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An appraisal of the disc–macula distance to disc diameter ratio in the assessment of optic disc size
Ophthal. Physiol. Opt. Vol. 19, No. 5, pp. 365±375, 1999 # 1999 The College of Optometrists. Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain 0275-5408/99 $20.00 + 0.00
PII: S0275-5408(99)00018-6
An appraisal of the disc±macula distance to disc diameter ratio in the assessment of optic disc size D. B. Barr, C. R. Weir and A. T. Purdie Tennent Institute of Ophthalmology, Gartnavel General Hospital, 1053 Great Western Road, Glasgow G12 0YN, UK Summary The disc±macula distance to disc diameter ratio (DM:DD ratio) has been advocated as a method of supporting the diagnosis of optic nerve hypoplasia. A DM:DD ratio of 3.00 has been claimed to be a satisfactory threshold value for this purpose. This study has critically evaluated the above claim and found a value of 3.00 to be too low. The threshold DM:DD ratio values for the diagnosis of unequivocal ONH for an adult population, 5 and 2 years of age were found to be respectively 4.20, 3.93 and 3.70, the values for the diagnosis of mild ONH being 3.68, 3.44 and 3.23. Lower computed values reduce the predictive power. The method of computation of the DM:DD ratio was modified to abolish potential error due to disc rotation and foveal displacement. In an adult population, there was no correlation between the DM:DD ratio and amblyopia or disc ovalness. There was a trend of increasing DM:DD ratio towards myopia and decreasing DM:DD ratio towards hypermetropia; the DM:DD ratio may therefore be falsely high in high myopia. DM:DD ratio values below threshold should therefore be interpreted with care until formal optic disc biometry can be performed. # 1999 The College of Optometrists. Published by Elsevier Science Ltd. All rights reserved
Introduction
ring of hypo- or hyperpigmentation (double ring sign), accompanied by a variable thinning of the peripapillary nerve ®bre layer (Brodsky, 1994; Hoyt and Good, 1992). Diagnosis in less extreme degrees of hypoplasia, however, may be much more dicult when the above signs are less pronounced. Moreover, the rationale of diagnosis of ONH based solely on optic disc size may be erroneous since large discs may be axonally de®cient and small discs do not preclude normal visual function (Brodsky, 1994). Furthermore, ONH may be segmental in which the optic disc is not diusely diminished in size and often appears pink and otherwise healthy (Hoyt and Good, 1992). There are several modalities of investigation which may help in the diagnosis of ONH. Red-free examination of the nerve ®bre layer (FriseÂn and Holmegaard, 1978), A-scan (Acers, 1981) and B-scan (Boynton et al., 1975) ultrasonography, axial tomography of the orbital optic nerve (Acers, 1981) and optic canal (Boynton et al., 1975) and electrophysiological testing (Kriss and Russell-Eggitt, 1992) have all yielded characteristic abnormalities when the degree of hypo-
In the evaluation of a child or adult with subnormal visual function, the dierential diagnosis includes optic nerve hypoplasia (ONH). This is a developmental anomaly of the optic nerve in which the number of axons within the nerve is subnormal, whilst the mesodermal and glial elements are normal (Whinery and Blodi, 1963). ONH is not uncommon (Hoyt, 1986) and has been cited as the commonest optic disc anomaly encountered in clinical practice (Brodsky, 1994). The eect of ONH on visual function is variable (Peterson and Walton, 1977; FriseÂn and Holmegaard, 1978) with acuities ranging from normal to no light perception (Brodsky, 1994). Diagnosis on fundoscopic appearance may be easy in extreme cases (Hoyt and Good, 1992) in which the optic disc is overtly small, grey or pale in colour with a mottled peripapillary halo bordered by a Correspondence and reprints requests to: Dr D.B. Barr Received: 25 June 1997 Revised form: 5 February 1999
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plasia is extreme. They are, however, less useful in more subtle cases (Hoyt and Good, 1992). In addition, there are several methods by which the absolute dimensions of the optic disc may be estimated (optic disc biometry). The slit lamp may be employed in such estimations using a Goldmann contact lens (Franceschetti and Bock, 1950) or the +90 dioptre (D) condensing lens (Ruben, 1994), the latter of which may underestimate optic disc dimensions (Barr, 1995a). Fundus photogrammetric estimations from a 30-degree transparency (Romano, 1989) or from a photograph (Littmann, 1982; Bennett et al., 1994) have been employed, the accuracy of the latter having already been appraised (Barr, 1995b). Other methods involving stereoscopic video images (Caprioli et al., 1986), an indirect ophthalmoscopic system (Montgomery, 1991) and the confocal scanning laser ophthalmoscope (Nasemann and Burk, 1990) have been used. A photogrammetric method in which relative measurements from a fundus photograph are made has been advocated as reliable supportive evidence for the diagnosis of ONH (Wakakura and Alvarez, 1987; Alvarez et al., 1988; Zeki et al., 1991). These relative measurements are computed to give the disc±macula distance to disc diameter ratio (DM:DD ratio). The DM:DD ratio has been de®ned (Wakakura and Alvarez, 1987) as the sum of the distance from the fovea to the temporal margin of the optic disc and half the horizontal disc diameter divided by the mean disc diameter (Figure 1). It can be seen that when the distance between the fovea and the disc centre is a
Figure 2. Fundus photographic measurements (F, fovea; c1, distance between vertical meridians through the fovea and the disc centre; c2, vertical foveal displacement; a1, major elliptical axis; a2, minor elliptical axis).
constant, the DM:DD ratio increases as the disc becomes smaller. Therefore, the greater the DM:DD ratio, the greater the liklihood that the disc is dimensionally hypoplastic. It is this assumption that forms the basis of its use as a simple aid in the diagnosis of ONH when formal optic disc biometry is either not available or not tolerated. From a study of 18 normal children and ®ve with clinically diagnosed ONH (Alvarez et al., 1988), it was concluded that a DM:DD ratio >3.00 was highly suggestive of ONH and that false positives were unlikely with a value of >3.04. The purposes of this study were as follows: the measurement of the DM:DD ratio in a large population of adult subjects and comparison of the distribution obtained with other studies of the DM:DD ratio in paediatric and adult populations; an assessment of method accuracy and inter-observer variation; the examination of any correlation between the DM:DD ratio and ametropia, amblyopia and disc ovalness; a ®nal comment, based on the above ®ndings, on the likely usefulness of the DM:DD ratio as a clinical aid in the diagnosis of ONH, both in paediatric and adult populations.
Subjects and methods
Figure 1. Computation of the DM:DD ratio (Wakakura and Alvarez, 1987). (F, fovea; b, distance from the fovea to the temporal disc margin along an horizontal meridian; a1, vertical disc diameter; a2, horizontal disc diameter).
The case records of 125 subjects (107 male, 18 female; mean age 27.6 2 7.4 years, range 18±58 years) were obtained who attended between 1980 and 1992 for routine examination prior to the use of laser equipment at the University of Glasgow. Each patient had been refracted and best corrected Snellen acuities recorded. Motility, anterior segments, intra-ocular pressures and fundi had been examined and a fundus photograph obtained for each eye. Subjects were excluded from the study if there was any history of ocular disease and if any disease had been detected at the time of the original examination. Fundus photographs were excluded if the quality or ®eld of view prevented accurate measurements. Measurements were
Disc±macula distance to disc diameter ratio: appraisal: D. B. Barr et al. performed by two masked observers. Data from only one observer were used in all computations, except in the assessment of inter-observer variation. The fundus photographs were projected onto a screen and linear measurements made using callipers. Optic discs were assumed to be elliptical and the lengths of the major (a1, Figure 2) and minor (a2, Figure 2) axes measured. The distances from the centre of the disc (intersection of the major and minor elliptical axes) to the fovea along the horizontal meridian (c1, Figure 2) and from the fovea to this meridian along the vertical meridian (c2, Figure 2) were also measured. Computations were carried out for each eye as follows: (1) DM:DD ratio = 2c1/(a1 + a2) (2) disc ovalness = a1/a2 (3) foveal displacement (a) = tanÿ1 (c2/c1) The rationale behind the departure from the previous method of computing the DM:DD ratio will be explained subsequently. Adopting a previously described convention (Williams and Wilkinson, 1992), the angle (a) between the horizontal meridian passing through the disc centre and a line joining the disc centre with the fovea was recorded as positive when the fovea was below the horizontal meridian, and negative when it was above. Data were processed to compute a DM:DD ratio distribution and examine correlations between DM:DD ratio and ametropia, amblyopia and disc ovalness. The DM:DD ratio distribution for this population of 125 subjects was calculated by using data from one fundus photograph chosen at random from 104 subjects for whom two fundus photographs were suitable and from 21 subjects for whom there was only one fundus photograph. In examining correlation between the DM:DD ratio and ametropia, the data from the right eyes of 125 subjects were used and DM:DD ratio distributions were calcu-
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lated for six arbitrarily de®ned classes of ametropia spherical equivalent (S) (S Rÿ7.00D; ÿ7.00 < S Rÿ5.00; ÿ5.00 < S Rÿ3.00; ÿ3.00 < S Rÿ1.00; ÿ1.00 < S R+ 1.00; S> + 1.00). Inter-class correlations were assessed by paired twotailed t-tests and by one-way analysis of variance. Correlation between DM:DD ratio and amblyopia was examined by calculating the intereye DM:DD ratio dierence in 104 subjects for whom two satisfactory fundus photographs were available. Within this cohort, amblyopes were identi®ed as those subjects in whom there was a greater than or equal to two lines of best corrected Snellen visual acuity dierence. Intereye DM:DD ratio dierence distributions were calculated for the amblyopic and non-amblyopic groups and compared using a paired two-tailed t-test. Disc ovalness, de®ned as the ratio of the major and minor elliptical axes, was calculated for 229 eyes and its relationship with the DM:DD ratio assessed by regression analysis. In order to assess inter-observer variation, for each of 193 fundus photographs, data from one observer were compared with those from a second observer, the second observer performing measurements at a dierent time and masked as to the results of the ®rst observer. The paired DM:DD ratio means and dierences were then used to assess the bias and con®dence intervals of the measurements between the two observers.
