A Mathematical Approach
to Medical Diagnosis
Application
Homer R.
to
Congenital
Heart Disease
Warner, M.D., Ph.D., Alan F. Toronto, M.D., L. George Veasey, M.D.,
and Robert Stephenson, Ph.D., Salt Lake City
DIAGNOSIS of disease from clinical data
is considered by the medical profession as a
art which may be mastered only after years
of careful study and extensive personal experience.
THE
subtle
Although rapid advances are being made in the
development of new and improved methods for
acquiring objective information from a patient
concerning his illness, similar progress has not been
made in analyzing and improving the logical proc¬
ess by which a diagnosis is deduced from this in¬
formation. The present study was undertaken to
find an explicit mathematical expression for this
logical process, with the hope that such an expres¬
sion might (1) improve the accuracy of diagnosis
in certain fields of medicine, (2) lead to a more
scientific approach to the teaching of medical diag¬
nosis, and (3) provide a means, with the help
of an electronic computer, for relieving the physi¬
cian of the task of storing and recalling for practical
in diagnosis an ever-increasing mass of statis¬
tical data. The derivation of such an equation is
herein presented and its usefulness illustrated in
its application to the diagnosis of congenital heart
disease from clinical data.
An equation of conditional probability
is derived to express the logical process
used by a clinician in making a diagnosis
from clinical data. Solutions of this equation take the form of a differential diagnosis. The probability that each disease
represents the correct diagnosis in any
particular patient may be calculated.
Sufficient statistical data regarding the
incidence of clinical signs, symptoms, and
electrocardiographic findings in patients
with congenital heart disease have been
assembled to allow application of this
approach to differential diagnosis in this
field. This approach provides a means by
which electronic computing equipment
may be used to advantage in clinical
medicine.
use
Theory
That the logical process involved in medical
as a problem in con¬
diagnosis could be expressed
ditional probability ' was suggested by Ledley and
Lusted.2 The problem consists of estimating the
likelihood or probability of event yi occurring in
the presence of another event, x. In this paper the
event y, is one disease among a series of diseases
Read before the Scientific Sessions of the American Heart Association,
St. Louis, Oct. 24, 1960.
From the Cardiovascular Laboratory of the Latter-day Saints Hospital and the departments of physiology and electrical engineering of
the University of Utah. Dr. Warner is an Established Investigator of
the American Heart Association.
yk, assumed to be mutually exclusive,
yi, y2,
and the event x is a set of clinical findings Xi, x2,
Xj, which will here be called symptoms even
though physical signs and electrocardiographic
findings are included. The probabihty of yi is de¬
.
.
.
.
.
.
fined by
N
Equation
where N
occur
in
yi
a
1:
P
^j
(yi' y2'
yi
•
•
•
yk)
is the number of times disease yi would
large random sample of
patients with diseases yi, y2,
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=
.
.
.
yk-
N,
(y%, y2,
yk)
s
.
•.
(P ) is simply
yi
the incidence of disease yi in this subpopulation
consisting only of people having one of these dis¬
eases. The probability of symptoms x occurring in
a patient with disease yi is given by
of individual symptoms let us consider the case of
two symptoms, xa and xb. It might be argued that
for xa to be truly independent of xb, the probability
of xa must not be influenced by the presence of xtJ;
that is
N
Equation 2:
xy
=_1
N
P
x|y
1
where N
yi also
patients with disease
is the number of
xy
having symptoms
Equation 8:
y.
*
Dividing the numer¬
right-hand term by
x.
denominator of the
N,
the size of the population
1 *
ator and
(yi> y2'
•
•
results in
\
V
„
P
Equation
xy
—_L
P
P
xiy,
3:
as
where Px is the
.
P
=
x
Equation
6:
P
=
expressed
In order to
=
x|yi
clarify
the
P
P
.
..
P
xjly!
...
yi xi!yi
P
(x
y
,x
112
,...x.)>
P
xjlyi
...P
VP P P
L
*k
v*k
x2,yk
xjlyk
all k
•
possible
'
xj
to
'
calculate the
) that each disease yi,
•
•
•
.
.
.
Xllyk' X2lyk'
.
xi,yi x2ly!
P P
for the
;|yi
P
Equa¬
tion under consideration and the incidence of each
of the patients' symptoms in each of these diseases
P
(P
etc.). These statistical data, required
in disease \\ must be
7:
may rewrite
yi
P P
y, x|y
VP P
in
Equation
we
expanded form as
V2,
yk exists in the presence of symptoms Xi,
x2,
Xj from statistical information concerning
the incidence of each disease P in the popula-
kyk XIYk
.
an
Equation 7,
of
use
yi:xi' v
k
the product of
the probabilities of the individual symptoms which
make up the set occurring in disease yi. This is
occurring
.
probability (P
.
which is an expression of Bayes' rule for the prob¬
ability of causes. Now, in fact, any symptom com¬
plex (x) may be represented as a series of inde¬
Xj. Thus, the
pendent symptoms \\, x2,
conditional probability (P ) of symptom complex
.
