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From the Department of Ophthalmology, University of British Columbia, Vancouver, Canada.
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Abstract |
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 Top Abstract IntroductionMethodsResultsDiscussionAppendix 1References |
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METHODS. The shape of the ONH was modeled by a smooth two-dimensional surface with a shape described by 10 free parameters. Parameters were adjusted by least-squares fitting to give the best fit of the model to the image. These parameters, plus others derived from the image using the model as a basis, were used to discriminate between normal and abnormal images. The method was tested by applying it to ONH topography images, obtained with the Heidelberg Retina Tomograph, from 100 normal volunteers and 100 patients with glaucomatous visual field damage.
RESULTS. Many of the parameters derived from the fits differed significantly between normal and glaucomatous ONH images. They included the degree of surface curvature of the disc region surrounding the cup, the steepness of the cup walls, the goodness-of-fit of the model to the image in the cup region, and measures of cup width and cup depth. The statistics of the parameters were analyzed and were used to construct a classifier that gave the probability, P(G), that each image came from the glaucoma population. Images were classified as abnormal if P(G) > 0.5. The probabilities assigned to each image were in most cases close to 0 (normal) or 1 (abnormal). Eighty-seven percent of the sample was confidently classified with P(G) < 0.3 or P(G) > 0.7. Within this group, the overall classification accuracy was 92%. The overall accuracy of the method (the mean of sensitivity and specificity, which were similar) in the whole sample was 89%.
CONCLUSIONS. ONH images can be classified objectively and dependably by an automated procedure that does not require prior manual outlining of disc boundaries.
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Introduction |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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The introduction of the confocal scanning laser ophthalmoscope, such as the Heidelberg Retina Tomograph (HRT; Heidelberg Engineering, Heidelberg, Germany), which is able to obtain accurate three-dimensional images of the surface topography of the optic nerve head (ONH),7 8 9 10 11 12 13 14 15 16 17 18 offers a promising means for early detection of glaucoma. A number of studies have shown that morphologic indices calculated from images of the ONH differ significantly between normal eyes and eyes with glaucomatous visual field defects.19 20 21 22 23 24 25 26 27 28 29 30 31 32 Parameters calculated from combinations of these indices can be used to diagnose the presence of glaucomatous field loss, within the populations from which normative values were obtained, with sensitivities and specificities that are typically in the range of 80% to 90%.
These methods all rely on shape parameters that are calculated by software after an initial stage in which a technician or clinician uses a computer mouse to manually outline the edge of the optic disc. This outlining process has been controversial, because different observers do not always agree where the disc margins should be placed, and this introduces an element of uncontrolled variability into the morphologic analysis.33 A solution to this problem is to develop automated image processing algorithms that do not require manual intervention. We propose such a method and show that the diagnostic accuracy of the parameters extracted by it is comparable to that of current methods based on prior manual outlining. The technique is based on parametric mathematical modeling of ONH shape, and it works by finding, for each image, those model parameters that produce the greatest degree of similarity between the model and the image. The parameter values are then used as descriptors of ONH morphology and as a basis for further morphologic analysis. To validate the technique, we applied it to a database of 100 images from eyes screened to exclude the presence of glaucoma, and to 100 images from eyes deemed, on the basis of visual field testing, to show early glaucomatous visual field damage.
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Methods |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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Criteria for Subject Selection
Normal Subjects.
The method of obtaining images from eyes in the normal group was as
follows:
Glaucoma Subjects.
Images from glaucoma subjects were selected by first reviewing the
files of approximately 415 patients (of FSM) for whom HRT image data
were available. The files were reviewed consecutively until images from
100 eyes had been obtained, satisfying the following inclusion
criteria: 1) open angles; 2) 20 visual fields (Humphrey or SITA
30-2; Humphrey Instruments) indicative of glaucomatous damage based on
the criteria of Mikelberg et al.,21
i.e., the presence of
(a) three adjacent points down by 5 dB with one of the points being
down by at least 10 dB, (b) two adjacent points down by 10 dB, or (c)
three adjacent points just above or below the nasal horizontal meridian
down by 10 dB. None of the points could be edge points except those
immediately above or below the horizontal meridian. The field taken
closest in time to the HRT scan was used for evaluation. In all except
one case, this was within 6 months of examination by the HRT; 3)
absence of eye disease other than glaucoma, such as cataract, vascular
occlusion or hemorrhage likely to interfere with visual field tests; 4)
less than a 7-D refractive error; and 5) HRT images obtained as the
mean of three separate scans with an SD less than 50 µm. Patients
with a mean deviation (MD) less than -10 dB were excluded.
