 m (t 0+ t )

(�X�)), mt 0(t0))

The function F is defined as a fuzzy aggregation operator

F : [0, 1]m+1� [0, 1]

that represents a linguistic relation between the trends matched in each fuzzysegment. Note that each sub-template may be defined on different sequences ofdata from the data set �X�.

Alternatively, if both belief and plausibility information are required, then theymay be calculated as functions of the beliefs and plausibilities of the template’ssegment, and those of the sub-templates. For a template with no sub-templates

Bel(T0 (t))=Bel(T0 (t)��X�)= supmt 0

(t 0)\0prod(Bel(S0 (t0+ t)), mt 0

(t0))

Pl(T0 (t))=Pl(T0 (t)��X�)= supmt 0

(t 0)\0prod(Pl(S0 (t0+ t)), mt 0

(t0))

Note here that the prod (product) function implicitly assumes that memberships tothe fuzzy segment and time of occurrence are independent. The resulting belief andplausibility are therefore measures of the amount of evidence that an event(represented by the template) occurred at a particular point in time (the time ofapplication). More generally, (if other evidential concepts were more appropriate)then equations of the form

Bel(T0 (t))= supmt 0

(t 0)\0FBel(Bel(S0 (t0+ t)), Pl(S0 (t0+ t)), mt 0

(t0))

Pl(T0 (t))= supmt 0

(t 0)\0FPl(Bel(S0 (t0+ t)), Pl(S0 (t0+ t)), mt 0

(t0))

could be used, where FBel and FPl are aggregation functions defined from [0, 1]3 to[0, 1]. These functions should be derived in accordance with the mathematicaltheory of evidence. Similar relations can be written to calculate the belief andplausibility of templates with sub-templates, except that the functions FBel and FPl


need to be extended to include the belief and plausibility measures of the sub-templates

Bel(T0 (t))= supmt 0

(t 0)\0FBel(Bel(S0 (t0+ t)), Pl(S0 (t0+ t)), mt 0

(t0), Bel(T0 1(t0+ t)),

Pl(T0 1(t0+ t)), ... , Bel(T0 m(t0+ t)), Pl(T0 m(t0+ t))

Pl(T0 (t))= supmt 0

(t 0)\0FPl((Bel(S0 (t0+ t)), Pl(S0 (t0+ t)), mt 0

(t0), Bel(T0 1(t0+ t)),

Pl(T0 1(t0+ t)), ... , Bel(T0 m(t0+ t)), Pl(T0 m(t0+ t)))

4. Results from anaesthesia monitoring

Having described the formulation of fuzzy trend templates, this section describesresults from their application to anaesthesia monitoring. The objective was toidentify the existence of seven specific problems (described in Table 1) that canoccur during anaesthesia.

This is a diagnostic task—that is, we wish to identify whether a problem exists.It is therefore more appropriate that we use the belief/plausibility capabilities offuzzy trend templates. In this way individual anaesthetists can tune the sensitivity ofthe resulting alarms to their particular needs. For example, one anaesthetist mightwish to be alerted when a problem is highly plausible, even though there is littleevidence that a problem has occurred (that is, belief is low). Another anaesthetistmay prefer that both belief and plausibility are high before an alarm is triggered.In a similar manner, it is possible to generate different types of alarms—forexample, ‘critical’ and ‘warning’ alarms—based on the dual belief and plausibilitymeasures.

Table 1Description of diagnoses to be made using fuzzy trend templates

DescriptionProblem

Inadequate analgesia (IA) Significant increases in heart rate and systolic blood pressure overabout half a minute; fall in pulse volume over the same periodSignificant rises in heart rate and end-tidal CO2 concentration overMalignant hyperpyrexia (MH)a number of minutes, followed by low peripheral saturation and ahigh core body temperature

Increased intracranial pressure Short term decrease in heart rate and simultaneous rise in systolic(IICP) blood pressure

Pulmonary shunt (PS) Low peripheral saturationSudden drop in end-tidal CO2 concentration and systolic bloodCardiac output failure (COF)pressure; followed by desaturation

Absolute hypovolaemia Fall in systolic blood pressure, and pulse volume; corresponding(AHV) increase in heart rate

Relative hypovolaemia (RHV) Fall in systolic blood pressure; increase in pulse volume and heartrate


