- Email: [email protected]
Evaluation of biopsy data in human enamel fluoride studies
Archs oral Viol. Vol. 16, pp. 1413-1426, 1971. Pcrgamon Press. Printed in Great Britain.
EVALUATION
OF BIOPSY DATA IN HUMAN FLUORIDE STUDIES R.
AASENDEN
ENAMEL
and E. C. MORENO
Forsyth Dental Center, Boston, Mass. 02115, U.S.A. Summary-Atreatment of biopsy data is given that permits comparisons of enamel F in experimental groups when the biopsies are taken to varying depths and their weight distributions are different for the groups under comparison. It is also shown how the mean F concentration profile can be reconstructed from biopsy data. Enamel F was compared in maxillary contralateral central incisors, lateral incisors, and first molars. It was found that the enamel F in the central incisors was equal, both in content and distribution with depth, and it was significantly different in the case of the lateral incisors. There was a marked trend for the F content of the right molars to be higher than that of the antimeres, but a level of significance was not reached. It was concluded that, with the biopsy method used, only central incisors can definitely be used interchangeably and as matched pairs in comparative studies. The observed differences in enamel F were most likely due to difficulties in utilizing equal sampling areas on contralateral teeth. INTRODUCTION
PRESENTknowledge of the F content of enamel has been derived mainly from layer analyses of extracted teeth. Recently, biopsy methods have been reported (BRUDEVOLD, MCCANN and GRIZIN,1968; HOTZ, MUHLEMANNand SCHAIT, 1970) by which F in surface enamel can be determined in uiuo; thus population samples involving substantial numbers of subjects can now be studied. Comparisons of F levels between groups have usually been made on the basis of the group mean F concentrations of enamel samples. Since there is an F gradient in enamel (BRUDEVOLD,GARDNERand SMITH, 1956; BRUDEVOLD, MCCANN and GRIN, 1968), such comparisons would be strictly valid only if all the samples were taken to a uniform depth. So far, there is no method available to attain complete uniformity in sampling depth, neither of in vitro nor of in vivo samples. A procedure using graphical interpolation in order to make comparisons at the same depth between samples taken to variable depths has been reported (MUHLEMANN,SCHAITand KONIG, 1964; MELLBERGet al., 1968). This procedure can be applied only when successive samples are taken from each tooth. Furthermore, linearity of the F gradient within the sampled layer must be assumed; unless the sampled layers are extremely thin, this assumption is questionable in the light of the results reported here. In this paper, a method for treatment of data is given that permits comparisons between groups when the samples are taken to varying depths and their weight distributions are different for the groups under comparison. By means of this method, meaningful comparisons can be made on the basis of a quantity referred to hereafter as “fluoride parameter”. The method involves subgrouping of the data, and, therefore, it requires fairly large population samples. For this reason, it is applicable mainly to 1413
1414
R. AASENDENAND E. C. MORENO
biopsy data. Clarification of the nature of the relationship between enamel F and caries calls for not only exact evaluation of differences in enamel F contents between experimental groups, but also for information on its distribution with depth. It will be shown here that biopsy data can be used to reconstruct the mean enamel F concentration profiles in experimental groups. In the biopsy method developed by BRUDEVOLD et al. (1968), the sampling area is not determined. Therefore, in comparative studies, the underlying assumption is that the mean sampling areas of the groups involved are equal. This condition may be closely fulfilled when comparisons are made on the basis of biopsies taken by the same operator from the same tooth in all subjects. This design was used in the surveys reported by AASENDEN et al. (1971). In F uptake studies testing topical agents, sampling area variations have been minimized by assessing the F increment as the difference in the F contents of samples taken from the same teeth before and after treatment (BRUDEVOLD et al., 1969). Successive samples are, however, not physically independent, a circumstance that makes questionable the application of usual statistical tests to the data. An alternative design that would be advantageous in comparative studies of this type would be the use of contralateral teeth as matched pairs. A critical study of this possibility is included in the present paper. It will be shown that, of the pairs studied, only the upper central incisors lend themselves for such a use with the method employed. EXPERIMENTAL
METHODS
Calculation of sampling depth In the biopsy method presently used the area, S, of the sampled surface, is not determined; thus, only relative sample depths (inversely proportional to the sampling area) can be calculated. The weight, IV, of each enamel sample is calculated from the corresponding analytical figure for its calcium content. It is assumed that the calcium content and the density of enamel are 37 per cent (BRUDEVOLDand &REMARK, 1967) and 2.95 g/cm3 (KARLSTRBM,1931), respectively. Therefore, assuming uniform abrasion of the sampling area, the depth, D, in pm (micrometers), to which a sample was taken is wx loo0 D = S x 2.95 where W is given in milligrams and S is given in square millimeters. As an example, for W = 0.10 mg, the sampling depth would be approximately 34/S pm. Grouping of enamel samples The calculated weights of the enamel samples are grouped into at least three classes according to their distribution plot. Limits for the class intervals are selected so that (a) enough samples are comprised within each category to compensate for variation in sampling areas and other relevant factors such as age and F exposure, and (b) an even distribution of samples is obtained around the midpoint of the class limits. To fulfill these criteria, it is often necessary to discard a few samples falling in scattered tails in the distribution plot. When the sample weight-distribution permits the selection of equal class intervals, the calculation of the fluoride parameter (and its standard error) is substantially simplified. If the enamel samples displayed a perfect weight distribution around the midpoint of each class interval. the functional relationship between mean F concentration and sample weight would be the same whether the midpoint or the actual mean weight of the intervals were considered as the variable. In real cases, however, the values of the actual mean weight in each class are often found to depart slightly from the values of the class interval midpoints. This departure is illustrated in Table 1 for biopsies of the left central incisors from a non-fluoridated area (see section on homologous teeth for
EVALUATIONOFBIOPSYDATAINHUMAN
ENAMELFLUORIDE
1415
STUDIES
25-
--
Mean weight
I
0.18 s_! S
1
Weight, Depth.
mg pm
FIG. 1. Mean F concentration of enamel samples, grouped by weight, as a function of sample weight (depth to which the samples were taken). Biopsies from the maxillary left central incisor in 64 subjects from a non-fluoridated area. Vertical bars represent standard errors.
clinical and analytical procedures). Consequently, as illustrated in Fig. 1, the dependence of the mean F concentration, c, on weight differs depending on which of the two variables is selected. It can be shown, however, that relatively small differences as those depicted in Fig. 1 do not affect significantly the calculated value of the fluoride parameter, provided the same weight interval is considered. Thus, for most practical purposes, the use of class interval midpoints is amply justified. TABLE
~.MEAN
Sample weight (mg) Interval midpoint Class mean weight N Mean [Fl (ppm)
F
CONCENTRATIONSOFENAMELSAMPLES,GROUPED BY WEIGHT,FROMTHE MAXILLARY LEFT CENTRALINCISORIN 68 SUBJECTS
0*09-<0*16 0.125 0.126 21 2213 + 170*
0*16-co.23 0.195 0.187 25 1971 f 113
0*23-
>0.30
4
* Standard error. Comparison of groups 1. Calculation offluoride parameter. The fluoride parameter, FP, is detined as the area enclosed by the two end ordinates, the x-axis and a curve fitted to the class mean F concentrations in plots such as those in Fig. 1. This area can be determined readily by approximate integration, using the Newton’s interpolation formula (MELLOR, 1955), when equal class intervals are used, i.e. when the midpoints of equal class intervals are used as the variable. For cases where three or four classes are used in the grouping of the enamel samples, the fluoride parameter is given by equations (1) and (2) respectively.
in which c” is the mean F concentration of the nth class and h is the value of the weight interval which is the same for all the classes. The variance, V., in the fluoride parameter can be obtained by propagation of errors on the equation used for the calculation of the area. In general,
1416
R.