The results of HRT scans and/or other types of ONH examination were excluded as criteria in making the classification of glaucoma, to make the prediction of visual field test results on the basis of ONH morphology more objective. However, abnormal ONH appearance was often a reason for the initial referral of the patient to the clinic. IOP was not used as a criterion for exclusion or inclusion, because it can be normal in glaucoma (often as a result of ongoing treatment). In cases in which both eyes satisfied the criteria for inclusion, the eye showing the lesser degree of visual field damage was chosen. If no other criteria applied, eyes were chosen to equalize the number of left and right eyes in the sample.
Table 3 lists the patient and normal subject demographics for the two groups. The average MD in the glaucoma group was -4.9 dB, which is similar to that in the patient groups studied by Mikelberg et al. (-5.5 dB),21 Wollstein et al. (-3.6 dB),32 and Brigatti et al. (-4.5dB).23
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Classification
The method used to classify images as normal or glaucomatous is
based on discriminant function analysis (DFA), and is described in
detail in the Appendix. Seven of the parameters were used in the
classification. They included the horizontal and vertical components of
image curvature, cup radius, maximum cup depth, the temporal cup
gradient measure, the fit of the parabolic function, and the fit in the
central region. These parameters passed the one-sample
Kolmogorov–Smirnov test for normality in both the normal and glaucoma
groups. They were used to calculate, for each image, the probability
that it came from the glaucoma group, denoted P(G). Cases
were classified as normal if P(G) < 0.5 and as
glaucoma if P(G) 
0.5.
Comparison with Standard HRT Parameters
To compare the method with the accuracy of classification obtained
using the standard HRT parameters, all the images were outlined using
standard software provided with the HRT (version 2.01), and the
resultant 14 global shape parameters were entered into a spreadsheet.
The DFA performed by Mikelberg et al.21
was then repeated
for comparison with the present method. This was accomplished in two
ways: by analyzing the original DFA formula values, as incorporated in
the HRT software, and by running a separate analysis to generate a new
DFA formula with SPSS (ver. 7.5; SPSS, Chicago, IL).
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Results |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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Figure 2 shows examples of 10 real images and the corresponding model images chosen at random: five from the normal group and five from the glaucoma group.
Analysis of Parameter Values
Table 4
gives the means ± SD of the model parameters, and Table 5
gives corresponding values for the morphologic indices derived from
them, for both the normal and glaucoma groups. The tables also show a
statistical measure of the differences between the groups
(d') which is the difference between the means of the two
groups divided by the average of the SDs. The rows in each table are in
decreasing order of d'—i.e., decreasing statistical
difference between the measures in the two groups. A one-way analysis
of variance (ANOVA) showed that in almost all cases the differences
were statistically significant (P < 0.001). The final
column, on the right side of each table, shows the Pearson correlation
coefficient (r) between each parameter and age, measured in
the normal group. Figure 4 shows one-dimensional model profiles, taken along the x-axis
at y = y0, calculated
using the averages of the parameters for the normal (solid line) and
glaucoma (dashed line) groups.
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The two model parameters showing the greatest difference between the two groups were horizontal and vertical image curvature, which measure the curvature of the image, minus the model cup, in the x (horizontal or nasotemporal) and y (vertical or superior–inferior) directions, respectively. In most of the normal images both curvature values were positive. It seems likely that these values reflect rim volume (i.e., the increase in thickness of the retinal nerve fiber layer as the axons converge toward the center of the disc). Our measurements show that curvature is greater in the horizontal axis than it is in the vertical axis and that this horizontal component was substantially reduced in the glaucoma group. In normal subjects the vertical component of curvature was smaller than the horizontal component, and in a minority of normal subjects, its value was negative. This component was also substantially reduced in the glaucoma group, where the mean value was negative.
