Fuzzy Sets and System~ 30 (1989) 121-133
North-Holland
121
A N O R T H O D O N T I C E X P E R T SYSTEM
.Ion SIMS WILLIAMS, Andrew MATTHEWMAN
Engineering Mathematics Department and Information Technology Research Centre, University
of Bdstol, Bdstol BS8 ITR, U.K.
David BROWN
Department of Child Dental Health, University of Bristol, Bristol BS8 ITR, U.K.
Received May 1987
Revised January 1988
Abstract: A Fuzzy Relational Inference Language, FRIL has been used to generate an expert
system to help nolo-specialist dentists with ol'thodontic problems. The use of fuzzy relations,
descriptors and numbers has been found useful in modelling the thought processes of
orthodontic specialists.
Keywords: Expert systems; dentistry; fuzzy logic; fuzzy linguistic quantifiers; FRIL.
L |ntroducdon
In this paper we illustrate how a fuzzy logic programming language, FRIL
(Fuzzy Relational Inference Language), has been used to model some of the
expertise of an orthodontic consultant.
Orthodontics is the branch of dentistry concerned with preventing and
correcting irregularities of the teeth, l'he reduction in the incidence of dental
caries in children in recent years has led to more emphasis being placed on
orthodontic problems. Large numbers of dentists were trained, however, when
orthodontics was a relatively ,eglectod discipline and although orthodontic
courses are now being made available to General Dental Practitioners (GDPs)~
there are long waiting lists for treatment by orthodontic specialists.
These long waiting lists are very undesirable, not least because some problems
become increasingly difficult to treat in older patients.
Furthermore, all waiting lists will contain a proportion of patients for whom an
earlier course of treatment by the patient's own GDP has failed to obtain a
satisfactory result. There will also be some who could, in fact, be competently
treated by a GDP rather than a specialist. Lastly, there will be those who are
either too young for treatment when seen by the specialist or, as has been said,
would have been better treated at an earlier age.
Members of the Department of Child Dental Health and the Information
Technology Research Centre at Bristol University have combined together to
generate an expert system which we hope will help to reduce the number of
patients on waiting lists by providing GDPs with an easily accessible advice
system.
0165-0114189153.50© 1989, Elsevier Science Publishers B.V. (North-Holland)
122
7. Sims Williams et ai.
In each jaw the teeth are arranged in a curve known as the dental arch.
Specifically, irregular teeth may be:
(a) lingually displaced- the tooth tilts towards the tongue from the line of the
dental arch;
(b) labially/buccaHy displaced- the tooth tilts towards the lips/cheeks from the
line of the dental arch;
(c) bodily displaced- the tooth is displaced lingually or labially from the dental
arch without tilting;
(d) mesially inclined-the tooth tilts along the line of the arch towards the
front of the mouth;
(e) distally inclined - the tooth tilts along the line of the arch towards the back
of the mouth;
(f) absent or malformed;
(g) late in erupting.
The objective of orthodontic treatment is to align irregular teeth so that the
appearance and functional efficiency of the dentition are improved.
2. A narrower objective for the f~rst stage of the expert system design
The process of treatment planning is as follows:
(i) Examine the lower arch and decide the treatment required to realign the
teeth.
(ii) Examine the upper arch and design the treatment required to realign these
teeth.
(ii~) Consider how the teeth in both the upper and lower arches will fit together
whc:n both sets of teeth h~ve completed their planned treatment and readjust the
plan for the upper arch if the fit is likely to be unsatisfactory.
It is eventually intended to develop an expert system capable of assessing
concurrently, for any case, these three stages of treatment planning.
However, as every case assessment begins with decisions about the lower arch
and is in most cases done without regard to the upper arch, it seemed sensible to
concentrate only on the first stage of treatment planning for our first attempt at
producing an expert system for orthodontic advice.
Our initial objective was to see whether it would be possible to incorporate an
orthodontic specialist's knowledge of the lower arch into computerized form. For
the first stage it was assumed that tooth displacement is a result of crowding.
