Vol.2, No.3, 211-224 (2010)
doi:10.4236/health.2010.23031
Copyright © 2010 SciRes Openly accessible at http://www.scirp.org/journal/HEALTH/
Health
Employing fuzzy logic in the diagnosis of a clinical case
G. Licata
Dipartimento FIERI, University of Palermo, Viale delle Scienze, Palermo, Italy; ninnilicata@yahoo.it
Received 20 October 2009; revised 9 December 2009; accepted 14 December 2009.
ABSTRACT
Fuzzy logic is a logical calculus which operates
with many truth values (while classical logic
works with the two values of true and false).
Since fuzzy logic considers the truth of scien-
tific statements like something softened, it is
fruitfully applied to the study of biological phe-
nomena, biology is indeed considered the field
of complexity, uncertainty and vagueness. In
this paper fuzzy logic is successfully applied to
the clinical diagnosis of a patient who suffers
from different diseases bound by a complex
causal chain. In this work it is presented a
mathematical foundation of fuzzy logic (with
connectives and inference rules) and then the
application of fuzzy reasoning to the study of a
clinical case. Probabilistic logic is widely con-
sidered the unique logical calculus useful in
clinical diagnosis, thus the usefulness of fuzzy
logic and its relation with probabilistic logic is
here explored. The presentation of the case is
supplied with all the features necessary to affect
a clinical diagnosis: physical exam, anamnesis
and tests.
Keywords: Fuzzy Logic; Probabilistic Logic; Clini-
cal Diagnosis; Biological Phenomena; Truth
1. INTRODUCTION
In this work fuzzy logic was applied to clinical diagnosis.
Fuzzy logic is a multi-valued logic, i.e. a logic which
works with many (finite or infinite) truth-values. Differ-
ently from classical logic, which works with two truth-
values (true or false, 1 or 0), fuzzy logic allows for de-
grees of truth. Fuzzy logic considers the truth of scientific
statements like something softened, then it is fruitfully
applied to the study of biological phenomena because
biology and medicine are considered the field of com-
plexity, uncertainty and vagueness. Fuzzy logic empha-
sizes the precision of clinical data: it includes scalar
quantities in the argumentations, providing conclusions
which give partial truth in results. This means that fuzzy
diagnosis, achieving the quantitative precision of clinical
signs, symptoms and laboratory tests, is capable to show
the vagueness of diagnostic argumentations. Classical
logic derives from absolutely true premises absolutely
true conclusions, but in the whole science the absolute
truth is an illusion: human science always works with
uncertainty. In medicine, and in particular in medical
practice, this problem is quite evident. Fuzzy and prob-
abilistic logic treats vagueness and uncertainty, but
nowadays only probabilistic logic is employed in clinical
diagnosis. The central aim of this work is to demonstrate
that fuzzy diagnosis may be an improvement and a comple-
tion of probabilistic diagnosis. Fuzzy logic, differently from
classical logic, not only shows the uncertainty in the argu-
mentations, but also measures it; and, differently from
probabilistic logic, treats a kind of uncertainty which prob-
ability does not consider. Many scholars are persuaded by
the work of de Finetti [1] that only one kind of mathe-
matical method is needed to treat uncertainty: probabil-
istic logic. If that was the case fuzzy logic would be not
necessary. On the other hand, Kosko argues that prob-
abilistic logic is a sub-theory of fuzzy logic, and that
probability handles only one kind of uncertainty. He also
claims to have proven a derivation of Bayes’ theorem from
the concept of fuzzy ‘subsethood’ [2]. Zadeh, the creator of
fuzzy logic, argues that fuzzy logic is different in char-
acter from probability, and it is not a substitute of prob-
abilistic logic. Which is the relation between fuzzy and
probabilistic logic? The clinical diagnosis is a good field
to look for the answer.
A lot of clinical phenomena, symptoms, signs and
laboratory tests are quantitative; whether their quantity is
wholly scalar, as in the case of lab test, or whether it is
determinable only by linguistic adjectives, as in the case
of symptoms, there is however a great advantage in the
use of fuzzy logic. In the “Methods” section of this work
it was furnished a mathematical-logic formulation of
fuzzy logic and of fuzzy set theory. In the “Results” sec-
tion fuzzy logic was applied to a real clinical case.
2. METHODS: ELEMENTS OF FUZZY
LOGIC
2.1. Set Theory
A set is a collection of objects which satisfies one particular
condition. The objects contained in the set are called
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“elements” of the set. The elements of a set can also be
sets. The term “set” is not defined: it is sufficient to give
an intuitive exposition of set theory. Following a usual
symbolization, the sets are denoted with capital letters A,
B, X, Y, and the elements of the sets with miniscule letters
a, b, x, y,
To denote that a is an element of A, it is used
a A (1)
To denote that a is not an element of A, it is used
a A (2)
To denote that A is the set which has as elements a, b, c,
it is used
A = {a, b, c, …} (3)
Definition 1. For two sets A and B, if each element of A
is an element of B, then A is a subset of B:
A B or B A (4)
To denote that A is not a subset of B, it is used
A B (5)
Definition 2. For two sets A and B, A B shall denote
the set constituted by the elements which belong to A or to
B. The set A B is called the “union of A and B”.
Definition 3. For two sets A and B, A B shall denote
the set constituted by the elements which belong to A and
to B. The set A B is called the “intersection of A and B”.