Vertical slant did not differ significantly from zero and did not differ significantly between the two groups. Nasotemporal slant averaged -0.093 mm/mm (-5.3°) in normal subjects, and this was significantly different from zero. In our sample of glaucoma subjects this difference disappeared and, on average, slant values did not differ significantly from zero. One interpretation of this difference is that there is a relatively greater loss of axons entering the temporal side of the disc in glaucoma. Other interpretations will be discussed later.
The measures of cup depth (zm) and radius (r0) were both increased, as would be expected, in the glaucoma group, although these differences were not as large as those for curvature and nasotemporal slant. The model’s measure of the slope of the cup walls (s, which is inversely related to the steepness) was slightly smaller in the glaucoma group, reflecting an increased steepness of the walls of the cup. However the difference was not statistically significant, and in the following section we show that other measures of wall steepness were more significantly affected by glaucoma.
The values of fp, the goodness of fit to the image of a model without a cup (i.e., a parabolic surface), were also analyzed. The fits were relatively poor in most cases, with the exception of images (almost always from normal subjects) in which the cup was poorly defined. Values of fp differed significantly between the two groups and were found to increase the accuracy with which images could be classified. The value of fp was lowest in normal eyes in which a cup was barely detectable or absent. Because it can be calculated without the need for initial guesses of parameter values, a low value of fp (e.g., <0.075 mm) can be used to identify images without a cup. Although it was not done here, these images can safely be assumed to be normal and can be excluded from further processing.
Although the model describes normal and abnormal discs relatively well and many of the model parameters differed significantly between groups, it was clear that it failed to describe some significant features of glaucomatous discs, in particular the notably increased steepness of the cup walls. We therefore calculated additional morphologic indices from the images, using the model parameters as a framework for the calculations. The center of the model cup, its radius, and slope were used to define a central circular region of the image, denoted R, which just enclosed the cup and its walls (Fig. 3) . Visual checks were made, and it was found that in no case did any part of a cup appear to fall outside this region, nor, in most cases, did the region greatly exceed the cup in size. Three measures of steepness were calculated, one for the whole region (gr) one for the nasal half of the region (grN), and one for the temporal half (grT). Table 5 shows that, in the normal group, the nasal gradient measure was larger than the temporal gradient measure, possibly because of the greater number of blood vessels on the nasal side. However the temporal measure differed more between the normal and the glaucoma groups.
An additional measure of goodness of fit was made: fR measured the fit of the model function within R. This value was significantly larger, on average, in images from the glaucoma group (Table 5) .
The measure of cup depth (zm) may be relatively insensitive to small local excavations in the bottom of the cup, which may be indicative of glaucoma. Therefore, we took as a depth measure the average of the 500 deepest values measured within region R (which typically contained 2500–3000 pixels). This measure (z500) with d' = 0.71, differed more between the two groups than did zm, for which d' = 0.50.
Effects of Age
The effect of age on the parameters was examined in the normal
group by calculating the Pearson correlation between each parameter and
age. The results are shown in Tables 4
and 5
. Age had a significant
effect on cup depth, where the correlation (r =
-0.372, slope = -0.0068 mm/y) indicates a decrease in depth with
increasing age. A smaller positive correlation between age and the
horizontal and vertical components of curvature was also found. In
almost every case, the effects of age, although small in magnitude,
were in the direction opposite those of glaucoma. This should result in
an increasing dissimilarity between normal and glaucomatous discs with
increasing age and, at least in theory, should make the detection of
glaucoma easier in older subjects.
Classification
Figure 5
shows the distribution of P(G) values (i.e., the probability
that an image comes from the glaucoma population) for the two
populations. It shows that most cases were correctly classified with
high confidence levels (i.e., P > 0.9 or < 0.1).
Table 6
shows the distribution of probability values and of classification
mistakes. The overall classification accuracy was 89% (specificity,
89%; sensitivity, 88%). As might be expected, the accuracy varied
with the confidence level of the classification. When the confidence
level was low (i.e., for P between 0.4 and 0.6; leftmost
column in Table 5
), the accuracy was 67% (4/6 cases). For high
confidence levels (i.e., P > 0.9 or < 0.1;
rightmost column in Table 6 ), the overall accuracy was higher at 96%
(118/123 cases). Eighty-seven percent of cases were classified with
P > 0.7 or < 0.3. Within this group, the overall
accuracy was 92%.
