 APPROXIMATE REASONING IN CANCER SURGERY
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
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
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
.THEN) )))NOT(THEN))AND
AND(NOT(IF(ELSE)THEN)AND
AND(IF((AND)ANDAND( (IF
1
11
qqp
pqp
ppp
p
p
p
"
""
(1)
In accordance with the generalized law modus
ponens (Zadeh, 1973) we interpret (1) as a statement
(IF(
p
pp "
1
)AND((IF(
p
pp "
1
)THEN
q)ELSE(IF(NOT(
p
pp
"
1
))THEN (NOTq))))
THEN q`,
(2)
provided that the semantic meaning of p
i
and p
i
`, i =
1,…, p, (q and q` respectively) is very close.
In (2) let p
i
and p
i
` be mapped in fuzzy sets P
i
and P
i
` in the universes X
i
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
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
c
2011 SCITEPRESS (Science and Technology Publications, Lda.) Let S
1
,…,S
p
denote clinical markers possessing
the decisive power in the evaluation of the operation
chance. We regard S
i
, i = 1,…, p, like the symptoms
whose growth levels are assimilated with codes.
Values of the codes form the universes X
i
= “S
i
’s
levels” = {1,…,k
i
,…,n
i
}. Assume that level 1 is
associated with the slightly heightened symptom
values whereas level n
i
indicates the dangerous
Universe Y consists of words describing
operation chance priorities. We set Y = “operation
chance priorities” = {L
1
= “none”, L
2
= “very little”,
L
3
= “little”, L
4
= “moderate”, L
5
= “promising”, L
6
= “very promising”, L
7
= “totally promising”}
assuming that Y is experimentally restricted to seven
chance priorities.
We assign p
i
`, p
i
, q and q` to sentences (Rakus-
p
i
` = “symptom S
i
is found in patient on level k
i
, i =
1,…,p”,
p
i
= “lower levels of S
i
are essential for a positive
operation outcome”,
q = “operation chance can be estimated on the basis
of S
1
and…and S
p
and
q` = “patient with the k
1
-level of S
1
and…and the k
p
-
level of S
p
gets an estimated operation chance as this
L
l
, which has the highest degree in Q`, l = 1,…, 7”,
Rule (2) will thus become a scheme
(IF (“symptom S
1
is found in patient on level k
1
and…and “symptom S
p
is found in patient on level
k
p
=
p
PP "
1
) AND ((IF “lower levels of S
1
are essential for a positive operation outcome”
and…and “lower levels of S
p
are essential for a
positive operation outcome” =
p
PP "
1
THEN
“operation chance can be estimated on the basis of
S
1
and…and S
p
= Q) ELSE (IF it is not true that
“lower levels of S
1
are essential for a positive
operation outcome” and…and “lower levels of S
p
are
essential for a positive operation outcome” =
)(
1 p
PPC "
THEN “operation chance cannot be
estimated on the basis of S
1
and…and S
p
= CQ)))
THEN “patient with the k
1
-level of S
1
and…and the
k
p
-level of S
p
gets an estimated operation chance as
this L
l
, which has the highest degree in Q`, l = 1,…,
7” = Q`.
)(
1 p
PPC "
and CQ are complements of
p
PP "
1
and Q.
3 DATA SETS IN X AND Y
The decision model, sketched in Section 2, includes
operations on fuzzy sets P
i
`, P
i
, Q and Q`. First we
design fuzzy sets in X
i
, i = 1,...,p, as structures
),...},2(),,1(
),1,(),,1(),,2({...,
))}(,()),...,(,()),...,1(,1{(
21
12
i
i
i
i
i
i
i
i
iii
n
n
i
n
n
i
i
n
n
i
n
n
i
i
P
ii
P
i
P
i
kk
kkk
nμnkμkμ
P
++
=
=
(3)
and
)},(),...,,(),...,,1{(
))}(,()),...,(,()),...,1(,1{(
1
)1(
ii
ii
i
i
iii
n
i
n
kn
i
n
n
iPiiPiP
i
nk
nμnkμkμ
P
=
=
(4)
referring to S
i
due to the definitions of p
i
and p
i
`.
The set Q is sophisticated to be stated as a fuzzy
set since its support consists of other fuzzy sets L
l
, l
= 1,…,7, defined in a symbolic chance reference set
Z =[z
min
,z
max
]= [0,1]. To find restrictions of L
l
we
study the technique of Rakus-Andersson (2010).
Suppose that L
1
,…,L
m
are included in the
linguistic list, where m is an odd positive integer
greater or equal to 5. Supports of the restrictions
)(z
l
L
μ
, 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
l
in three groups,
namely, a family of “leftmost” sets L
1
,…,
2
1m
L
, the
set
2
1+m
L
in the middle” and a collection of
rightmost” sets
2
3+m
X
,…,L
m
.