Definition 4. For two sets A and B, a function f of A in B
is a law which associates to each element of A an element
of B. To denote this function it is used f: A B.
An “indicator function” or a “characteristic function”
is a function defined on a set X that indicates the mem-
bership of an element x in a subset A of X. The indicator
function of a subset A of a set X is a function
1A: X {0,1}
defined as
1 if x A
1
A(x) =
0 if x A
2.2. Fuzzy Set
The fuzzy set theory is an extension of classical set theory.
As the classical sets are used in classical two-valued logic,
fuzzy sets are used in fuzzy logic. In classical set theory
the membership of elements in relation to a set is assessed
in binary terms according to a crisp condition: an element
either belongs (1, true) or does not belong (0, false) to the
set. By contrast, fuzzy set theory permits a gradual
membership of the elements in relation to a set. This fact
is described with the aid of a membership function
μ[0,1]. The membership function of a fuzzy set is a
generalization of the indicator function of classical sets.
In fuzzy logic, it represents the degree of truth as an ex-
tension of valuation. For any crisp set X, a membership
function on X is any function from X to the real unit in-
terval [0, 1]. Membership functions on a crisp set X rep-
resent fuzzy subsets of X. It is remarkable that fuzzy sets
are defined as subsets of a classical set; this is the reason
why the fuzzy set theory is considered an extension of
classical set theory. The membership function represent-
ing a fuzzy set is usually denoted by μÃ. For an element x
of X, the value μà (x) is called the membership degree of x
to the fuzzy set Ã, which is a subset of X. The membership
degree μà (x) quantifies the grade of membership of the
element x to the fuzzy set Ã. The value 0 means that x is
not a member of the fuzzy set; the value 1 means that x is
fully a member of the fuzzy set. The values between 0 and
1 characterize members which belong to the fuzzy set
only partially. Usually membership functions with values
in [0, 1] are called [0, 1]-valued membership functions. In
Figure 1, the function μà (x) of an element of the fuzzy
subset à of X is represented by the curved line, while the
function μA (x) of an element of the crisp subset A of X is
represented by the broken line.
For A X Ã
2.3. Fuzzy Logic
Fuzzy logic, derived from fuzzy set theory, is an useful
calculus which represents a reasoning that is approximate
rather than precisely deduced as in classical predicate or
propositional logic. It can be thought of as the application
side of fuzzy set theory dealing with well thought out real
world. Fuzzy logic admits set membership values to
range (inclusively) between 0 and 1, and, in its linguistic
form, admits imprecise concepts like “slightly”,
“enough”, “very”, “not completely” and so on.
2.3.1. Connectives
As the classical logic, but differently from probabilistic
logic, fuzzy logic is truth-functional. The AND (), OR
(), and NOT () operators of classical logic exist also in
fuzzy logic; they are usually defined as the minimum,
maximum, and complement. When they are defined in this
way they are called “Zadeh operators”, indeed they were
Figure 1. The function μà (x) of an element of the fuzzy subset Ã
of X is represented by the curved line, the function μA (x) of an
element of the crisp subset A of X is represented by the broken line.
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first defined in Zadeh’s original papers. The most important posed in the past for fuzzy logic, such as those of Sugeno
[3], Dubois and Prade [4] and Yager [5]. Each of these
provides a way to vary the “gain” on the function so that it
can be very restrictive or very permissive. Here I follow
the Gödel T-norm (min) and Gödel T-conorm (max) as
defined in Hajek [6].
thing to understand about fuzzy logical reasoning is the
fact that it is a superset of standard Boolean logic. If we
keep the fuzzy values at their extremes of 1 (completely
true), and 0 (completely false), the laws of classical logic
will be valid. Consider, for example, the truth tables of
classical logic (Figure 2). 2.3.2. Fuzzy Implication: If-Then Rules
Knowing that in fuzzy logic the truth of any sentence is a
matter of degree, these truth tables must be defined through
others connectives. Input values can be real numbers be-
tween 0 and 1. One function which will preserve the results
of the AND truth table (for example) and also extend to all
real numbers between 0 and 1 is the min operation. This
operation resolves the sentence A AND B, where A and B
are limited to the range (0, 1), by using the function min (A,
B). Using the same reasoning, it is possible to replace the
OR operation with the max function, so the sentence A OR
B, where A and B are limited to the range (0, 1), becomes
equivalent to max (A, B). Finally, the operation NOT A
becomes equivalent to the operation 1 A. The truth
function of negation has to be non-increasing (and assign 0
to 1 and vice versa); the function 1 A (Łukasiewicz ne-
gation) is the best known candidate. Let’s consider Figure
3: the truth table in Figure 2 is completely unchanged by
this substitution.
Fuzzy sets are the subjects and predicates of fuzzy logic.
The If-Then rule statements are used to formulate the
conditional statements that comprise fuzzy logic. A single
fuzzy If-Then rule assumes the canonical form
1) if x is A then y is B
or, in fuzzy propositional logic,
2) if p then q
In 1) A and B are linguistic values defined by fuzzy
sets on the ranges (universes of discourse) X and Y, re-
spectively. The if-part of the rule, “x is A” (or p), is called
the antecedent or premise, while the then-part of the rule,
y is B” (or q), is called the consequent or conclusion. An
example of such a rule might be.
If service is good then tip is average.
