APPROXIMATE REASONING IN CANCER SURGERY

Elisabeth Rakus-Andersson

Department of Mathematics and Science, Blekinge Institute of Technology, Valhallavägen 1, 37179 Karlskrona, Sweden

Keywords: Approximate reasoning, Deterministic operation chance, Parametric membership functions.

Abstract: The compositional rule of inference, grounded on the modus ponens law, is one of the most effective fuzzy

systems. We modify the classical version of the rule (Zadeh, 1973, 1979) to propose an original model, which

concerns determining an operation chance for gastric cancer patients. The operation prognosis will be

dependent on values of biological markers indicating the progress of the disease.

1 INTRODUCTION

One of the early systems, evolved by Zadeh as an

approach to decision making in vague

circumstances, was the technique of approximate

reasoning (Zadeh, 1973, 1979). The compositional

rule of inference found adherents who adapted the

primary foundations of the theory to own models

(Baldwin and Pilsworth, 1979; Mizumoto and

Zimmermann, 1982; Zimmermann, 2002).

Some trials of technical use of approximate

reasoning have been made, but it is still difficult to

find a medical application based on the inference

rule. The rule was once tested by the author in order

to make decisions concerning operation chances for

gastric cancer patients (Rakus-Andersson, 2009).

The decisions were based on values of one

biological marker C-reactive proteins CRP, regarded

by physicians as the essential index of cancer

progress.

In the current paper we wish to extend the

number of clinical symptoms in the model. In

practice we want to add a value of age to CRP-value

(Do Kyong-Kim et al., 2009) to deduce a verbal

evaluation of the operation chance for post-surgical

survival in cancer diseases.

We discuss the approximate reasoning structures

in Section 2. Fuzzy sets, taking place in the model,

will be created in Section 3. Section 4 is added as a

presentation of the algorithm prognosis made for an

individual patient.

2 RULE OF INFERENCE

Surgical decisions are made with the highest

thoughtfulness in the case of patients suffering from

cancer. The physician wants to prognosticate the

operation role positively; we therefore introduce the

concept “operation chance” to determine the

outcome of a surgery.

The most decisive clinical markers CRP and age

found in an individual patient will constitute the

input data in the approximate reasoning model to

evaluate the operation chance for survival.

Let us state a logical compound tautology

(Rakus-Andersson, 2009)

.THEN) )))NOT(THEN))AND

AND(NOT(IF(ELSE)THEN)AND

AND(IF((AND)ANDAND( (IF

qqp

pqp

ppp

(1)

In accordance with the generalized law modus

ponens (Zadeh, 1973) we interpret (1) as a statement

(IF(

′

∧∧

′

pp "

)AND((IF(

pp ∧∧"

)THEN

q)ELSE(IF(NOT(

∧

))THEN (NOTq))))

THEN q`,

(2)

provided that the semantic meaning of p

and p

`, i =

1,…, p, (q and q` respectively) is very close.

In (2) let p

and p

` be mapped in fuzzy sets P

and P

` in the universes X

and let q and q` be

expressed by fuzzy sets Q and Q` in the universe Y.

We now make a feedback to the medical task

previously outlined to evaluate the operation

chances as verbal expressions.

466

Rakus-Andersson E..

APPROXIMATE REASONING IN CANCER SURGERY.

DOI: 10.5220/0003642204660469

In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (FCTA-2011), pages 466-469

ISBN: 978-989-8425-83-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

Let S

,…,S

denote clinical markers possessing

the decisive power in the evaluation of the operation

chance. We regard S

, i = 1,…, p, like the symptoms

whose growth levels are assimilated with codes.

Values of the codes form the universes X

= “S

’s

levels” = {1,…,k

,…,n

}. Assume that level 1 is

associated with the slightly heightened symptom

values whereas level n

indicates the dangerous

symptom intensity (Rakus-Andersson, 2009).

Universe Y consists of words describing

operation chance priorities. We set Y = “operation

chance priorities” = {L

= “none”, L

= “very little”,

= “little”, L

= “moderate”, L

= “promising”, L

= “very promising”, L

= “totally promising”}

assuming that Y is experimentally restricted to seven

chance priorities.

We assign p

`, p

, q and q` to sentences (Rakus-

Andersson, 2009)

` = “symptom S

is found in patient on level k

, i =

1,…,p”,

= “lower levels of S

are essential for a positive

operation outcome”,

q = “operation chance can be estimated on the basis

of S

and…and S

”

and

q` = “patient with the k

-level of S

and…and the k

level of S

gets an estimated operation chance as this

, which has the highest degree in Q`, l = 1,…, 7”,

Rule (2) will thus become a scheme

(IF (“symptom S

is found in patient on level k

”

and…and “symptom S

is found in patient on level

” =

′

∩∩

′

PP "

) AND ((IF “lower levels of S

are essential for a positive operation outcome”

and…and “lower levels of S

are essential for a

positive operation outcome” =

PP ∩∩"

THEN

“operation chance can be estimated on the basis of

and…and S

” = Q) ELSE (IF it is not true that

“lower levels of S

are essential for a positive

operation outcome” and…and “lower levels of S

are

essential for a positive operation outcome” =

)(

1 p

PPC ∩∩"

THEN “operation chance cannot be

estimated on the basis of S

and…and S

” = CQ)))

THEN “patient with the k

-level of S

and…and the

-level of S

gets an estimated operation chance as

this L

, which has the highest degree in Q`, l = 1,…,

7” = Q`.