=
With this expression it is
hVPykx|yk
«Ilk
and 5 results in
P
y,ix
=
,
Equation 9:
x
all k
x
.
xyi
P
*
Combining equations 3, 4,
a
-^—
probability
5:
=
possible.
of symptoms x occur¬
in
with
of those diseases. If
one
any
patient
ring
these diseases are considered mutually exclusive,
it follows that
Equation
.
X
the non-uniform distribution of xb in diseases yi,
yk and not due to a direct causal relation¬
ship between xa and xb. In the selection of symp¬
toms to be used in a particular field care must be
taken to adhere to this criterion as closely as
With
=
P
can be true only if xb is uniformly
throughout the population. This means
P
P
1 and that x„ is
xblyi
xbly2
xb,yk
of no diagnostic value. For this reason Equation 8
must be an inequality. In spite of this, these symp¬
toms for present purposes are truly independent
of each other as long as this inequality is due only
distributed
that Pv
tion 6 in
P
P
y,ix
1
|X
ab
However, this
y2,
By the same reasoning the probability of disease yi
occurring in the presence of symptom complex x
Equation 4:
=
X
to
yi
may be written
P
.
meaning of independence
right-hand term of Equation 9, may be
compiled and stored in a form (punched cards,
punched paper tape, or magnetic tape) that will
make them readily available for an electronic digi¬
tal computer to extract the pertinent numbers (de¬
pending upon the symptoms presented by the
patient) for carrying out the calculation called for
by the equation.
Because Equation 9 uses only probabilities in¬
volving those symptoms actually present in the
patient under consideration, the absence of a par¬
ticular symptom does not influence the diagnosis.
Thus, in order to make use of the fact that the ab¬
sence of a symptom may have a bearing on the
probability of a given disease being present, Equa¬
tion 9 is modified to give
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Equation
Congenital Heart Disease
Because the accuracy of a diagnosis of congenital
Application
10:
p p
yil(VY
"
V
(1
X8lyi
Xjlyi
V
(1 "P. ..)... P
X8lyk
yk Xllyk
Xj'yk
«11 k
yl
~
x
iy
il
where the bar above x8 in the left-hand term
indicates that symptom x8 is not present in
the patient under consideration. Because P
X8lyi
heart disease from clinical symptoms may be
checked by cardiac catheterization and/or findings
surgery, and because relatively objective clinical
may be easily obtained, this field was
chosen for a pilot study. The list of symptoms and
at
findings
ing symbols,
—
Symptoms
Code'
BW
{S
BW
BW
BW
BW
BW
BW
BW
I X4
age 1 mo. to 1 yr.
age 1 to 20 yr.
=
I X«
=
Ut
=
BW
BW
BW
BW
BW
BW
BW
B
t
diastolic
thrill loudest in left 2nd
diastolic
X25 =
murmur loudest in left 2nd interspace
¡continuóos murmur loudest in left 2nd
2nd interspace
murmur loudest in
¡diastolic murmur loudest in
2nd interspace
chest
systolic murmur heard best over
continuous murmur heard best over posterior chest
¡systolic
right
right
BW
BW
BW
BW
BW
ECG axis less than 0°
¡R wave greater than 1.2 mv in lead Vi
X-A7- R' or qR pattern in lead Vi
¡R wave greater than 2.0 mv in lead Vb
¡T wave In lead Va inverted (no digitalis)
diastolic murmur loudest at apex
t' \X41 early
late diastolic murmur loudest at apex
holo-systolic murmur loudest in left 4th interspace
I X48: mid-systolic murmur loudest in left 4th interspace
holo-diastolic murmur loudest in left 4th interspace
I X45 early diastolic murmur loudest in left 4th interspace
1X85=
M
X3«:
•|X40:
w
X4 7=
+
<
X4S=
w
XiO-
BW
X50=
mid-systolic murmur with thrill loudest in 2nd
interspace
¡holo-systolic murmur with thrill loudest in 2nd
interspace
¡mid-systolic murmur without thrill loudest in
left interspace
¡holo-systolic murmur without thrill loudest in
left interspace
¡murmur louder than gr 3/6
left
left
2nd
2nd
tlndicates presence of mutually exclusive symptoms within brackets
as special cases (see text).
*B: symptom used on brown check-off sheet; W: symptom used on
which must be handled
white check-off sheet.