Crowding occurs when the sizes of a child's teeth are too large for the space
available in the arch, with the result that erupting teeth are squeezed out of
alignment. It is by far the major cause of tooth displacement.
While there is a useful role for this expert system in training dentists, we have
concentrated on its future use by General Dental Practitioners who would be
unable to carry out complex orthodontic techniques. The range of recommendations for treatment is thus:
1. Accept the existing alignment because crowding is either very mild or nonexistent.
An orthodontic expert system
123
2. Extract the lower first premolars, 4-'['4", or IA and R4 in the expectation
that crowding will resolve spontaneously and residual spaces fully close down of
their own accord.
3. As in 2, but use an applicance to ensure that adequate space remains while
the anterior teeth are aligning.
4. Defer a decision on the grounds that it is too early to make one.
5. Refer the patient to a consultant for advice or treatment.
3. An exmnp~e of the thought pa~ern of the odheden~e censu|t~t
The way the consultant works and thinks is not normally in terms of verbal
statements. However, for the transmission of knowledge to dentists in training, a
verbal approach is used and this has allowed the consultant to generate a set of
rules describing how he practices. It is not our intention here to discuss the
complete set of rules, but rather to show by illustrative example how FRIL can be
used to model the consultant's thought process.
The central problem addressed in the present expert system is one of lower
arch crowding. Crowding may be relieved by extracting teeth. ']'he following rules
apply as the crowding becomes more severe:
1. If there is no significant crowding, no treatment is necessary.
2. If there is very mild crowding then too much space will be made available by
removing the lower first premolars, 4-'[4, and hence the patient should be referred
to a specialist for treatment in order to ensure excess space is closed down.
3. If there is moderate crowding then extract the first premolars and allow the
second premolars, 5-~, and molars, 76]67 to come forwards to take up excess
space.
4. If there is severe crowding then extract the first premolars and fit a space
maintainer which will stop the second premolars and the molars from moving
forwards.
5. If there is very severe crowding then refer to a specialist for treatment.
These are cases where the extraction of first premolars gives insumcient space to
resolve crowding.
This simple set of rules depends on a rather more complex concept of
crowding. The level of crowding is a function C(r, a, s, m) where
r is the additional space that is required if all the teeth are to be correctly
aligned, given as a percentage of the width of the first premolars;
a is the patient's age;
s is the patient's sex; and
m is a measure of whether the second molars ~
have erupted or not.
The orthodontic consultant does not work in terms of a model in four variables
but rather in two variables, C(r, d) where d is a developmental state. The
* Dentists use a simple numbering systen~ as a shorthand to specify teeth or sets of teeth. The
symbol - ' [ - represents the lower arch with -] a~d_fl__indicating the right and left sides respectively.
The teeth are numbered from the centre line so 21 ] 12 represents the four |ower incisors.
J. Sims Williams et 02.
124
100
10
'11
l: ~
I ~1
14
15
16
AGE/YEARS
Fig. 1. The fuzzy definitions of less than and greater than 12[ and 14.
developmental state is the patient's (fuzzy) membership of four groups defined on
the (a, s, m) space:
Group A consists of males of 14 years or more and females of 12.5 years or
more whose second molars ~ have erupted or are erupting.
Group B consists of males of 14 years or more and females of 12.5 years or
more whose second molars are unerupted.
Group C consists of males of less than 14 years and females of less than 12.5
years whose second molars, have erupted or are erupting.
Group D consists of males of less than 14 years and females of less than 12.5
years whose second molars are unerupted.
The developmental state of a patient's jaw does not suddenly change when he
reaches the age of 14 (or 12.5 if female), so the fuzzy truth value of the patient
being over 14 is a ramp and not a step function; see Figure 1. The numbers in the
specification of the groups are thus fuzzy numbers.
Readers who are interested in more detail of the dental aspects of this work
will find this in [5].
4. The | a n p a g e FRIL
FRIL, Fuzzy Relational Inference Language, is a high level automatic
inference language like PROLOG except that instead of being based on Predicate
Logic it is based on the mathematics of relations and incorporates a fuzzy
inference capability through fuzzy relations.