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We examined, retrospectively, those images that had been confidently misclassified by the procedure (i.e., those with P < 0.1 or P > 0.9), as well as the corresponding visual fields. This included three normal and two glaucoma subjects (Fig. 6) . The clinical interpretation of the normal cases was that two of them had large discs, suggesting that the large cups were a consequence of large disc size. The interpretation of the appearance of the third disc was that it was suspicious, although the visual field was normal. In both confident false-negative glaucoma cases, the clinical interpretation was of normal disc appearance despite glaucomatous visual fields. The observation that the false-positive cases had large discs suggested that this might account for some of the other false positive results. Analysis of disc area (i.e., the HRT parameter ag) showed that disc area in the 11 false-positive cases was 3.07 ± 0.570 mm2, and this was significantly larger (P < 0.001) than the area in the correctly classified normal subjects, which was 2.335 ± 0.578 mm2.
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We calculated the correlations, across both normal and glaucoma groups, between the model parameters and the standard HRT parameters, and with the visual field MD. Table 7 shows the values for some selected HRT and model parameters. As might be expected, the HRT measure of disc area (ag) did not correlate strongly with any of the model parameters, because the model does not provide an explicit estimate of disc area. The highest correlation (r = 0.54) was with the fit of the parabolic function (fp). However, there was also a strong correlation (r = 0.52) with the model’s measure of cup radius (r0) which can be explained because cup area and disc area are known to be strongly correlated in normal discs.37 38 39 There was a strong correlation (r = 0.94) between the model’s measure of cup depth (z500) and the corresponding HRT measure (mdg). The fit of the parabolic surface (fp) also correlated strongly (r = 0.94) with mdg, because its value is small when a cup is absent, and therefore its value largely reflects cup depth. Both z500 and r0 showed weak correlations with HRT parameters hvc and var, on which the shape of the cup would be expected to have little effect.
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We calculated the correlation between each parameter and the visual field MD. These values are also given in Table 7 (second row from the bottom). The parameter showing the highest correlation with MD was horizontal image curvature (c; r = 0.55). The HRT indices showing the strongest correlations were abr (r = -0.43), var (r = 0.48), and mhc (r = -0.41).
For comparison with previous studies, we subjected all the HRT and model parameters to a receiver operating characteristic (ROC) analysis, using the methods described in Iester et al.27 Some of these values are given in the bottom row of Table 7 . The parameter showing the highest area under the curve (a measure of discrimination between two groups) was the horizontal curvature measure (c, area = 0.93). The best HRT measure was mhc (area = 0.91); csm did less well, with an area = 0.77.
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Discussion |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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Limitations of the Method
The particular mathematical model that we have chosen is unlikely
to be the only one that could be used. Its two main components, a
circularly symmetric cup placed on a background with parabolic
curvature, are both arguably unrealistic choices that could be
improved, given that the goal is to find a model that can accurately
reproduce the range of ONH shapes encountered in normal subjects.
First, real cups are often not circularly symmetric. Although modeling
an asymmetric cup would require introduction of additional shape
parameters, some of them might turn out to be informative. Second, the
parabolic curvature of the background (i.e., the rim and disc margins)
is unrealistic, because it leads to depth values that increase as the
square of the distance from the center of the cup, which obviously does
not happen in reality. The function probably only works because the
images, which are 10° x 10° in size, do not extend outside the
region in which the retinal nerve fiber layer is becoming increasingly
thick as fibers converge toward the optic nerve. This suggests that the
method would work less well on larger (e.g., 15° x 15°) images.
Finding the best-fitting model parameters for each image requires the application of iterative nonlinear least-squares optimization, which is not guaranteed to work in all cases.34 Although we chose a method (Levenburg–Marquardt) that is believed to be one of the most efficient, it, as in all similar procedures, requires a good initial estimate of the parameters that are to be adjusted if it is to work properly. It is not hard to make these estimates, however, and in our sample of images the method almost always (198 of 200 cases) converged on an acceptable solution. The two images in which the method failed were both from normal eyes in which a cup was barely perceptible. Such cases can be detected either by visual inspection or by first fitting a function without a cup—that is, by calculating fp. If this value is low (e.g., <0.075 mm) the image can, on the basis of the present results, be safely classified as normal (only one of the glaucoma images had a value of fp < 0.10 mm, and this image was classified as normal by the subsequent analysis in any case).