Results For the 125 subjects in this study, the DM:DD ratio (mean 2 standard deviation) was 2.819 2 0.394. In measuring projected fundus images using callipers, the accuracy of so doing is 20.5 mm. The mean disc diameter for all measurements was 86.6 mm and so the accuracy of measurement was 20.6%.
Table 1. Ametropia classes and correlations with DM:DD ratio Ametropia classes (spherical equivalent (S), dioptres) SRÿ7.00 Number in class Ametropia mean Ametropia range DM:DD ratio ÿ7.00 < SRÿ5.00 ÿ5.00 < SRÿ3.00 ÿ3.00 < SRÿ1.00 ÿ1.00 < SR + 1.00 S> + 1.00
ÿ7.00 < S Rÿ5.00
ÿ5.00 < S Rÿ3.00
ÿ3.00 < S Rÿ1.00
ÿ1.00 < S R + 1.00
S> + 1.00
6 7 11 22 73 6 ÿ8.42 2.4 ÿ5.5 20.5 ÿ4.1 2 0.6 ÿ1.8 2 0.5 ÿ0.12 0.2 +2.6 2 2.0 ÿ12.75 to ÿ7.00 ÿ6.25 to ÿ5.00 ÿ4.75 to ÿ3.00 ÿ2.75 to ÿ1.00 ÿ0.875 to +0.125 +1.25 to +6.50 3.22 0.7 2.7 20.3 3.1 20.5 2.8 2 0.3 2.82 0.4 2.7 20.3 0.153 0.722 0.089 0.124 0.138
0.126 0.540 0.623 0.671
Inter-class DM:DD ratio correlations (p values) 0.103 0.128 0.816 0.097 0.322 0.364 (one-way analysis of variance, F = 2.272)
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For each class of ametropia spherical equivalent, the ametropia distribution, ametropia range, DM:DD distribution and inter-class correlations, as assessed by paired two-tailed t-tests, are shown in Table 1. It was found that, whilst there was a tendency for the DM:DD ratio to be lower in hypermetropia and higher in myopia, none of the inter-class correlations achieved statistical signi®cance. However, one-way analysis of variance indicated that the dierence between the DM:DD ratio means of the ametropia classes almost achieved statistical signi®cance (F = 2.272, p = 0.052). For the 104 subjects for whom two fundus photographs were available, the intereye DM:DD ratio dierence for the amblyopic group (n = 9) was 0.207 2 0.212 and for the non-amblyopic group (n = 95), the value was 0.209 2 0.183. The dierence between these two groups is not signi®cant (p = 0.979). There was found to be no signi®cant correlation between the DM:DD ratio and disc ovalness (r = 0.124, p = 0.061). Vertical foveal displacement, with respect to the disc, was +6.1082 3.8428. In assessing inter-observer variation, the means of 193 paired DM:DD ratios from two observers were plotted against the dierence between the same pair of measurements (Figure 3). The dierence between measurements of the DM:DD ratio was found to increase (absolutely) as the mean DM:DD ratio increased, and so bias and con®dence intervals were
Figure 4. Effect of disc rotation on computation of the DM:DD ratio. (F, fovea; y, angle of disc rotation).
calculated for mean DM:DD ratios of R3.00 (n = 142) and >3.00 (n = 51). There was negligible observer bias in the two groups, the mean dierences being respectively ÿ0.016 (95% con®dence interval ÿ0.064 to 0.032) and ÿ0.079 (95% con®dence interval ÿ0.236 to 0.078). The upper limit of agreement for mean DM:DD ratios R3.00 was 0.57 (95% con®dence interval 0.49 to 0.65) and the lower limit of agreement for this group was ÿ0.60 (95% con®dence interval ÿ0.69 to ÿ0.52). For mean DM:DD ratios of >3.00, the upper limit of agreement was 1.06 (95% con®dence interval 0.78 to 1.33), and the lower limit of agreement ÿ1.21 (95% con®dence interval ÿ1.48 to ÿ0.94).
Discussion
Figure 3. Inter-observer variation in DM:DD ratio estimation. The mean of a pair of DM:DD ratio measurements is plotted against the difference. (ÐÐ mean; --- upper and lower limits of agreement; - - - - - - boundaries of 95% confidence intervals for the means and limits of agreement).
In our study, the DM:DD ratio was calculated from: 2c1/(a1 + a2) (Figure 2), a departure from the original method which used: ((a2/2) + b)/((a1 + a2)/2) (Figure 1). The latter method may incur error due to disc rotation or vertical foveal displacement. Disc rotation (Figure 4) has been reported as 82.7 2 27.58 (Williams and Wilkinson, 1992). When the disc is rotated, unless elliptical axes are estimated, the disc diameters are measured by the tangent lengths a1' and a2'. The distance from the fovea to the temporal disc margin may be measured by the distances b1, b2 or b3 incurring mean errors of respectively 1+ 6%, +2% and +2% (Appendix A). The distance from the fovea to an horizontal meridian passing through the disc centre has been reported to be 0.58 2 0.30 mm (Williams and Wilkinson, 1992), corresponding to an angle of foveal displacement of +6.11 2 3.328. More
Disc±macula distance to disc diameter ratio: appraisal: D. B. Barr et al. extreme degrees of vertical foveal displacement (heterotopia) are uncommon (Cohen and Weisberg, 1950) but occur occasionally in craniofacial disorders associated with the tilted disc syndrome (Margolis et al., 1977, 1984; Margolis and Seigel, 1979). In our study, vertical foveal displacement was found to be +6.108 23.8428, in close agreement with the above study. In the presence of vertical foveal displacement, the distance from the fovea to the disc centre is always greater than the distance along an horizontal meridian through the disc centre. The overestimation (D) of the DM:DD ratio is: D = (1/cosa)ÿ1 (a = vertical foveal displacement) which is 13% for an extreme vertical foveal displacement of 148. Errors due to disc rotation and/or vertical foveal displacement would therefore seem to be insigni®cant. However, the DM:DD ratio has been assessed previously in an adult population (Williams and Wilkinson, 1992) and was 2.85 20.35 using the minor axis of the optic nerve head (as was done in our study) and 2.59 20.28 using the horizontal diameter of the optic nerve head, which may incur error when there is disc rotation. These two values are signi®cantly dierent (p < 0.001) which suggests that the theoretically small error introduced by disc rotation is signi®cant in practice. Errors due to both disc rotation and vertical foveal displacement are, however, eectively abolished by measuring the fovea±disc centre distance (c1, Figure 4) and computing the DM:DD ratio as: 2c1/ (a1 + a2). The DM:DD ratio distribution in the above study (Williams and Wilkinson, 1992) was 2.85 2 0.35 and in our study was 2.819 2 0.394. Both studies computed the DM:DD ratio using the same method and the resultant DM:DD ratio distributions are in close agreement. Having determined the optimal method of computing the DM:DD ratio, it is desirable to examine its predictive power in correctly classifying an optic disc as being hypoplastic. As stated, the DM:DD ratio is used to aid in the diagnosis of ONH when optic disc biometry is unavailable or not tolerated. It has therefore been used most as an aid in the diagnosis of ONH in children. A statement on the predictive power of the DM:DD ratio in this regard must apply both to paediatric and adult populations. The DM:DD ratio distribution in a paediatric population was 2.62 2 0.21 (Alvarez et al., 1988) which is signi®cantly dierent from adult distributions of 2.85 20.35 (Williams and Wilkinson, 1992) and 2.819 2 0.394 (our study). Since there are only two parameters used to compute the DM:DD ratio, there must be a signi®cant dierence in the fovea±disc centre distance and/or mean disc area between paediatric and adult populations to account for this dierence in DM:DD ratio distributions.
