An Application of Fuzzy Integral to Medical Diagnosis

Copyright © IFAC Theory and Application of Digital Control New Delhi, India 1982

AN APPLICATION OF FUZZY INTEGRAL TO MEDICAL DIAGNOSIS M. M. Gupta, P. N. Nikiforuk, Y. Tsukamoto* and R. Martin-Clouaire Cybernetics Research Laboratory, College of Engineering, University of Saskatchewan, Saskatoon, Saskatchewan S7N OWO, Canada

* Visiting from Tokyo Institute of Technology, Japan. Abstract. A new medical diagnostic method designed to deal with fuzzy information is proposed. The extensive definition of fuzzy integral provides a tool to aggregate physician's knowledge, patient's state and attitudes for subjective evaluation. The introduction of the product space consisting of symptoms provides a wide field where phYSicians can express their medical knowledge, particularly concerning with the interrelations among symptoms. The solution representing the assessment of the occurrence of each disease, then, can be obtained from view-points of both credibility and plausibility. Finally the effectiveness of this method is demonstrated by conSidering two rather simplified cases of heart disease. Keywords. Medical knowledge representation, interrelation among symptoms, fuzzy integral, plausible expectation, credible expectation, heart diseases, fuzzy set theory. 1.

INTRODUCTION

One of the most important and crucial tasks in medical sciences is the diagnosis of the diseases . The logic or algorithms which are actually used consciously or unconsciously by physicians are very complicated because of the wide variety of situations encountered. Any theory for developing diagnostic method should enable physicians to fully represent their medical knowledge. It would be very helpful to design a model reflecting the sense as well as knowledge possessed by phys~c~ans. With such a view in mind, a fuzzy set theoretic approach to computer aided diagnosis is investigated. The relationship between signs and diseases is often described qualitatively rather than quantitatively. This is due to the absence of sharp boundaries characterizing the attributes which may describe a given disease. The relative importance assigned to the different combinations of symptoms is more or less subjective. Most of the time, the observations suffer from imprecision arising from measurements and/or with a subjective appreciation in the case of non-measurable symptoms. Furthermore, the diagnosis assessment might be context dependent,and due to their nature the attitudes to be chosen in subjective evaluation may be well expressed using the fuzzy concepts. In the situation described above, the results must be inherently fuzzy.

a certain disease occurs. In this pape~ the model of medical knowledge also contains another kind of relationship laying emphasis on the interrelations among symptoms. This is done by introducing the product space consisting of various combinations of symptoms. The present method is based upon an exten~ sive definition of Sugeno's fuzzy integral (1977) and is implemented on POP 11/ 60 computer; this approach gives an evaluation of the occurrence of each disease . By virtue of the flexibility of A-fuzzy measure, the result can be obtained from two different points of view: plausibility and credibili ty, and it has the form of a fuzzy set of the interval [0, 11. This fuzzy set can be decoded into a linguistic statement which is more suitable to follow the natural process in medicine. In Section 2, the various theoretical concepts and techniques needed in the method are developed. The application to simplified cases of heart diseases including results of simulation is described in Section 3, in which the evaluations from the two points of view are shown to be very important and useful for medical diagnosis. 2.

MATI-lEMATlCAL PRELIMINARIES

In this section some mathematical tools which are relevant to our development of me~ ical diagnostic methods are stated. For a detailed mathematical theory of fuzzy sets

In previous studies, Sanchez et al. (1980) represented medical knowledge by means of a symptom-disease relation indicating the possible states of symptoms on the occasion when

449

M. M. Gupta et aL.

450

and fuzzy integrals, the reader is referred to Zadeh (1975) and Sugeno (1977). The mathematical formulation of medical diagnosis as developed by Sanchez et al. (1980) is included in the present method. Additionally, the new mathematical tools proposed in this paper are ~A-transform by which probability measure is transformed into A-fuzzy measure, and the extensive definition of fuzzy integral in the sense of Zadeh's extension principle. Let {y., j = 1, ..• , n} be the collection of names 6f the attributes to be associated with symptoms and let Pj denote the following fuzzy statement.