The adjective good can be represented as a number
between 0 and 1, thus the antecedent is an interpretation
that gives a single number between 0 and 1. The adjective
average is represented as a fuzzy set, and so the conse-
quent is an assignment that assigns the entire fuzzy set B
to the output variable y. In the If-Then rule, the word “is”
is used in two entirely different ways depending on
whether it appears in the antecedent or the consequent. In
general, the input of an If-Then rule is the current value
for the input variable (in this case, service), while the
With these three functions it is possible to resolve any
construction using fuzzy sets and the fuzzy logical op-
erations AND, OR, and NOT. Clearly, it is only defined
here one particular correspondence between two-valued
and multi-valued logical operations for AND, OR, and
NOT. This correspondence is not unique. Not only min
and ma x but several kind of connectives have been pro-
Figure 2. Truth tables of classical logic.
Figure 3. Truth tables of fuzzy logic.
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214
output is an entire fuzzy set (in this case, average). This
set will later be “defuzzyfied, assigning a numerical
value to the output. The defuzzyfication is the process of
producing a numerical result in fuzzy logic. Typically, a
fuzzy system will have a number of tables of conversion
that transform some variables (scalar or linguistic) into
numerical results: the result is described in terms of
membership into fuzzy sets. Interpreting an If-Then rule
involves distinct parts: first evaluating the antecedent
(which involves fuzzyfying the input variable and then
defuzzyfying it), second applying that result to the con-
sequent (implication). In the case of two-valued or clas-
sical logic, If-Then rules don’t present difficulty. If the
premise is true, then the conclusion is true. If we relax the
restrictions of two-valued logic and assume that the an-
tecedent is a fuzzy statement, this shall reflect on the
conclusion. If the antecedent is true to some degree of
membership, then the consequent is also true to that same
degree. In other words:
Classical logic p q (p and q are either true or false)
Fuzzy logic 0.5 p 0.5 q (partial antecedents provide
partial implication)
The antecedent of a rule can have multiple parts, for
example:
if sky is gray and wind is strong and barometer is fal-
ling, then...
All parts of the antecedent must be defuzzyfied, cal-
culated simultaneously and resolved to a single number
using the logical operators described in the preceding
section. The consequent of a rule can also have multiple
parts. For example:
if temperature is cold then hot water valve is open and
cold water valve is closed.
The consequent is affected by the antecedent and all
consequents are affected equally by the result of the an-
tecedent. The consequent assigns a fuzzy set to the output,
then the implication function modifies that fuzzy set to
the degree specified by the antecedent. The If-Then rules
which were used in the case report (Section 3) gave to (all
parts of the) the consequent the same fuzzy value which
was given to the antecedent. Using the If-Then rules is a
four-part process:
1) Fuzzifycation of variables: Provide tables of con-
version that transform some variables (scalar or linguistic)
into fuzzy sets, the tables show that the degree of mem-
bership to the fuzzy set correspond to numerical values
between 0 and 1 (0 and 100%).
2) Defuzzifycation: Resolve all fuzzy statements in the
antecedent giving a precise degree of membership be-
tween 0 and 1 (0 and 100%) to each part. If there is only
one part into the antecedent, this is the degree of support
for the rule.
3) Apply fuzzy operators to multiple part antecedents:
If there are multiple parts in the antecedent, apply fuzzy
logic operators (connectives) and resolve the antecedent
to a single number between 0 and 1. This is the degree of
support for the rule.
4) Apply implication method: Use the degree of sup-
port for the entire rule to shape the output fuzzy set. The
consequent of a fuzzy rule assigns an entire fuzzy set to
the output. If the antecedent is only partially true, (i.e., is
assigned a value less than 1), then the output fuzzy set is
truncated according to the implication method.
The general structure of the If-Then rules, with their
modifications with connectives, is: if p * p’ then q * q’
where * denotes the binary operations of conjunction and
disjunction. The inference rule that was used in the case
report (Section 3) to provide the If-Then statements is the
generalized modus ponens (GMP), which has the form:
((p q) p’) q
where p’ is always a quantitative modification of p and q
is always a quantitative modification of q (through the
degree of membership). This rule allows quantitative
implications as
p q
0.60 p
0.60 q
3. RESULTS: CASE REPORT
Maria C., a 67-years-old white woman, was admitted to
our unit because of dyspnoea, fatigue, leg oedema and
abdominal enlargement.
The patient was quite well until two weeks ago, when
she complained of increasing weakness, abdominal en-
largement and foot enlargement. Moreover, she felt in-
creasingly tired and suffered breathing difficulties, in
particular after physical activity or at the end of the day.
Physical exam: on admission the patient was pale and
dyspnoeic at rest. On clinical examination the patient had
leg oedema and ascites.
Blood
Pressure 160/80 mm/Hg
Breathing
rate Hyperpnoea: 22 cycle/min
Heart
Rate 140 b/min
Weight 68.8 Kg
Heart
Percussion of heart area estimate enlargement of
heart size. Normal heart sounds with systolic heart
murmur 3/6 Levine. Tachycardia and arrhythmias.
Thorax
Reduced chest expansibility during inspiration and
expiration. An abolished fremitus on the right
pulmonary base and a reduced fremitus on the other
part of the right chest. The chest percussion reveals
the absence of the “clear lung sound”. On auscul-
tation of the chest the normal breathing is impaired
and is revealed the presence of rales on the right
part of the chest and on the medium-basal left part
of the chest.