Diseases
yi =normal
septal defect without pulmonary stenosis or pulmonary
hypertension*
defect with pulmonary stenosis
¡=atrial
septal
ys
y4 ¡=atrial septal defect with pulmonary hypertension*
endocardial
cushion defect (A-V commune)
—complete
y5
yo =partial anomalous pulmonary venous connections (without a trial
septal defect)
y7 =total anomalous pulmonary venous connections (supradiaphragmatic)
ys =¡tricuspid atresia without transposition
=Ebstein's
anomaly of tricuspid valve
y9
yio=ventricular septal defect with valvular pulmonary stenosis
11=ventricular
septal defect with infundibular stenosis
y
yi2=pulmonary stenosis, valvular (with or without probe-patent fora¬
men ovale)
yis=pulinonary stenosis, infundibular (with or without probe-patent
foramen ovale)
yii=pulmonary atresia
y i.-»—pulmonary artery stenosis (peripheral)
yio=pulmonary hypertension,* isolated
yi7=aortic-pulmonary window
yis=patent ductus arteriosus without pulmonary hypertension*
yi9=pulmonary arteriovenous fistula
y20=mitral stenosis
y2i=primary myocardial disease
y22=anomalous origin of left coronary artery
y23=aortic valvular stenosis
y24=subaortic stenosis
y25=coarctation of aorta
yso—truncus arteriosus
y27—transposed great vessels
y28=corrected transposition
ya =atrial
y20=absent
aortic arch
yso=ventricular septal defect without pulmonary hypertension*
y3i=ventrieular septal defect with pulmonary hypertension*
y8s=patertt ductus arteriosus with pulmonary hypertension*
y8a=tricuspid atresia with transposition
^Pulmonary hypertension is defined
systemic arterial pressure.
as
pulmonary artery
pressure
^
interspace
posterior
I X2'
J X2: accentuated 2nd heart sound in left 2nd interspace
1X2!
^X2ft= diminished 2nd heart sound in left 2nd interspace
xso= ¡right ventricular hyperactivity by palpation
X3i: ¡forceful apical thrust
:pulsatile liver
:absent or diminished femoral pulsation
I X341 ECG axis more than 110°
\X2'
BW
BW
w
systolic
systolic
space
¡systolic murmur without
interspace
X24 =
BW
BW
w
syncope
,X23 =
BW
W
X14:
i
BW
BW
W
chest
—
BW
W
W
W
W
easy
Xii:
X12:
X13:
systolic
f
Lxifl—
B
BW
BW
fatigue
orthopnea
pain
repeated respiratory infections
Xio:
murmur loudest at apex
murmur loudest at apex
murmur loudest in left 4th interspace
X17
< X18= diastolic murmur loudest in left 4th interspace
continuous murmur loudest in left 4th interspace
murmur with thrill loudest in left 2nd inter¬
B
BW
BW
BW
BW
cyanosis, intermittent
cyanosis, differential
squatting
dyspnea
X15:
XlÖ =
B
B
>20 yrs.
cyanosis, mild
cyanosis, severe (with clubbing)
=
JX5
concerning the incidence of each
Table 2.—List of Diseases Included
in Differential Diagnosis
S 1
Table 1.—List of Symptoms to Be
Evaluated by Physician
is
tical information
the
probability of symptom x8 not occurring
in a patient with disease yi. Thus, the absence of a
symptom is treated as a discrete event when Equa¬
tion 10 is used, and the probability that a symptom
is absent can be obtained directly from the proba¬
bility figure for the presence of the symptom.
resent
study, with their correspond¬
displayed in Tables 1 and 2. Statis¬
diseases used in this
represents the probability of symptom x8 occurring
complement (1 P ) must rep-
in disease yi, its
to
of these symptoms in each of these diseases is
presented in Table 3. The numbers in the first
column represent the incidence of each disease (Py)
in the subpopulation made up of patients referred
to this laboratory and suspected of having congen¬
ital heart disease. The rest of Table 3 is a matrix
with symptoms along the horizontal axis and dis¬
eases listed vertically. For instance, the number
0.02 at the intercept of symptom x4 and disease V2
the probability or incidence of
represents P
,
mild cyanosis occurring in a patient with atrial
septal defect without pulmonary hypertension. (In
this study pulmonary hypertension is arbitrarily
defined as pulmonary artery pressure equal to or
greater than aortic pressure.)
Several things about this symptom-disease matrix
require explanation. Listed among the diseases is a
category called normal. The incidence of normal
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(P )
study
in this
referred
is 0.10 since 10% of the
Since the patient can belong in only one of the
three age groups, these three "symptoms" cannot
be considered independent of one another. Thus
if the patient's age is between 1 and 20 years, P
patients
laboratory for heart catheterization
by physiologic studies, which include
dye-dilution curves. The figures in the incidence
column and symptoms xt, x-2, and x:i (age) may vary
from one population to the next, while the other
data, which express the probability of each symp¬
are
to
this
normal
is used in
probability
used in this case.