The language has been described before [1, 2]. In 1984 a working version of the
language was implemented on an IBM-PC [4] and it is this version that has been
An orthodontic expert system
125
Table 1
likes person 1
Ann
Jill
person 2
X
Henry
Jane
50
95
used for the work described. Since then various improvements to the system have
been described [3]. FRIL has many of the procedural capabilities of languages
like PASCAL as well as being capable of representing knowledge in relational
form and permitting the operations of a relational database query language. In a
relational database the existence of a tuple in a r¢!ation constitutes a statement
that the tuple satisfies the relation (i.e. with truth value 100%). in FRIL however
most relations associate a truth value, CHI or X ¢ [0, 100] with each tuple.
There are several ways of representing knowledge in FRIL. The simplest way
uses base relations e.g. as in Table 1. Relations can contain attributes which are
themselves relations creating structures potentially of great complexity.
Relations can be defined in terms of other relations as in PROLOG:
e.g.
friends(x, y):-likes(x, y), likes(y, x),
but written
friends((x, y) likes(x, y) & likes(y, x))
and called virtual relations.
Sets of rules of the form
Take action A for reasons B and C and D . . . if r
can be modelled in FRIL.
We set up a base relation as in Table 2. Here r is a procedure which is called in
order to evaluate the value to be associated with the tuple (A, B, C, D , . . . ) . The
B, C, D, etc. are not parameters to the procedures, but parameters in the choice
of procedure. The procedures are self-contained and draw their information from
the FRIL knowledge base.
Having evaluated all the values in a rule set, we can regard the resulting fuzzy
set as the set of appropriate actions to take (with an indication of why, since the
same action can be advisable for different reasons). A decision about which
action should be taken can now be made by choosing the action A which has the
highest value, this being the 'a=lost appropriate' of the set of actions.
Table 2
Rule-set
head body 1 body 2
body 3
A1
A2
DI
D2
BI
B2
C1
C2
. . . procedure
rl
r2
126
J. S/ms Wi///ams eta/.
This can be done by the statewent
which((head) max(rule_set(head, body 1, b o d y 2 , . . . , procedure(x))).
The which statement is a query which returns a relation whose attributes are
specified on a (target list). Both which statements and virtual relations have the
same form
( relation name) ((target list ) (clause) ),
( which ) ( ( target list ) (clause) ),
where (target list) is a list of variables, e.g. (x, y), and
(clause) :---: (relation) J (relation) (operator) (clause),
(relation) :=: (virtual relation) j (base relation),
(operator) :--: (&, for conjunction) [ (OR, for disjunction)
J(&&, for sequenced conjunction).
Other aspects of FRIL necessary to understand the implementation of the expert
system will be explained as required.
We hope to show, by describing our orthodontic application, that the FRILL
mechanism described above can be a powerful tool for modelling human thought
processes.
5. Expert systemd~|p
This expert system has been designed for dentists to use when assessing what to
do about a child's lower teeth when these teeth are not well aligned. The system
provides him with feedback so he can be reassured that the data has been
correctly entered and gives him the reason or reasons for the recommendation he
is given. The system also provides him with warnings if the expecteO end result of
treatment is less than ideal.
In Section 2 we list the five principal recommendations that the system is
designed to discrimin~,te. There will not be a single correct recommendation, but
after weighing the evidence we are able to put forward a CHI or truth value
associated with each recommendation. Normally the dentist would expect to
adopt the recommendation with the highe~t CHI value.
The relational calculus in FRIL operates all tuples simultaneously rather than
doing a depth-first search as in PROLOG. There are a large number of reasons
why a child's teeth may not be suitable for treatment by his loca~ dentist and may
need a referral to a consultant so, rather than assess all possibilities in parallel, we
have divided the expert system analysis into two phases. The first phase considers
the simple reasons why the patient may need to be referred to a consultant or
why it may be too early to do anything. This is done by generating the truth
values, CHI, associated with the relation given in Table 3.