AASENDEN
AND E. C. MORENO
in which EE. is the standard error in the mean F concentration of the nth class. The variances and standard errors in the fluoride parameters when 3 or 4 classes are used in the grouping of samples are obtained by applying equation (3) to equations (1) and (2), respectively. Thus
E3
=
v, =
+
(EdI
tv’43
2$
[E*E, + 9E*-,, + 9E*e, + EKES] = g
. #4(EE) (5)
The use of the fluoride parameter de&red in equation (1) or (2) is restricted to those cases where the class limits are common to the groups under comparison. When such is not the case, an approximate integration of the cumulative curves can be made by the use of the trapezoid rule. This geometrical procedure, although not as accurate as the integration using Newton’s interpolation formula, offers the simplest method to calculate the FP when the classes are unequally spaced or when the groups to be compared do not have common class limits. When the class limits are not common to both groups, the FP is calculated using limits encompassed by both cumulative curves, i.e. without using extrapolation of either curve. The standard error is then calculated by applying the general equation (3) to the case in question. By way of illustration, the fluoride parameter corresponding to the set of data in Table 1 can be calculated by the use of equation (1) and its standard error by equation (4). The calculations yield the values FP = 0*2T3 and E = 0*O12. The subscript numerals are not significant digits; however, they have been carried on in the statistical tests. The F concentrations in Table 1, also, are given with more significant digits than justified by their uncertainties for the purpose of calculation verification. 2. Reconstruction of fluoride concentration profile. The functions plotted in the example of Fig. 1 should be clearly distinguished from the functions representing the F concentration profiles in the enamel. Whereas each point in the cumulative curves of Fig. 1 contains “integrated” information on the F content from the surface of the enamel to the depth (or weight) plotted, a point on a concentration protie represents a “differential” property, namely the F concentration in a plane at that particular depth. The relationship between these two kinds of functions is treated here first with a hypothetical example and then with actual biopsy data. In Fig. 2 is shown a hypothetical F profile (curve 1) in enamel; arbitrary units are used in the coordinates. This curve is described by the equation Cr = 9.853 - 4.428 W + 0.879 W* - 0.059 W3
(6)
with a degree of determination of O-998. A biopsy taken to a weight (depth) of W,, units will contain a total amount, F,, of fluoride given by W, F. =
s 0
C, dW = 4 (W.).
(7)
Corresponding to this value of F. there is a point in the cumulative plot (curve II) which represents the mean F concentration, c., of the biopsy sample taken to weight (depth) W.. This mean concentration is given by c _ &W”) n (8) K and it is plotted at the abscissa W.. It is seen then that, in this particular example, the cumulative function is c = 9.853 - 2.214 W + 0.293 W* - 0.015 W3. (9)
EVALUATION
OF BIOPSY
I
I
DATA
/
2
IN HUMAN
ENAMEL
STUDIES
1417
I
I
3
FLUORIDE
4
Weightor depth,
5
6
7
arbitrary units
FIG. 2. Hypothetical F concentration profile in enamel, curve I, and its corresponding cumulative curve, curve II. Solid curves I and II plotted according to equations (8) and (1 l), respectively. Points on curve I calculated from points in the cumulative curve by the procedure given in this paper. Vertical bars represent standard errors.
It becomes clear that, if the c function were known in actual cases, the function describing the real profile could be easily obtained multiplying C?by Wand differentiating with respect to W. In practice, however, it is unlikely that the form of the c function can be determined adeauatelv. The concentrations calculated from chemical analyses on biopsy samples correspond then to points on the cumulative curve. We can simulate actual conditions by dividing the weight range in the cumullative curve into classes and considering various arbitrary points (individual samples) within each class. In the present example, six classes were selected and from four to six points were considered in each class. The mean fluoride concentration, c?,,, in the nth class is
in which F1 and W, are the total fluoride and the weight of the ith sample in the class, respectively. As shown in Fig. 2, these mean concentrations are plotted at the actual mean weights of the respective classes, that is, at Wn = (7w)/(N.). There is a standard error associated with each C?,,.In the present example, the standard error stems only from the presence of a gradient within each class; in actual cases the error also reflects differences in sampled surface area in addition to the individual variation in F concentrations. In Table 2 are given the I?,,values calculated by equation (10) (fourth column) for the various classes. These figures are to be compared with those in the third column which were calculated by equation (9) using the values of W given in the second column. It is apparent that the differences in the two sets of values are negligible. Thus, the assumption of a linear gradient within each class, tacitly made in equation (IO), does not introduce a significant error. The total mean fluoride, Fo,“, sampled from the surface to a mean weight W, is obtained by multiplying the figures in the fuurth column of Table 2 by their corresponding mean weights (second column); the results of these operations are given in column 5. Now we proceed to the reconstruction of the fluoride profile.
R.
1418
AA~ENDEN AND
E. C. MORENO
TABLE 2. MEAN F CONCENTRATIONS AND MEAN F WEIGHTS FOR WEIGHT INTERVALS IN THE CUMULATIVE CURVE OF FIG. 2
Weight class
Class mean weight, w
Cmfrom (9)
l-2 2-3 3-4 4-5 5-6 6-7
1.37 2.51 3.56 4.53 5.57 6.55
7.330 5.908 5.018 4.464 4.061 3.775
C” from (10) 7.338 5.923 5.019 4.469 4.067 3.775
i f f f rt &
Mean total F sampled from surface to W'. 10.090 14.867 17.868 20.245 22.651 24.728
o-183* 0.149 0.116 0.088 0.073 0.034
f 0.251 f O-375 ho.413 + 0.398 ztO.407 & 0.224
* Standard error. The total mean fluoride, F., n--l, present in the interval between the mean weights W. and W.-r is simply F., n--l = Fo,. - Fo,n-1 (11) If we assume a linear gradient between the mean weights r” and kp”_, in the fluoride profile, we can define a concentration, CF: F CF = W” L-i”_; for a point in the profile having an abcissa W = W,,_, + (W,- W&/2. The profile concentrations calculated by equation (12) have standard errors that can be estimated by propagation. Thus ECF
=
d(wn’ E 2Cn+ w*,-l EZ,,,). W” - W”4
The results of the calculations of mean total fluorides and profile concentrations by the use of equations (11) and (12), respectively, are shown in Table 3. The profile concentrations and their standard errors are plotted in Fig. 2. Clearly, the calculated points represent very closely the concentrations of the true profile. The relatively large magnitude of the standard errors relates to (a) the small number of samples within each class considered in the present example, and (b) the small differences between quantities of the same magnitude (equation 11) for which the errors are of an additive nature, as shown in equation (13). TABLE 3. MEAN F IN WEIGHT INTERVALS AND F CONCENTRATIONS IN PROFILE OF FIG. 2.
Mean weight interval (curve II)
Plotted weight (curve I)
O-l.38 1.38-2.51 2.51-3.56 3.56-4-53 4.53-5.57 5.57-6.55
0.69 1.94 3.03 4.05 5.05 6.06
* Standard error.
Mean F in weight interval
IO.090 4.777 3.001 2.377 2.406 2,077
f 0*251* ho.451 f 0.558 f 0.574 f 0.569 f 0.464
Profile concentrations
7.338 4.209 2.858 2.450 2.314 2.119
ztO.183 f 0.397 f 0.531 & 0.591 + 0.547 + 0.474
EVALUATlONOFBIOPSYDATAINHUhfAN
I
I
ENAMELFLUORIDESTUDIES
I
0.10
O-20
Weight,
34
Se
Depth,
S
1419
S
mg pm
FIG. 3. Cumulative F curve, A, and reconstructed F concentration profile, B, obtained from enamel biopsies of the maxillary left central incisor in 64 subjects from a nonfluoridated area. Vertical bars represent standard errors. The foregoing procedure for the profile reconstruction was applied to the example in Fig. 1. The reconstructed profile is shown in Fig. 3 in which the cumulative curve is also included for comparative purposes. It is apparent that the standard errors associated with the reconstructed profile are much larger than those associated with the cumulative curve. The possibility of using reconstructed profiles for statistical evaluation in comparative studies and the limitations of such an approach are commented on in the Discussion. 3. Statistical evaluation. Comparison of two independent groups on the basis of the fluoride parameter can easily be done by the use of the t-test. Through the computation of the group FP, n degrees of freedom (corresponding to the n number of classes into which the group was divided) are lost; thus, the appropriate number of degrees of freedom associated with the t-value is Ni + Nz - 2p1, N1 and Nz being the total number of samples in Group I and Group II, respectively. Analysis of variance likewise can be applied to test for differences when more than two groups are involved. The full advantage of matched pairs cannot be realized in biopsy studies unless the paired enamel samples are taken to the same depth. The use of a t-test for independent samples, as it is done in the present study, constitutes a conservative statistical criterion. Thus, whereas significant differences can be validly judged on the basis of such a test, only a cautious conclusion can be reached when the differences do not attain the significance level associated with the test. Application of the data treatment to studies on homologous teeth The data treatment was applied to two separate studies. Study I comprised 68 subjects from a nonfluoridated area (Hyannis, Massachusetts, water F < 0.1 ppm); Study II comprised 75 subjects from a fluoridated area (Salem, Massachusetts, water F: I ppm). The subjects were 9-l 1 yr old. These groups constituted part of the population samples described in detail by AASENDEN et al. (1971). Enamel biopsies were taken from the intact labial surfaces of the four maxillary incisors and from the areas mesial to the buccal grooves on the buccal surfaces of the maxillary first molars. All the biopsies were taken by one operator who was informed of the results of Study I before undertaking the sampling in Study II. Prior to the enamel sampling, the tooth was cleaned with a rubber cup and insoluble sodium metaphosphate in order to remove soft deposits. The enamel sample was obtained following the procedure of BRUDEVOLD et al. (1968). By this procedure enamel is polished off with a rotating midget felt cone impregnated with silicon carbide. The cone is coated with glycerine which traps and forms a slurry with the ground enamel particles. The slurry and the felt cone are transferred to a plastic test tube and subjected to F and Ca analyses. The technique was aimed at utilization of the largest possible sampling area, which of necessity was smaller than the defined area available for sampling; felt worn off the cone occupied several square millimeters adjacent to the gingiva and
1420
R.