Furthermore it is important that care be taken
not to include in the list of symptoms any two
symptoms which invariably occur together, since
this strongly suggests interdependence and a
causal relationship between them. For instance,
disease, should remain constant from
one laboratory to the next. Each of these probabil¬
ities in the matrix was determined by us from (1)
a careful review of published data of others, par¬
tom in each
ticularly
Keith and co-workers,3
(2)
10 but the complement of the
for the other two age groups is not
Equation
review of data
Table 3
—
Diseases
yi
.
Incidence
xi
xa
x:i
xi
x-,
xo
X7
xs
x¡>
xio
xu
X12
0.100
01
10
40
60
0(1
01
(Kl
05
02
20
25
10
01
05
00
01
35
03
01
01
01
01
10
50
10
70
30
50
01
02
20
30
50
00
7(1
05
05
01
80
01
40
00
50
00
00
01
01
15
(Il
(Kl
10
(Il
00
00
00
22
30
40
80
80
75
75
00
00
(Il
50
01
00
00
00
00
00
80
01
01
01
01
50
90
y2
y3
.081
.005
311
60
60
yt
.001
10
20
y5
y«
yys
y»
.027
20
10
50
yio
yn
yi2
yia
yi4
yir,
yi«
yi7
yis
yia
y2o
y21
y22
y2a
y24
y25
y2o
y27
y2s
y2o
yao
.006
.001
.018
.001
.054
.063
.045
.013
.014
.001
.013
.001
.072
.002
.(108
.013
.001
.(136
.009
.054
.005
.063
.001
.001
.252
20
50
1(1
40
40
20
20
40
70
48
45
55
55
10
02
45
05
10
15
10
05
01
01
65
10
05
30
22
25
30
44
25
30
1(1
10
01
01
00
01
00
00
00
90
09
05
10
10
01
10
90
05
45
50
01
01
10
45
«0
45
10
40
40
50
00
01
(Il
00
01
01
(Il
01
01
01
45
01
(Il
01
01
30
01
05
01
45
01
01
00
(Il
10
10
(Il
01
(Il
01
01
00
01
01
01
01
00
01
20
10
(Il
30
20
30
01
00
6(1
60
05
01
01
01
01
01
(Il
50
01
20
30
20
20
2(1
70
7(1
10
10
10
50
90
30
60
yai
ys2
.081
.008
15
30
30
yai
.009
40
70
70
30
50
29
29
80
80
70
40
10
30
39
70
00
30
01
15
01
01
(Il
(Il
60
10
40
55
30
01
30
01
05
50
01
01
05
10
01
(Il
00
(10
10
00
80
00
10
00
05
10
50
00
obtained from 1,035 patients referred to this labo¬
ratory for diagnostic catheterization, and (3) esti¬
mates based upon the pathologic physiology of the
defect in the case of rare defects in which adequate
statistics were not available.
Notice that each patient is classified according
to age into one of three categories, 1 month to 1
year, 1 year to 20 years, and over 20 years of age.
The patient's age is treated as a symptom. For in¬
stance, the number 0.70 occurring at the intercept
of x-2 and ylf! indicates that this symptom (age 1 to
20 years) will occur in 70 out of 100 patients with
infundibular pulmonary stenosis coming to this
laboratory. In this way, then, the fact is recognized
that the age of the patient does influence the prob¬
ability that he has any given diagnosis.