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Using least squares non-linear regression analysis of optic disc measurements performed at autopsy on 95 patients whose age ranged from 4.8 months gestational age to 21.9 years (Rimmer et al., 1993), it was concluded that 95% of the adult optic disc size is achieved by the age of 12 months and, by inspection of the data, 99% is achieved by the age of 18 months. This is supported by data which suggests that growth of the posterior scleral opening has ®nished at about 2 years of age (Quigley, 1982). Any dierence in DM:DD ratio distribution between paediatric (>18 months of age) and adult populations therefore cannot be due to a dierence in the mean disc diameter, and must be due to a dierence in the fovea±disc centre distance. After birth, the axial length of the globe (as measured by A-scan ultrasonography) increases in a triphasic pattern (Larsen, 1971; Gordon, 1985). Linear growth occurs between birth (16.61 20.60 mm) and 2 years (20.41 2 0.26 mm), between 2 years and 5 years (21.45 2 0.54 mm) and between 5 years and approx. 14 years (23.22 20.74 mm) when the globe has reached adult size (pooled data (Larsen, 1971; Gordon, 1985; Harayama et al., 1981)). There is therefore a biphasic stepwise linear increase in axial length after the age of 2 years, at which time the optic disc has reached >99% of adult size. The axial length is therefore 188% of adult size at 2 years of age and 194% at 5 years. Assuming uniform expansion of the globe to include the area between the fovea and the optic disc, and taking a chord distance between the fovea and the optic disc (that which is actually measured in computing the DM:DD ratio), this percentage change in axial length results in a change in the magnitude of the DM:DD ratio by precisely the same amount. We may therefore expect the DM:DD ratio distribution in a paediatric population to be 112% lower (2 years of age) and 16% lower (5 years of age) than that of an adult population (>14 years of age). Ametropia may have an indirect in¯uence on the DM:DD ratio, the distribution of ametropia being dierent in paediatric and adult populations. It may exert its in¯uence via its correlation with either axial length or disc area. Ametropia is not correlated with disc area (Bengtsson, 1980; Britton et al., 1987; Jones et al., 1988; Chi et al., 1989; Varma et al., 1994) and so this may be discounted. Paremmetropic ametropia is not correlated with axial length (Sorsby et al., 1962; Bennett and Rabbetts, 1984) (which, assuming uniform expansion of the globe, would be directly proportional to the fovea±disc centre distance). High refractive errors, however, are correlated with axial length, high myopia being correlated with a longer axial length and high hypermetropia correlated with a shorter axial length. For constant disc size, the eect of axial length in high ametropia may therefore be expected to result
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in a falsely high DM:DD ratio in high myopia (leading to misclassi®cation of a normal optic disc as one that is hypoplastic), the converse occurring in high hypermetropia. There was no inter-class correlation between ametropia and the DM:DD ratio in our study (Table 1) although there was a trend of increasing DM:DD ratio with increasing myopia. One-way analysis of variance indicated that the dierences in the means between the ametropia sub-classes was almost signi®cant. To calculate the predictive power of the DM:DD ratio in correctly classifying an optic disc as being hypoplastic, normal distributions of the mean disc diameter and fovea±disc centre distance must be known. From a meta-analysis (Appendix B) of studies that have presented data on disc area (Britton et al., 1987; Varma et al., 1994; Caprioli and Miller, 1987; Chihara and Chihara, 1994; Jonas and KoÈnigsreuther, 1994), in which all magni®cation corrections were made according to Littmann's method (Littmann, 1982), the disc area in an adult population is 2.60 2 0.50 mm2. Assuming circularity of the disc, the mean disc diameter (d) is calculated from: d = 2Z(A/p), where A is the disc area. This assumed disc circularity must be justi®ed. Ovalness (f), de®ned as the ratio of the vertical and horizontal disc diameters, in an adult population, is 1.127 20.108 (pooled data (Williams and Wilkinson, 1992; Chihara and Chihara, 1994; Williams, 1987)). A large proportion (99.73%) of the population will therefore have a disc ovalness of R1.451. The ratio (D) of mean disc diameter for an oval disc (f>1) and that for a circular disc (f = 1) is
independent of absolute disc area and is calculated from: ÿ p D f 1 1=f =2: For f = 1.451, D = 1.0188, which represents only 12% increase in the mean disc diameter for this extreme degree of ovalness. It is therefore considered justi®able to ignore disc ovalness when calculating mean disc diameter from disc area. The term `microdisc' has been used to de®ne unequivocal dimensional disc hypoplasia and is regarded as an optic disc whose area is less than or equal to the mean disc area minus two standard deviations (Jonas et al., 1989). For 99.73% of the population, the mean disc diameter will be between 1.18 and 2.29 mm. Unequivocal ONH may therefore be de®ned as one whose area is 1.60 mm2 or less, corresponding to a mean disc diameter of 1.43 mm. One of the authors (DBB) has performed optic disc biometry on 13 patients (aged between 5 and 87 years) with phenotypic ONH and it is accepted in our department that mild ONH is present when the disc area is 2.00 mm2 or less, corresponding to a mean disc diameter of 1.60 mm. The normal distribution of the fovea±disc centre distance is also required for the calculation of predictive power of the DM:DD ratio. From data acquired in a previous study (Williams and Wilkinson, 1992; Williams, 1995), this distance is 4.89 2 0.33 mm; 99.73% of the population will therefore have a fovea± disc centre distance between 3.90 and 5.88 mm. A DM:DD ratio of r3.00 (Alvarez et al., 1988) later
Figure 5. Relationship between the DM:DD ratio, mean disc diameter and the fovea±disc centre distance (c1).
Disc±macula distance to disc diameter ratio: appraisal: D. B. Barr et al. revised down to 2.94 (Zeki et al., 1991), has been advocated as supportive evidence for the diagnosis of ONH. Using the normal distributions for mean disc diameter and fovea±disc centre distance, the relationship between the DM:DD ratio, mean disc diameter and fovea±disc centre distance can now be expressed graphically (Figure 5). This shows the false positive and false negative proportions when a DM:DD ratio of 3.00 is chosen as the threshold value. The apparently high proportion of false positives is misleading: a more realistic impression may be gained from a threedimensional representation (Figure 6). The false positive (FP) and false negative (FN) proportions are calculated as shown in Appendix C. The true positive proportion (TP) proportion is calculated from: TP = 0.0213ÿFN and true negative (TN) from: TN = 0.9733ÿFP. Sensitivity (N) is calculated from: N = TP/(TP + FN) and speci®city (S) from: S = TN/ (TN + FP). The predictive power (V+) of the DM:DD ratio may now be calculated from: V + = (((Sÿ1)(Pÿ1)/NP) + 1)ÿ1, where P = prevalence (Spilker, 1991). The predictive power of the DM:DD ratio for unequivocal and mild ONH as a function of age is shown in Figure 7. In using the DM:DD ratio for the diagnosis of unequivocal ONH for an adult population, a DM:DD ratio of r4.20 will identify the condition with almost complete certainty, the values being 3.93 and 3.70 for 5 years and 2 years of age respectively. Intermediate age values may be estimated from
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the nomogram. Similarly, for the diagnosis of mild ONH for an adult population, a DM:DD ratio of r3.68 will identify the condition with almost complete certainty, the values being 3.44 and 3.23 for 5 years and 2 years of age respectively. As previously stated, it has been claimed (Alvarez et al., 1988) that a DM:DD ratio of >3.00 is highly suggestive of ONH and that false positives are unlikely with a value of >3.04. From Figure 7, it can be seen that, for the diagnosis of unequivocal ONH, a threshold DM:DD ratio value of 3.00 is too low, the predictive power is 140% for 2 years of age falling to 110% for an adult population indicating that >60% of patients identi®ed as having ONH (based on a value of 3.00) would have a disc area of >1.60 mm2 (false positives). However, the predictive power of a threshold value of 3.00 is better for the diagnosis of mild ONH, having a power of >80% for R6 years of age dropping to 150% for adults. This study therefore disagrees with the claim that false positives for the diagnosis of unequivocal or mild ONH are unlikely with a DM:DD ratio of >3.04. Inter-observer variation, in this study, was found to increase as the DM:DD ratio increases. For a mean DM:DD ratio of R3.00 between the two observers, the 95% con®dence interval was 20.59 which represents moderately good agreement but for a mean DM:DD ratio of >3.00, the 95% con®dence interval was 21.13 representing rather poorer agreement. Thus interobserver variation is greater, and therefore method reliability is poorer, when the patient has ONH. The two
Figure 6. Three-dimensional representation of the relationship between the DM:DD ratio, mean disc diameter and the fovea±disc centre distance (c1). Dark shading represents false positive proportion. Population percentage is arbitrary.
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Figure 7. Nomograms of the relationship between the DM:DD ratio, predictive power (V+) for diagnosis of unequivocal/mild ONH and age.
possible sources of this variation lie in identi®cation of the disc margins and the foveal position. It was not possible to determine which of the two contributed to the inter-observer variation for higher values of the DM:DD ratio since the two observers performed measurements at dierent times with dierent magni®cations of the fundus photographs. It is highly likely, however, that variation in interpretation of the foveal position from the fundus photograph is the more signi®cant contributor to inter-observer variation in DM:DD ratio calculation. Diculty in interpretation of the foveal position is likely to be a problem when there is foveal hypoplasia (in which the anatomical boundaries of the fovea are ill-de®ned) coincident with ONH which may occur in albinism (Spedick and Beauchamp, 1986; Oetting et al., 1994) and aniridia (Layman et al., 1974; Nelson et al., 1984).
Conclusions Estimation of the DM:DD ratio from a fundus photograph may be a simple, reproducible and useful clinical aid in the diagnosis of ONH. Computation of the DM:DD ratio is sensitive to the method employed and should be: fovea±disc centre distance/mean disc diameter (major and minor elliptical axes). This is theoretically more accurate and of practical importance. Low ametropia (24.00 D) is irrelevant but care should be taken in high myopia in which the DM:DD ratio may be falsely high. Reproducibility is poorer
with higher values of the DM:DD ratio; diculty in interpretation of the foveal position is likely to in¯uence reproducibility. Threshold DM:DD ratio values of 4.20, 3.93 and 3.70 for adults, 5 and 2 years of age respectively diagnose unequivocal ONH with almost complete certainty as do respective values of 3.68, 3.44 and 3.23 for the diagnosis of mild ONH. A threshold DM:DD ratio value of 3.00 is too low for the con®dent diagnosis of unequivocal ONH in any age group, but is better for the diagnosis of mild ONH, especially in younger patients. These limitations of the usefulness of the DM:DD ratio in the diagnosis of ONH should be appreciated until the patient is suciently old or capable of tolerating formal optic disc biometry. Acknowledgements The authors would like to thank Professor TD Williams, University of Waterloo, Ontario, Canada for providing data essential for this study and Messrs John McCormick and Iain Smith for illustrative work.
References Acers, T. E. (1981). Optic nerve hypoplasia: septo-opticpituitary dysplasia syndrome. Trans. Am. Ophthalmol. Soc. 79, 425±457. Alvarez, E., Wakakura, M., Khan, Z. and Dutton, G. N. (1988). The disc±macula distance to disc diameter ratio: a new test for con®rming optic nerve hypoplasia in young children. J. Pediatr. Ophthalmol. Strab. 25, 151±154.