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Abdomen
The abdomen is taut, but no discomfort is elicited
during palpation. The percussion of the abdomen
reveals a dull sound about 2 centimetres below to
the navel.
Liver: enlargement liver: it can be palpated 3 cen-
timetres below to the costal margin. Normal con-
sistency, surface and tenderness.
Spleen: normal
Other Presence of legs oedema, with impressive oedema
of the ankles.
Anamnesis:
The patient was born in Italy and she lived with her
son;
She did not use alcohol or tobacco;
Diagnosis of hypertension when she was in her 40s;
Open-heart surgery, to repair a little atrial septal defect,
when she was in her 50s;
Total hysterectomy for cancer when she was in her 53s;
Diagnosis of diabetes mellitus when she was in her
55s;
Diagnosis of atrial fibrillation when she was in her 59s;
The patient’s current medications were metformin,
digoxin and ramipril.
3.1. Pre-test Hypotheses
On the basis of the clinical history of the patient, of the
presence of Dyspnoea at rest (a), Oedema (b), Tachyar-
rhythmias (c), Epatomegaly (d), Ascites (e), and Pleural
effusion (f), it is possible to formulate the following di-
agnostic hypotheses in agreement with clinical probability:
Hepatic Cirrhosis (t)
Nephrosic Syndrome (u)
Pneumonia (w)
Myocardial Ischemia (x)
Congestive Heart Failure (y)
Worsening of the Supraventricular Arrhythmias (z)
Using the If-Then rules the inferences have the form:
1) If a and b and c and d and e and f then t or u or w or x
or y or z
1’) If a and b and c and d and e and f then t and u and w
and x and y and z
In particular, the clinical probability suggests the fol-
lowing inferences:
2) If a and b and c and d and e and f then y
3) If a and f then w or y
3’) If a and f then w and y
4) If b and e and f then u
5) If a and c then z or x or y
5’) If a and c then z and x and y
6) If b and d and e and f then t
Tables of fuzzy conversion of signs and symptoms.
Tables 1-6 provide a correspondence function between
the intensity of the clinical phenomena, measured on the
usual clinical parameters, and the fuzzy values between 0
and 1 (degree of membership to a fuzzy set). Defuzzifica-
tion of data: The following values result from the con-
version of symptoms and signs found in our patient, Ma-
ria C., in defuzzyed values.
Fuzzy value of Dyspnoea (a) = 0,95
Fuzzy value of Oedema (b) = 0,70
Fuzzy value of Tachyarrhythmia I = 0,75
Fuzzy value of Epatomegaly (d) = 0,75
Fuzzy value of Ascites (e) = 0,75
Fuzzy value of Pleural Effusion (f) = 0,75
3.2. Fuzzy Evaluation of Pre-test Hypotheses
Quantifying the “if” parts of the inferences (sings and
symptoms), it is possible to obtain the quantification of
“then” parts of the inferences (diagnosed diseases). This
is the application of generalized modus ponens (GMP). In
this step of diagnostic process the quantification of di-
agnosed diseases could seem premature, but the GMP is
employed since now to show the increasing usefulness of
fuzzy logic in the diagnosis progress. Let’s rewrite the
diagnostic hypotheses of Subsection 3.1. with fuzzy values:
2) If 0,95 a and 0,70 b and 0,75 c and 0,75 d and 0,75 e
and 0,75 f then 0,70 y
3) If 0,95 a and 0,75 f then 0,75 w or 0,75 y
3’) If 0,95 a and 0,75 f then 0,75 w and 0,75 y
4) If 0,70 b and 0,75 e and 0,75 f then 0,70 u
5) If 0,95 a and 0,75 c then 0,75 z or 0,75 x or 0,75 y
5’) If 0,95 a and 0,75 c then 0,75 z and 0,75 x and 0,75 y
6) If 0,70 b and 0,75 d and 0,75 e and 0,75 f then 0,70 t
First turn of tests
Kidney function
Urea nitrogen (mg/dl) 58 v.n. 10-50
Creatinine (mg/dl) 0.8 v.n. 0.5-1.1
Na+ (mEq/l) 133 v.n. 133-145
K+ (mEq/l) 4 v.n. 3.3-5.1
Ca++ (mg/dl) 8.3 v.n. 8.5 -10.2
Proteinuria 24-hour urine sample 0.4 < 0.6 g/die
Liver function
AST (U/L) 34 v.n. < 31
ALT (U/L) 42 v.n. < 31
Bilirubin 1.11 v.n. 0.2-1.1
Gamma-GT 90 v.n. 5-36
Fosfatase alkaline 191 v.n. 35-104
Serum Albumin 4.02 v.n. 3.48-5.39
Serum Gamma Globulin 1.65 v.n. 0.67-1.56