01
01
leía
05
X14
xi-,
xie
X17
xis
xid
xao
X21
xzz
03
05
01
70
80
10
10
(Il
02
05
57
05
01
02
02
30
05
15
90
10
10
40
01
40
20
05
05
10
02
20
60
05
05
05
80
10
30
75
20
15
90
90
(¡5
05
05
05
20
15
15
10
10
05
05
20
05
01
01
05
99
05
01
95
01
01
01
10
01
02
02
02
15
02
02
05
25
02
02
02
02
07
02
02
02
(12
02
10
05
05
05
00
02
02
05
05
02
20
40
10
10
20
20
(15
10
05
40
20
01
30
80
20
80
90
70
01
70
10
20
1(1
50
40
50
50
30
30
20
05
(15
01
05
(Il
15
05
20
15
60
30
30
30
30
70
01
10
20
(Il
30
01
05
01
01
10
50
20
(¡0
30
70
20
80
30
90
05
05
20
10
20
20
05
01
2(1
15
20
20
05
15*
20
clubbing
arate
01
01
01
01
01
01
2(1
01
10
01
01
15
30
05
01
05
10
22
20
25
10
01
10
10
(Il
01
15
20
10
30
of the
20
02
15
05
10
65
95
20
20
10
10
25
02
02
02
05
02
02
05
01
05
20
05
65
65
05
01
02
25
(Il
01
05
05
01
05
10
05
05
05
01
00
50
02
80
05
20
15
05
10
10
10
05
01
02
10
10
15
02
20
02
30
20
20
02
02
01
35
20
35
05
10
20
02
10
02
01
10
10
10
(12
01
02
02
02
02
02
02
30
05
05
02
05
02
05
02
01
05
05
01
01
01
(Il
02
10
10
03
10
1(1
01
02
02
05
05
02
02
05
70
02
02
02
05
30
02
50
02
1)2
05
02
02
30
10
05
20
02
95
10
10
10
30
02
02
05
05
05
20
05
02
02
01
20
05
02
02
10
13
10
10
02
02
02
02
02
20
02
05
02
05
02
02
10
20
20
20
25
20
01
05
25
70
70
01
4(1
10
01
(Il
40
02
10
10
05
02
02
02
90
01
10
20
02
02
05
02
02
20
02
35
(Il
01
(Il
02
05
04
01
05
(Il
1(1
(Il
10
05
15
01
02
00
05
02
05
fingers
symptom since it
01
02
10
was
7(1
50
50
10
10
10
70
05
not
occurs
02
02
02
included
in the
05
as a
same
05
25
10
10
sep¬
patients
heart disease who exhibit severe
it is included as a part of the
definition of severe cyanosis. Inclusion of redun¬
dant (interdependent) symptoms would result in an
unreal increase in the probability of those diseases
having a high incidence of these symptoms when
these symptoms are present and a falsely low prob¬
ability when these symptoms are absent.
There are other symptoms in the list which are
mutually exclusive. For instance, the existence of
x,-, excludes by definition x4, x6, and x7. Thus, it
would be an error to consider the absence of X4, x«,
and x7 as additional pieces of information once x-,
is known to be present. On the other hand, the abwith
congenital
cyanosis. Instead,
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of
sence
through
x4
x7 in
particular
a
case
prepared from a check-off list of symptoms on
which the physician, from his examination of the
patient, has marked those symptoms presented by
the patient. (X-ray data are not presented in this
paper but are being evaluated for inclusion in the
symptom list at the present time.)
From this information, the computer then cal¬
culates, with use of Equation 9 or 10, the probabil¬
ity of each of the 33 congenital heart diseases being
(no
important fact and must be recog¬
cyanosis)
nized by using in Equation 10 the complement of
the sum of the probabilities of each of these symp¬
toms occurring in the distase in question, which is
-P
-P
1-Px
-Px
x
x
is
an
.
4
|y1
5
|y
6
1
|y1
7
|y1
Groups of mutually exclusive symptoms are indi¬
cated by brackets in Table 1, and a complete list
of mutually exclusive symptoms, together with
instructions about what data should be used in
solving Equation 10 in any particular case, is
given in Table 4.
Symptom-Disease
present in the patient under consideration. The dis¬
with probability greater than 1% are printed
out at the end of the calculation together with their
respective probabilities. Two symptom lists are
eases
Matrix
Symptoms
X23
X24
X2S
X28
Xn
X28
X39
X30
X31
X:i;
X33
X;U
X35
X36
X37
X.ts
X39
X40
X41
Xli
X43
X44
X4S
05
02
03
01
01
01
01
01
00
01
05
01
15
02
02
02
02
02
04
05
05
85
05
20
70
02
02
02
02
30
05
02
01
03
20
05
95
02
10
85
02
10
02
10
15
02
15
02
02
02
02
02
02
20
02
20
02
20
01
01
01
02
02
02
01
01
01
10
02
90
02
02
01
05
01
10
10
01
35
60
60
10
20
20
01
10
02
02
85
60
10
10
02
02
02
02