Disc±macula distance to disc diameter ratio: appraisal: D. B. Barr et al. Barr, D. B. (1995a). Estimation of optic disc size (letter). Br. J. Ophthalmol. 79, 298±299. Barr, D. B. (1995b). An appraisal of the accuracy of Littmann's method of determining the real dimension of a retinal object. Acta Ophthalmol. Scand. , 73 (Suppl. 216). Bengtsson, B. (1980). The alteration and asymmetry of cup and disc diameters. Acta Ophthalmol. 58, 726±732. Bennett, A. G. and Rabbetts, R. B. (1984). Clinical Visual Optics, Butterworths. Bennett, A. G., Rudnicka, A. R. and Edgar, D. F. (1994). Improvements on Littmann's method of determining the size of retinal features by fundus photography. Graefe's Arch. Clin. Exp. Ophthalmol. 232, 361±367. Boynton, J. R., Pheasant, T. R. and Levine, M. R. (1975). Hypoplastic optic nerves studied with B-scan ultrasonography and axial tomography of the optic canals. Can. J. Ophthalmol. 10, 473±481. Britton, R. J., Drance, S. M., Schulzer, M., Douglas, G. R. and Mawson, D. K. (1987). The area of the neuroretinal rim of the optic nerve in normal eyes. Am. J. Ophthalmol. 103, 497±504. Brodsky, M. C. (1994). Congenital optic disc anomalies. Surv. Ophthalmol. 39, 89±112. Caprioli, J. and Miller, J. M. (1987). Optic disc rim area is related to disc size in normal subjects. Arch. Ophthalmol. 105, 1683±1685. Caprioli, J., Klingbeil, U., Sears, M. and Pope, B. (1986). Reproducibility of optic disc measurements with computerised analysis of stereoscopic video images. Arch. Ophthalmol. 104, 1035±1039. Chi, T., Ritch, R., Stickler, D., Pitman, B., Tsai, C. and Hsieh, F. Y. (1989). Racial dierences in optic nerve head parameters. Arch. Ophthalmol. 107, 836±839. Chihara, E. and Chihara, K. (1994). Covariation of optic disc measurements and ocular parameters in the healthy eye. Graefe's Arch. Clin. Exp. Ophthalmol. 232, 265±271. Cohen, I. J. and Weisberg, H. K. (1950). Vertical heterotopia of the macula. Arch. Ophthalmol. 44, 419±423. Franceschetti, A. and Bock, R. H. (1950). Megalopapilla: a new congenital anomaly. Am. J. Ophthalmol. 33, 227±234. FriseÂn, L. and Holmegaard, L. (1978). Spectrum of optic nerve hypoplasia. Br. J. Ophthalmol. 62, 7±15. Gordon, R. A. and Donzis, P. B. (1985). Refractive development of the human eye. Arch. Ophthalmol. 103, 785±789. Harayama, K., Amemiya, T. and Nishimura, H. (1981). Development of the eyeball during fetal life. J. Pediatr. Ophthalmol. Strab. 18, 37±40. Hoyt, C. S. (1986). Optic nerve hypoplasia: changing perspective. Aust. N.Z. J. Ophthalmol. 14, 325±331. Hoyt, C. S. and Good, W. V. (1992). Do we really know the dierence between optic nerve hypoplasia and atrophy? Eye 6, 201±204. Jonas, J. B. and KoÈnigsreuther, K. A. (1994). Optic disk appearance in ocular hypertensive eyes. Am. J. Ophthalmol. 117, 732±740. Jonas, J. B., Gusek, G. C. and Naumann, G. O. H. (1988). Optic disc, cup and neuroretinal rim size, con®guration and correlations in normal eyes. Invest. Ophthalmol. Vis. Sci. 29, 1151±1158. Jonas, J. B., Koviszewski, G. and Neumann, G. O. H. (1989). Morning glory syndrome and Handmann's anomaly in congenital macropapilla. Extreme variants of con¯uent optic pits. Klin. Mbl. Augenheilk 195, 371±374.
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Kriss, A. and Russell-Eggitt, I. (1992). Electrophysiological assessment of visual pathway function in infants. Eye 6, 145±153. Larsen, L. (1971). The sagittal growth of the human eye. IV. Ultrasonic measurement of the axial length of the eye from birth to puberty. Acta Ophthalmologica 49, 873±886. Layman, P. R., Anderson, D. R. and Flynn, J. T. (1974). Frequent occurrence of hypoplastic optic discs in patients with aniridia. Am. J. Ophthalmol. 77, 513±516. Littmann, H. (1982). Zur Bestimmung der wahren GroÈûe eines Objektes auf dem Hintergrund des lebenden Auges. Klin. Mbl. Augenheilk 180, 286±289. Margolis, S. and Siegel, I. M. (1979). The tilted disc syndrome in craniofacial diseases. ed. J. L. Smith. In: Neuro-ophthalmology Focus 1980, Masson, New York, pp. 97±116. Margolis, S., Siegel, I. M., Choy, A. and Breinin, G. M. (1977). Oculocutaneous albinism associated with Apert's syndrome. Am. J. Ophthalmol. 84, 830±839. Margolis, S., Aleksic, S., Charles, N., McCarthy, J., Greco, A. and Budzilovich, G. 1984. Retinal and optic nerve ®ndings in Goldenhar±Gorlin syndrome. Ophthalmology 91, 1327±1333. Montgomery, D. M. I. (1991). Measurement of optic disc and neuroretinal rim areas in normal and glaucomatous eyes: a new clinical method. Ophthalmology 98, 50±59. Nasemann JE, Burk ROW (Eds) 1990. Scanning laser ophthalmoscopy. Quintessenz, Munich. Nelson, L. B., Spaeth, G. L. and Nowinski, T. S. (1984). Aniridia: a review. Surv. Ophthalmol. 28, 621±642. Oetting, W. S., Summers, C. G. and King, R. A. (1994). Albinism and the associated ocular defects. Metabol. Ophthalmol. 17, 5±9. Peterson, R. A. and Walton, D. S. (1977). Optic nerve hypoplasia with good visual acuity and visual ®eld defects: a study of children of diabetic mothers. Arch. Ophthalmol. 95, 254±258. Quigley, H. A. (1982). Childhood glaucoma: results with trabeculectomy and study of reversible cupping. Ophthalmology 89, 219±226. Quigley, H. A., Brown, A. E., Morrison, J. D. and Drance, S. M. (1990). The size and shape of the optic disc in normal human eyes. Arch. Ophthalmol. 108, 51±57. Rimmer, S., Keating, C., Chou, T., Farb, D. M., Christenson, P. D., Foos, R. Y. and Bateman, J. B. (1993). Growth of the human optic disc and nerve during gestation, childhood and early adulthood. Am. J. Ophthalmol. 116, 748±753. Romano, P. F. (1989). Simple photogrammetric diagnosis of optic nerve hypoplasia. Arch. Ophthalmol. 107, 824±826. Ruben, S. (1994). Estimation of optic disc size using indirect biomicroscopy. Br. J. Ophthalmol. 78, 363±364. Sorsby, A., Leary, G. A. and Richards, M. J. (1962). Correlation ametropia and component ametropia. Vision Res. 2, 309±313. Spedick, M. J. and Beauchamp, G. R. (1986). Retinal vascular and optic nerve abnormalities in albinism. J. Pediatr. Ophthalmol. Strabismus 23, 58±63. Spilker, B. (1991). Guide to clinical trials 1991, Raven Press Ltd. New York, pp. 278±279. Varma, R., Tielsch, J. M., Quigley, H. A., Hilton, S. C., Katz, J., Spaeth, G. L. and Sommer, A. (1994). Race-, age-, gender-, and refractive error-related dierences in the normal optic disc. Arch. Ophthalmol. 112, 1068±1076. Wakakura, M. and Alvarez, E. (1987). A simple clinical method of assessing patients with optic nerve hypoplasia.
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The disc±macula distance to disc diameter ratio. Acta Ophthalmol. (Kbh) 65, 612±617. Whinery, R. D. and Blodi, F. C. (1963). Hypoplasia of the optic nerve: a clinical and histopathological correlation. Trans. Am. Acad. Ophthalmol. Otol. 67, 733±738. Williams, T. D. (03 Oct 1995). (personal communication). Williams, T. D. (1987). Elliptical features of the human optic nerve head. Am. J. Optom. Physiol. Opt. 64, 172±178. Williams, T. D. and Wilkinson, J. M. (1992). Position of the fovea centralis with respect to the optic nerve head. Optom. Vis. Sci. 69, 369±377. Zeki, S. M., Dudgeon, J. and Dutton, G. N. (1991). Reappraisal of the ratio of disc to macula/disc diameter in optic nerve hypoplasia. Br. J. Ophthalmol. 75, 538±541.
where m = mean, s = standard deviation and n = number of observations, the pooled mean, mp, may be calculated from: mp
m X r1
mr nr =
m X
nr
r1
and the pooled standard deviation, sp, from: ! ! m m X X 2 2 2 2 mr nr ÿ 1 mr nr ÿ mp nr = sp r1
ÿ1
r1
m X
nr
r1
Appendix A Errors in the DM:DD ratio due to disc rotation (Figure 4) The errors represent the ratio of the DM:DD ratio for the non-rotated disc and the DM:DD ratio calculated taking tangent distances a1' and a2' and the speci®ed distance from the fovea to the temporal disc margin: ÿ D b1 a2 h 2 c1 ÿ a2 sin y f 1 = ÿ 2 c1 g h ÿ ÿ D b2 a2 h 2 c1 ÿ a2 f=g f 1 = ÿ 2 c1 g h D b3 f 1= g h where: a2 = elliptical minor axis of disc; f = major axis/minor axis; c1 = fovea±disc centre distance; y = angle of disc rotation; g = Z(f2y + cos2y); h = Z(f2cos2y + sin2y). The parameter ranges for error calculations are: a2 = 1.69 2(3 0.23) mm (pooled data (Varma et al., 1994; Jonas and KoÈnigsreuther, 1994; Quigley et al., 1990)), f = 1.127 2 (3 0.08) (pooled data (Williams and Wilkinson, 1992; Chihara and Chihara, 1994; Williams, 1987)), c1 = 4.89 2(3 0.33) mm (Williams, 1995) y = 82.7 2(3 27.5)8 (Williams and Wilkinson, 1992).