Table 2Parametric definitions of some fuzzy trend templates in the form (a, b, c, d)

Fuzzy template off-Fuzzy segment du-Template Fuzzy course Fuzzy courseset t0 (s)origin k0slope m (s−1) ration d0 (s)

20, 60, 20, 0IA (heart rate) −30, −10, 0, 00.5, 1, 0.3, 0.3 0, 0, 0, 0bpm

0, 0, 0, 0IA (sys b.p.) 20, 60, 20, 01, 1.3, 0.5, 1.3 −5, 5, 5, 5mmHg

20, 60, 20, 0 −15, 5, 5, 5IA (pulse vol.) 0, 0, 0, 0−3.3, −10, −1.7, −6.7

0, 0, 0, 0 300, 600, 60, 0MH (heart 0.13, 0.23, 0.13, −300, −100, 0, 00.13rate)

300, 600, 60, 0 0, 0, 30, 30MH (EtCO2)% 0.005, 0.013, 0, 0, 0, 00.005, 0.007

38, 42, 1.5, 4MH (Temp.) 300, 600, 60, 00, 0, 0, 0 30, 120, 10, 20°C

10, 280, 5, 20300, 600, 60, 0MH (Peri. 80, 94, 10, 40, 0, 0, 0Sat.)%

Given the linguistic problem descriptions in Table 1 translation of the linguisticrules into fuzzy trend templates was found to be relatively straightforward. Need-less to say, in order to correctly interpret the rules some additional, contextual(medical domain) knowledge was necessary. Parameters of some of the fuzzy trendtemplates are given in Table 2. The fuzzy membership functions used weretrapezoidal, as shown in Fig. 2. Linear fuzzy courses, of the form given in Eq. (2)have been used throughout this study. Notice that the anaesthetists have providedlinguistic rules that use relative changes. In the implementation of such rules thefirst point of each signal template was shifted to the actual value of the signal atthat time. That is, the fuzzy course offset k0 in Eq. (2) is given by

k0 =x [t0]

where x is the signal to which the fuzzy course is related and t0 is the crisp timeorigin of the fuzzy segment containing the fuzzy course—see Eq. (4).

This system was tested using data recorded from :70 h of operations atAuckland Hospital—a regional general hospital. No particular type of surgicalprocedure was excluded from this study and so the cases represent a reasonablecross-section of anaesthetics provided to adult patients (children are catered for bya separate facility). Some additional data for rarely occurring problems wascollected from simulations run at the New Zealand National Patient SimulationTraining Centre (using the Human Patient Simulator version 1.3, Medical Educa-tion Technologies Inc., University of Florida).

The input variables specified in the anaesthetists’ rules—�X� in Eq. (11)—weretaken directly from an anaesthesia monitor (Datex Engstrom AS/3) using anRS-232 serial interface. Such numeric data is equivalent to the information avail-able to the anaesthetist on the anaesthesia monitor screen. The measured signals


were smoothed and filtered for artifacts using an application-generic statisticalmoving average filter [28]. For this study, physiological data was sampled at 0.1Hz, and the belief and plausibility measures of the templates were recalculatedfor each new time-step. It should be noted that fuzzy trend templates can bedesigned independently of sampling rate. The fuzzy trend template system wasprogrammed in C+ + and runs on 32-bit Microsoft Windows platforms. Per-formance of the implementation on an Intel Pentium Pro 200 MHz-based com-puter was more than adequate in terms of speed—off-line evaluation of theseven templates was :40 times faster than real-time using non-optimised code.

With regard to diagnostic performance, our current implementation of thesystem correctly diagnosed :95% of the problems encountered during surgeryand noted in the recorded data by the administering anaesthetist. Of theseevents, 80% were considered by the system to be worthy of an alarm (belief andplausibility both ‘high’), the rest generating a ‘prompt’ (belief ‘medium’, plausi-bility ‘high’). These rules were implemented fuzzily according to the preferencesof an anaesthetist. Significantly, of the events detected by the fuzzy trend tem-plates but not annotated by the anaesthetist, about 88% generated prompts, notalarms. The overall rate of false positive alarms was:35%, which is a signifi-cantly better rate than for simple threshold alarms [18,33].