AASENDEN
AND
E. C. MORENO
margins of approximately 1 mm were left unsampled mesially and distally in order to avoid contamination from neighbouring teeth. Fluoride in the enamel sample was determined by means of a lanthanum membrane electrode (FRANTand Ross, 1966) following the method described by MCCANN (1968). Calcium was determined by atomic absorption spectrophotometry. The F concentration in the sample was calculated on the basis of the sample weight which, in turn, was calculated from the calcium analysis (see calculation of sampling depth). The uncertainties in the fluoride and calcium determinations were estimated as rt2 and f3 per cent of the amounts analysed, respectively. In all cases, some samples at the tails of the weight distributions were discarded in order to achieve grouping of the enamel samples with common classes for the two teeth under comparison. The mates of the samples discarded in one tooth group were identified and excluded also in the other tooth group. This exclusion was not necessary for the statistical treatments, since t-tests for independent samples were applied. However, the inherent pairing of the two sets of data eliminates variations in several factors, such as age, sex and tooth size. RESULTS
Cumulative F plots for the central incisors in Study I are shown in the upper portion of Fig. 4. The curves represent results obtained with 57 pairs. Eleven pairs had to be discarded because of the dissimilarity in the weight distributions of the two groups. It is apparent that the cumulative curves for both teeth follow a similar
,
Weight, mg Depth,
pm
FIG. 4. Cumulative F curves, A, and reconstructed F concentration profiles, B, obtained from enamel biopsies of the maxillary central incisors in 57 subjects from a non-fluoridated area. Vertical bars representing standard errors are shifted for clarity.
EVALUATlON
30
t
OF BIOPSY
DATA
--__
IN HUMAN
ENAMEL
STUDIES
1421
--_a
-_
Sample
FLUORIDE
weight,
mg
FIG. 5. Weight frequency distributions of enamel samples taken from maxillary lateral incisors and first molars in 68 subjects from a non-fluoridated area.
pattern and that the mean fluoride in the enamel displays a rather steep gradient. Indeed, the reconstructed profiles in the lower portion of Fig. 4 show that the mean F concentration for both teeth changes from about 2000 ppm to about 1000 ppm within the weight interval 0*06-0*23 mg. An approximate value for the sampling area in these teeth is 40 mm2. Thus, the decrease in fluoride concentration previously mentioned occurs within an approximate depth interval of 0.5-2 pm. The fluoride parameters and their standard errors for the two central incisors were calculated according to equations (1) and (4), respectively. The difference between the two values, 0.25, f 0*013 for the right and 0~26~ f 0.01, for the left tooth, was non-significant. The weight distributions of the enamel samples from the two other pairs of contralateral teeth in Study I were so dissimilar that it was impossible to achieve a grouping of samples conducive to meaningful comparisons. The frequency distribution curves shown in Fig. 5 clearly illustrate the situation. Any attempt to group the samples within the common weight ranges would result in classes having so few samples that statistical evaluation would be meaningless. The cumulative F plots for the central incisors in Study II are shown in the upper portion of Fig. 6. In this case the criteria used in the grouping of samples were best fulfilled by the use of four classes. Only three pairs had to be discarded in the grouping process. The two cumulative curves, as in the case of the central incisors in Study I, are very close to each other; consequently, the two fluoride parameters, calculated by equation (2), are practically identical. The closeness of the points in the cumulative plots is indicative of a similar distribution of fluoride in enamel with depth. Such similarity is clearly shown by the reconstructed F concentration profiles in the lower portion of Fig. 6. The two profiles display steep concentration gradients similarly to the profiles for the central incisors in Study I shown in Fig. 4. The higher F concentrations in the profiles of Study II reflect the origin of the experimental subjects, namely their living in a fluoridated area. A detailed study on the differences in enamel F of the two population samples is reported by AASENDEN er al. (1971).
R. AASENDENAND E. C. MORENO
1422
I-
I 0.10 34
s
I 0.20
se
S
:
, Weight, mg Depth, pm
FIG. 6. Cumulative F curves, A, and reconstructed F concentration profiles, B, obtained from enamel biopsies of the maxillary central incisors in 72 subjects from a fluoridated area. Vertical bars representing standard errors are shifted for clarity.
The weight distributions of the biopsies from left and right central incisors in Study II were quite similar. Thus, in this particular case, comparison can be made on the basis of the grand means of the F concentrations for the two teeth. The grand means and their standard errors were 2670 f 70 ppm and 2680 f 70 ppm for the right and left tooth, respectively. Comparisons on the basis of these grand means or on the basis of the fluoride parameters yield equivalent results. In contrast to the results in Study I, the sample weights from the lateral incisors and first molars in Study II permitted their grouping into common classes for each pair. The cumulative plots for the two pairs are shown in Fig. 7. In the case of the lateral incisors, the difference in the fluoride parameters, O-35, f 0.Olr for the right and 0~31~ & O*Olo for the left tooth, is significant at the O-02 confidence level. It is not clear whether this difference relates to an actual difference in the F contents of the two matched teeth or to a difference in the group mean sampling areas. This matter will be treated in the Discussion. The weight distributions of the samples from the paired molars required a grouping with unequal intervals in order to achieve common mean weight classes as shown in the lower portion of Fig. 7. For this reason, the fluoride parameters were calculated
EVALUATION
OF BIOPSY
DATA
IN HUMAN
ENAMEL
FLUORIDE
STUDIES
1423
Fparameter
-i-
I
I
0 IO
0 20
Weight, mg
34 s
68 s
Depth, pm
FIG. 7. Cumulative F curves obtained from enamel biopsies of 71 paired maxillary lateral incisors and of 69 paired maxillary first molars in subjects from a fluoridated area. Vertical bars representing standard errors are shifted for clarity.