Appendix B Meta-analysis: data pooling For m groups: m1 ; s1 ; n1 ; m2 ; s2 ; n2 ; m3 ; s3 ; n3 ; . . . mm ; sm ; nm
Appendix C Calculation of false positive and false negative proportions for selected DM:DD ratio (for diagnosis of unequivocal ONH) Graphical representations of the intersection of two normal distributions are shown in Figures 5 and 6. The false positive (FP) and false negative (FN) proportions are calculated by a volume analysis of this threedimensional structure. The equations are inexact but close approximations to the actual volumes. ÿ ÿ mc 3 sc ÿ md ÿ 2 sd k =0:1 sc X n1 ÿ ÿ1 2 %1 ÿ 4 m p ln =4 s p p 2 FP 2 pÿ 1=2 eÿ zp =2 dzp ÿ2 ÿ md ÿ 2 sd k n 0:1 sc 2 sÿ1 2 pÿ 1=2 eÿ cÿmc =sc =2 dc c ÿ ÿ md ÿ 2 sd k n ÿ 1 0:1 sc where: %1= ÿ m ÿ 2 s k=0:1 s n s d c c d ÿ ln md ÿ 2 sd n 0:1 sc =k ÿ 1 ÿ ÿ md ÿ 2 sd k=0:1 sc n ÿ 1 sc ÿ ln md ÿ 2 sd n ÿ 1 0:1 sc =k
Disc±macula distance to disc diameter ratio: appraisal: D. B. Barr et al. 375 ÿ m ÿ 2 sd k ÿ m 3 sc =0:1 sc d c where: %2= X ÿ md ÿ 2 sd k=0:1 sc ÿ n ÿ 1 sc ÿ n1 ln md ÿ 2 sd ÿ n ÿ 1 0:1 sc =k ÿ ÿ2 ÿ md ÿ 2 sd k=0:1 sc ÿ n sc ÿ ÿ 1=2 ÿ z2p =2 ln md ÿ 2 sd ÿ n 0:1 sc =k ÿ 1 2 p e dzp ÿ ÿ1 2 FN In the above equations, k = DM:DD ratio; mc = mean %2 ÿ 4 m p ln =4 s p p ÿ of fovea±disc centre distance; sc = standard deviation ÿ md ÿ 2 sd k ÿ n ÿ 1 0:1 sc of fovea±disc centre distance; md = mean of pooled mean disc diameter; sd = standard deviation of pooled sÿ1 2 pÿ 1=2 eÿ cÿmc =sc 2 =2 dc mean disc diameter; mp = mean of pooled disc area; c = standard deviation of pooled disc area; s ÿ p md ÿ 2 sd k n 0:1 sc zp = standard normal variable of pooled disc area.
PII: S0275-5408(99)00018-6
An appraisal of the disc±macula distance to disc diameter ratio in the assessment of optic disc size D. B. Barr, C. R. Weir and A. T. Purdie Tennent Institute of Ophthalmology, Gartnavel General Hospital, 1053 Great Western Road, Glasgow G12 0YN, UK Summary The disc±macula distance to disc diameter ratio (DM:DD ratio) has been advocated as a method of supporting the diagnosis of optic nerve hypoplasia. A DM:DD ratio of 3.00 has been claimed to be a satisfactory threshold value for this purpose. This study has critically evaluated the above claim and found a value of 3.00 to be too low. The threshold DM:DD ratio values for the diagnosis of unequivocal ONH for an adult population, 5 and 2 years of age were found to be respectively 4.20, 3.93 and 3.70, the values for the diagnosis of mild ONH being 3.68, 3.44 and 3.23. Lower computed values reduce the predictive power. The method of computation of the DM:DD ratio was modified to abolish potential error due to disc rotation and foveal displacement. In an adult population, there was no correlation between the DM:DD ratio and amblyopia or disc ovalness. There was a trend of increasing DM:DD ratio towards myopia and decreasing DM:DD ratio towards hypermetropia; the DM:DD ratio may therefore be falsely high in high myopia. DM:DD ratio values below threshold should therefore be interpreted with care until formal optic disc biometry can be performed. # 1999 The College of Optometrists. Published by Elsevier Science Ltd. All rights reserved
Introduction
ring of hypo- or hyperpigmentation (double ring sign), accompanied by a variable thinning of the peripapillary nerve ®bre layer (Brodsky, 1994; Hoyt and Good, 1992). Diagnosis in less extreme degrees of hypoplasia, however, may be much more dicult when the above signs are less pronounced. Moreover, the rationale of diagnosis of ONH based solely on optic disc size may be erroneous since large discs may be axonally de®cient and small discs do not preclude normal visual function (Brodsky, 1994). Furthermore, ONH may be segmental in which the optic disc is not diusely diminished in size and often appears pink and otherwise healthy (Hoyt and Good, 1992). There are several modalities of investigation which may help in the diagnosis of ONH. Red-free examination of the nerve ®bre layer (FriseÂn and Holmegaard, 1978), A-scan (Acers, 1981) and B-scan (Boynton et al., 1975) ultrasonography, axial tomography of the orbital optic nerve (Acers, 1981) and optic canal (Boynton et al., 1975) and electrophysiological testing (Kriss and Russell-Eggitt, 1992) have all yielded characteristic abnormalities when the degree of hypo-
In the evaluation of a child or adult with subnormal visual function, the dierential diagnosis includes optic nerve hypoplasia (ONH). This is a developmental anomaly of the optic nerve in which the number of axons within the nerve is subnormal, whilst the mesodermal and glial elements are normal (Whinery and Blodi, 1963). ONH is not uncommon (Hoyt, 1986) and has been cited as the commonest optic disc anomaly encountered in clinical practice (Brodsky, 1994). The eect of ONH on visual function is variable (Peterson and Walton, 1977; FriseÂn and Holmegaard, 1978) with acuities ranging from normal to no light perception (Brodsky, 1994). Diagnosis on fundoscopic appearance may be easy in extreme cases (Hoyt and Good, 1992) in which the optic disc is overtly small, grey or pale in colour with a mottled peripapillary halo bordered by a Correspondence and reprints requests to: Dr D.B. Barr Received: 25 June 1997 Revised form: 5 February 1999
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plasia is extreme. They are, however, less useful in more subtle cases (Hoyt and Good, 1992). In addition, there are several methods by which the absolute dimensions of the optic disc may be estimated (optic disc biometry). The slit lamp may be employed in such estimations using a Goldmann contact lens (Franceschetti and Bock, 1950) or the +90 dioptre (D) condensing lens (Ruben, 1994), the latter of which may underestimate optic disc dimensions (Barr, 1995a). Fundus photogrammetric estimations from a 30-degree transparency (Romano, 1989) or from a photograph (Littmann, 1982; Bennett et al., 1994) have been employed, the accuracy of the latter having already been appraised (Barr, 1995b). Other methods involving stereoscopic video images (Caprioli et al., 1986), an indirect ophthalmoscopic system (Montgomery, 1991) and the confocal scanning laser ophthalmoscope (Nasemann and Burk, 1990) have been used. A photogrammetric method in which relative measurements from a fundus photograph are made has been advocated as reliable supportive evidence for the diagnosis of ONH (Wakakura and Alvarez, 1987; Alvarez et al., 1988; Zeki et al., 1991). These relative measurements are computed to give the disc±macula distance to disc diameter ratio (DM:DD ratio). The DM:DD ratio has been de®ned (Wakakura and Alvarez, 1987) as the sum of the distance from the fovea to the temporal margin of the optic disc and half the horizontal disc diameter divided by the mean disc diameter (Figure 1). It can be seen that when the distance between the fovea and the disc centre is a
Figure 2. Fundus photographic measurements (F, fovea; c1, distance between vertical meridians through the fovea and the disc centre; c2, vertical foveal displacement; a1, major elliptical axis; a2, minor elliptical axis).
constant, the DM:DD ratio increases as the disc becomes smaller. Therefore, the greater the DM:DD ratio, the greater the liklihood that the disc is dimensionally hypoplastic. It is this assumption that forms the basis of its use as a simple aid in the diagnosis of ONH when formal optic disc biometry is either not available or not tolerated. From a study of 18 normal children and ®ve with clinically diagnosed ONH (Alvarez et al., 1988), it was concluded that a DM:DD ratio >3.00 was highly suggestive of ONH and that false positives were unlikely with a value of >3.04. The purposes of this study were as follows: the measurement of the DM:DD ratio in a large population of adult subjects and comparison of the distribution obtained with other studies of the DM:DD ratio in paediatric and adult populations; an assessment of method accuracy and inter-observer variation; the examination of any correlation between the DM:DD ratio and ametropia, amblyopia and disc ovalness; a ®nal comment, based on the above ®ndings, on the likely usefulness of the DM:DD ratio as a clinical aid in the diagnosis of ONH, both in paediatric and adult populations.
Subjects and methods
Figure 1. Computation of the DM:DD ratio (Wakakura and Alvarez, 1987). (F, fovea; b, distance from the fovea to the temporal disc margin along an horizontal meridian; a1, vertical disc diameter; a2, horizontal disc diameter).