Myocardial specifics enzymes
CK (U/L) 170 v.n. <190
CK Mb-Massa (ng/ml) 1.96 v.n. <2.4
Troponin I (ng/ml) 0.055 v.n. <0.08
Mioglobin (ng/ml) 127 v.n. <120
Others laboratory tests
Glucose 150 v.n. 80-125
Glycosilated Haemoglobin 7.9 % v.n. <7
Cholesterol (mg/dl) 109 < 200
HDL (mg/dl) 33 < 60
LDL (mg/dl) 65.6 < 130
Triglycerides (mg/dl) 52 < 150
Uric Acid (mg/dl) 4.1 < 7
ESR 72 < 15
PCR 0.6 Absent
LDH (U/L) 422 v.n. 250-480
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216
Table 1. Fuzzyfication of Dyspnoea (a).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Dyspnoea
(a) Paroxysmal nocturnalfor strenuous exercisefor light exerciseat rest
Table 2. Fuzzyfication of Oedema (b).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Oedema (b) +/++++ ++/++++ +++/++++ ++++/++++
Table 3. Fuzzyfication of Tachyarrhythmias I (c).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Tachy-
arrhythmias I Slow Moderate Fast
Table 4. Fuzzyfication of Epatomegaly (d).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Epatomegaly
(d)
1-3 cm below to the
costal margin
3-5 cm below to the
costal margin
Over 5 cm below to the
costal margin
Openly accessible at
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217
Table 5. Fuzzyfication of Ascites (e).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Ascites (e) More 3 cm below
the navel
1-3 cm below
the navel
1-3 cm above
the navel
More 3 cm above
the navel
Table 6. Fuzzyfication of Pleural Effusion (f).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Pleural
Effusion (f)Basal pleural effusionMedium-Basal pleural effusionApical pleural effusion
Urinalysis
Color yellow
pH 7
Glucose (mg/dl) 0.0
Protein (mg/dl) 70
Haemoglobin > 1
Ketones (mg/dl) 0.0
Bilirubin (mg/dl) 0.0
Urobilinogen 0.2
Specific gravity 1009.0
Red blood cells (n°/uL) 2041
White blood cells (n°/uL) 48
Epitelial cells (n°/uL) 1
Bacterial (n°/uL) 1345
Miceti (n°/uL) 0
Complete blood count
Red cell count (mm3) 3.810.000 v.n. 4.000.000-6.000.000
Haemoglobin (g/dl) 9.7 v.n. 12-17
Haematocrit (%) 30.2 % v.n. 26-50 %
Mean corpuscular
volume (pg) 79.3 v.n. 80-99
White cell count
(mm3) 5770 v.n. 4.000-10.000
Differential count (%)
Neutrophils
Lymphocytes
Monocytes ,Basophils,
Eosinophils
73.1 %
16.8 %
10.1 %
Platelet count (mm3) 300.000 v.n. 150.000-450.000
Aptt (sec) 33.5 v.n. 24-36
Fibrinogen (mg/dl) 299.4 v.n. 150-450
Instrumental tests
Electrocardiography (ECG): Atrial Flutter. Bundle
branch block Strain.
Chest radiography: Redistribution of blood flow to the
nondependent portions of the lungs. Perihilar and lower-
lobe airspace filling with the confluent opacities. Medium
-basal bilateral pleural effusions. Cardiomegaly.
Abdomen Ultrasonography: Hepatomegaly. Increased
liver echogenicity like moderate steatosis. Absence of
enlargement of hepatic veins. Normal aspect and echo-
genicity of gallbladder without stones. ICV and Portal vein:
moderate enlargement of the diameter and poor respiratory
variation. Normal pancreas and spleen size and echo-
genicity. Normal size and sonographic appearance of the
right and left kidney with normal hypoechoic appearance
of the medullary pyramids. Presence of right pleural effu-
sion and ascites.
Tables of fuzzy conversion of laboratory and instru-
mental tests. Tables 7-17 provide a correspondence func-
tion between the scalar quantities of the laboratory tests,
the instrumental tests measured on the usual clinical
parameters, and the fuzzy values between 0 and 1 (degree
of membership to a fuzzy set).
Defuzzyfication of data. The following values result
from the conversion in defuzzyfied values of the results
of laboratory and instrumental tests (measured on the
usual clinical parameters) found in our patient, Maria C.
Fuzzy value of AST (34 U/L) (h’) = 0,48
Fuzzy value of ALT (42 U/L) (i’) = 0,52
Fuzzy value of Total bilirubin (1,11 mg/dl) (j’) = 0,48
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218
Table 7. Fuzzyfication of AST (h’).