)(

1 p

PPC ∩∩"

and CQ are complements of

PP ∩∩"

and Q.

3 DATA SETS IN X AND Y

The decision model, sketched in Section 2, includes

operations on fuzzy sets P

`, P

, Q and Q`. First we

design fuzzy sets in X

, i = 1,...,p, as structures

),...},2(),,1(

),1,(),,1(),,2({...,

))}(,()),...,(,()),...,1(,1{(

iii

kkk

nμnkμkμ

−−

−−=

′

′′′

(3)

and

)},(),...,,(),...,,1{(

))}(,()),...,(,()),...,1(,1{(

)1(

iii

iPiiPiP

nμnkμkμ

−−

(4)

referring to S

due to the definitions of p

and p

The set Q is sophisticated to be stated as a fuzzy

set since its support consists of other fuzzy sets L

, l

= 1,…,7, defined in a symbolic chance reference set

Z =[z

min

max

]= [0,1]. To find restrictions of L

study the technique of Rakus-Andersson (2010).

Suppose that L

,…,L

are included in the

linguistic list, where m is an odd positive integer

greater or equal to 5. Supports of the restrictions

)(z

, l = 1,…,m, will cover parts of the reference

set Z = [0,1]. We introduce E to be the length of Z.

We divide all expressions L

in three groups,

namely, a family of “leftmost” sets L

,…,

1−m

, the

set

1+m

“in the middle” and a collection of

“

rightmost” sets

3+m

,…,L

The “

leftmost” family is given by

()

⎪

⎩

⎪

⎨

⎧

−+≥

−+≤≤−+

⎟

⎠

⎞

⎜

⎝

⎛

−+≤≤−+

⎟

⎠

⎞

⎜

⎝

⎛

−

−+≤

−−

−−−−

−+−

−−−

−+−

−

−−

−

111)1(2

)1(

1)1(21

min

)1(

min

)1(for0

,)1()1(

for2

,)1()1(

for21

,)1(for1

)(

min

tzz

tzt

tztz

tzz

(5)

for parameter

t, t = 1,…,

1−m

To implement the “

rightmost” functions we use

APPROXIMATE REASONING IN CANCER SURGERY

467

()

()()

()

()()

(

)

()

⎪

⎩

⎪

⎨

⎧

−+−≥

−+−≤≤−+−

⎟

⎠

⎞

⎜

⎝

⎛

−

−+−≤≤−+−

⎟

⎠

⎞

⎜

⎝

⎛

−+−≤

−

−−−

−+−−

−−−−

−+−−

−−

−

−−

+−

min

1)1(2

)1(

1)1(211

)1(

)1(for1

,)1()1(

for21

,)1()1(

for2

,)1(for0

)(

min

tzEz

tEz

tzEz

tzEztE

tEztE

tEz

(6)

for

t = 1,…,

1−m

. The t-values are set in the formula

(6) in the reverse order to generate the sequence

3+m

,…,L

The function of

1+m

L is constructed as

⎪

⎩

⎪

⎨

⎧

≥

≤≤

⎟

⎠

⎞

⎜

⎝

⎛

≤≤

⎟

⎠

⎞

⎜

⎝

⎛

−

≤≤

⎟

⎠

⎞

⎜

⎝

⎛

−

≤≤

⎟

⎠

⎞

⎜

⎝

⎛

≤

−

.for0

,for2

,for21

,for2

,for0

)(

)1(2

)1(

)1(2

)1(

)1(2

)1(22

2)1(2

)2(

)1(2

)2(

)1(2

)3(

)1(2

)3(

)1(2

)1(

)1(2

)3(

(7)

For the list “

operation chance priorities” we

state

min

= 0, m = 7 and E =1. Fuzzy sets L

, l = 1,…,

7, are depicted in Fig. 1.