The attribute 'decision' contains the name of the procedure that must be called
to evaluate the CH! value for that tuple. If the maximum CHI value in
An orthodontic expert system
127
Table 3
Initial_decisions
action
reason
decision
refer
refer
refer
refer
refer
refer
refer
refer
dental_delay
not_present
abnormal_form/bad_prognosis
bodily_displaced
inclination
rotation
asymmetry
insufficient_eruption
dec_dd
dec_np
dec_af/bp
dec_bd
dec_i
dec_r
dec_a
dec_ie
initial_decision exceeds 70 then the corresponding action is recommended to the
user complete with the reason, otherwise the program proceeds to evaluate
possible treatments as given by the re|ation in Table 4.
This relation has the same structure a~ Initial_decisions and is evaluated so that
a report can be generated recommending the actions and giving the reasons
complete with warnings if necessary.
See Section 6 for an example of the sort of report the system generates.
The evaluation of possible treatments is done by the command in FRIL:
update( treatment(A B) treatment_decisions(A B • decisions(x)))
where the asterisk is just a convention to remind the programmer that the third
attribute of treatment_decisions is a procedure call.
The procedure is called to evaluate the CHI value associated with each tuple
and the FRIL command 'update' creates a relation with the copied attributes and
the calculated CHI value. Thus for example, the tuple
action
extract
reason
moderate
decision
dec_m
calls the procedure dec_m which is
dec_m((x) mode(add/multiply) &
(which((x) dummy(x) & group(grp) &
room_needed( - both room ) & group_m(grp • grp_m(room )) )) ).
As we have explained in Section 3, crowding in the consultant's model is a
Table 4
Treatment_decisions
action
reason
decision
refer
refer
do_nothing
extract
extract_and_brace
very_mild
very_severe
not crowded
moderate
severe
dec_vm
dec_vs
dec_nc
dec_m
dec_s
J. Sims Wdlimm et aL
128
function C(r, d) where r is the room needed and d is the developmental state as
given by groups A, B, C, D. The patient may not fit exactly into any group and a
FRIL procedure 'find-group' has been used to generate a fuzzy relation 'group'
which states the patient's membership of the different ~roups. The definition of
the groups is given in Section 3. The definition implies there are binary states less
than 14 years old or not and similarly less than 12.5 years old or not but we have
set up fuzzy definitions of these states so that patients' memberships of the groups
add up to 100%.
Rule 3 can be stated as two rules which correspond closely to the
implementation:
extract ~
if moderate crowding,
moderate crowding if (group = A and (room_needed e room_M_A))
or (group = B and (room_needed ¢ room_M_B))
or (group - C and (room_needed e room_M_C))
or (group = D and (room_needed ¢ room_M_D)),
where room_M..X for X = A, B, C, D are fuzzy sets on the room_needed space.
The procedure dec_m sets the mode switch so that, for the evaluation of CHI,
ADD and MULTIPLY are used for disjunctions and conjunctions instead of the
default settings of MAX and MIN.
This then gives a correctly weighted CHI value for the truth of 'moderate
crowding' as shown by the following example. 'Moderate crowding' is a fuzzy
description of the crowding as is illustrated in Figure 2.
Suppose, for the patient under consideration, we have the base relations of
Table 5.
We also have Table 6, where M_A, M_B, M_C, and M_D are/-type relations
which specify the truth that a patient's lower arch is moderately crowded given r,
NOT
CROWDED
VERY
MILD
HODERATE
SEVERE
VERY
SEVERE
~.00
4S
0
.
-10
0
12
20
40
70
85
95
105
110
Room r e q J d / ~ 4 ~
Fig, 2, The variation of crowding condition with percentage of ~
crowding for patients in group C.
width required to resolve the
An orthodontic expert system
129
Tabie 5
Group
group
X
B
D
65
35
Room_needed
side
room
X
right
left
both
6"?
73
70
100
100
100
Table 7
Table 6
Group_M
group
/.type
X
A
B
C
D
M_A
M_B
M_C
M_D
100
100
100
100
M_B
room
X
200
85
75
45
25
-10
0
0
100
100
0
0
the room needed, and the group A, B, C or D. Figure 3 shows the graphical form
of these relations. /-type relations are relations of one numeric attribute which
carry out linear interpolataon of values between listed tuples to provide values
for tuples which are unlisted. For example the/-type relation M - B is shown in
Table 7.