The model does not provide any estimate that can be directly related to disc area. Some of its parameters correlate with the HRT measure of area (Table 7) , but these correlations seem likely to be indirect. Disc area itself is not affected in glaucoma29 40 ; however, cup area and disc area are strongly correlated in normal eyes37 38 39 and the model’s measure of cup size would probably be more useful if it could be accompanied by an estimate of disc area and be re-expressed as a ratio. This is supported by the observation that disc area tends to be larger in the misclassified normal subjects. It remains to be seen whether such an estimate can be provided by an automated method (perhaps by analyzing the reflectivity image, using the determination of cup size and position by the present method as a guide).
The classification method we used is based on the assumption that the data values in each group are distributed according to a multivariate normal distribution. It is related to the methods used in DFA,41 but we have also used the advantage of providing an estimate of the probability that an image belongs to one of the two groups. We have not so far tested other classification methods that could be used (e.g., back-propagation neural nets23 42 ) that might perform better than the present method.
Limitations of the Study
Validation of the accuracy of the classification method depends
critically on the selection of subjects for the control normal and
glaucoma groups. Ideally, the normal subject group should be an
unbiased sample of the glaucoma-free population, and the subjects in
the glaucoma group should also be an unbiased sample of the glaucoma
population and have early visual field damage. We cannot guarantee that
either of these conditions has been met in the present study. Although
we took pains to exclude the presence of glaucoma in our normal group
by means of visual field testing and IOP measurements, an unusually
high proportion of the volunteers (33/100) reported a family history of
glaucoma. Often, this was a reason for volunteering. This raises the
possibility that some of them may either have had early glaucoma or
that they may have had a higher proportion of congenital disc
abnormalities predisposing them to glaucoma than would be found in the
normal population. This would tend to decrease rather than increase the
classification accuracy of the study. As it turned out, the proportion
of false-positive cases within the family history group (3/33) was
slightly less than it was in the rest of the group (8/67). It can be
argued that it is not necessarily bad that screening methods be tested
with a population at risk for glaucoma (e.g., elderly people with a
family history), provided glaucoma has been excluded as best it can
without ONH examination, because it is persons in this population who
are most likely to seek screening.
It is possible that optic disc morphology in our glaucoma sample was more abnormal than it is in the glaucoma population as a whole, because an abnormal disc morphology may have been one of the reasons for initial referral to the glaucoma clinic. (The implication is that glaucoma in combination with a relatively normal disc appearance exists and often goes undiagnosed.) This could artifactually increase the accuracy of our classification. There seems little we can do about this source of bias that will persist even though we were careful to apply only visual field criteria in selecting patients for inclusion or exclusion.
We attempted to match the samples for age, excluding normal subjects less than 25 years of age, to make the distributions match more closely. The normal group was nevertheless approximately 8 years younger (approximately 0.6 SDs) than the glaucoma group. Age was found to have a significant effect on some of the parameters used in the classification. This could bias the results, because age alone would then produce a difference in the parameter values between the two groups. However, when we examined the effect of age on normal discs we found that in almost every case the effects were in the direction opposite those found in the glaucoma group. In other words, in normal subjects, discs become less glaucomatous in appearance as they age, and discrimination between the two groups should be easiest in older subjects. The reasons for this do not seem clear. The difference in age between the two groups is likely, however, at least in theory, to have made classification more difficult.