02
02
02
02
02
02
45
20
20
60
60
90
02
00
02
02
01
01
01
85
85
10
02
02
10
90
02
02
01
01
01
30
02
40
02
05
02
02
02
01
02
01
01
10
02
02
01
01
01
30
02
02
02
70
03
05
01
01
01
05
05
20
20
10
10
01
01
50
05
50
20
01
01
20
20
20
10
30
20
20
10
50
02
01
15
15
01
01
10
05
02
02
02
25
30
50
01
02
05
25
10
02
02
90
02
75
01
80
01
25
01
02
60
01
01
01
01
05
01
70
40
15
05
02
01
15
15
02
01
20
01
20
05
02
01
05
70
02
70
01
05
01
02
70
85
85
02
01
01
01
85
70
01
01
01
01
01
01
01
15
00
01
01
01
05
05
01
40
50
40
03
01
01
01
01
02
15
60
30
10
01
01
01
01
01
01
20
02
20
10
01
01
01
01
01
15
10
01
01
01
01
01
20
20
10
40
20
20
90
30
90
10
01
90
05
05
05
05
10
02
02
02
02
02
02
02
02
02
20
02
02
20
02
02
85
02
02
05
02
02
01
02
10
01
95
01
02
02
01
05
95
15
05
10
02
05
05
05
01
01
01
01
05
02
02
01
01
01
01
10
80
05
05
02
02
02
02
01
02'
05
01
02
02
02
02
02
05
02
01
01
01
02
01
01
02
02
70
01
01
02
01
85
10
10
10
10
01
10
95
30
Use of
02
10
80
01
01
01
10
01
01
20
20
01
01
01
01
01
01
01
40
40
01
05
01
01
30
10
02
02
05
05
02
01
10
01
02
01
95
02
01
95
02
01
01
01
01
01
01
01
95
95
95
10
95
02
01
02
15
02
02
60
05
10
02
10
10
02
02
50
05
02
05
05
50
05
05
06
02
02
10
05
02
02
40
05
05
02
40
20
02
02
90
90
10
02
20
02
02
20
02
01
01
10
10
10
01
05
15
15
05
10
02
02
02
70
02
02
02
01
02
01
02
02
01
01
01
40
30
05
01
01
01
01
01
20
02
50
05
40
10
01
10
10
30
20
10
02
02
10
40
05
10
05
01
01
30
02
30
05
02
10
30
01
05
20
01
05
05
30
02
90
02
02
10
02
01
01
05
05
99
01
02
01
10
05
05
30
40
02
02
02
02
10
10
85
85
05
02
02
02
20
40
30
20
10
10
70
05
10
05
05
90
80
05
01
30
01
70
01
70
01
02
Computer
Because of the large number of calculations re¬
quired to make each diagnosis in the example
(congenital heart disease) used in this paper, it is
necessary to use a digital computer if Equation 10
is to be solved in a practical way. This equation
can be solved by almost any general-purpose elec¬
tronic digital computer which has the capability of
"floating decimal point" operation. The incidence
of each symptom in each disease shown in the
matrix is transferred to punch cards. These disease
cards, together with cards which contain the pro¬
gram telling the computer what operations to per¬
form, are transferred into the computer memory by
a card-reading machine. Another punched card is
75
75
02
10
05
10
05
05
15
15
02
70
15
15
40
20
20
10
10
15
10
04
05
05
10
90
05
10
10
05
05
05
02
02
02
02
20
01
02
10
01
02
02
02
02
02
01
02
02
02
02
05
40
30
30
30
92
30
10
30
15
45
05
05
10
10
10
X46 Xli
00
05
60
05
00
01
01
20
01
20
02
01
20
15
15
02
02
60
05
05
68
68
01
02
01
05
00
15
05
02
05
02
02
02
02
02
05
25
02
02
02
01
02
02
10
30
02
25
02
05
60
01
02
02
01
01
01
01
02
02
02
02
05
02
01
Xlt»
Xr.o
80
90
05
01
10
60
38
01
01
20
70
30
20
05
02
02
20
05
25
05
90
25
25
25
02
25
05
10
20
10
05
01
01
02
02
90
40
20
60
80
70
50
50
80
02
75
10
85
10
30
10
10
70
05
05
01
10
10
90
90
02
02
01
01
20
20
10
10
20
40
30
30
30
10
02
02
02
02
05
10
02
02
02
05
05
02
01
01
02
02
02
02
10
03
05
10
10
03
05
10
01
01
10
01
02
01
02
02
02
05
02
05
05
10
10
01
20
30
30
10
30
80
05
10
01
02
40
80
20
60
20
05
02
01
05
10
02
XlH
01
05
05
10
10
10
01
01
05
10
10
30
30
30
30
10
05
20
30
65
40
50
60
20
85
50
20
50
checked off by the clinician after his examination
of each patient. On one list (brown sheet) murmurs
are described only as to timing and location, while
on the other list (white sheet) the time-course of
intensity of the murmurs is included (see Table 1).
Equation 10 is solved with each of these sets of
symptoms and the resulting differential diagnoses
are compared. Although the calculation based on
the white sheet often gave a higher probability to
the correct diagnosis, this was not consistently the
case, particularly in instances in which classifica¬
tion of the time-course of murmur intensity was
difficult even with the help of a phonocardiogram.