Fuzzy measure
1
0,95
0.90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
AST
U/L (h’) 37 70 100 200 400 600 800 1000
Table 8. Fuzzyfication of ALT (i’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
ALT
U/L (i’) 37 70 100 200 400 600 800 >1000
Table 9. Fuzzyfication of Total bilirubin (j’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Total bilirub-
bin mg/dl (j’) 1,5 35 10 15 20 25 >30
Table 10. Fuzzyfication of Gamma GT (k’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Gamma
GT U/L (k’) 60 80 100 120 140 160 180 >200
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219
Table 11. Fuzzyfication of Fosfatase Alkaline (l’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Fosfatase
Alkaline
U/L (l’)
105 125 145 165 185 205 225 >225
Table 12. Fuzzyfication of Platelets (m’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Platelets
n/mm3(m’) 150000 100000 50000 10000
Table 13. Fuzzyfication of CK Mb Massa (e’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
CK Mb Massa
(e’) 2,5 3,5 4,5 5,5
Table 14. Fuzzyfication of Troponina I (f’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Troponin I
ng/ml
(f’)
0,08 0,1 0,2 0,3
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220
Table 15. Fuzzyfication of Supraventricular Arrhythmias ECG (a’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Supraventricular
Arrhythmias
ECG
(a’)
Atrial Fibril-
lation Moder-
ate HR
Atrial Fibril-
lation Slow
HR
Atrial
Fibrillation
Fast HR
Atrial
Flutter
Slow
Atrial
Flutter
Fast
Table 16. Fuzzyfication of Pulmonary Congestion (d’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Pulmonary
Congestion
signs (d’)
Basal Sporadic
Rales Basal RalesMedium-Basal
Rales Pulmonary Oedema
Table 17. Fuzzyfication of Pleural Effusion RX (c’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Pleural
Effusion RX
(c’)
Opacity of
costophrenic angles
Basal pleural
effusion
Medium-Basal
pleural effusion
Apical
pleural
effusion
Fuzzy value of Gamma GT (90 U/L) (k’) = 0,60
Fuzzy value of Fosfatase alkaline (91 U/L) (l’) = 0,40
Fuzzy value of Platelets (300000 n/mm3) (m’) = 0,40
Fuzzy value of CK Mb-Massa (1,96 ng/ml) (e’) = 0,40
Fuzzy value of Troponin I (1, 96 ng/ml) (f’) = 0,45
Fuzzy value of Supraventricular Arrhytmias ECG
(atrial flutter) (a’) = 0,90
Openly accessible at
Fuzzy value of Pulmonary congestion signs (d’) = 0,80
Fuzzy value of Pleural effusion RX (c’) = 0,60
3.3. Refused Hypotheses
On the basis of the first round of tests it is possible to
eliminate some hypotheses among those considered in
Subsection 3.2. The values of white blood cells, PCR, the
thorax RX and the absence of fever permit to exclude the
suspect of Pneumonia (w). The values of Creatinine,
Urinalysis, 24h Proteinuria and Serum albumin permit to
exclude the suspect of Nephrosic Syndrome (u). Thus it is
possible to eliminate the following hypotheses, because
the suspect of Congestive Heart Failure (y) shall be con-
sidered in other inferences:
3) If 0,95 a and 0,75 f then 0,75 w or 0,75 y
3’) If 0,95 a and 0,75 f then 0,75 w and 0,75 y
4) If 0,70 b and 0,75 e and 0,75 f then 0,70 u
3.4. Congestive Heart Failure Hypothesis
Now it is possible to add the laboratory and instrumental
tests data in the inference (2) of the four remaining di-
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221
agnostic hypotheses of Subsection 3.2.
2) If 0,95 a and 0,70 b and 0,75 c and 0,75 d and 0,75 e
and 0,75 f then 0,70 y
Anamnesis, symptoms and signs permit to suspect a
moderate Congestive Heart Failure. Now it is possible to
consider the following data to reach a more precise di-
agnosis:
Fuzzy value of Dyspnoea = 0,95 (a); Fuzzy value of
Supraventricular Arrhythmias ECG = 0,90 (atrial flutter)
(a’); Fuzzy value of Oedema = 0,70 (b); Fuzzy value of
Pulmonary Congestion signs = 0,80 (d’). These findings
indicate with sufficient certainty the presence of Con-
gestive Heart Failure. I shall consider, as specifically
suggestive of the seriousness of Congestive Heart Failure
of our patient, the fuzzy values of Dyspnoea (0,95), of
Supraventricular Arrhythmias ECG (Atrial flutter) (0,90)
and of Pulmonary Congestion signs (0,80), so the fol-
lowing inference will be valid:
2’) If 0,95 a and 0,90 a’ and 0,80 d’ then 0,80 y
This part of diagnostic route which regards the Con-
gestive Heart Failure is completed: the patient suffers
from a moderate-severe (0,80) Congestive Heart Failure
(y).
3.5. Supraventricular Arrhythmias
Worsening or Myocardial Ischemia
Hypothesis
Now it is possible to add the laboratory and instrumental
data in the inference (5) of the diagnostic hypotheses
considered in Subsection 3.2.
5) If 0,95 a and 0,75 c then 0,75 z or 0,75 x or 0,75 y
Since it is well known that our patient suffered from
Congestive Heart Failure, it is possible to eliminate, in
this inference, the disjunction which regards the Conges-
tive Heart Failure (y). Symptoms and signs permit to
suspect a moderate Supraventricular Arrhythmias Wors-
ening (z) or a moderate Myocardial Ischemia (x). Now it
is possible to consider the following data to reach a more
precise diagnosis:
Fuzzy value of CK Mb-Massa (1, 96 ng/ml) = 0,40 (e’);
Fuzzy value of Troponin I (1, 96 ng/ml) = 0,45 (f’);
Fuzzy value of Supraventricular Arrhythmias ECG = 0,90
(a’).
5.1) If 0,40 e’ and 0,45 f’ then 0,40 x
5.2) If 0,95 a and 0,90 a’ then 0,90 z
The inference 5.1 eliminates the hypotheses of Myo-
cardial Ischemia, but the presence of a strong Dyspnoea
(fuzzy value of 0,95) and of Supraventricular Arrhyth-
mias in ECG with fuzzy value of 0,90 (Atrial Flutter),
indicates in 5.2 an acute Supraventricular Arrhythmias
Worsening (z).