Interpretation of the Shape Changes
The two parameters showing the largest difference (in statistical
terms) between the two groups were horizontal and vertical image
curvature (parameters c and d, Table 4
). Positive
values of these indices mean that the neuroretinal rim region around
the cup is convexly curved, causing it to bulge upward into the
vitreous. The degree of curvature seems likely to be determined at
least in part by geometric factors. As axons converge toward the center
of the disc the nerve fiber layer will become, of necessity,
increasingly thick. Only as axons leave the disc through the optic
nerve can the layer start to decrease in height. On geometric grounds
curvature would be predicted to be proportional to the total number of
ganglion cell axons and inversely proportional to the diameter of the
nerve. Our results bear this out, inasmuch as the curvature values were
greatly reduced in the glaucoma patients (suggesting a substantial loss
of axons), whereas a negative correlation between c and
d and disc area (ag; r = -0.34
and -0.30, respectively) was also observed (Table 7)
. These
considerations suggest that it may be of interest to perform a more
exact mathematical analysis of the way in which geometric factors
determine ONH surface topography.
The horizontal component of curvature was greater than the vertical component in normal ONH images, and this implies that the rim region tends to be elongated vertically. In images in which the horizontal component is positive and the vertical component negative (some normal images, and most glaucoma ones) the rim region is saddle-shaped and can be pictured as two parallel ridges on the nasal and temporal sides of the cup. That both values were decreased by relatively large amounts in the glaucoma group (Table 4) suggests that axons had been lost from many different regions of the retina. The observation of negative curvature values along the vertical axis in the glaucoma group suggests a relatively large loss of axons in the superior and inferior regions of the ONH; however, that the horizontal component changes more in absolute terms than the vertical component seems to suggest the opposite—that is, a greater number of axons entering the nerve on the nasal and temporal sides has been lost. It is possible that this loss is less apparent because there were more axons in these regions to begin with.
Previous morphologic studies of the ONH in glaucoma do not appear to have attempted to quantify the steepness of the cup walls. The measure we devised (gr; Fig. 3 ) showed a statistically large difference between groups (Table 5) . We defined gr in terms of the component of the gradient measured in a direction radial to the center of the cup (Appendix), because our initial measurements showed that although similar noncomponent measures were significantly greater in the glaucoma group, measures based on the radial component differed more. It is in this direction that the pressure gradient across the cup is steepest,43 and our findings are consistent with the expectation that pressure is one of the sources of damage to the nerve. However, an increase in the steepness of the cup walls may be the result of glaucoma, as well as a cause of it. For example, focal loss of ganglion cells originating in one region of the retina would cause the disappearance of a bundle of axons originating from that region and may be manifest as a notch in the rim, which in turn could lead to a steepening of the cup wall at that location. Alternatively, congenitally steep regions of the cup could predispose toward glaucomatous damage, given that the steeper the gradient the smaller the radius of curvature of the axons as they enter the nerve. Such sharp bends may be especially vulnerable to pressure-induced damage. It is possible that both mechanisms are at work, leading to a vicious cycle of damage.
We further divided the gradient measure into nasal and temporal components. Our results showed that in normal images the nasal component was greater on average than the temporal component (Table 5) . However, the temporal component was more severely affected by glaucoma. It is possible that the nasal component is more influenced by blood vessels, which are more numerous on the nasal side and which would probably not be affected by glaucoma. The temporal component would then be a better measure, whether or not it was in fact more severely affected by glaucoma. We have not so far performed a more detailed sector-based examination of gradient measures, but this could be worthwhile.
Other measures showed smaller differences between the two groups, although they were also useful in classification. The measures r0 and zm were both increased in the glaucoma group, although, as has been observed previously,44 they are relatively poor indicators on their own. When analyzing cup depth measures, we found that zm differed less between the groups than did the alternative measure (z500), which is an average of the 500 most extreme depth values within the cup region. This measure would be expected to be sensitive to the presence of localized excavations in the cup—namely, the pitting that is observed in glaucoma.45
The interpretation of the goodness-of-fit parameter fR (root mean square difference between the model and the image in the cup region) is less clear. Errors in the determination of topography by the HRT are likely to contribute only a small amount to the values, because these errors (the pixel variability we observed on repeat imaging) average approximately ±0.025 mm, as has been shown by others36 and in the present results, and the values of fR were almost always several times greater than this (Table 5) . Although it seems clear that glaucomatous ONH images show shape variations that are not captured by the model and that are not present in normal images, we have not identified what these differences are. There are many possibilities, including notches and variations in the height of the rim that are not captured by the curvature measures, asymmetric cup shapes, excavations in the walls and floor of the cup, and irregularities caused by blood vessels. Further characterization of these variations should be possible, and may be more informative than the goodness-of-fit measures.