= 1,

p. c:. "y . is R .", J J J

(1)

... , n

where Rj is a fuzzy set of Uj which is characteri zed by a mapping, )JR · : U· -+ [0, 1]. When it is necessary to empha~ize lhat the statement Pj is related to ith diagnosis denoted by Xi' It will be written as "Yj is Rj(i)". Furthermore, let lP · t: "y. is R~" J -

,

J

I, ... , n

J'

(3)

Each of the elements of Z corresponds to some fuzzy statement such that: "Yn is on"

Yk

= Min

(Height (OJ

(4)

where OJ takes either Ri or Rj '. Also the suffix k, as in (4), inaicates an index for representing one of all the possible combinations of (01, ... , on). This conjunctive fuzzy statement (4) may also be written as:

n Qj))

(9)

j

Assuming that N different statements as given by (5) form an exhaustive statement, the probabilistic measure defined on 2 Z can be derived from {Yk' k = 1, ... N}. That is, define: N

rk

= Yk

~

/

k=l

Yk ,

(10)

I, ••• , N

k

Then, {rk; k = 1, ... , N} means the discrete probability which can yield r: 2Z -+ [0, ll, having the same properties as those of a probability measure de!ined on the discrete measurable space (Z, 2). Tsukamoto et al. (1980) has shown that the fuzzy measures subject to A-rule could be deduced from the pr~ bability measure. Definition 1

~A:

[0, II -+ [0, 1]

~A(r) ~ ((1 + A)r_l)/A, A£ 1 -1,

(2)

where Rj stands for the complement of Rj. Now let us consider the product set defined as:

AND --- AND

It can be easily shown that:

Lemma 1

~A

(11)

oc[

is a function that is one-one and onto.

Now let G(.) be A-fuzzy measure on (Z, 2 Z). Theorem 1 For a given A, there exists a bijective mapping transforming r into G. (Proof)

For A C Z let b. = ~A(r(A)),

G(A)

A£ ]-1,

(12)

oc[

Then it follows by Lemma 1 that

~\l (G(A))

reA)

(13)

log(l+A) (1 + H(A))

"Y is "'k", k

1, .•• , N

(5)

where N = 2n. In this paper, Pk's are called the referential fuzzy statements which are considered to represent the states of the attributes on the occasion when some disease occurs. On the other hand, the patient's state is expressed by the following fuzzy proposition. 0j @ "Y j is Qj"

j

= 1,

... , n

(6)

where Qj is a fuzzy set of Uj. This also leads to the conjunctive propositions, called the observational proposition.

o

=

"Y is Q"

where Q

= Ql

(7)

re,,)

reAl

U

A2)

reAl) +

=

re A2)

log(l+ A) (l+ A G(Al

= log(l+A)

U

Let us now consider the consistency degrees of a given observational proposition with all the referencial statements. Define Yk as the possibilistic consistency degree given by:

(1 + A(G(A l ) + G(A;p

+ AG(A l ) G(A 2)))

(14)

Thus A-rule concerning G(.) is derived as (c') G(Al U A2) = G(A l ) + G(A2) (IS) + AG(A l ) G(A 2) , A£ l - l, oc[

Conversely it is easily shown by (13) that (a') -+ (a) and (b') -+ (b). Further we have from (13) and (IS), for

k

=

1, ... , N

(8)

=",

A2))

(a'), (b') and (c') indicate that G(.) is A-fuzzy measure.

x ... x Qn·

Yk ~ Height ("'k n Q),

First it is shown that (Z, 2Z, r)-+(Z, 2 Z, G). Since it holds that (a) r(Z) = I, (b) =0 and (c) forVAl, VA2c Z such that Aln A2 reAl u A2) = reAl) + r(A 2) , it follows from (12) that (a~ G(Z) = 1 and (b~ G(") = 0 where " stands for the empty set. Further, by (13) we have

An V AI' VA2 C Z,

G(Al u A ) 2 G(A l )

+

Al

Applicati o~.

n A2 = IJ

((1 + A/(A l

=

G(A 2 )

+

((1

A)r(A l )

+

of Fuzzy Integral to Medical Diagnosis application of the ixtension principle to the function EH 0 C • the fuzzy set v is transformed into some fuzzy set of [0, 11 whose base variable is t. That is, for 'It£[O , 1),

A2) - l)/ A

U

AG(A l )G(A 2 ) r(A 2) _ l)/ A

+

and, thereby, (c) follows.

lJ ~ (v)

(z ~A r k )

= $A

k

Z -+ [0, 1].