by approximate integration using the trapezoid rule. The difference between the two fluoride parameters, 0.21, f 0*010 for the right and 0.19, i_ 0.01, for the left molar, is not statistically significant. DISCUSSION
The treatment of biopsy data used in this paper for comparative studies on homologous teeth has two salient features: the grouping of samples (and their F concentrations) according to their weights, and the use of a fluoride parameter which provides a useful combination of the data for statistical evaluations. The marked F gradient in the outer few microns of enamel reported by other investigators (BRUDEVOLD et al., 1968) and in the present paper (see Figs. 4 and 6) invalidates, in our opinion, comparisons between groups based on the grand mean F concentrations of samples taken to depths varying within a relatively wide range. An exception to this statement, as mentioned in the results, is when the biopsy depth distributions of the groups under comparison are quite similar. In survey work, however, when the biopsies from different groups are taken at different times and under different conditions, dissimilarity in weight distributions may be the rule rather than the exception (AASENDENet al., 1971). The results reported in this paper show that even when the biopsies of contralateral teeth are taken at the same time, the weight distributions for the two teeth may differ greatly. Indeed, of the 6 pairs of contralateral teeth studied, only one pair (central incisors in Study II) displayed similar weight distributions. Indiscriminate use of grand means for comparative purposes may lead to erroneous conclusions; for example, on this basis, a comparison between the two central incisors
1424
R. AASENDEN
AND
E. C. MORENO
in Study I results in a highly statistical significant difference whereas a comparison between the molars in Study II yields a non-significant difference. Such conclusions are certainly not warranted when the dependence of F concentration on weight is taken into account as demonstrated by the cumulative curves and the reconstructed profiles (and standard errors) in Fig. 4 and by the cumulative curves in the lower portion of Fig. 7. The grouping of samples into common classes for the groups under comparison compensates for the differences in weight distributions. Comparisons between groups could of course be done on a class by class basis. For this purpose, it is necessary to select the class intervals in such a way that the actual mean weights of corresponding classes in the different groups are quite comparable; deviations of the class mean weights from the interval midpoints are of no concern. The shortcoming of a class by class comparison is the drastic reduction in degrees of freedom as contrasted when all the samples are used in the statistical test. Thus, as reported by AASENDEN et al. (1971), even in cases where there are substantial and consistent differences all along the cumulative curves, a class by class statistical analysis may not yield the conclusion that the population samples differed significantly. This difficulty is overcome by combining the data as done in the fluoride parameter. When comparing results obtained from different teeth in the same subjects, the combination of data becomes imperative, because the different enamel samples from a subject are only seldom found in a common class. The parameter introduced here has the following distinct advantages: (a) it combines the data considering the F concentration dependence on weight in samples taken to any depth within the experimental range, (b) its numerical value depends on both the mean F content and its distribution in the enamel of the given group, (c) it lends itself to statistical analyses, and (d) it can be obtained by simple calculations. Other combinations of data such as a simple sum of the class mean F concentrations do not display features (a) and (b) above. Conventional methods such as analysis of covariance could be applied to biopsy data if the form of the relationship between the biopsy weight and the F concentration could be determined adequately from the individual data. In the populations studied here, the variability in F concentration was so great that no formal relationship could be determined with the necessary reliability for adjustment of data. If, in other population samples, the coefficients in the function describing the F concentration dependence on biopsy weight can be assessed reliably, evaluation of differences between groups could also be made on the basis of a direct comparison between the cumulative curves. Furthermore, in such cases, the reconstructed profile may have advantages other than furnishing direct information on the mean F distribution in a population sample; comparisons could then be made on the basis of total mean F within any specified weight (depth) limits. The mean total F would be given by a simple analytical integration, as indicated in equation (7). A possible advantage of such a procedure is that it would allow to establish empirical correlations between caries scores and mean total F present up to various depths in the enamel. Thus, it would provide means to clarify whether the critical location of F in enamel is at the very surface, or if its cariostatic effect derives from enhanced concentrations up to a
EVALUATION
OF BIOPSY
DATA
IN HUMAN
ENAMEL
FLUORIDE
STUDIES
1425
given depth. Presently, this procedure would be impaired by the large standard errors in the calculated profile concentration as found in this work. The results reported here on comparison of homologous teeth show that the F contents and its distributions are Amost identical in the two maxillary central incisors. There is no reason to expect actual differences in the fluoride status of paired central incisors; therefore, it must be concluded that equivalent sampling areas were used by the operator on the left and on the right teeth. In contralateral teeth other than the central incisors, post-eruptive factors, such as toothbrushing, could bring about differences in the enamel fluoride. However, it is difficult to conceive that in young individuals, as the participants in the present study, such post-eruptive factors could account for large differences. Therefore, it seems reasonable to conclude that the observed difference between the paired lateral incisors is due to utilization of a larger sampling area on the left than on the right teeth. It is recognized that the difference found between the molars did not reach the level of significance. However, in view of the conservative statistical test used here, and the marked trend apparent in Fig. 7, the statistical result should be taken cautiously. Differences in the sampling areas of contralateral teeth are most probably caused by the different position of the operator relative to the tooth surfaces in the left and in the right side of the jaw. The results reported in this paper indicate that, because of operative inconsistencies difficult to overcome, only the central incisors can definitely be used interchangeably and as matched pairs in comparative studies. It is likely, however, that by prolonged training of an operator or by standardizing the sampling area other homologous teeth would lend themselves for such a use. Acknowledgement-The cooperation of Barnstable County Health Department and Salem Board of Health is gratefully acknowledged. The work was supported by U.S.P.H. Grant DE 2183. R&run&-Un traitement des resultats de biopsies permet d’effectuer des comparaisons du fluor de l’email de groupes exp&imentaux, lorsque les biopsies sont realis& a diverses profondeurs et leur r&partition pond&ale est ditRrente pour Ies groupes Studies. La facon de retrouver le pro6l de concentration moyen du F, a partir de resultats de biopsies, est indique. Le fluor de P&nail est compare entre des incisives centrales superieures symetriques, des incisives laterales et des premieres molaires. Le F de l’email dans les incisives centrales est identique, a la fois en concentration et en distribution, selon la profondeur, et ce contenue est significativement different dans les incisives la&ales. Le. contenu en F des molaires droites a tendance &Ctre plus eleve que c&i des molaires gauches, mais la difI6rence n’est pas significative. I1 apparait qu’avec la methode de biopsie utilis&, settles les incisives cedntrales peuvent Ctre utilisees dans des etudes comparatives. Les differences observees sont probablement en rapport avec des difficultes de recueillir des &hantillons identiques sur des dents controlaterales. Zusammenfassung-Bioptische Daten erlauben nach geeigneter Behandlung den Vergleich von Fluorid im Schmelz, such wenn die einzelnen Biopsien bis zu verschiedener Tiefe entnommen werden und wenn sich ihre Gewichtsverteilungen zwischen den zu vergleichenden Gruppen unterscheiden. Es wird such gezeigt, wie das Protil der mittleren Fluoridkonzentration aus bioptischen Daten rekonstruiert werden kann. Schmelzlluorid wurde in den kontralateralen mittleren Schneidetihnen, seitlichen Schneideziihnen und ersten Molaren des Oberkiefers verglichen. In den mittleren Schneidezihnen war der Schmerlziluoridgehalt gleich, sowohl in der Gesamtmenge A.O.B. 16/12--o
1426
R. AASENDENAND E. C. MORENO als such in der Verteihmg zur Tiefe, und es unterschied sich signskant im Fall der lateralen SchneidezSihne. Es gab einen deutlichen Trend zu haherem Fluoridgehalt der rechten Molaren gegeniiber ihren antimeren, ohne jedoch signifikante Werte zu erreichen. Mit der verwendeten Biopsiemethode kbnnen deshalb nur mittlere Schneideziihne wechselweise und als gemischte Paare in VergleicMstudien verwendet werden. Die im Schmetzfluoridgehalt beobachteten Unterschiede waren vor allem auf Schwierigkeiten zuriickzufiihren, an den kontrolateralen ZBhnen gleiche Bezirke zuc Entnahme der Probe zu benutzen. REFERENCES
AASENDEN,R., ALLUKIAN,M., BRUDEVOLD,F. and WELLOCK,W. D. 1971. An in vivo study on enamel fluoride in children living in a fluoridated and in a non-fluoridated area. Archs oral Biol. 16, 1399-1411. BRUDBVOLD,F., GARDNER,D. E. and SMITH,F. A. 1956. The distribution of fluoride in human enamel. J. dent. Res. 35, 420-429. BRUDEVOLD,F. and SPIREMARK, R. 1967. Chemistry of the mineral phase of enamel. In: Structural and Chemical Organization of Teeth, Vol. 2. Chav. 18. Academic Press Inc.. New York. BRUDEVOLD,F., MCCANN, fi. G. and GR& P. 1968. An enamel biopsy method for determination of fluoride in human teeth. Archs oral Biol. 31,877-885. BRUDEVOLD,F., AASENDEN,R., MCCANN, III, H. G. and MCCANN, H. G. 1969. Use of an enamel biopsy method for determination of in vivo uptake of fluoride from topical treatments. Caries Res. 3,119-133. FRANT, M. S. and Ross, J. W., JR. 1966. Electrode for sensing fluoride ion activity in solution. Science 154,1553-1554. HOTZ, P., MUHLEMANN,H. R. and SCHAIT,A. 1970. A new method of enamel biopsy for fluoride determination. Helv. odont. Acta 14, 1-5. KARLSTR~M,S. 1931. Physical, Physiological and Pathological Studies of Dental Enamel. Fahlkranz Boktryckeri, Stockholm. MCCANN, H. G. 1969. Determination of fluoride in mineralized tissues using the fluoride ion electrode. Archs oral Biol. 13,475-477. MELLBERO,J. R., NICHOLSON,C. R., MILLER, B. G. and ENGLANDER,H. R. 1968. Acquisition of fluoride in vivo by enamel from repeated topical sodium fluoride applications in a fluoridated area: A preliminary report. J. dent. Res. 47, 733-736. MELL~R, J. W. 1955. Higher Mathematics for Students of Chemistry and Physics, pp. 311-312 and 335-340, 4th edn. Dover Publications, New York. MUHLEMANN;H. R., SCHAIT, A. and KO&IG, K. G. 1964. A chemical method for the removal of enamel surface layers and its suitability for fluoride analysis. Helv. odont. Acta 8,147-153.
