Fuzzy Sets and System~ 30 (1989) 121-133 North-Holland
121
AN 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.
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 recommenda- tions for treatment is thus:
1. Accept the existing alignment because crowding is either very mild or non- existent.
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.
124 J. Sims Williams et 02.
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 |anpage 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
Table 1
likes person 1 person 2 X
Ann Henry 50 Jill Jane 95
125
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 . . . procedure
A1 BI C1 DI rl A2 B2 C2 D2 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
(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 system d~|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
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
As we have explained in Section 3, crowding in the consultant's model is a
Table 4
Treatment_decisions action reason decision
refer very_mild dec_vm refer very_severe dec_vs do_nothing not crowded dec_nc extract moderate dec_m extract_and_brace severe dec_s
128 J. Sims Wdlimm et aL
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,
~.00
4S
NOT VERY VERY CROWDED MILD HODERATE SEVERE SEVERE
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 ~ width required to resolve the crowding for patients in group C.
B 65 D 35
Table 6
An orthodontic expert system
Tabie 5
Room_needed side room X
right 6"? 100 left 73 100 both 70 100
129
I00
Table 7
Group_M group /.type X
A M_A 100 B M_B 100 C M_C 100 D M_D 100
M_B room X
200 0 85 0 75 100 45 100 25 0
-10 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,
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.}
130 J. Sims Williams et al.
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.
Age
Y
ears
M
onth
s
Sex
Une
rupt
ed t
eeth
Cro
wdi
ng
dist
ribu
tion
Can
ine
incl
inat
ion
Too
th
rota
tion
Cro
wdi
ng
leve
l
Tab
le 8
. T
he d
ata
give
n to
the
exp
ert
syst
em
Pat
ient
num
ber
1 2
3 4
5 6
7 8
9 1
0
Inci
sor/
cani
ne
crow
ding
in
righ
t qu
adra
nt
Inci
sor/
cani
ne
crow
ding
in
left
qua
dran
t P
rem
olar
c~o
wdi
ng
in r
ight
qua
dran
t P
rem
olar
cro
wdi
ng
in l
eft
quad
rant
3-3
1-1
ff 2-3
r~
37
r~
4-1
perc
enta
ge
requ
ired
pe
rcen
tage
re
quir
ed
10
12
12
11
13
7 1
9 0
3
F F
M
M
M
75--I
- 75
-g1
N
75-1
- Y
Y
Y
Y
Y
Y
N
N
N
N
Y
C
B
C
C
H
H.
F F
G
F G
G
G
G
F
G
14
29
0 14
12
13
13
10
11
11
3 5
10
9
M
F F
M
M
[7 75
[57
--
7--
~.-
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
N
N
N
N
Y
N
N
N
N
Y
N
N
N
N
Y
N
A
C
B
C
D
B
B
C
B
C
B
B
D
C
D
D
F G
H
F
G
F J
F G
G
I
F G
H
F
F H
F
G
H
G
H
! G
F
F G
G
G
G
F
G
H
F G
F
F G
F
F G
F
G
G
F F
F F
63
25
35
0 14
24
50
13
63
63
25
13
14
14
35
50
J |" t i t-
a
Ila
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 No. of treatment Actqal treatment
1 Refer due to distal tilt & excessive rotation of 3-]
2 Refer due to very mild crowding in arch
3 Refer due to excessive rotation of
4 Extract 4-=~, warning about slight distal tilt of 3~
5 Refer due to excessive rotation of [1
6 No Treatment
7 Refer due to distal tilt of
g Refer due to very mild crowding in arch
9 Refer due to excess=re rotation of 2-~ & distal tilt of
10 Refer due to distal tilt of
Extraction of 4--']'4, pl_us fixed appliance to correct tilt and rotation of 37 (::~ Refer.)
Extraction of 4-~, fixed appliance later fitted to close down excess space ( =~ Refer.)
Fixed appliance used to correct rotation of ['i'. ( ~ Refer,)
Extraction of 4--'~. Result good
Fixed appliance used to correct rotation of 1-~ & to close, down excess space (=~ Refer.)
No Treatment
Extraction of 4-T4. Fixed appliance later fitted to upright 3 ~ and close down space ( =~ Refer.)
Extraction of 4-'~. Space closure very slow so fixed appliance fitted (=~ Refer.)
Fixed appliance used to correct rotation of ( =~ Refer.)
Extraction of 4-'~. Result good, but 3 ~ are a little distally inclined
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. the 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.
S m m y
An orthodontic expert system 133
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, FRIL-An 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.