Definition 2 The fuzzy integral of H with respect to G over Z is defined by:

fHdG

~

sup ex£ [0,11

(17)

Hex ~ {zk; H(zk) ~ ex}

where

(18)

Lemma 2 If G(.) is given by (16) it holds that for YAl' 'V A2 £ 1-1, oo [ such that Al
_

((1 + A) 8 - 1

A

)'

A

=

8 ( 1 - 8) 2

;!

"'8£[0,11

0, V 8 e;( 0, 11

that is, f CA ) ~ 0 for \>" A£ l-l ,

0> [

(Q.E.D.)

In view of the such dependency of EA upon the value of A, the extensive definition of the fuzzy integral may be considered in the sense of Zadeh's extension principle. Definition 3

t; : 1-1, oo [

-+

10, 1

log Cl + A) ) Ilog (1 + A) I

v = t; ( A)

/::, 1 (1

Lemma 3

t; is a continuous one-one function.

2

Definition 4 t Lemma 4

= EH

1

+

1-1,

EH CA )

+

/::,

f

00

[-+

(20)

[0, 11

H d $A (r)

MEDICAL DIAGNOSIS

Medical Knowledge Representation and Observation

Two kinds of relation between signs and diseases are considered. Their representations through the two tables described below relate the medical knowledge.

It can be shown that f CA ) takes at A= 0 the maximum value such as : max f CA)

3. 3.1

fCA) - aT G(Hex )

=~ dA

In the above mathematics, we have described the mathematical formalization of the present medical diagnostic method. The clear distinction should be drawn between four fuzzy quantities, i.e. R, Q, H and v. R and Hare related to medical knowledge possessed by physicians, and v expresses the attitute in subjective evaluation . The medical oriented interpretation of these quantities will now be given in the following section.

(19)

For ex£ [O, 11 let d

!:.

(23) The fuzzy set EH will be referred to as fuzzy fuzzy expectation which can be viewed as a linguistic value of fuzzy integral, and which is to be found out as the solution of medical diagnosis.

Now let us define a measurable function

E (H)

(22)

where

(16)

where r k means r({zk}) for zk £ Z. H:

t;(EH-l(t))

(Q.E.D.)

Thus it is concluded that if N referential propositions and an observational one are given, A-fuzzy measures for "'A c Z can be calculated by : G(A)

451

(21)

EH is a non-increasing function of A.

Let v be a fu zzy set of [0, 1), then by the

The first relation is concerned with the patterns expressing linguistic relationship between symptoms and diagnoses in a give~ pathology, Table 1. This form of model was first proposed by Sanchez et al. (1980) . The element Rj(i) characterizes the possible state of the attribute Yj for the disease i, which can express a 'property given in the literature by such statement: "for the disease i, the attribute y . may appear with s lightly increased inten~i ty". It should be noted that each relationship carries the significant possible feature of an attribute. But even though an attribute measured on a patient is not consistent with its relevant fuzz y distribution, one cannot necessarily conclude that the associate possible disease does not occur. This problem leads to the necessity of the construction of another relation defined and discussed next . In order to concentrate on the structure and interrelations of signs, a second relation linking the different possible collections of symptoms with the diagnosis, is now introduced. For each possible disease, the physician is asked to score a value between