EVALUATION
OF BIOPSY DATA IN HUMAN FLUORIDE STUDIES R.
AASENDEN
ENAMEL
and E. C. MORENO
Forsyth Dental Center, Boston, Mass. 02115, U.S.A. Summary-Atreatment of biopsy data is given that permits comparisons of enamel F in experimental groups when the biopsies are taken to varying depths and their weight distributions are different for the groups under comparison. It is also shown how the mean F concentration profile can be reconstructed from biopsy data. Enamel F was compared in maxillary contralateral central incisors, lateral incisors, and first molars. It was found that the enamel F in the central incisors was equal, both in content and distribution with depth, and it was significantly different in the case of the lateral incisors. There was a marked trend for the F content of the right molars to be higher than that of the antimeres, but a level of significance was not reached. It was concluded that, with the biopsy method used, only central incisors can definitely be used interchangeably and as matched pairs in comparative studies. The observed differences in enamel F were most likely due to difficulties in utilizing equal sampling areas on contralateral teeth. INTRODUCTION
PRESENTknowledge of the F content of enamel has been derived mainly from layer analyses of extracted teeth. Recently, biopsy methods have been reported (BRUDEVOLD, MCCANN and GRIZIN,1968; HOTZ, MUHLEMANNand SCHAIT, 1970) by which F in surface enamel can be determined in uiuo; thus population samples involving substantial numbers of subjects can now be studied. Comparisons of F levels between groups have usually been made on the basis of the group mean F concentrations of enamel samples. Since there is an F gradient in enamel (BRUDEVOLD,GARDNERand SMITH, 1956; BRUDEVOLD, MCCANN and GRIN, 1968), such comparisons would be strictly valid only if all the samples were taken to a uniform depth. So far, there is no method available to attain complete uniformity in sampling depth, neither of in vitro nor of in vivo samples. A procedure using graphical interpolation in order to make comparisons at the same depth between samples taken to variable depths has been reported (MUHLEMANN,SCHAITand KONIG, 1964; MELLBERGet al., 1968). This procedure can be applied only when successive samples are taken from each tooth. Furthermore, linearity of the F gradient within the sampled layer must be assumed; unless the sampled layers are extremely thin, this assumption is questionable in the light of the results reported here. In this paper, a method for treatment of data is given that permits comparisons between groups when the samples are taken to varying depths and their weight distributions are different for the groups under comparison. By means of this method, meaningful comparisons can be made on the basis of a quantity referred to hereafter as “fluoride parameter”. The method involves subgrouping of the data, and, therefore, it requires fairly large population samples. For this reason, it is applicable mainly to 1413
1414
R. AASENDENAND E. C. MORENO
biopsy data. Clarification of the nature of the relationship between enamel F and caries calls for not only exact evaluation of differences in enamel F contents between experimental groups, but also for information on its distribution with depth. It will be shown here that biopsy data can be used to reconstruct the mean enamel F concentration profiles in experimental groups. In the biopsy method developed by BRUDEVOLD et al. (1968), the sampling area is not determined. Therefore, in comparative studies, the underlying assumption is that the mean sampling areas of the groups involved are equal. This condition may be closely fulfilled when comparisons are made on the basis of biopsies taken by the same operator from the same tooth in all subjects. This design was used in the surveys reported by AASENDEN et al. (1971). In F uptake studies testing topical agents, sampling area variations have been minimized by assessing the F increment as the difference in the F contents of samples taken from the same teeth before and after treatment (BRUDEVOLD et al., 1969). Successive samples are, however, not physically independent, a circumstance that makes questionable the application of usual statistical tests to the data. An alternative design that would be advantageous in comparative studies of this type would be the use of contralateral teeth as matched pairs. A critical study of this possibility is included in the present paper. It will be shown that, of the pairs studied, only the upper central incisors lend themselves for such a use with the method employed. EXPERIMENTAL
METHODS
Calculation of sampling depth In the biopsy method presently used the area, S, of the sampled surface, is not determined; thus, only relative sample depths (inversely proportional to the sampling area) can be calculated. The weight, IV, of each enamel sample is calculated from the corresponding analytical figure for its calcium content. It is assumed that the calcium content and the density of enamel are 37 per cent (BRUDEVOLDand &REMARK, 1967) and 2.95 g/cm3 (KARLSTRBM,1931), respectively. Therefore, assuming uniform abrasion of the sampling area, the depth, D, in pm (micrometers), to which a sample was taken is wx loo0 D = S x 2.95 where W is given in milligrams and S is given in square millimeters. As an example, for W = 0.10 mg, the sampling depth would be approximately 34/S pm. Grouping of enamel samples The calculated weights of the enamel samples are grouped into at least three classes according to their distribution plot. Limits for the class intervals are selected so that (a) enough samples are comprised within each category to compensate for variation in sampling areas and other relevant factors such as age and F exposure, and (b) an even distribution of samples is obtained around the midpoint of the class limits. To fulfill these criteria, it is often necessary to discard a few samples falling in scattered tails in the distribution plot. When the sample weight-distribution permits the selection of equal class intervals, the calculation of the fluoride parameter (and its standard error) is substantially simplified. If the enamel samples displayed a perfect weight distribution around the midpoint of each class interval. the functional relationship between mean F concentration and sample weight would be the same whether the midpoint or the actual mean weight of the intervals were considered as the variable. In real cases, however, the values of the actual mean weight in each class are often found to depart slightly from the values of the class interval midpoints. This departure is illustrated in Table 1 for biopsies of the left central incisors from a non-fluoridated area (see section on homologous teeth for
EVALUATIONOFBIOPSYDATAINHUMAN
ENAMELFLUORIDE
1415
STUDIES
25-
--
Mean weight
I
0.18 s_! S
1
Weight, Depth.
mg pm
FIG. 1. Mean F concentration of enamel samples, grouped by weight, as a function of sample weight (depth to which the samples were taken). Biopsies from the maxillary left central incisor in 64 subjects from a non-fluoridated area. Vertical bars represent standard errors.
clinical and analytical procedures). Consequently, as illustrated in Fig. 1, the dependence of the mean F concentration, c, on weight differs depending on which of the two variables is selected. It can be shown, however, that relatively small differences as those depicted in Fig. 1 do not affect significantly the calculated value of the fluoride parameter, provided the same weight interval is considered. Thus, for most practical purposes, the use of class interval midpoints is amply justified. TABLE
~.MEAN
Sample weight (mg) Interval midpoint Class mean weight N Mean [Fl (ppm)
F
CONCENTRATIONSOFENAMELSAMPLES,GROUPED BY WEIGHT,FROMTHE MAXILLARY LEFT CENTRALINCISORIN 68 SUBJECTS
0*09-<0*16 0.125 0.126 21 2213 + 170*
0*16-co.23 0.195 0.187 25 1971 f 113
0*23-
>0.30
4
* Standard error. Comparison of groups 1. Calculation offluoride parameter. The fluoride parameter, FP, is detined as the area enclosed by the two end ordinates, the x-axis and a curve fitted to the class mean F concentrations in plots such as those in Fig. 1. This area can be determined readily by approximate integration, using the Newton’s interpolation formula (MELLOR, 1955), when equal class intervals are used, i.e. when the midpoints of equal class intervals are used as the variable. For cases where three or four classes are used in the grouping of the enamel samples, the fluoride parameter is given by equations (1) and (2) respectively.
in which c” is the mean F concentration of the nth class and h is the value of the weight interval which is the same for all the classes. The variance, V., in the fluoride parameter can be obtained by propagation of errors on the equation used for the calculation of the area. In general,
1416
R.