The point to be made here is that in applying this
approach to diagnosis a compromise must be
reached between two alternatives: (1) the desir-
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Table 4.—Terms to Be Used in Equation 10 in Cases of
Mutually Exclusive Symptoms
Check
Sheet Symptoms
B/W
XI—X3
B/W
X4—XT
Symptom
Present
X2
X3
PX3
XI
Pxi
X5
PX5
Pxe
PX7
XG
X7
none
B/W
X20—X27
X26
X27
neither
B/W
XS8—X21)
X34—XSB
X30—X37
PX29
X34
X30
X37
X17—X19
none
B
X20—X23
X22
X23
PX23
X20
X21
X20, X22
X21, X22
none
W
Xl», X42—X45
X19
X42
X43
X44
X45
X42, Xl4
X43, Xl4
X42, X45
X43, X45
none
W
X40—X41
W
PXl9
PX42
PX42 PX44
PX43 PX44
PX42 PX45
PX43 PX45
(1—PxisHl—PX42—Px4s)
(1—PX44—PX4B)
X22
Xi«
X48
X49
Xi«, X22
X47, X22
Xl8, X22
X49, X22
X23
none
(1—PX40—PX41)
(1—PXlO—PX47—PX46—PX4»)
(1-PX22)
(1—PX22)
(1—PX22)
(1 —PX22)
PX22
PXlO
PX47
PX48
PX49
PX40
PX22
PX47 PX22
PX4S PX22
PX49 PX22
PX23
(1—PX22XI—PX23)
(1—PXlO —PXlT—PX48—PX40)
of using as much information
(2) the limitations in accuracy
ability
as
possible,
with which
the more detailed information can be observed in
the patient and the necessary statistical data can
be obtained.
Example.—The case shown in Table 5 illustrates
the effect of using both positive and negative in¬
formation in making a diagnosis. The list of symp¬
toms indicates that the patient was over 20 years
of age and complained of easy fatigue and orthopnea; his pulmonary second sound was diminished;
his electrocardiogram exhibited an axis greater than
Disease
Prob-
No.
ability
yn
yio
yie
0.33
.28
x29
.
yia
.11
.14
X34
.
yi3
.04
X30
.
.
Diagnosis With
Equation 10
Disease
No.
y12
yia
yio
yn
yio
Prob-
ability
0.62
Diagnosis With
Equation 10
and Without Xn
Disease
No.
.21
yi2
yi3
.07
yio
Prob-
ability
0.73
.24
.02
.04
.03
tion 10 is used, the probability of isolated pulmo¬
nary stenosis became 0.83 while the probability of
tetralogy
of Fallot was only 0.11. This patient was
to have y]2 both by physiologic studies
and at surgery.
Also illustrated in this table is the way in which
this approach may be used to evaluate the contri¬
bution made toward a diagnosis by any given
symptom. Here the calculation has been carried
out with and without the symptom of orthopnea
( xn ). Had this patient not complained of orthopnea
the probability of isolated pulmonary stenosis (yl2
and yi3) would have been 0.97 as compared to 0.83
when orthopnea was considered present. This, of
course, results from the fact that orthopnea rarely
occurs in patients with yi2 or y13. Since the presence
or absence of just one symptom may make a real
difference in the differential diagnosis, as in this
instance, it is apparent that each symptom on the
list must be accurately evaluated in every case if
the correct probabilities are to be calculated. For
this reason, only the most objective symptoms
should be included in the definition of the original
list for any study, even if this must be done at some
sacrifice of detail.
In the case of the present study we are under
the impression from our experience to date that
symptoms 10 through 14 detract from the ac¬
curacy of diagnosis as often as they contribute,
because of the difficulty involved in assessing their
actual presence or absence in many cases plus the
inaccuracy of the available statistical data regard¬
ing the incidence of these symptoms in each of
these diseases. These 5 symptoms might well be
eliminated from the list.
later found
(1—PX44—PX46)
PX43 (1—PX44—PX4B)
PX44 (1—PX42—PX13)
P.X46 (1—PX42—PX13)
PXll
X47
and
(1—Px2o— PXii)(l—PX22)(1—PX23)
PX40
X22—X23
Xn
PX21 PX22
X40
X4U—X49
xio.
PX2Ü PX22
Xli
neither
Symptoms
x 3.