Fig. 3 shows belief and plausibility output of the system in response to a caseof inadequate analgesia. This particular problem occurs over quite a short periodof time and primarily is a test of the speed of diagnosis of fuzzy trend templates.It can be seen that both plausibility and belief rise sharply in response to anincreasing heart rate and decreasing pulse volume. The absence of a bloodpressure signal (non-invasive blood pressures are taken at relatively long inter-vals) is automatically accounted for by the theory of evidence. That is, eventhough increasing blood pressure is not present as a symptom (thus our beliefthat blood pressure is increasing is zero), the plausibility is one and so thediagnosis is not affected. In comparison, a system based only on fuzzy member-ships (that is, beliefs—see Eq. (6)) would limit the diagnosis to the least mem-bership of the constituent trends, and would not make the diagnosis.

A diagnosis made over a much longer time-scale (and thus requiring a muchhigher degree of fuzziness in template parameters) is shown in Fig. 4. In thiscase the patient has succumbed to malignant hyperpyrexia, a rarely occurringdisorder whose symptoms are often misinterpreted. In fact, the definitive symp-tom is an increase in body temperature, but a temperature probe is only occa-sionally used in normal surgery. In this case no temperature information wasavailable until after the diagnosis had been made by the anaesthetist and atemperature probe inserted (not shown). Nevertheless, it can be seen that thefuzzy trend templates generate a diagnosis at a much earlier stage. Note that thefuzzy trend template allows the treatment of desaturation without compromisingthe diagnosis through the use of large vagueness when specifying when desatura-tion will occur relative to the start of the increase in heart rate (the relevantparameters are given in Table 2).

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Fig. 3. Diagnosis of two cases of inadequate analgesia.

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Fig. 4. Diagnosis of malignant hyperpyrexia.


5. Conclusions

Given that the original intention of this research was to create a system to helpthe anaesthetist in the interpretation of data, we feel that the results presented hereindicate that fuzzy trend templates are a promising technique for this application.The characteristics of temporal relations in the rules used in this application can beadequately represented by the fuzzy trend template formalism. This follows fromthe fact that, in theory at least, fuzzy trend templates can represent any vaguetemporal relationship between two events. It remains to be seen, however, whetherfuzzy trend templates give similarly good results in other diagnostic domains.

In testing fuzzy trend templates in anaesthesia monitoring we have found them tobe an efficient mechanism for implementing linguistic time-series pattern matchingrules in computer systems. They would seem to be especially useful in applicationswhere the implementation must preserve the transparency of the experts’ linguisticrules. Fuzzy set theory as the basis for representing vagueness provides a strongtheoretical foundation and could encompass the implementation of other fuzzytechnologies such as machine learning and on-line adaptation [3].

Finally, we feel that the ability of fuzzy trend templates to generate meaningfulbelief and plausibility measures is a significant advantage over techniques basedpurely in fuzzy set theory. Diagnostic applications in particular could benefit fromthe integration of temporal pattern matching and evidential reasoning.

Acknowledgements

The authors thank Dr Brian Robinson (Programme Director, National PatientSimulation Training Centre), and Dr Graham van Renen (Royal Adelaide Hospi-tal) and Dr Neil Pollock (Palmerston North) for their help in the provision of data.The primary author also wishes to acknowledge the support of the University ofAuckland through the award of a University of Auckland Doctoral Scholarship.

Appendix A

We wish to prove that Pl(S0 )]Bel(S0 ). It is obvious from Eqs. (5) and (10) that

ms%(�x�n)]mS0 (�x�n)

Now the function n(�) is monotonic and decreasing, so we can also write

n(mS0 %(�x�n))5n(mS0 (�x�n))

Substituting this into Eq. (8) gives

mS0( (�x�n)5n(mS0 (�x�n))


but we also know from Eq. (9) that

n(mS0 (�x�n))51−mS0 (�x�n)

from which we conclude that

mS0( (�x�n)51−mS0 (�x�)

Rearranging gives

1−mS0( (�x�n)]mS0 (]�x�n)

and substitution using the definitions given in Eqs. (6) and (7)

Bel(S0 )=mS0 (�x�n)

Pl(S0 )=1−mS0( (�x�n)

results in the required relation:

Pl(S0 )]Bel(S0 ).

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Diagnostic monitoring in anaesthesia using fuzzy trend templates for matching temporal patterns