One other measure, slant in the nasotemporal axis, showed a significant difference between the two groups (Table 4) . Normal images tended to be slanted, by approximately 6° (-0.098 mm/mm), on average, in such a way that the temporal (foveal) side is higher than the nasal side. This slant was usually absent in the glaucoma group. The slant in normal subjects may be explained by the fact that more axons enter the disc on the temporal than on the nasal side, causing an overall negative slant of the surface in the nasal direction. This effect would be offset by the greater number of blood vessels on the nasal side of the disc that may increase the overall measure of height on that side.46 Although the decrease in slant observed in the glaucoma group suggests a relatively greater loss of fibers entering the nerve on the temporal side (i.e., on the side closest to the fovea), the difference should be interpreted cautiously. Slant in the image can be introduced by changes in the angle of the ophthalmoscope relative to the optical axis of the eye18 and this angle is chosen by the operator. The images from the normal group were all obtained by one operator, who was not one of those who obtained the images in the glaucoma group. Systematic differences in the operator settings, although small, may have caused artifactual differences between the groups. Against this, it could be argued that when imaging is performed through undilated pupils (as was the case in this study) there is very little scope for changing the viewing angle of the ophthalmoscope. In practice, once the subject is fixating, the angle is determined almost entirely by the requirement that the disc be close to the center of the image.
This study, like others before it,21 23 27 28 31 32 42 44 has shown that ONH profiles of patients with glaucoma can be distinguished from those of normal subjects with a high degree of reliability. Although it has identified several new shape parameters that are substantially altered by glaucoma, it has not shown what the pattern of change is in the very earliest stages of glaucoma, before visual field changes have occurred. It is plausible that those factors that are most changed by glaucoma are the ones that change earliest, but this is not necessarily the case. The early pattern of change may be different and may be harder to detect reliably. Loss of surface curvature, steepening of the cup walls, greater surface irregularity, and pitting, widening, and deepening of the cup are all different types of change that may not all occur at the same time or to the same degree in different patients. Automated mathematical analysis of ONH shape, because it does not depend on subjective estimates of disc boundaries, offers a promising method for identifying and reliably quantifying these different changes in long-term studies in individual patients.
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Appendix 1 |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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 | (1) |
 | (1A) |
For each image, the 10 free parameters of the model were adjusted to
give the best fit of the model to the image. Fit (f) was
defined as the root mean square of the difference between the image and
the model, measured in millimeters
 | (2) |
and ß, which are
normally equal, scale pixel indices in the i and
j directions to millimeters in the x and
y directions, respectively, and 
scales the
one-byte-per-pixel value in the image (0–255) to millimeters in the
depth (z) dimension. Fitting was performed in two stages: first, we made initial estimates of the parameter values and second, we refined the values to minimize f, using an iterative nonlinear least-squares fitting procedure. The initial estimates were made as follows: 1) we calculated a least-squares fit to the image of just the last five terms of equation 1 (i.e., of the parabolic surface). Because this function is linear in its five parameters, it is possible to calculate the best fitting parameter values explicitly using standard methods (Gauss–Jordan elimination).34 2) This function was then subtracted from the raw image, to obtain one in which the cup should be the major feature. The average of the positions and values of the 50 largest (i.e., deepest) pixel values in this image was then calculated to obtain estimates of cup position (x0,y0) and depth (zm), respectively. A region of pixels, equal to one tenth of the image width, along the edges of the image was excluded when searching for deep pixels. 3) The fit of the parabolic surface was then repeated, this time excluding the region of the image likely to contain the cup—namely, a region within a distance of 0.5 mm from the estimated center of the cup. This fit was used to obtain the initial estimates of a, b, c, d, and z0. The initial estimates of cup radius and slope were fixed at 0.5 mm and 0.1, respectively. After this, the parameter estimates were further refined using the Levenburg–Marquardt optimization technique.34
These procedures were applied to 10° x 10° images extracted from
databases created by the operating software (version 2.01) provided
with the HRT. The program HRTCOMP was used to extract the images, and
the program DBSCALES was used to extract the appropriate scaling
parameters (, ß, and ) for each image. Because the edges of the
images sometimes contain artifacts, a region typically 10 pixels wide
along each edge of the image was excluded from analysis. To decrease
processing time, the 256 x 256-pixel images were reduced in size
by averaging over blocks of 4 x 4 pixels, to give typically
60 x 60-pixel images. Averaging over smaller blocks, or not
averaging at all, made little difference to the estimated parameter
values.