3.6. Supraventricular Arrhythmias
Worsening, Myocardial Ischemia and
Congestive Heart Failure Hypothesis
Clinical signs and symptoms suggest the presence of
Supraventricular Arrhythmias Worsening, Myocardial
Ischemia and Congestive Heart Failure, according with
the following inference:
5’) If 0,95 a and 0,75 c then 0,75 z and 0,75 x and 0,75 y
The former analysis confirms the presence of a strong
Supraventricular Arrhythmias Worsening (0,90) and of an
heavy Congestive Heart Failure (0,80 y). The 5.1 infer-
ence excludes the presence of Myocardial Ischemia (0,40
x). Thus it is possible to conclude that the inference 5’ was
partial right, but the 5.2 and the 2’ inferences, supported
by laboratory and instrumental tests, furnish a more pre-
cise description of the seriousness of Supraventricular
Arrhythmias Worsening and of Congestive Heart Failure.
3.7. Cirrhosis Hypothesis
According with inference (6) symptoms and signs permit
to suspect a moderate Cirrhosis (0, 70 t):
6) If 0,70 b and 0,75 d and 0,75 e and 0,75 f then 0,70 t
Now it is possible to add the laboratory and instru-
mental findings to obtain a more precise diagnosis:
Fuzzy value of AST (34 U/L) = 0,48 (h’); Fuzzy value
of ALT (42 U/L) = 0,52 (i’); Fuzzy value of Total
bilirubin (1,11 mg/dl) = 0,48 (j’); Fuzzy value of Gamma
GT (90 U/L) = 0,60 (k’); Fuzzy value of Fosfatase alka-
line (91 U/L) = 0,40 (l’ ); Fuzzy value of Platelets (300000
n/mm3) = 0,40 (m’); Fuzzy value of Ascites = 0,75 (e).
6’) If 0,48 h’ and 0,52 i’ and 0,48 j’ and 0,60 k’ and 0,40
l’ and 0,40 m’and 0,75 e then 0,40 t
The fuzzy values of Platelets (0,40), AST (0,48), ALT
(0,52), Total Bilirubin (0,48) and of Fosfatase alkaline
(0,40) permit to exclude the hypothesis of Cirrhosis, but
fuzzy values of Gamma GT (0,60), of Ascites (0,75) and
of Epatomegaly (0,75) permit to hypothesize in our pa-
tient a mild Congestive Hepatopathy (z’ ). Then the fol-
lowing inference will be valid:
6’’) If 0,60 k’ and 0,75 e and 0,75 d then 0,60 z’
Second turn of tests
The hypothesis of a mild Congestive Hepatopathy
needs to be confirmed with an echocardiography.
Cardiac Ultrasonography Findings: Normal size of the
aorta. Calcifications of the aortic and mitralic valves.
Normal left ventriclular myocardium. Sistolic function
reduced at rest. Ventricular Ejection fraction: 42 %. Right
sections enlargement. Paradox movement of the inter-
ventricular septum. Colour doppler imaging: light mitral
stenosis, moderate mitral regurgitation, moderate-severe
Tricuspidal regurgitation (PAPs 42 mm/Hg). ICV (infe-
rior cava vein) diameter greater than 2.3 cm with poor
respiratory variation.
Tables of fuzzy conversion of laboratory and instru-
mental findings (II turn). Tables 18-21 provide a corre-
spondence function between the results of the instru-
mental tests, measured on the usual clinical parameters,
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222
Table 18. Fuzzyfication of Ejection fraction (k’’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Ejection fraction
%
(k’’)
54 41 27 18
Table 19. Fuzzyfication of Right sections enlargement (l’’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Right
sections
enlargement
(l’’)
Light Moderate
Severe (with
movement paradox of
the septum)
Table 20. Fuzzyfication of Tricuspidal regurgitation (n’’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Tricuspidal
regurgitation
(n’’)
Light Moderate Severe
Table 21. Fuzzyfication of Enlargement of inferior cava vein (o’’).
Fuzzy measure
1
0,95
0,90
0,85
0,80
0,75
0,70
0,65
0,60
0,55
0,50
Enlargement
of inferior
cava vein
(o’’)
Enlargement with
normal respiratory
modulation
Enlargement with
poor respiratory
modulation
Enlargement
without respiratory
modulation
Openly accessible at
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223
and the fuzzy values between 0 and 1 (degree of mem-
bership to a fuzzy set).
Defuzzification of data. The following values result
from the conversion in defuzzyfied values of the results of
instrumental tests (measured on the usual clinical pa-
rameters) found in our patient, Maria C.
Fuzzy value of Ejection fraction 42 % (k’’) = 0,60
Fuzzy value of Right sections enlargement (l’’) = 0,90
Fuzzy value of moderate-severe Tricuspidal regurgita-
tion (n’’) = 0,80
Fuzzy value of Enlarged inferior cava vein with poor
respiratory modulation (o’’) = 0,80
3.8. Congestive Hepatopathy Hypothesis
Fuzzy values of Gamma GT = 0,60 (k’), of Ascites = 0,75
(e) and of Epatomegaly = 0,75 (d) permit to hypothesize
in our patient a Congestive Hepatopathy (z’). Then the
following inference will be valid:
6’’) If 0,60 k’ and 0,75 e and 0,75 d then 0,60 z’
Now it is possible to consider the following data to
obtain a more precise diagnosis:
Fuzzy values of Right Sections enlargement = 0,90 (l’’),
moderate-severe Tricuspidal regurgitation = 0,80 (n’’),
Enlarged inferior cava vein with poor respiratory modu-
lation = 0,80 (o’’). Thus the following inference will be
valid:
6’’’) If 0,90 l’’ and 0,80 n’’ and 0,80 o’’ then 0,80 z’
The fuzzy values of Right Sections enlargement = 0,90
(l’’), of moderate-severe Tricuspidal regurgitation = 0,80
(n’’) and of Enlarged inferior cava vein with poor respi-
ratory modulation = 0,80 (o’’) permit the diagnosis of a
moderate-severe (0,80) Congestive Hepatopathy (z’).