The “
leftmost” family is given by
()
()
+
++
++
+
=
+
+
11
111)1(2
2
)1(
1)1(21
min
2
)1(
1
min
)1(for0
,)1()1(
for2
,)1()1(
for21
,)1(for1
)(
1
11
1
1
min
m
E
m
E
m
E
m
E
m
E
m
E
tz
m
E
m
E
m
E
tzz
m
E
L
tz
tzt
tztz
tzz
z
m
E
m
E
m
E
m
E
m
E
t
μ
(5)
for parameter
t, t = 1,…,
2
1m
.
To implement the “
rightmost” functions we use
APPROXIMATE REASONING IN CANCER SURGERY
467 ()
()()
()
()
()()
(
)
()
()
+
++
++
+
=
+
+
+
1
min
1
min
1)1(2
2
)1(
1)1(211
2
)1(
11
)1(for1
,)1()1(
for21
,)1()1(
for2
,)1(for0
)(
1
1
min
1
11
1
m
E
m
E
m
E
m
E
tzEz
m
E
m
E
m
E
m
E
tEz
m
E
m
E
L
tzEz
tzEztE
tEztE
tEz
z
m
E
m
E
m
E
m
E
m
E
tm
μ
(6)
for
t = 1,…,
2
1m
. The t-values are set in the formula
(6) in the reverse order to generate the sequence
2
3+m
L
,…,L
m
.
The function of
2
1+m
L is constructed as
=
+
+
+
+
.for0
,for2
,for21
,for21
,for2
,for0
)(
)1(2
)1(
)1(2
)1(
)1(2
2
)1(22
2
2)1(2
)2(
2
)1(2
)2(
)1(2
)3(
2
)1(2
)3(
1
)1(2
)1(
1
2
1
2
1
)1(2
)3(
2
1
m
mE
m
mE
m
Em
z
m
Em
E
z
E
m
mE
z
m
mE
m
mE
z
m
mE
L
z
z
z
z
z
z
z
m
E
m
mE
m
E
E
m
E
E
m
E
m
mE
m
μ
(7)
For the list “
operation chance priorities” we
state
z
min
= 0, m = 7 and E =1. Fuzzy sets L
l
, l = 1,…,
7, are depicted in Fig. 1.
When defuzzifying the sets
L
l
we consider z-
coordinates of the intersection points between
1)( =z
l
L
μ
and 1)( <z
l
L
μ
. We denote the z-values
by
z(L
1
) = 0, z(L
2
) = 0.166, z(L
3
) = 0.338, z(L
4
) = 0.5,
z(L
5
) = 0.668, z(L
6
) = 0.834, z(L
7
) = 1 and we let
them represent
L
1
,…,L
7
.
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
z
µ(z)
Figure 1: Fuzzy sets L
1
L
7.
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
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
1
,…,k
p
) X
1
×…×X
p
we determine
the intersections
)))}(),...,(min(),,...,((
))),...,(),...,(min(),,...,((
))),...,1(),...,1(min(),1,...,1{((
11
11
1
1
1
1
p`P`Pp
p`P`Pp
`P`P
p
nμnμnn
kμkμkk
μμ
PP
p
p
p
=
"
(9)
and
)))}.(),...,(min(),,...,((
))),...,(),...,(min(),,...,((
))),...,1(),...,1(min(),1,...,1{((
11
11
1
1
1
1
pPPp
pPPp
PP
p
nμnμnn
kμkμkk
μμ
PP
p
p
p
=
"
(10)
In conformity with Zadeh (1973) and
Zimmermann, (2002) we introduce the matrix
R
being a mathematical expression of the implication
(IF
p
PP
"
1
THEN Q)
ELSE (IF
)(
1 p
PPC
"
THEN CQ).
The membership function of
R is yielded by
)))(),...,((
)),(),...,((,1min(
)),,...,((
1...
1)...(
1
1
1
lCQpPP
lQpPPC
lpR
Lkk
Lkk
Lkkμ
p
p
μμ
μμ
+
+
=
(11)
for all (
k
1
,…,k
p
) X
1
×…×X
p
and all L
l
Y.
Set
Q` will be formed as (Zadeh, 1973)
RPPQ`
p
D)...(
1
= .
(12)
Q` is designated by the membership function
))).),,...,((
),,...,((min(max)(
1
1
),...,(
´
1
1
1
lpR
p
PP
XX
kk
lQ
Lkk
kkLμ
p
p
p
μ
μ
××
=
"
"
(13)
By comparing magnitudes of membership
degrees in set
Q` with respect to all L
l
, l = 1,…,7, we
select this chance priority
L
l
, 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
1
, P
2
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{(
21
2
1
=
=
=
PP
P
P
(14)
while P
1
`, P
2
` 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{(
21
2
1
=
=
=
PP
P
P
(15)
provided that X
1
= {1,2,3,4} and X
2
= {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
""
##
""
##
""
#
#
""
R
LLL
.
(16)
We in6sert R given by (16) and
21
PP
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
5
= “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
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
in the subject of cancer surgery.
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Mizumoto, M., Zimmermann, H. J., 1982. Comparison of
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APPROXIMATE REASONING IN CANCER SURGERY
469 