(1-Pxi7)(l—Px18)(l—PX19)
PX17 PX18
PX20 (1—PX22)
PX21 (1—PX22)
PX22 (1—PX20—PX2l)
X17, X18
greater than 1.2 mv in lead Vi;
Diagnosis With
Equation 9
PX36
PX37
PX19
wave
Table 5.—Test Case Illustrating Effect
of Including Negative Information
(1—PX34—Pxsb)
X19
R
by phonocardiogram he had a midsystolic mur¬
without a thrill, which was of equal intensity
in the pulmonary (2nd left interspace) and the pre¬
cordial (4th left interspace) area. Calculation of the
probabilities for each disease with use only of the
positive information (Equation 9) resulted in a
higher probability for tetralogy of Fallot (yl0 and
yn) than for isolated pulmonary stenosis (y12 and
y 13). However, when both positive and negative in¬
formation were taken into account, as when Equa-
PX34
PX3B
Xl8
Xl7
an
mur,
(1—Px2s—PX29)
(1—Pxs«—PX37)
PX17 (1—PX18)
PX18 (1—PX17)
neither
B
(1—Px2e—PX27)
X2B
neither
B/W
(1—PX4—Pxs—Pxo—Px7)
PX2S
X35
Equation 10
PX28
PX27
X28
neither
B/W
Term Used in
Pxi
Px2
xi
110° and
and
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Experience to Date
differential diagnosis obtained
Evaluation of
with
Because the
this approach represents an estimation of probabili¬
ties in which the statistical data of Table 3 are
used, it is impossible from a limited number of
cases to evaluate its accuracy. However, it is ap¬
parent from our experience to date with 36 cases
that the most probable diagnosis estimated with
Equation 10 agrees with the actual diagnosis made
by physiologic studies and observation at surgery
at least as often as does the most probable diagno¬
sis estimated by 3 experienced cardiologists from
the same clinical information. Furthermore, the
differential diagnosis resulting from solution of the
frequently more complete and, in retro¬
spect, often appears more logical to the clinicians
than the differential diagnosis listed by each of
them prior to seeing the equation's prediction.
It must be emphasized that Equation 10 was
derived directly from the definition of conditional
probability. Thus, any evaluation of the accuracy of
the predictions made by this approach should be
considered as testing the adequacy of the matrix
equation
of the diagnosis and what symptoms are unimpor¬
tant?
A solution of Equation 10 for any given set of
symptoms provides an objective, reproducible stand¬
ard against which students may check the accu¬
racy of their own deductions from these same symp¬
toms. The effect of modifying the symptom set in
any desired fashion on the differential
may be readily observed.
approach to the teaching of diagnosis of
congenital heart disease is in current use at this
hospital and has met with enthusiastic acceptance
by medical students.
This
Appendix
is
of statistical data and not of the equation. Given
the correct original data matrix and accurate ob¬
servations of the patient, the calculated probabili¬
ties will be correct. Final refinement of the present
data matrix must await the accumulation of suffi¬
cient data for calculation of new probabilities
(P ). Since the presence or absence of each
xilyk
and follow-up
information almost invariably yields the diagnosis
with certainty, the data for satisfactory recalcula¬
tion of symptom incidence are routinely accumu¬
lating. The computer will be used to recalculate
its own data matrix when the amount of data is
sufficient.
symptom is determined in each
Aids to
case,
Teaching
That an explicit expression of the logic used in
medical diagnosis has potential usefulness as a tool
for teaching diagnosis to medical students and phy¬
sicians seems apparent. The approach here pre¬
sented provides a framework within which any
diagnostic problem may be formulated and criti¬
cally analyzed.
Often the very act of attempting to formulate
the problem in terms required for application of
Equation 10 results in new insight by providing
answers to such questions as:
1. What is the exact definition of each symptom
and each disease?
2. Are certain symptoms interdependent and oth¬
ers mutually exclusive?
3. What symptoms are important determinants
diagnosis
use of Equation 10, consider the
of
a
simple
population consisting of just 2 dis¬
eases (yi and y2) and 3 independent symptoms (xi,
x2, and X3). The relative incidence of these 2 dis¬
eases and the probability of each symptom in each
disease are shown in the matrix below.
To illustrate the
case
Incidence
yi
.
y2
.
0.23
0.77
Xi
xa
x3
0.1
0.8
0.7
0.2
0.6
0.5
If the patient to be diagnosed presents with
symptoms Xi and x3, Equation 10 would be solved
with use of the following numbers to make the
diagnosis:
p
y
t
I(x
, x ,
2*
1
1.
-
x
=
3
_0.23(0.1)(l-0.7)(0.6)_ =0.016
0.23(0.1X1-0.7K0.6)+0.77(0.8X1-0.2K0.5)
and
y2,(VVX3)
=
_0.77(0.8X1-0.2X0-5)__ft q84
0.23(0.1X1-0.7X0.6)+0.77(0.8X1-0.2X0.5)
325
Eighth Ave., Salt Lake City
3
(Dr. Warner).
The computer program may be obtained from
Corporation, 6071 2nd Ave., Detroit 32.
Burroughs
This work was supported by the Burroughs Corporation
and by a grant from the National Institutes of Health.
References
Feller, W., An Introduction to Probability Theory and
Application, New York: John Wiley & Sons, Inc., 1960.
2. Ledley, R. S., and Lusted, L. B., Use of Electronic
Computers to Aid in Medical Diagnosis, Proc Inst Radio
Engineers 47:1970-1977 (Nov.) 1959.
3. Keith, J. D., and others: Heart Disease in Infancy and
Childhood, New York: The Macmillan Co., 1958.
1.
Its
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