AASENDEN
AND E. C. MORENO
in which EE. is the standard error in the mean F concentration of the nth class. The variances and standard errors in the fluoride parameters when 3 or 4 classes are used in the grouping of samples are obtained by applying equation (3) to equations (1) and (2), respectively. Thus
E3
=
v, =
+
(EdI
tv’43
2$
[E*E, + 9E*-,, + 9E*e, + EKES] = g
. #4(EE) (5)
The use of the fluoride parameter de&red in equation (1) or (2) is restricted to those cases where the class limits are common to the groups under comparison. When such is not the case, an approximate integration of the cumulative curves can be made by the use of the trapezoid rule. This geometrical procedure, although not as accurate as the integration using Newton’s interpolation formula, offers the simplest method to calculate the FP when the classes are unequally spaced or when the groups to be compared do not have common class limits. When the class limits are not common to both groups, the FP is calculated using limits encompassed by both cumulative curves, i.e. without using extrapolation of either curve. The standard error is then calculated by applying the general equation (3) to the case in question. By way of illustration, the fluoride parameter corresponding to the set of data in Table 1 can be calculated by the use of equation (1) and its standard error by equation (4). The calculations yield the values FP = 0*2T3 and E = 0*O12. The subscript numerals are not significant digits; however, they have been carried on in the statistical tests. The F concentrations in Table 1, also, are given with more significant digits than justified by their uncertainties for the purpose of calculation verification. 2. Reconstruction of fluoride concentration profile. The functions plotted in the example of Fig. 1 should be clearly distinguished from the functions representing the F concentration profiles in the enamel. Whereas each point in the cumulative curves of Fig. 1 contains “integrated” information on the F content from the surface of the enamel to the depth (or weight) plotted, a point on a concentration protie represents a “differential” property, namely the F concentration in a plane at that particular depth. The relationship between these two kinds of functions is treated here first with a hypothetical example and then with actual biopsy data. In Fig. 2 is shown a hypothetical F profile (curve 1) in enamel; arbitrary units are used in the coordinates. This curve is described by the equation Cr = 9.853 - 4.428 W + 0.879 W* - 0.059 W3
(6)
with a degree of determination of O-998. A biopsy taken to a weight (depth) of W,, units will contain a total amount, F,, of fluoride given by W, F. =
s 0
C, dW = 4 (W.).
(7)
Corresponding to this value of F. there is a point in the cumulative plot (curve II) which represents the mean F concentration, c., of the biopsy sample taken to weight (depth) W.. This mean concentration is given by c _ &W”) n (8) K and it is plotted at the abscissa W.. It is seen then that, in this particular example, the cumulative function is c = 9.853 - 2.214 W + 0.293 W* - 0.015 W3. (9)
EVALUATION
OF BIOPSY
I
I
DATA
/
2
IN HUMAN
ENAMEL
STUDIES
1417
I
I
3
FLUORIDE
4
Weightor depth,
5
6
7
arbitrary units
FIG. 2. Hypothetical F concentration profile in enamel, curve I, and its corresponding cumulative curve, curve II. Solid curves I and II plotted according to equations (8) and (1 l), respectively. Points on curve I calculated from points in the cumulative curve by the procedure given in this paper. Vertical bars represent standard errors.
It becomes clear that, if the c function were known in actual cases, the function describing the real profile could be easily obtained multiplying C?by Wand differentiating with respect to W. In practice, however, it is unlikely that the form of the c function can be determined adeauatelv. The concentrations calculated from chemical analyses on biopsy samples correspond then to points on the cumulative curve. We can simulate actual conditions by dividing the weight range in the cumullative curve into classes and considering various arbitrary points (individual samples) within each class. In the present example, six classes were selected and from four to six points were considered in each class. The mean fluoride concentration, c?,,, in the nth class is
in which F1 and W, are the total fluoride and the weight of the ith sample in the class, respectively. As shown in Fig. 2, these mean concentrations are plotted at the actual mean weights of the respective classes, that is, at Wn = (7w)/(N.). There is a standard error associated with each C?,,.In the present example, the standard error stems only from the presence of a gradient within each class; in actual cases the error also reflects differences in sampled surface area in addition to the individual variation in F concentrations. In Table 2 are given the I?,,values calculated by equation (10) (fourth column) for the various classes. These figures are to be compared with those in the third column which were calculated by equation (9) using the values of W given in the second column. It is apparent that the differences in the two sets of values are negligible. Thus, the assumption of a linear gradient within each class, tacitly made in equation (IO), does not introduce a significant error. The total mean fluoride, Fo,“, sampled from the surface to a mean weight W, is obtained by multiplying the figures in the fuurth column of Table 2 by their corresponding mean weights (second column); the results of these operations are given in column 5. Now we proceed to the reconstruction of the fluoride profile.
R.
1418
AA~ENDEN AND
E. C. MORENO
TABLE 2. MEAN F CONCENTRATIONS AND MEAN F WEIGHTS FOR WEIGHT INTERVALS IN THE CUMULATIVE CURVE OF FIG. 2
Weight class
Class mean weight, w
Cmfrom (9)
l-2 2-3 3-4 4-5 5-6 6-7
1.37 2.51 3.56 4.53 5.57 6.55
7.330 5.908 5.018 4.464 4.061 3.775
C” from (10) 7.338 5.923 5.019 4.469 4.067 3.775
i f f f rt &
Mean total F sampled from surface to W'. 10.090 14.867 17.868 20.245 22.651 24.728
o-183* 0.149 0.116 0.088 0.073 0.034
f 0.251 f O-375 ho.413 + 0.398 ztO.407 & 0.224
* Standard error. The total mean fluoride, F., n--l, present in the interval between the mean weights W. and W.-r is simply F., n--l = Fo,. - Fo,n-1 (11) If we assume a linear gradient between the mean weights r” and kp”_, in the fluoride profile, we can define a concentration, CF: F CF = W” L-i”_; for a point in the profile having an abcissa W = W,,_, + (W,- W&/2. The profile concentrations calculated by equation (12) have standard errors that can be estimated by propagation. Thus ECF
=
d(wn’ E 2Cn+ w*,-l EZ,,,). W” - W”4
The results of the calculations of mean total fluorides and profile concentrations by the use of equations (11) and (12), respectively, are shown in Table 3. The profile concentrations and their standard errors are plotted in Fig. 2. Clearly, the calculated points represent very closely the concentrations of the true profile. The relatively large magnitude of the standard errors relates to (a) the small number of samples within each class considered in the present example, and (b) the small differences between quantities of the same magnitude (equation 11) for which the errors are of an additive nature, as shown in equation (13). TABLE 3. MEAN F IN WEIGHT INTERVALS AND F CONCENTRATIONS IN PROFILE OF FIG. 2.
Mean weight interval (curve II)
Plotted weight (curve I)
O-l.38 1.38-2.51 2.51-3.56 3.56-4-53 4.53-5.57 5.57-6.55
0.69 1.94 3.03 4.05 5.05 6.06
* Standard error.
Mean F in weight interval
IO.090 4.777 3.001 2.377 2.406 2,077
f 0*251* ho.451 f 0.558 f 0.574 f 0.569 f 0.464
Profile concentrations
7.338 4.209 2.858 2.450 2.314 2.119
ztO.183 f 0.397 f 0.531 & 0.591 + 0.547 + 0.474
EVALUATlONOFBIOPSYDATAINHUhfAN
I
I
ENAMELFLUORIDESTUDIES
I
0.10
O-20
Weight,
34
Se
Depth,
S
1419
S
mg pm
FIG. 3. Cumulative F curve, A, and reconstructed F concentration profile, B, obtained from enamel biopsies of the maxillary left central incisor in 64 subjects from a nonfluoridated area. Vertical bars represent standard errors. The foregoing procedure for the profile reconstruction was applied to the example in Fig. 1. The reconstructed profile is shown in Fig. 3 in which the cumulative curve is also included for comparative purposes. It is apparent that the standard errors associated with the reconstructed profile are much larger than those associated with the cumulative curve. The possibility of using reconstructed profiles for statistical evaluation in comparative studies and the limitations of such an approach are commented on in the Discussion. 3. Statistical evaluation. Comparison of two independent groups on the basis of the fluoride parameter can easily be done by the use of the t-test. Through the computation of the group FP, n degrees of freedom (corresponding to the n number of classes into which the group was divided) are lost; thus, the appropriate number of degrees of freedom associated with the t-value is Ni + Nz - 2p1, N1 and Nz being the total number of samples in Group I and Group II, respectively. Analysis of variance likewise can be applied to test for differences when more than two groups are involved. The full advantage of matched pairs cannot be realized in biopsy studies unless the paired enamel samples are taken to the same depth. The use of a t-test for independent samples, as it is done in the present study, constitutes a conservative statistical criterion. Thus, whereas significant differences can be validly judged on the basis of such a test, only a cautious conclusion can be reached when the differences do not attain the significance level associated with the test. Application of the data treatment to studies on homologous teeth The data treatment was applied to two separate studies. Study I comprised 68 subjects from a nonfluoridated area (Hyannis, Massachusetts, water F < 0.1 ppm); Study II comprised 75 subjects from a fluoridated area (Salem, Massachusetts, water F: I ppm). The subjects were 9-l 1 yr old. These groups constituted part of the population samples described in detail by AASENDEN et al. (1971). Enamel biopsies were taken from the intact labial surfaces of the four maxillary incisors and from the areas mesial to the buccal grooves on the buccal surfaces of the maxillary first molars. All the biopsies were taken by one operator who was informed of the results of Study I before undertaking the sampling in Study II. Prior to the enamel sampling, the tooth was cleaned with a rubber cup and insoluble sodium metaphosphate in order to remove soft deposits. The enamel sample was obtained following the procedure of BRUDEVOLD et al. (1968). By this procedure enamel is polished off with a rotating midget felt cone impregnated with silicon carbide. The cone is coated with glycerine which traps and forms a slurry with the ground enamel particles. The slurry and the felt cone are transferred to a plastic test tube and subjected to F and Ca analyses. The technique was aimed at utilization of the largest possible sampling area, which of necessity was smaller than the defined area available for sampling; felt worn off the cone occupied several square millimeters adjacent to the gingiva and
1420
R.