When defuzzifying the sets

we consider z-

coordinates of the intersection points between

1)( =z

and 1)( <z

. We denote the z-values

z(L

) = 0, z(L

) = 0.166, z(L

) = 0.338, z(L

) = 0.5,

z(L

) = 0.668, z(L

) = 0.834, z(L

) = 1 and we let

them represent

,…,L

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

µ(z)

Figure 1: Fuzzy sets L

–L

Then we build the set “numerical operation

chance

”, which gets the constraint s(z, 0, 0.5, 1) over

[0,1]. We compute the degrees of

z(L

) via this

constraint to obtain set Q in the form

)}.1,(),945.0,(),78.0,(

),5.0,(),22.0,(),055.0,(),0,{(

765

4321

LLL

LLLLQ

(8)

Further, for all (

,…,k

) ∈ X

×…×X

we determine

the intersections

)))}(),...,(min(),,...,((

))),...,(),...,(min(),,...,((

))),...,1(),...,1(min(),1,...,1{((

p`P`Pp

`P`P

nμnμnn

kμkμkk

μμ

′

∩∩

′

(9)

and

)))}.(),...,(min(),,...,((

))),...,(),...,(min(),,...,((

))),...,1(),...,1(min(),1,...,1{((

pPPp

nμnμnn

kμkμkk

μμ

∩

(10)

In conformity with Zadeh (1973) and

Zimmermann, (2002) we introduce the matrix

being a mathematical expression of the implication

(IF

∩

THEN Q)

ELSE (IF

)(

1 p

PPC

∩

THEN CQ).

The membership function of

R is yielded by

)))(),...,((

)),(),...,((,1min(

)),,...,((

1...

1)...(

lCQpPP

lQpPPC

lpR

Lkk

Lkkμ

μμ

∩∩

(11)

for all (

,…,k

) ∈ X

×…×X

and all L

∈ Y.

Set

Q` will be formed as (Zadeh, 1973)

RPPQ`

D)...(

′

∩∩

′

= .

(12)

Q` is designated by the membership function

))).),,...,((

),,...,((min(max)(

),...,(

lpR

Lkk

kkLμ

′

∩∩

′

××

∈

(13)

By comparing magnitudes of membership

degrees in set

Q` with respect to all L

, l = 1,…,7, we

select this chance priority

, which assists the

largest value of

)(

´ lQ

Lμ .

4 CHANCE DETERMINATION

The CRP-value and age are decisive markers of the

prognosis in cancer surgery (Do-Kyong Kim et al.,

2009).

FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications

468

The heightened values of CRP (measured in

milligrams per liter) are discerned in levels

1 = “almost normal” for CRP < 10,

2 = “heightened” if 10

≤ CRP ≤ 20,

3 = “very heightened” if 20

≤ CRP ≤ 25,

4 = “dangerously heightened” for CRP > 25.

The age borders are decided as

1 = “not advanced for surgery” if “age” < 60,

2 = ”advanced for surgery” if 60 ≤ ”age” ≤ 80,

3 = “dangerous for surgery” if “age” > 80.

Suppose that in a seventy-year-old patient the

CRP-value is measured to be 18.

Due to (4) and (10) sets P

, P

and their

intersection are expressed as

)}25.0),3,4((

),...,5.0),2,3((),...,1),1,1{((

)},34.0,3(),66.0,2(),1,1{(

)},25.0,4(),5.0,3(),75.0,2(),1,1{(

=∩

(14)

while P

`, P

` and their cut are computed, with

respect to (3) and (9), as

)}5.0),3,4((

),...,75.0),2,3((),...,66.0),1,1{((

)},66.0,3(),1,2(),66.0,1{(

)},5.0,4(),75.0,3(),1,2(),75.0,1{(

′

∩

′

(15)

provided that X

= {1,2,3,4} and X

= {1,2,3}.

Matrix R, found in compliance with (11), is

expanded as a two-dimensional table

⎥

⎦

⎤

⎢

⎣

⎡

25.075.075.0

5.015.0

15.00

)3,4(

)2,3(

)1,1(

741

LLL

(16)

We in6sert R given by (16) and

′

∩

′

determined by (15) in (12) in order to estimate

)}.66.0,(),715.0,(),88.0,(),84.0,(

),72.0,(),715.0,(),66.0,{(`

7654

321

LLLL

LLLQ =

(17)

The largest membership degree in (17) points out

chance L

= “promising” for a result of the operation

on the elderly patient whose CRP-index is evaluated

on the second growth level.

5 CONCLUSIONS

We have adapted approximate reasoning as a

deductive algorithm to introduce the idea of

evaluating the operation chance for patients with

heightened values of biological indices in cancer

diseases.

The formulas of membership functions in data

sets have been expanded by applying a formal

mathematical design invented by the author. The

data sets involve parametric families of functions,

which allow preparing a computer program. We

have tested a large sample of patient data to get the

results mostly converging to the physicians’

prognoses. This confirms reliability of the system.

ACKNOWLEDGEMENTS

The author thanks the Blekinge Research Board in

Karlskrona – Sweden for the grant funding this

research. The author is grateful to Medicine

Professor Henrik Forssell for all helpful hints made

in the subject of cancer surgery.

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