The case records of 125 subjects (107 male, 18 female; mean age 27.6 2 7.4 years, range 18±58 years) were obtained who attended between 1980 and 1992 for routine examination prior to the use of laser equipment at the University of Glasgow. Each patient had been refracted and best corrected Snellen acuities recorded. Motility, anterior segments, intra-ocular pressures and fundi had been examined and a fundus photograph obtained for each eye. Subjects were excluded from the study if there was any history of ocular disease and if any disease had been detected at the time of the original examination. Fundus photographs were excluded if the quality or ®eld of view prevented accurate measurements. Measurements were
Disc±macula distance to disc diameter ratio: appraisal: D. B. Barr et al. performed by two masked observers. Data from only one observer were used in all computations, except in the assessment of inter-observer variation. The fundus photographs were projected onto a screen and linear measurements made using callipers. Optic discs were assumed to be elliptical and the lengths of the major (a1, Figure 2) and minor (a2, Figure 2) axes measured. The distances from the centre of the disc (intersection of the major and minor elliptical axes) to the fovea along the horizontal meridian (c1, Figure 2) and from the fovea to this meridian along the vertical meridian (c2, Figure 2) were also measured. Computations were carried out for each eye as follows: (1) DM:DD ratio = 2c1/(a1 + a2) (2) disc ovalness = a1/a2 (3) foveal displacement (a) = tanÿ1 (c2/c1) The rationale behind the departure from the previous method of computing the DM:DD ratio will be explained subsequently. Adopting a previously described convention (Williams and Wilkinson, 1992), the angle (a) between the horizontal meridian passing through the disc centre and a line joining the disc centre with the fovea was recorded as positive when the fovea was below the horizontal meridian, and negative when it was above. Data were processed to compute a DM:DD ratio distribution and examine correlations between DM:DD ratio and ametropia, amblyopia and disc ovalness. The DM:DD ratio distribution for this population of 125 subjects was calculated by using data from one fundus photograph chosen at random from 104 subjects for whom two fundus photographs were suitable and from 21 subjects for whom there was only one fundus photograph. In examining correlation between the DM:DD ratio and ametropia, the data from the right eyes of 125 subjects were used and DM:DD ratio distributions were calcu-
367
lated for six arbitrarily de®ned classes of ametropia spherical equivalent (S) (S Rÿ7.00D; ÿ7.00 < S Rÿ5.00; ÿ5.00 < S Rÿ3.00; ÿ3.00 < S Rÿ1.00; ÿ1.00 < S R+ 1.00; S> + 1.00). Inter-class correlations were assessed by paired twotailed t-tests and by one-way analysis of variance. Correlation between DM:DD ratio and amblyopia was examined by calculating the intereye DM:DD ratio dierence in 104 subjects for whom two satisfactory fundus photographs were available. Within this cohort, amblyopes were identi®ed as those subjects in whom there was a greater than or equal to two lines of best corrected Snellen visual acuity dierence. Intereye DM:DD ratio dierence distributions were calculated for the amblyopic and non-amblyopic groups and compared using a paired two-tailed t-test. Disc ovalness, de®ned as the ratio of the major and minor elliptical axes, was calculated for 229 eyes and its relationship with the DM:DD ratio assessed by regression analysis. In order to assess inter-observer variation, for each of 193 fundus photographs, data from one observer were compared with those from a second observer, the second observer performing measurements at a dierent time and masked as to the results of the ®rst observer. The paired DM:DD ratio means and dierences were then used to assess the bias and con®dence intervals of the measurements between the two observers.
Results For the 125 subjects in this study, the DM:DD ratio (mean 2 standard deviation) was 2.819 2 0.394. In measuring projected fundus images using callipers, the accuracy of so doing is 20.5 mm. The mean disc diameter for all measurements was 86.6 mm and so the accuracy of measurement was 20.6%.
Table 1. Ametropia classes and correlations with DM:DD ratio Ametropia classes (spherical equivalent (S), dioptres) SRÿ7.00 Number in class Ametropia mean Ametropia range DM:DD ratio ÿ7.00 < SRÿ5.00 ÿ5.00 < SRÿ3.00 ÿ3.00 < SRÿ1.00 ÿ1.00 < SR + 1.00 S> + 1.00
ÿ7.00 < S Rÿ5.00
ÿ5.00 < S Rÿ3.00
ÿ3.00 < S Rÿ1.00
ÿ1.00 < S R + 1.00
S> + 1.00
6 7 11 22 73 6 ÿ8.42 2.4 ÿ5.5 20.5 ÿ4.1 2 0.6 ÿ1.8 2 0.5 ÿ0.12 0.2 +2.6 2 2.0 ÿ12.75 to ÿ7.00 ÿ6.25 to ÿ5.00 ÿ4.75 to ÿ3.00 ÿ2.75 to ÿ1.00 ÿ0.875 to +0.125 +1.25 to +6.50 3.22 0.7 2.7 20.3 3.1 20.5 2.8 2 0.3 2.82 0.4 2.7 20.3 0.153 0.722 0.089 0.124 0.138
0.126 0.540 0.623 0.671
Inter-class DM:DD ratio correlations (p values) 0.103 0.128 0.816 0.097 0.322 0.364 (one-way analysis of variance, F = 2.272)
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For each class of ametropia spherical equivalent, the ametropia distribution, ametropia range, DM:DD distribution and inter-class correlations, as assessed by paired two-tailed t-tests, are shown in Table 1. It was found that, whilst there was a tendency for the DM:DD ratio to be lower in hypermetropia and higher in myopia, none of the inter-class correlations achieved statistical signi®cance. However, one-way analysis of variance indicated that the dierence between the DM:DD ratio means of the ametropia classes almost achieved statistical signi®cance (F = 2.272, p = 0.052). For the 104 subjects for whom two fundus photographs were available, the intereye DM:DD ratio dierence for the amblyopic group (n = 9) was 0.207 2 0.212 and for the non-amblyopic group (n = 95), the value was 0.209 2 0.183. The dierence between these two groups is not signi®cant (p = 0.979). There was found to be no signi®cant correlation between the DM:DD ratio and disc ovalness (r = 0.124, p = 0.061). Vertical foveal displacement, with respect to the disc, was +6.1082 3.8428. In assessing inter-observer variation, the means of 193 paired DM:DD ratios from two observers were plotted against the dierence between the same pair of measurements (Figure 3). The dierence between measurements of the DM:DD ratio was found to increase (absolutely) as the mean DM:DD ratio increased, and so bias and con®dence intervals were
Figure 4. Effect of disc rotation on computation of the DM:DD ratio. (F, fovea; y, angle of disc rotation).
calculated for mean DM:DD ratios of R3.00 (n = 142) and >3.00 (n = 51). There was negligible observer bias in the two groups, the mean dierences being respectively ÿ0.016 (95% con®dence interval ÿ0.064 to 0.032) and ÿ0.079 (95% con®dence interval ÿ0.236 to 0.078). The upper limit of agreement for mean DM:DD ratios R3.00 was 0.57 (95% con®dence interval 0.49 to 0.65) and the lower limit of agreement for this group was ÿ0.60 (95% con®dence interval ÿ0.69 to ÿ0.52). For mean DM:DD ratios of >3.00, the upper limit of agreement was 1.06 (95% con®dence interval 0.78 to 1.33), and the lower limit of agreement ÿ1.21 (95% con®dence interval ÿ1.48 to ÿ0.94).
Discussion
Figure 3. Inter-observer variation in DM:DD ratio estimation. The mean of a pair of DM:DD ratio measurements is plotted against the difference. (ÐÐ mean; --- upper and lower limits of agreement; - - - - - - boundaries of 95% confidence intervals for the means and limits of agreement).
In our study, the DM:DD ratio was calculated from: 2c1/(a1 + a2) (Figure 2), a departure from the original method which used: ((a2/2) + b)/((a1 + a2)/2) (Figure 1). The latter method may incur error due to disc rotation or vertical foveal displacement. Disc rotation (Figure 4) has been reported as 82.7 2 27.58 (Williams and Wilkinson, 1992). When the disc is rotated, unless elliptical axes are estimated, the disc diameters are measured by the tangent lengths a1' and a2'. The distance from the fovea to the temporal disc margin may be measured by the distances b1, b2 or b3 incurring mean errors of respectively 1+ 6%, +2% and +2% (Appendix A). The distance from the fovea to an horizontal meridian passing through the disc centre has been reported to be 0.58 2 0.30 mm (Williams and Wilkinson, 1992), corresponding to an angle of foveal displacement of +6.11 2 3.328. More
Disc±macula distance to disc diameter ratio: appraisal: D. B. Barr et al. extreme degrees of vertical foveal displacement (heterotopia) are uncommon (Cohen and Weisberg, 1950) but occur occasionally in craniofacial disorders associated with the tilted disc syndrome (Margolis et al., 1977, 1984; Margolis and Seigel, 1979). In our study, vertical foveal displacement was found to be +6.108 23.8428, in close agreement with the above study. In the presence of vertical foveal displacement, the distance from the fovea to the disc centre is always greater than the distance along an horizontal meridian through the disc centre. The overestimation (D) of the DM:DD ratio is: D = (1/cosa)ÿ1 (a = vertical foveal displacement) which is 13% for an extreme vertical foveal displacement of 148. Errors due to disc rotation and/or vertical foveal displacement would therefore seem to be insigni®cant. However, the DM:DD ratio has been assessed previously in an adult population (Williams and Wilkinson, 1992) and was 2.85 20.35 using the minor axis of the optic nerve head (as was done in our study) and 2.59 20.28 using the horizontal diameter of the optic nerve head, which may incur error when there is disc rotation. These two values are signi®cantly dierent (p < 0.001) which suggests that the theoretically small error introduced by disc rotation is signi®cant in practice. Errors due to both disc rotation and vertical foveal displacement are, however, eectively abolished by measuring the fovea±disc centre distance (c1, Figure 4) and computing the DM:DD ratio as: 2c1/ (a1 + a2). The DM:DD ratio distribution in the above study (Williams and Wilkinson, 1992) was 2.85 2 0.35 and in our study was 2.819 2 0.394. Both studies computed the DM:DD ratio using the same method and the resultant DM:DD ratio distributions are in close agreement. Having determined the optimal method of computing the DM:DD ratio, it is desirable to examine its predictive power in correctly classifying an optic disc as being hypoplastic. As stated, the DM:DD ratio is used to aid in the diagnosis of ONH when optic disc biometry is unavailable or not tolerated. It has therefore been used most as an aid in the diagnosis of ONH in children. A statement on the predictive power of the DM:DD ratio in this regard must apply both to paediatric and adult populations. The DM:DD ratio distribution in a paediatric population was 2.62 2 0.21 (Alvarez et al., 1988) which is signi®cantly dierent from adult distributions of 2.85 20.35 (Williams and Wilkinson, 1992) and 2.819 2 0.394 (our study). Since there are only two parameters used to compute the DM:DD ratio, there must be a signi®cant dierence in the fovea±disc centre distance and/or mean disc area between paediatric and adult populations to account for this dierence in DM:DD ratio distributions.