3.9. Other Findings
The echocardiographic findings permit to confirm the
Congestive Heart Failure with Congestive Hepatopathy
and to establish a plausible pathophysiologic link between
these two diseases. Fuzzy value of EF 42 % = 0,60 (k’’)
confirms our diagnosis of Congestive Heart Failure,
moreover all echocardiographic findings permit to affirm
that the Congestive Heart Failure of our patient is a Global
Congestive Heart Failure.
Now it is possible to consider the following data to
obtain a more precise diagnosis:
Fuzzy value of Right sections enlargement = 0,90 (l’’)
and Enlarged inferior cava vein with poor respiratory
modulation = 0,80 (o’’). The following inference will be
valid:
7) If 0,90 l’’ and 0,80 o’’ then 0,80 y C 0,80 z’
Where C indicates the causal relation between Con-
gestive Heart Failure (y) and Congestive Hepatopathy
(z’).
Final diagnosis
Maria C., affected by diabetes mellitus, Hypertension
and Atrial fibrillation, was admitted to our unit because of
dyspnoea, fatigue, leg oedema and abdominal enlargement.
She suffers from the following diseases:
1) A moderate-severe (0,80) Global Congestive Heart
Failure;
2) A severe (0,90) Worsening of the Supraventricular
Arrhythmia (Atrial Flutter);
3) A moderate-severe (0,80) Congestive Hepatopathy;
4) A causal relation between Congestive Heart Failure
and Congestive Hepatopathy.
A moderate-severe (0,80) Congestive Heart Failure
(CHF) is a condition in which the heart can no longer
pump enough blood to the rest of the body. It is almost
always a chronic, long-term condition, although it can
sometimes develop suddenly. In our case, probably, the
decompensed CHF (0,80) was caused by the onset
worsening of the Supraventricular Arrhythmia (0,90).
With heart failure, many organs don’t receive enough
oxygen and nutrients, which damages them and reduces
their ability to function properly. Decompensed right
ventricular or global Heart Failure (0,80) causes trans-
mission of elevated central venous pressures directly to
the liver; venous congestion impedes efficient drainage of
sinusoidal blood flow and sinusoidal stasis results in ac-
cumulation of deoxygenated blood and, ultimately, in a
moderate-severe (0,80) Congestive Hepatopathy and
cardiac cirrhosis. As the heart’s pumping action is lost (as
in our case), blood may go back up into other areas of the
body, including: the liver (0,80 Congestive Hepatopathy),
the gastrointestinal tract (0,75 Ascites) the extremities
(0,70 leg oedema) and the lungs (0,95 Dyspnoea, 0,60
Pleural Effusion and 0,80 Pulmonary Congestion Signs).
4. CONCLUSIONS
Is fuzzy logic an improvement of probabilistic logic for
clinical diagnosis? In what sense is it an improvement?
The success of fuzzy diagnosis in its application to a real
clinical case shows that fuzzy diagnosis is an improve-
ment of probabilistic logic. The accuracy of final diagno-
sis, with pathophysiological explanation within the de-
scription of the causes, of sings and of symptoms, is a
proof that fuzzy logic describes the pathological phe-
nomena with a precision that probabilistic diagnosis
cannot reach. Nevertheless, fuzzy diagnosis seems to be
impossible without the contribution of probabilistic rea-
soning. Indeed the physician, in the differential diagnosis,
chooses the hypotheses to follow on the basis of prob-
ability. The differential diagnoses of Subsection 3.1 ob-
tain their precision with the quantitative inferences of
fuzzy logic, but the preference of some diseases more than
others, is probabilistic. Finally, it is now possible to an-
swer to the question on the relation between probabilistic
and fuzzy logic in the diagnostic algorithm. At the be-
ginning of diagnosis, on the basis of signs, symptoms and
anamnesis, a probabilistic reasoning must lead the re-
search, while, in the progress of diagnosis, the more and
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224
more rich plenty of data requires the use of fuzzy infer-
ences. In this way it is possible to see that some mean-
ingful fuzzy values (laboratory and instrumental data) are
able to address the probabilistic choice of the diseases to
consider. At the end of diagnostic process the fuzzy in-
ferences can identify the nature and the seriousness of
diseases found. Thus, fuzzy logic is not a substitute of
probabilistic logic, but its natural complement.
REFERENCES
[1] Finetti, B. de (1989) La logica dell’incerto, A Cura Di M.
Mondadori, Il Saggiatore, Milano.
[2] Kosko, B. (1993) Fuzzy thinking: the new science of fuzzy
logic, Hyperion, New York.
[3] Sugeno, M. (1977) Fuzzy measures and fuzzy integrals: A
survey, in fuzzy automata and decision processes, M.M.
Gupta et al., North Holland, New York, 89-102.
[4] Dubois, D. and Prade, H. (1980) Fuzzy sets and systems:
Theory and applications, Academic Press.
[5] Yager, R. (1980) On a general class of fuzzy connectives,
Fuzzy Sets and Systems, 4, 235-242.
[6] Hajek, P. (2006) Fuzzy logic, Voice of Stanford Encyclo-
Pedia of Philosophy.