So M_B(70)ffi 100 and M_B(35)- 50. Since M_D(70)--75, the expression in
dec_m,
dummy(x) & group(grp) & room_needed( = both room)
& group_m(grp • grpm(room))
I00
i
D,
/
0
-I0
0
15
10 25 30 35 40 45 50
65 70 75 80 85
100%
Room req'd
Fig. 3. The truth that a patient's lower arch is moderately crowded given the room req'd. To resolve
crowding as a percentage of the -'1-- width and the developmental group d ¢ {A, B, C, D.}
J. Sims Williams et al.
130
will evaluate to the relation
x
group
room
X
ABCDE
B
70
65 ~---(65 * 100)/100
ABCDE
D
70
26*--(35* 75)/100
The '=both' in the room_needed relation picks out the average of the room
needed in ~the left and right segments of the arch, and the conjunctions produce a
'join' of the relations with a value obtained by multiplying the values of the
original relations.
When the match to the target of the 'which' statement is made, the attributes
grp and room are projected out and the ADD rule for disjunction gives the
relation
x
X
ABCDE
91
This is the value passed by dec_m for the truth of moderate crowding used in
Rule 3.
6. Results
The expert system described in this paper is undergoing clinical trials in which
dentists, who are not orthodontic consulta~.s, are using the system to assess sets
of patients' teeth in the Bristol Dental Hospi;,-.~ ,'Y~e res;z!ts produced by the
dentists are being compared with consultants" ~dvice. "I'nis is an important test of
the system as we need to know if a non-specialist dentist can accurately take the
required data from a set of teeth moulds and input this data into the system.
However, before the system was used by GPDs. It was important to make sure
that it could reach the same conclusions as an orthodontic consultant would have
done. We therefore tested the system using an orthodontic consultant to look at
teeth moulds and input the data. The teeth moulds used were those of patients
who had been treated previously. We compared the recommendation of the
expert system with treatment actually given. A set of data given to the expert
system is set out in Table 8 and the results in Table 9. Details concerning the
degree of tooth rotation and canine inclinati~m as listed in Table 8 can be found in
[5]. The recommended course of treatment shown in the results is a considerable
abbreviation of the output given to the dentist by the expert system. We show in
Table 10 the output generated for a particularly unfortunate and fictitious patient.
In Table 9 of the results the actual treatment is the treatment that has already
been given. The fitting of fixed aprliances is something that cannot be done by
GDPs in their surgeries, so our expert system should recommend a referral for
such cases. Notice that the expert system veers on the conservative side in case
number 10. The results suggest that an expert system can give advice comparable
to the clinician upon whose knowledge it is based when he himself is entering the
necessary clinical data. The next step is obviously to see whether a General
Dental Practitioner would have obtained the same recommendations.
Years
Months
H.
F
F
G
G
G
29
14
H
F
G
G
G
F
14
0
4-1 percentage
required
percentage
required
Crowding
level
B
C
1-1
ff
2-3
r~
37
r~
Y
N
Tooth
rotation
N
N
C
C
N
Y
3-3
Y
Y
Y
Y
Canine
inclination
Y
Y
75-g1
75--I-
Incisor/canine
crowding in
right quadrant
Incisor/canine
crowding in
left quadrant
Premolar c~owding
in right quadrant
Premolar crowding
in left quadrant
N
F
G
G
F
F
F
F
25
63
63
63
C
C
A
B
F
G
H
F
H
G
Y
N
75-1
M
N
N
M
11
0
F
12
9
12
1
4
10
7
Crowding
distribution
Unerupted teeth
Sex
Age
3
2
1
Patient number
25
35
H
I
G
G
G
G
B
B
N
N
Y
Y
-
M
13
3
5
Table 8. The data given to the expert system
F
75[57
Y
M
[7
Y
13
0
F
F
H
G
F
G
C
B
N
N
14
14
G
G
G
G
F
F
D
D
N
Y
Y
13
3
12
11
Y
7
6
14
24
F
H
H
G
G
F
B
C
N
N
Y
Y
--
F
13
5
8
35
50
J
F
!