AASENDEN
AND
E. C. MORENO
margins of approximately 1 mm were left unsampled mesially and distally in order to avoid contamination from neighbouring teeth. Fluoride in the enamel sample was determined by means of a lanthanum membrane electrode (FRANTand Ross, 1966) following the method described by MCCANN (1968). Calcium was determined by atomic absorption spectrophotometry. The F concentration in the sample was calculated on the basis of the sample weight which, in turn, was calculated from the calcium analysis (see calculation of sampling depth). The uncertainties in the fluoride and calcium determinations were estimated as rt2 and f3 per cent of the amounts analysed, respectively. In all cases, some samples at the tails of the weight distributions were discarded in order to achieve grouping of the enamel samples with common classes for the two teeth under comparison. The mates of the samples discarded in one tooth group were identified and excluded also in the other tooth group. This exclusion was not necessary for the statistical treatments, since t-tests for independent samples were applied. However, the inherent pairing of the two sets of data eliminates variations in several factors, such as age, sex and tooth size. RESULTS
Cumulative F plots for the central incisors in Study I are shown in the upper portion of Fig. 4. The curves represent results obtained with 57 pairs. Eleven pairs had to be discarded because of the dissimilarity in the weight distributions of the two groups. It is apparent that the cumulative curves for both teeth follow a similar
,
Weight, mg Depth,
pm
FIG. 4. Cumulative F curves, A, and reconstructed F concentration profiles, B, obtained from enamel biopsies of the maxillary central incisors in 57 subjects from a non-fluoridated area. Vertical bars representing standard errors are shifted for clarity.
EVALUATlON
30
t
OF BIOPSY
DATA
--__
IN HUMAN
ENAMEL
STUDIES
1421
--_a
-_
Sample
FLUORIDE
weight,
mg
FIG. 5. Weight frequency distributions of enamel samples taken from maxillary lateral incisors and first molars in 68 subjects from a non-fluoridated area.
pattern and that the mean fluoride in the enamel displays a rather steep gradient. Indeed, the reconstructed profiles in the lower portion of Fig. 4 show that the mean F concentration for both teeth changes from about 2000 ppm to about 1000 ppm within the weight interval 0*06-0*23 mg. An approximate value for the sampling area in these teeth is 40 mm2. Thus, the decrease in fluoride concentration previously mentioned occurs within an approximate depth interval of 0.5-2 pm. The fluoride parameters and their standard errors for the two central incisors were calculated according to equations (1) and (4), respectively. The difference between the two values, 0.25, f 0*013 for the right and 0~26~ f 0.01, for the left tooth, was non-significant. The weight distributions of the enamel samples from the two other pairs of contralateral teeth in Study I were so dissimilar that it was impossible to achieve a grouping of samples conducive to meaningful comparisons. The frequency distribution curves shown in Fig. 5 clearly illustrate the situation. Any attempt to group the samples within the common weight ranges would result in classes having so few samples that statistical evaluation would be meaningless. The cumulative F plots for the central incisors in Study II are shown in the upper portion of Fig. 6. In this case the criteria used in the grouping of samples were best fulfilled by the use of four classes. Only three pairs had to be discarded in the grouping process. The two cumulative curves, as in the case of the central incisors in Study I, are very close to each other; consequently, the two fluoride parameters, calculated by equation (2), are practically identical. The closeness of the points in the cumulative plots is indicative of a similar distribution of fluoride in enamel with depth. Such similarity is clearly shown by the reconstructed F concentration profiles in the lower portion of Fig. 6. The two profiles display steep concentration gradients similarly to the profiles for the central incisors in Study I shown in Fig. 4. The higher F concentrations in the profiles of Study II reflect the origin of the experimental subjects, namely their living in a fluoridated area. A detailed study on the differences in enamel F of the two population samples is reported by AASENDEN er al. (1971).
R. AASENDENAND E. C. MORENO
1422
I-
I 0.10 34
s
I 0.20
se
S
:
, Weight, mg Depth, pm
FIG. 6. Cumulative F curves, A, and reconstructed F concentration profiles, B, obtained from enamel biopsies of the maxillary central incisors in 72 subjects from a fluoridated area. Vertical bars representing standard errors are shifted for clarity.
The weight distributions of the biopsies from left and right central incisors in Study II were quite similar. Thus, in this particular case, comparison can be made on the basis of the grand means of the F concentrations for the two teeth. The grand means and their standard errors were 2670 f 70 ppm and 2680 f 70 ppm for the right and left tooth, respectively. Comparisons on the basis of these grand means or on the basis of the fluoride parameters yield equivalent results. In contrast to the results in Study I, the sample weights from the lateral incisors and first molars in Study II permitted their grouping into common classes for each pair. The cumulative plots for the two pairs are shown in Fig. 7. In the case of the lateral incisors, the difference in the fluoride parameters, O-35, f 0.Olr for the right and 0~31~ & O*Olo for the left tooth, is significant at the O-02 confidence level. It is not clear whether this difference relates to an actual difference in the F contents of the two matched teeth or to a difference in the group mean sampling areas. This matter will be treated in the Discussion. The weight distributions of the samples from the paired molars required a grouping with unequal intervals in order to achieve common mean weight classes as shown in the lower portion of Fig. 7. For this reason, the fluoride parameters were calculated
EVALUATION
OF BIOPSY
DATA
IN HUMAN
ENAMEL
FLUORIDE
STUDIES
1423
Fparameter
-i-
I
I
0 IO
0 20
Weight, mg
34 s
68 s
Depth, pm
FIG. 7. Cumulative F curves obtained from enamel biopsies of 71 paired maxillary lateral incisors and of 69 paired maxillary first molars in subjects from a fluoridated area. Vertical bars representing standard errors are shifted for clarity.