Morphologic Indices Derived Using the Model
After the function fits and derivation of parameter values,
additional morphologic indices were calculated. The selection and
definition of these was guided by their usefulness in discriminating
between normal and glaucomatous images. First, a set of pixels,
R, which included the cup was defined with a center position
(x0,
y0) and a radius =
r0 +
loge(9)s. Within this region, cup
depth is greater than 10% of its value at the center. The calculations
were performed on the raw 256 x 256-pixel images. As described,
pixels on a defined border along the edges of the image were excluded
from R. The following values were then calculated.
First, the goodness of fit was defined
 | (3) |
Second, an index of the steepness of the cup walls was determined.
Although parameter s (equation 1)
gives a measure of
steepness, we found that a more informative measure could be obtained
by summing the image gradient values within R. This was
performed with two further modifications: Only the radial component of
the gradient (i.e., the component measured in the direction pointing
toward the center of the cup) was used, and only gradient values with
large negative slope values less than -45° were included in the sum.
We define the gradient in terms of its x and y
components as
 | (4) |
The radial component Gr is given by
 | (5) |
Because depth is measured as a positive quantity, the steeper the
slope of the cup walls, the more negative are the radial gradient
values. The measure used in this calculation, defined as the positive
quantity gr, is given by
 | (6) |
Third, we calculated an index of maximum cup depth, defined as the
average of the 500 largest depth values, measured within region
R. Denoting this average as
I500, we define the index as
 | (7) |
Fourth, the fit to the image of a curved surface without a cup (i.e., equation 1 , but without the first term on the righthand side) was calculated, by analogy with equation 2 , and is denoted by fp. This value is low in normal images, particularly those in which the cup is small or absent.
Classification
For a given set of D parameters measured from each
image (the procedure for selecting these is described below), i.e., a
data vector x = (x1,
x2,...
xD), we calculated the probability that
the point came from the normal group—that is,
P(x|N), and the probability that it came from
the glaucoma group i.e., P(x|G). These
probabilities were calculated using the multivariate normal probability
density function35
:
 | (8) |
We then calculated the probability that the measurements were from an
eye with glaucoma, i.e., P(G|x). According to
Bayes’ theorem,35
this is
 | (9) |
Parameter Selection
Not all the described parameters were used for the purpose of
classifying images. We excluded those that showed little difference
between the groups (s, z0
and b). Some sets of parameters were obviously closely
related (e.g., zm and
z500; f and
fR; and
gr,
grN, and
grT) and for these
parameters we took the one showing the largest difference, measured by
d', between the two groups. One parameter, horizontal image
slope (a) was excluded because the difference between the
two groups could have been artifactual (see the Discussion section).
This resulted in a set of seven parameters, defined as
x = c, d,
z500,
grT,
fp,
fR, and
r0. These were used to calculate, for
each image, the probability that it came from the glaucoma group—that
is, P(G|x). Cases were classified as glaucoma
if P(G|x) > 0.5. Because
P(G|x) = 1 -
P(N|x) this is equivalent to the condition that
P(G|x) > P(N|x).
With this method, specificity (the percentage of normal cases correctly
classified) and sensitivity (the percentage of glaucoma cases correctly
classified) tend to be similar and equal to the overall classification
accuracy.
|
Acknowledgements |
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|
Footnotes |
|---|
Submitted for publication June 23, 1999; revised December 1, 1999; accepted January 18, 2000.
Commercial relationships policy: P(NVS, AC); N(GS, FSM).
Corresponding author: Nicholas V. Swindale, Department of Ophthalmology, University of British Columbia, 2550 Willow Street, Vancouver, BC V5Z 3N9, Canada. swindale{at}interchange.ubc.ca
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