F
F
F
B
D
Y
N
Y
Y
7--~.-
M
10
10
9
50
13
F
F
G
G
F
F
C
D
N
N
Y
Y
M
11
9
10
Ila
t-a
i
t
|"
J
132
Z Sims Williams et aL
Table 9. The expert system's recommendations for treating the cases shown in Table 8 and the actual
treatment undertaken
Recommended course
of treatment
Actqal treatment
1
Refer due to distal tilt &
excessive rotation of 3-]
Extraction of 4--']'4,pl_us fixed appliance to correct
tilt and rotation of 37 (::~ Refer.)
2
Refer due to very mild
crowding in arch
Extraction of 4-~, fixed appliance later fitted
to close down excess space ( =~ Refer.)
3
Refer due to excessive
rotation of
Fixed appliance used to correct rotation of ['i'.
( ~ Refer,)
4
Extract 4-=~, warning about
slight distal tilt of 3~
Extraction of 4--'~. Result good
5
Refer due to excessive
rotation of [1
Fixed appliance used to correct rotation of 1 - ~ &
to close, down excess space (=~ Refer.)
6
No Treatment
No Treatment
7
Refer due to distal tilt of
Extraction of 4-T4. Fixed appliance later fitted
to upright 3 ~ and close down space ( =~ Refer.)
g
Refer due to very mild
crowding in arch
Extraction of 4-'~. Space closure very slow so
fixed appliance fitted (=~ Refer.)
9
Refer due to excess=re
rotation of 2-~ & distal
tilt of
Fixed appliance used to correct rotation of
Refer due to distal tilt of
Extraction of 4-'~. Result good, but 3 ~ are a
little distally inclined
No.
10
( =~ Refer.)
Many more cases have been tested including cases where the treatment given
by the consultant was not successful. Since the expert system is based on the best
orthodontic expertise available, the system did not do any better than the
consultant. It does not however make 'silly errors' provided the data is correctly
entered.
Table 10. Josephine Soap
The patient is dentally delayed.
The r~ght first molar and right second molar are not present in the arch.
The ~ ~:ond m ~ r , ~s ~ t of n~rmal ~,~m and good prognosis.
t h e l~t central incisor is bodily displaced from the arch.
The inclination of the right canine is anfavourable. 3-5 contact would not be ideal if the 4 was
extracted and uprighting of the canine with a fixed appliance would be necessary to correct distal
tilt.
The right canine ar~ left lateral incisor are badly rotated.
The lower arch crowding is asymmetric.
The centre-line is probably significantly offset to the left and a fixed appliance will be needed to
correct this.
It is therefore advisable that the patient sees a specialist.
An orthodontic expert system
133
Smmy
An expert system has been generated which successfully models the advice that
orthodontic consultants give to General Dental Practitioners for the alignment of
crowded lower teeth. An eplarged expert system, if widely adopted, would
reduce the number of referrals to consultants and thus provide an improved
service to the public. The system shows promise of being able to be developed
into a training aid for specialists. FRIL has been found to be a very powerful tool
for generating expert systems.
References
[!] J.F. Baldwin and S.Q. Zhou, F R I L - A fuzzy relational inference language, Fuzzy Sets and
Systems 14 (1984) 155-174.
[2] J.F. Baldwin, F R I L - A n inference language based on fuzzy logic, Proc. Expert Systems 83
Conference, Cambridge (1984) 163-173.
[3] J.F. Baldwin, Automated fuzzy and probabilistic inference, Fuzzy Sets and Systems 18 (1986)
219-235.
[4] L.J. Pegrum, FRIL on the IBM-PC: User Guide to the System, Engineering Mathematics Dept.,
University of Bristol (1984).
[5] J.H. Sims Williams, C. Stephens, I.D. Brown and A. Matthewman, A computer-controlled expert
system for orthodontic advice, Brit. Dent. J. 163 (1987) 161-166.