by approximate integration using the trapezoid rule. The difference between the two fluoride parameters, 0.21, f 0*010 for the right and 0.19, i_ 0.01, for the left molar, is not statistically significant. DISCUSSION
The treatment of biopsy data used in this paper for comparative studies on homologous teeth has two salient features: the grouping of samples (and their F concentrations) according to their weights, and the use of a fluoride parameter which provides a useful combination of the data for statistical evaluations. The marked F gradient in the outer few microns of enamel reported by other investigators (BRUDEVOLD et al., 1968) and in the present paper (see Figs. 4 and 6) invalidates, in our opinion, comparisons between groups based on the grand mean F concentrations of samples taken to depths varying within a relatively wide range. An exception to this statement, as mentioned in the results, is when the biopsy depth distributions of the groups under comparison are quite similar. In survey work, however, when the biopsies from different groups are taken at different times and under different conditions, dissimilarity in weight distributions may be the rule rather than the exception (AASENDENet al., 1971). The results reported in this paper show that even when the biopsies of contralateral teeth are taken at the same time, the weight distributions for the two teeth may differ greatly. Indeed, of the 6 pairs of contralateral teeth studied, only one pair (central incisors in Study II) displayed similar weight distributions. Indiscriminate use of grand means for comparative purposes may lead to erroneous conclusions; for example, on this basis, a comparison between the two central incisors
1424
R. AASENDEN
AND
E. C. MORENO
in Study I results in a highly statistical significant difference whereas a comparison between the molars in Study II yields a non-significant difference. Such conclusions are certainly not warranted when the dependence of F concentration on weight is taken into account as demonstrated by the cumulative curves and the reconstructed profiles (and standard errors) in Fig. 4 and by the cumulative curves in the lower portion of Fig. 7. The grouping of samples into common classes for the groups under comparison compensates for the differences in weight distributions. Comparisons between groups could of course be done on a class by class basis. For this purpose, it is necessary to select the class intervals in such a way that the actual mean weights of corresponding classes in the different groups are quite comparable; deviations of the class mean weights from the interval midpoints are of no concern. The shortcoming of a class by class comparison is the drastic reduction in degrees of freedom as contrasted when all the samples are used in the statistical test. Thus, as reported by AASENDEN et al. (1971), even in cases where there are substantial and consistent differences all along the cumulative curves, a class by class statistical analysis may not yield the conclusion that the population samples differed significantly. This difficulty is overcome by combining the data as done in the fluoride parameter. When comparing results obtained from different teeth in the same subjects, the combination of data becomes imperative, because the different enamel samples from a subject are only seldom found in a common class. The parameter introduced here has the following distinct advantages: (a) it combines the data considering the F concentration dependence on weight in samples taken to any depth within the experimental range, (b) its numerical value depends on both the mean F content and its distribution in the enamel of the given group, (c) it lends itself to statistical analyses, and (d) it can be obtained by simple calculations. Other combinations of data such as a simple sum of the class mean F concentrations do not display features (a) and (b) above. Conventional methods such as analysis of covariance could be applied to biopsy data if the form of the relationship between the biopsy weight and the F concentration could be determined adequately from the individual data. In the populations studied here, the variability in F concentration was so great that no formal relationship could be determined with the necessary reliability for adjustment of data. If, in other population samples, the coefficients in the function describing the F concentration dependence on biopsy weight can be assessed reliably, evaluation of differences between groups could also be made on the basis of a direct comparison between the cumulative curves. Furthermore, in such cases, the reconstructed profile may have advantages other than furnishing direct information on the mean F distribution in a population sample; comparisons could then be made on the basis of total mean F within any specified weight (depth) limits. The mean total F would be given by a simple analytical integration, as indicated in equation (7). A possible advantage of such a procedure is that it would allow to establish empirical correlations between caries scores and mean total F present up to various depths in the enamel. Thus, it would provide means to clarify whether the critical location of F in enamel is at the very surface, or if its cariostatic effect derives from enhanced concentrations up to a
EVALUATION
OF BIOPSY
DATA
IN HUMAN
ENAMEL
FLUORIDE
STUDIES
1425
given depth. Presently, this procedure would be impaired by the large standard errors in the calculated profile concentration as found in this work. The results reported here on comparison of homologous teeth show that the F contents and its distributions are Amost identical in the two maxillary central incisors. There is no reason to expect actual differences in the fluoride status of paired central incisors; therefore, it must be concluded that equivalent sampling areas were used by the operator on the left and on the right teeth. In contralateral teeth other than the central incisors, post-eruptive factors, such as toothbrushing, could bring about differences in the enamel fluoride. However, it is difficult to conceive that in young individuals, as the participants in the present study, such post-eruptive factors could account for large differences. Therefore, it seems reasonable to conclude that the observed difference between the paired lateral incisors is due to utilization of a larger sampling area on the left than on the right teeth. It is recognized that the difference found between the molars did not reach the level of significance. However, in view of the conservative statistical test used here, and the marked trend apparent in Fig. 7, the statistical result should be taken cautiously. Differences in the sampling areas of contralateral teeth are most probably caused by the different position of the operator relative to the tooth surfaces in the left and in the right side of the jaw. The results reported in this paper indicate that, because of operative inconsistencies difficult to overcome, only the central incisors can definitely be used interchangeably and as matched pairs in comparative studies. It is likely, however, that by prolonged training of an operator or by standardizing the sampling area other homologous teeth would lend themselves for such a use. Acknowledgement-The cooperation of Barnstable County Health Department and Salem Board of Health is gratefully acknowledged. The work was supported by U.S.P.H. Grant DE 2183. R&run&-Un traitement des resultats de biopsies permet d’effectuer des comparaisons du fluor de l’email de groupes exp&imentaux, lorsque les biopsies sont realis& a diverses profondeurs et leur r&partition pond&ale est ditRrente pour Ies groupes Studies. La facon de retrouver le pro6l de concentration moyen du F, a partir de resultats de biopsies, est indique. Le fluor de P&nail est compare entre des incisives centrales superieures symetriques, des incisives laterales et des premieres molaires. Le F de l’email dans les incisives centrales est identique, a la fois en concentration et en distribution, selon la profondeur, et ce contenue est significativement different dans les incisives la&ales. Le. contenu en F des molaires droites a tendance &Ctre plus eleve que c&i des molaires gauches, mais la difI6rence n’est pas significative. I1 apparait qu’avec la methode de biopsie utilis&, settles les incisives cedntrales peuvent Ctre utilisees dans des etudes comparatives. Les differences observees sont probablement en rapport avec des difficultes de recueillir des &hantillons identiques sur des dents controlaterales. Zusammenfassung-Bioptische Daten erlauben nach geeigneter Behandlung den Vergleich von Fluorid im Schmelz, such wenn die einzelnen Biopsien bis zu verschiedener Tiefe entnommen werden und wenn sich ihre Gewichtsverteilungen zwischen den zu vergleichenden Gruppen unterscheiden. Es wird such gezeigt, wie das Protil der mittleren Fluoridkonzentration aus bioptischen Daten rekonstruiert werden kann. Schmelzlluorid wurde in den kontralateralen mittleren Schneidetihnen, seitlichen Schneideziihnen und ersten Molaren des Oberkiefers verglichen. In den mittleren Schneidezihnen war der Schmerlziluoridgehalt gleich, sowohl in der Gesamtmenge A.O.B. 16/12--o
1426
R. AASENDENAND E. C. MORENO als such in der Verteihmg zur Tiefe, und es unterschied sich signskant im Fall der lateralen SchneidezSihne. Es gab einen deutlichen Trend zu haherem Fluoridgehalt der rechten Molaren gegeniiber ihren antimeren, ohne jedoch signifikante Werte zu erreichen. Mit der verwendeten Biopsiemethode kbnnen deshalb nur mittlere Schneideziihne wechselweise und als gemischte Paare in VergleicMstudien verwendet werden. Die im Schmetzfluoridgehalt beobachteten Unterschiede waren vor allem auf Schwierigkeiten zuriickzufiihren, an den kontrolateralen ZBhnen gleiche Bezirke zuc Entnahme der Probe zu benutzen. REFERENCES
AASENDEN,R., ALLUKIAN,M., BRUDEVOLD,F. and WELLOCK,W. D. 1971. An in vivo study on enamel fluoride in children living in a fluoridated and in a non-fluoridated area. Archs oral Biol. 16, 1399-1411. BRUDBVOLD,F., GARDNER,D. E. and SMITH,F. A. 1956. The distribution of fluoride in human enamel. J. dent. Res. 35, 420-429. BRUDEVOLD,F. and SPIREMARK, R. 1967. Chemistry of the mineral phase of enamel. In: Structural and Chemical Organization of Teeth, Vol. 2. Chav. 18. Academic Press Inc.. New York. BRUDEVOLD,F., MCCANN, fi. G. and GR& P. 1968. An enamel biopsy method for determination of fluoride in human teeth. Archs oral Biol. 31,877-885. BRUDEVOLD,F., AASENDEN,R., MCCANN, III, H. G. and MCCANN, H. G. 1969. Use of an enamel biopsy method for determination of in vivo uptake of fluoride from topical treatments. Caries Res. 3,119-133. FRANT, M. S. and Ross, J. W., JR. 1966. Electrode for sensing fluoride ion activity in solution. Science 154,1553-1554. HOTZ, P., MUHLEMANN,H. R. and SCHAIT,A. 1970. A new method of enamel biopsy for fluoride determination. Helv. odont. Acta 14, 1-5. KARLSTR~M,S. 1931. Physical, Physiological and Pathological Studies of Dental Enamel. Fahlkranz Boktryckeri, Stockholm. MCCANN, H. G. 1969. Determination of fluoride in mineralized tissues using the fluoride ion electrode. Archs oral Biol. 13,475-477. MELLBERO,J. R., NICHOLSON,C. R., MILLER, B. G. and ENGLANDER,H. R. 1968. Acquisition of fluoride in vivo by enamel from repeated topical sodium fluoride applications in a fluoridated area: A preliminary report. J. dent. Res. 47, 733-736. MELL~R, J. W. 1955. Higher Mathematics for Students of Chemistry and Physics, pp. 311-312 and 335-340, 4th edn. Dover Publications, New York. MUHLEMANN;H. R., SCHAIT, A. and KO&IG, K. G. 1964. A chemical method for the removal of enamel surface layers and its suitability for fluoride analysis. Helv. odont. Acta 8,147-153.