M. M. Gupta et al.

452

°Thisand valuefor caneachbe combination of the form Pk' interpreted as a degree of 1

belief of the statement: "the combination Pk is sufficient to infer the concerned diagnosis". From a mathematical point of view, for each possible disease i, we define a mapping Hi: Z + [0, 1), Table 2. In the sequel, since each element of Z is unequivocally noticed by an index k in {I, ..• , N}, Hi(k) will stand for Hi(Pk)' Obviously, we have to assign the values 1 to Hi(N) (all attributes are in keeping with their characteristic corresponding shapes), and to Hi(l) (none of these attributes corresponds to their defined feature). This construction is obviously natural since this sort of hierarchy in the set of possible combinatiomdescribes a medical reality. The relative apparent presence of only some dominant symptoms can be sufficient to conclude reliably or at least to give an approximate information about the severity of the abnormal state of the patient.

°

Validity of data obtained on a patient is highly dependent upon the excellence of laboratory technique, including proper manipulation of equipment and environmental control. Thus an observation is a set of measures which are contaminated by imprecision. We shall represent them by fuzzy members Qj with appropriate bandwidth quantifying this Imprecision, see Sanchez et al. (1980). Therefore an observation will merely be given by a collection of fuzzy numbers Q = (Ql"" ,Qn). Disposing of a medical knowledge representation and data collected on a patient, one can enter the next stage of diagnostic process which is the subject of the following sub-section. 3.2

Context Dependent Assessment

Our goal is to evaluate how each disease occurs. Since we will repeat the same treatment for each possible diagnosis let us take only the ith one into consideration. The first step is to obtain Yk defined in (9), which is computed as follows: Yk (i)

MIN { SUP jdl, .•. ,n} u j E: U j IJ

Qj

[MIN (1J6~ (i) (u j ) , J

(u.)))}

= Hi(k l )

then EHi CA) where

~ Hi(kz)~ .•. ~ Hi(k N)

= MAX

°

The consideration about the attitute of evaluation seems to be very useful in medical diagnosis. For instance, in the case of routine examination (preventive medicine), the physician may be interested by an evaluation of the plausibility of the occurence of some diseases. Conversely, he may demand more credibility in the diagnosis of a patient in hospital before choosing the appropriate ther~py or the surgical intervention or adjunctive tests, because a safety reasoning is almost compulsory in such case. Examples with simulation results will now be discussed 3.3

Examples

Consider an example involving two possible diseases, myocardial infarction (MI) and pulmonary infarction (PI) and four attributes, GOT = transaminase, CPK = creatine phosphokinase, LDH = lactic dehydrogenase and SCAN, (Table 1 and Table 2). Note that the last attribute is of binary type. Consider the observation given in Figure 2. The plausible and credible expectations calculated are shown in Figures 3 and 4 respectively. One can see that the myocardial infarction (MI) is expected to occur from both points of view and it implies that the patient suffers from this disease. The pulmonary infarction (PI) is not expected to occur from both plausible and credible points of view, and one can conclude that this disease is not present. Consider now the observation shown in Figure

=°

{MIN(Hi(k ~ ), G(A ~ ))} k ~d 1, . .. ,'N)

A ~ = {Pk .... , Pk } 1 R.

[0, 1) is a non-increasing fUnction of v. Thus we can say that the fuzzy expectation corresponding to a value of v taken in the neighbourhood of is plausible, and conversely that a value of v close to 1 generates an appreciation of the credibility. In other words, v carries the information about the attitude of evaluation which is context dependent. Since such notion is a typical example of fuzzy concept, let us represent it by a fuzzy set of )0,1[, Figure 1. Then applying the extension principle as shown in (22), one obtains a fuzzy set giving a plausible or credible expectation, whose interpretation falls into the category of linguistic approximation problems.

(24 )

J

As shown in Section 2, a A-fuzzy measure G over Z can be generated by those consistency degrees. This allows Ofie to compute the fuzzy integral EHi of the measurable function Hi' Moreover, since Z is a finite set of N elements one can calculate EHi by the following device. For a given AE:)-l, 00[, rearrange Hi(k) in a decreasing order such that: 1

It can easily be shown that EH' 0<;-1: )0, l[ +

(25)

5, which is close to previous one but with a

much lower confidence level in the measurements. The plausible and credible expectations shown in Figures 6 and 7 respectively, indicate that the pulmonary infarction is disconfirmed but one cannot state that the myocardial infarction is indisputably present because the credible expectation is rather low. This results is due to the high imprecision in the measurements, Fig. 5. Such doubtful case requires further adjunctive tests. 4.