369
Using least squares non-linear regression analysis of optic disc measurements performed at autopsy on 95 patients whose age ranged from 4.8 months gestational age to 21.9 years (Rimmer et al., 1993), it was concluded that 95% of the adult optic disc size is achieved by the age of 12 months and, by inspection of the data, 99% is achieved by the age of 18 months. This is supported by data which suggests that growth of the posterior scleral opening has ®nished at about 2 years of age (Quigley, 1982). Any dierence in DM:DD ratio distribution between paediatric (>18 months of age) and adult populations therefore cannot be due to a dierence in the mean disc diameter, and must be due to a dierence in the fovea±disc centre distance. After birth, the axial length of the globe (as measured by A-scan ultrasonography) increases in a triphasic pattern (Larsen, 1971; Gordon, 1985). Linear growth occurs between birth (16.61 20.60 mm) and 2 years (20.41 2 0.26 mm), between 2 years and 5 years (21.45 2 0.54 mm) and between 5 years and approx. 14 years (23.22 20.74 mm) when the globe has reached adult size (pooled data (Larsen, 1971; Gordon, 1985; Harayama et al., 1981)). There is therefore a biphasic stepwise linear increase in axial length after the age of 2 years, at which time the optic disc has reached >99% of adult size. The axial length is therefore 188% of adult size at 2 years of age and 194% at 5 years. Assuming uniform expansion of the globe to include the area between the fovea and the optic disc, and taking a chord distance between the fovea and the optic disc (that which is actually measured in computing the DM:DD ratio), this percentage change in axial length results in a change in the magnitude of the DM:DD ratio by precisely the same amount. We may therefore expect the DM:DD ratio distribution in a paediatric population to be 112% lower (2 years of age) and 16% lower (5 years of age) than that of an adult population (>14 years of age). Ametropia may have an indirect in¯uence on the DM:DD ratio, the distribution of ametropia being dierent in paediatric and adult populations. It may exert its in¯uence via its correlation with either axial length or disc area. Ametropia is not correlated with disc area (Bengtsson, 1980; Britton et al., 1987; Jones et al., 1988; Chi et al., 1989; Varma et al., 1994) and so this may be discounted. Paremmetropic ametropia is not correlated with axial length (Sorsby et al., 1962; Bennett and Rabbetts, 1984) (which, assuming uniform expansion of the globe, would be directly proportional to the fovea±disc centre distance). High refractive errors, however, are correlated with axial length, high myopia being correlated with a longer axial length and high hypermetropia correlated with a shorter axial length. For constant disc size, the eect of axial length in high ametropia may therefore be expected to result
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in a falsely high DM:DD ratio in high myopia (leading to misclassi®cation of a normal optic disc as one that is hypoplastic), the converse occurring in high hypermetropia. There was no inter-class correlation between ametropia and the DM:DD ratio in our study (Table 1) although there was a trend of increasing DM:DD ratio with increasing myopia. One-way analysis of variance indicated that the dierences in the means between the ametropia sub-classes was almost signi®cant. To calculate the predictive power of the DM:DD ratio in correctly classifying an optic disc as being hypoplastic, normal distributions of the mean disc diameter and fovea±disc centre distance must be known. From a meta-analysis (Appendix B) of studies that have presented data on disc area (Britton et al., 1987; Varma et al., 1994; Caprioli and Miller, 1987; Chihara and Chihara, 1994; Jonas and KoÈnigsreuther, 1994), in which all magni®cation corrections were made according to Littmann's method (Littmann, 1982), the disc area in an adult population is 2.60 2 0.50 mm2. Assuming circularity of the disc, the mean disc diameter (d) is calculated from: d = 2Z(A/p), where A is the disc area. This assumed disc circularity must be justi®ed. Ovalness (f), de®ned as the ratio of the vertical and horizontal disc diameters, in an adult population, is 1.127 20.108 (pooled data (Williams and Wilkinson, 1992; Chihara and Chihara, 1994; Williams, 1987)). A large proportion (99.73%) of the population will therefore have a disc ovalness of R1.451. The ratio (D) of mean disc diameter for an oval disc (f>1) and that for a circular disc (f = 1) is
independent of absolute disc area and is calculated from: ÿ p D f 1 1=f =2: For f = 1.451, D = 1.0188, which represents only 12% increase in the mean disc diameter for this extreme degree of ovalness. It is therefore considered justi®able to ignore disc ovalness when calculating mean disc diameter from disc area. The term `microdisc' has been used to de®ne unequivocal dimensional disc hypoplasia and is regarded as an optic disc whose area is less than or equal to the mean disc area minus two standard deviations (Jonas et al., 1989). For 99.73% of the population, the mean disc diameter will be between 1.18 and 2.29 mm. Unequivocal ONH may therefore be de®ned as one whose area is 1.60 mm2 or less, corresponding to a mean disc diameter of 1.43 mm. One of the authors (DBB) has performed optic disc biometry on 13 patients (aged between 5 and 87 years) with phenotypic ONH and it is accepted in our department that mild ONH is present when the disc area is 2.00 mm2 or less, corresponding to a mean disc diameter of 1.60 mm. The normal distribution of the fovea±disc centre distance is also required for the calculation of predictive power of the DM:DD ratio. From data acquired in a previous study (Williams and Wilkinson, 1992; Williams, 1995), this distance is 4.89 2 0.33 mm; 99.73% of the population will therefore have a fovea± disc centre distance between 3.90 and 5.88 mm. A DM:DD ratio of r3.00 (Alvarez et al., 1988) later
Figure 5. Relationship between the DM:DD ratio, mean disc diameter and the fovea±disc centre distance (c1).
Disc±macula distance to disc diameter ratio: appraisal: D. B. Barr et al. revised down to 2.94 (Zeki et al., 1991), has been advocated as supportive evidence for the diagnosis of ONH. Using the normal distributions for mean disc diameter and fovea±disc centre distance, the relationship between the DM:DD ratio, mean disc diameter and fovea±disc centre distance can now be expressed graphically (Figure 5). This shows the false positive and false negative proportions when a DM:DD ratio of 3.00 is chosen as the threshold value. The apparently high proportion of false positives is misleading: a more realistic impression may be gained from a threedimensional representation (Figure 6). The false positive (FP) and false negative (FN) proportions are calculated as shown in Appendix C. The true positive proportion (TP) proportion is calculated from: TP = 0.0213ÿFN and true negative (TN) from: TN = 0.9733ÿFP. Sensitivity (N) is calculated from: N = TP/(TP + FN) and speci®city (S) from: S = TN/ (TN + FP). The predictive power (V+) of the DM:DD ratio may now be calculated from: V + = (((Sÿ1)(Pÿ1)/NP) + 1)ÿ1, where P = prevalence (Spilker, 1991). The predictive power of the DM:DD ratio for unequivocal and mild ONH as a function of age is shown in Figure 7. In using the DM:DD ratio for the diagnosis of unequivocal ONH for an adult population, a DM:DD ratio of r4.20 will identify the condition with almost complete certainty, the values being 3.93 and 3.70 for 5 years and 2 years of age respectively. Intermediate age values may be estimated from
371
the nomogram. Similarly, for the diagnosis of mild ONH for an adult population, a DM:DD ratio of r3.68 will identify the condition with almost complete certainty, the values being 3.44 and 3.23 for 5 years and 2 years of age respectively. As previously stated, it has been claimed (Alvarez et al., 1988) that a DM:DD ratio of >3.00 is highly suggestive of ONH and that false positives are unlikely with a value of >3.04. From Figure 7, it can be seen that, for the diagnosis of unequivocal ONH, a threshold DM:DD ratio value of 3.00 is too low, the predictive power is 140% for 2 years of age falling to 110% for an adult population indicating that >60% of patients identi®ed as having ONH (based on a value of 3.00) would have a disc area of >1.60 mm2 (false positives). However, the predictive power of a threshold value of 3.00 is better for the diagnosis of mild ONH, having a power of >80% for R6 years of age dropping to 150% for adults. This study therefore disagrees with the claim that false positives for the diagnosis of unequivocal or mild ONH are unlikely with a DM:DD ratio of >3.04. Inter-observer variation, in this study, was found to increase as the DM:DD ratio increases. For a mean DM:DD ratio of R3.00 between the two observers, the 95% con®dence interval was 20.59 which represents moderately good agreement but for a mean DM:DD ratio of >3.00, the 95% con®dence interval was 21.13 representing rather poorer agreement. Thus interobserver variation is greater, and therefore method reliability is poorer, when the patient has ONH. The two
Figure 6. Three-dimensional representation of the relationship between the DM:DD ratio, mean disc diameter and the fovea±disc centre distance (c1). Dark shading represents false positive proportion. Population percentage is arbitrary.
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Figure 7. Nomograms of the relationship between the DM:DD ratio, predictive power (V+) for diagnosis of unequivocal/mild ONH and age.
possible sources of this variation lie in identi®cation of the disc margins and the foveal position. It was not possible to determine which of the two contributed to the inter-observer variation for higher values of the DM:DD ratio since the two observers performed measurements at dierent times with dierent magni®cations of the fundus photographs. It is highly likely, however, that variation in interpretation of the foveal position from the fundus photograph is the more signi®cant contributor to inter-observer variation in DM:DD ratio calculation. Diculty in interpretation of the foveal position is likely to be a problem when there is foveal hypoplasia (in which the anatomical boundaries of the fovea are ill-de®ned) coincident with ONH which may occur in albinism (Spedick and Beauchamp, 1986; Oetting et al., 1994) and aniridia (Layman et al., 1974; Nelson et al., 1984).
Conclusions Estimation of the DM:DD ratio from a fundus photograph may be a simple, reproducible and useful clinical aid in the diagnosis of ONH. Computation of the DM:DD ratio is sensitive to the method employed and should be: fovea±disc centre distance/mean disc diameter (major and minor elliptical axes). This is theoretically more accurate and of practical importance. Low ametropia (24.00 D) is irrelevant but care should be taken in high myopia in which the DM:DD ratio may be falsely high. Reproducibility is poorer
with higher values of the DM:DD ratio; diculty in interpretation of the foveal position is likely to in¯uence reproducibility. Threshold DM:DD ratio values of 4.20, 3.93 and 3.70 for adults, 5 and 2 years of age respectively diagnose unequivocal ONH with almost complete certainty as do respective values of 3.68, 3.44 and 3.23 for the diagnosis of mild ONH. A threshold DM:DD ratio value of 3.00 is too low for the con®dent diagnosis of unequivocal ONH in any age group, but is better for the diagnosis of mild ONH, especially in younger patients. These limitations of the usefulness of the DM:DD ratio in the diagnosis of ONH should be appreciated until the patient is suciently old or capable of tolerating formal optic disc biometry. Acknowledgements The authors would like to thank Professor TD Williams, University of Waterloo, Ontario, Canada for providing data essential for this study and Messrs John McCormick and Iain Smith for illustrative work.
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Kriss, A. and Russell-Eggitt, I. (1992). Electrophysiological assessment of visual pathway function in infants. Eye 6, 145±153. Larsen, L. (1971). The sagittal growth of the human eye. IV. Ultrasonic measurement of the axial length of the eye from birth to puberty. Acta Ophthalmologica 49, 873±886. Layman, P. R., Anderson, D. R. and Flynn, J. T. (1974). Frequent occurrence of hypoplastic optic discs in patients with aniridia. Am. J. Ophthalmol. 77, 513±516. Littmann, H. (1982). Zur Bestimmung der wahren GroÈûe eines Objektes auf dem Hintergrund des lebenden Auges. Klin. Mbl. Augenheilk 180, 286±289. Margolis, S. and Siegel, I. M. (1979). The tilted disc syndrome in craniofacial diseases. ed. J. L. Smith. In: Neuro-ophthalmology Focus 1980, Masson, New York, pp. 97±116. Margolis, S., Siegel, I. M., Choy, A. and Breinin, G. M. (1977). Oculocutaneous albinism associated with Apert's syndrome. Am. J. Ophthalmol. 84, 830±839. Margolis, S., Aleksic, S., Charles, N., McCarthy, J., Greco, A. and Budzilovich, G. 1984. Retinal and optic nerve ®ndings in Goldenhar±Gorlin syndrome. Ophthalmology 91, 1327±1333. Montgomery, D. M. I. (1991). Measurement of optic disc and neuroretinal rim areas in normal and glaucomatous eyes: a new clinical method. Ophthalmology 98, 50±59. Nasemann JE, Burk ROW (Eds) 1990. Scanning laser ophthalmoscopy. Quintessenz, Munich. Nelson, L. B., Spaeth, G. L. and Nowinski, T. S. (1984). Aniridia: a review. Surv. Ophthalmol. 28, 621±642. Oetting, W. S., Summers, C. G. and King, R. A. (1994). Albinism and the associated ocular defects. Metabol. Ophthalmol. 17, 5±9. Peterson, R. A. and Walton, D. S. (1977). Optic nerve hypoplasia with good visual acuity and visual ®eld defects: a study of children of diabetic mothers. Arch. Ophthalmol. 95, 254±258. Quigley, H. A. (1982). Childhood glaucoma: results with trabeculectomy and study of reversible cupping. Ophthalmology 89, 219±226. Quigley, H. A., Brown, A. E., Morrison, J. D. and Drance, S. M. (1990). The size and shape of the optic disc in normal human eyes. Arch. Ophthalmol. 108, 51±57. Rimmer, S., Keating, C., Chou, T., Farb, D. M., Christenson, P. D., Foos, R. Y. and Bateman, J. B. (1993). Growth of the human optic disc and nerve during gestation, childhood and early adulthood. Am. J. Ophthalmol. 116, 748±753. Romano, P. F. (1989). Simple photogrammetric diagnosis of optic nerve hypoplasia. Arch. Ophthalmol. 107, 824±826. Ruben, S. (1994). Estimation of optic disc size using indirect biomicroscopy. Br. J. Ophthalmol. 78, 363±364. Sorsby, A., Leary, G. A. and Richards, M. J. (1962). Correlation ametropia and component ametropia. Vision Res. 2, 309±313. Spedick, M. J. and Beauchamp, G. R. (1986). Retinal vascular and optic nerve abnormalities in albinism. J. Pediatr. Ophthalmol. Strabismus 23, 58±63. Spilker, B. (1991). Guide to clinical trials 1991, Raven Press Ltd. New York, pp. 278±279. Varma, R., Tielsch, J. M., Quigley, H. A., Hilton, S. C., Katz, J., Spaeth, G. L. and Sommer, A. (1994). Race-, age-, gender-, and refractive error-related dierences in the normal optic disc. Arch. Ophthalmol. 112, 1068±1076. Wakakura, M. and Alvarez, E. (1987). A simple clinical method of assessing patients with optic nerve hypoplasia.
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where m = mean, s = standard deviation and n = number of observations, the pooled mean, mp, may be calculated from: mp
m X r1
mr nr =
m X
nr
r1
and the pooled standard deviation, sp, from: ! ! m m X X 2 2 2 2 mr nr ÿ 1 mr nr ÿ mp nr = sp r1
ÿ1
r1
m X
nr
r1
Appendix A Errors in the DM:DD ratio due to disc rotation (Figure 4) The errors represent the ratio of the DM:DD ratio for the non-rotated disc and the DM:DD ratio calculated taking tangent distances a1' and a2' and the speci®ed distance from the fovea to the temporal disc margin: ÿ D b1 a2 h 2 c1 ÿ a2 sin y f 1 = ÿ 2 c1 g h ÿ ÿ D b2 a2 h 2 c1 ÿ a2 f=g f 1 = ÿ 2 c1 g h D b3 f 1= g h where: a2 = elliptical minor axis of disc; f = major axis/minor axis; c1 = fovea±disc centre distance; y = angle of disc rotation; g = Z(f2y + cos2y); h = Z(f2cos2y + sin2y). The parameter ranges for error calculations are: a2 = 1.69 2(3 0.23) mm (pooled data (Varma et al., 1994; Jonas and KoÈnigsreuther, 1994; Quigley et al., 1990)), f = 1.127 2 (3 0.08) (pooled data (Williams and Wilkinson, 1992; Chihara and Chihara, 1994; Williams, 1987)), c1 = 4.89 2(3 0.33) mm (Williams, 1995) y = 82.7 2(3 27.5)8 (Williams and Wilkinson, 1992).
Appendix B Meta-analysis: data pooling For m groups: m1 ; s1 ; n1 ; m2 ; s2 ; n2 ; m3 ; s3 ; n3 ; . . . mm ; sm ; nm
Appendix C Calculation of false positive and false negative proportions for selected DM:DD ratio (for diagnosis of unequivocal ONH) Graphical representations of the intersection of two normal distributions are shown in Figures 5 and 6. The false positive (FP) and false negative (FN) proportions are calculated by a volume analysis of this threedimensional structure. The equations are inexact but close approximations to the actual volumes. ÿ ÿ mc 3 sc ÿ md ÿ 2 sd k =0:1 sc X n1 ÿ ÿ1 2 %1 ÿ 4 m p ln =4 s p p 2 FP 2 pÿ 1=2 eÿ zp =2 dzp ÿ2 ÿ md ÿ 2 sd k n 0:1 sc 2 sÿ1 2 pÿ 1=2 eÿ cÿmc =sc =2 dc c ÿ ÿ md ÿ 2 sd k n ÿ 1 0:1 sc where: %1= ÿ m ÿ 2 s k=0:1 s n s d c c d ÿ ln md ÿ 2 sd n 0:1 sc =k ÿ 1 ÿ ÿ md ÿ 2 sd k=0:1 sc n ÿ 1 sc ÿ ln md ÿ 2 sd n ÿ 1 0:1 sc =k
Disc±macula distance to disc diameter ratio: appraisal: D. B. Barr et al. 375 ÿ m ÿ 2 sd k ÿ m 3 sc =0:1 sc d c where: %2= X ÿ md ÿ 2 sd k=0:1 sc ÿ n ÿ 1 sc ÿ n1 ln md ÿ 2 sd ÿ n ÿ 1 0:1 sc =k ÿ ÿ2 ÿ md ÿ 2 sd k=0:1 sc ÿ n sc ÿ ÿ 1=2 ÿ z2p =2 ln md ÿ 2 sd ÿ n 0:1 sc =k ÿ 1 2 p e dzp ÿ ÿ1 2 FN In the above equations, k = DM:DD ratio; mc = mean %2 ÿ 4 m p ln =4 s p p ÿ of fovea±disc centre distance; sc = standard deviation ÿ md ÿ 2 sd k ÿ n ÿ 1 0:1 sc of fovea±disc centre distance; md = mean of pooled mean disc diameter; sd = standard deviation of pooled sÿ1 2 pÿ 1=2 eÿ cÿmc =sc 2 =2 dc mean disc diameter; mp = mean of pooled disc area; c = standard deviation of pooled disc area; s ÿ p md ÿ 2 sd k n 0:1 sc zp = standard normal variable of pooled disc area.