CONCLUSION

A new computer aided medical diagnosis based

An Application of Fuzzy Integral to Medical Diagnosis on the fuzzy integral and taking into account the subjective attitude of evaluation has been described. This method has proved its effectiveness but nevertheless its applicability and its success depend primarily on the construction of the two tables representing the medical knowledge . The restrictiveness imparted to the underlying model by this construction requires further studies before one can rely on the diagnostic suggestion resulting from this automatic procedure.

,

/

,

,, , I I

,I

I

I

, I

II

I I I

0 0.03 0.33 0.66 0.99 Fig. 1.

REFERENCES Sanchez, E., Gouvernet, J., Bartolin, R., Vovan, L. , (1980), Linguistic approach in fu zzy logi c of the WHO classification of dyslipoproteinemias, The International Congress on Applied Systems Research and Cybernetics , Acapulco. Sugeno, M. (1977), Fuzzy measures and fu z zy integrals: a survey, 89,102 in Fuzzy automata and decision processes, North Holland, M. Gupta, G. Saridis and B. Gaines Eds. Tsukamoto, Y., Gupta, M., Nikiforuk, P. , (1980), On density function of A-fu zzy measure, The International Congress on Applied Systems Researchand Cybernetics, Acapulco. Zadeh, L., (19 75), The concept of a linguistic variable and its application to approximate reasoning. Part 1, 2 and 3. In£. Sci. 8,199-249; 8,301-357; 9,43-80.

- FOR A

II

,,1,I

/

,

453

TABLE 2.

Pk

1 2 3 4 5 6 7 8 9 10

E

M. I.

p . I.

0 0 .3 .3 .4 .4 .8 .8 .3 .3 .5 .5 .6 .6 1 1

0 .5 .3 .7 .1 .5 .4 .7 .2 .6 .3 .8 .2 .6 .6 1

I

s ~ 5 3 54

5'

53 53 53 53 53 53

si si

t

12 13 14 15 16

II l (M . I.), H2(P.I.)

Z

st

51 I I 5 I 52 5 2 s~ 5 I 52 5 ' 52 51 5 2, 51 5 2, 51 5 2 5 I 52 51 5 2 5 1 52 51 5 2 51 5 2 51 5 2

11

Fuzzy sets v

Second Relation:

k

, ,

54 I 54 54 54 5 4, 54 54 54 54 54 54 54 54 54 54

si , si , 53 , , 53 si , si 53 , 53 53

PLAUSIBLE EXPECTATION FOR A CREDIBLE EXPECTATION

ACKNOWLEDGEMENT This work is supported by the NSERC of Canada under Grants A-56 25 and A-l080.

TABLE 1.

First Relation :

R

GOT(U/L}

CPK (U/U

LDH (U/U

SCAN

M.I. 30 50

200 220

o

30 50

1150

250

400

o

180 250 450 500

o

290

P.!' 10

30 50 120 170

454

M. M. Gupta et aL .

GOT .5

CPK

LDH

SCAN

253

490

o

BW'5

BW·5

54 Fig . 2.

Observation of Attributes (Example 1)

P.I.

P.I.

V

/

M.I.

)

o L-.JL.......L---Y--. 0.3 Fig . 3.

0.2

Plausible Expectation

CPK

LDH

SCAN

55

245

490

o

p.1.

Fig. 6.

Credible Expectation

GOT

Fig. 5 .

o

Fig. 4.

0.8 I

Observation of Attributes (Example 2)

p.1.

MJ.

o 0.3

Plausible Expectation

L---'----'---ji--.

0.3

0.7 Fig. 7.

M.I.

0.8 I

Credible Expectation