Abstract
purpose. To classify images of optic nerve head (ONH) topography obtained by
scanning laser ophthalmoscopy as normal or glaucomatous without prior
manual outlining of the optic disc.
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.
Glaucoma is a slow and irreversible neurodegenerative disease,
the onset of which is usually not detected by the patient. Diagnosis
may be based on a combination of variables,
1 2 but the
most dependable single index is probably the identification of a
characteristic pattern of visual field defects. However, these defects
may only appear after a substantial amount of retinal damage has
occurred.
3 4 5 6 There is a widely accepted need, therefore,
for a method that can reliably detect glaucomatous damage at an early
stage so that treatment to prevent further progression can be
instigated. Ideally, such a test would have high sensitivity and
specificity and be quickly and cheaply administered to large numbers of
persons in the normal population, especially those most likely to be at
risk of the disease, such as the elderly and those with a family
history of glaucoma.
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.
Methods
The Model
Images of normal and glaucomatous ONHs obtained with the confocal
scanning laser ophthalmoscope typically exhibit a central, roughly
circular depression of variable width and depth (the cup), superimposed
on a relatively smooth surface with a variable degree of curvature (the
rim region). This curvature is typically convex in normal subjects and
is caused by the layer of ganglion cell axons becoming, of geometric
necessity, increasingly thick as the axons converge toward the optic
nerve. These regularities in shape suggest the possibility of
description by a mathematical model. Such a model would ideally have a
small number of parameters, whose interpretation in anatomic terms
would be relatively straightforward. We devised and investigated a
model (described in detail in the Appendix) that has 10 free
parameters, each of which corresponds to a specific aspect of ONH
morphology
(Fig. 1) . A complete description of these parameters is given in
Table 1 . A standard nonlinear, least-squares fitting
technique
34 was used to adjust the values of the
parameters to produce the closest degree of similarity between the
model image and the particular ONH image under consideration.
Figure 2 shows examples of ONH images and corresponding best-fitting model
images.
After these initial fits, a number of morphologic indices were
calculated, using the best-fitting model image as a guide. The
selection and definition of these was guided by their usefulness in
discriminating between normal and glaucomatous images. The methods used
to calculate the indices are described in detail in the Appendix, and a
summary is given in
Table 2 . In brief, the model was first used to establish the coordinates of a
circular region that covered the cup
(Fig. 3) . Within this region we calculated (from the real image) 1) an overall
measure of the steepness of the cup walls
(
g r) and separate temporal
(
g r T) and nasal
(
g r N) components of this measure;
2) a measure of the goodness-of-fit of the model to the image in this
region (
f R): This value would be
expected to be large in images in which the cup was irregular in shape
and not well-described by the model; 3) a measure of the goodness of
fit of the image to a model without a cup—that is, a smooth parabolic
surface (
f p): This measure would be
large in images with large cups and small in images in which a cup was
small or absent; and 4) a measure of maximum cup depth
(
z 500) that was the average of the 500
largest depth values present for image pixels within the cup region.
This method was applied to a database of 100 images obtained from eyes
screened to exclude the presence of glaucoma and 100 images obtained
from eyes with open angles and showing visual field changes indicative
of glaucoma. Criteria for subject selection are described in detail in
the next section. The model fitting, analyses, and
classification
35 were implemented with the aid of a
batch-processing language that allowed the calculations to be performed
on each of the images automatically without user intervention. A
233-MHz Pentium II computer performed the computations, and the
total processing time for each image was approximately 3 to 6 seconds.
Criteria for Subject Selection
Normal Subjects.
The method of obtaining images from eyes in the normal group was as
follows:
-
Advertisements asking for volunteers were placed in local media and
in locations likely to be frequented by the elderly. Staff at the Eye
Care Center and their friends and relatives were also recruited.
Volunteers were told about the nature and purposes of the study and
asked some preliminary questions. They were excluded at this stage if
questioning revealed that they had eye disease or a history of eye
disease known to be related to glaucoma (e.g., pigmentary dispersion
syndrome), if they had a condition such as keratoconus or cataract
severe enough to interfere with scanning, if they did not have normal
corrected visual acuity, or if they had strabismus (which often causes
fixation difficulties during scanning).
-
If they passed the screening questions, volunteers visited the clinic
and were given a consent form to sign. A brief medical history,
including details of any relative(s) who had glaucoma was obtained, and
the following tests were performed: a Humphrey (San Leandro, CA) visual
field test (threshold 30-2) of both eyes, determination of intraocular
pressure (IOP) measured in both eyes by tonometry (Tono-Pen XL; Mentor,
Santa Barbara, CA), and HRT scans, 10° × 10° in size, of
each eye through undilated pupils. To obtain these images, at least
three separate scans of each ONH were obtained, and a single mean of
three images was calculated.
36 When more than
three scans were obtained, the set of three giving the lowest SD, as
reported by the HRT software, was chosen.
-
The following criteria were applied for inclusion in the normal group:
The visual field had to be within normal limits in both eyes as defined
by the Humphrey glaucoma hemifield test (some subjects whose results
were borderline on this test were included after further clinical
evaluation of their fields), there had to be less than 30% fixation
losses during testing, the IOP had to be 21 mm Hg or less, and the SD
obtained on averaging three separate HRT scans had to be 50 μm or
less. After performing scans and visual field tests on 107 volunteers,
two subjects less than 25 years old were excluded on grounds of age,
four subjects were excluded because of severe (less than −7 D) myopia,
and one subject was excluded because of abnormal visual field test
results. A family history of glaucoma was not used as a criterion for
exclusion. As it turned out, many (33/100) of the volunteers reported a
positive family history, and this was often the reason that subjects
gave for volunteering in the first place.
-
In the majority of cases, scans from both eyes of each subject were
available, and only one was chosen for inclusion in the final database.
This was done either at random, or in such a way as to make the numbers
of left and right eyes equal.
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
The study followed the tenets of the Declaration of Helsinki, the
subjects provided informed consent, and the project was approved by the
Clinical Research Ethics Board of the University of British Columbia.
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).
Results
Robustness of the Numerical Procedures
Nonlinear least-squares optimization of parameters is dependent on
good initial estimates and may fail to produce meaningful results if
the initial estimates are poor
34 or if the model itself is
not a good one. For each image, we therefore checked that the fitting
procedure had produced what seemed likely to be correct values,
particularly with respect to cup position
(
x 0 and
y 0) and cup depth
(
z m). This was achieved by visually
examining the model and real images to check that they seemed similar,
that cup position was close to the position of the real cup, and that
parameter values were within expected ranges. Acceptable fits and
parameter values were judged to have been obtained in 198 of the 200
images. The two images in which the automated procedure failed were
from normal eyes in which a cup was barely detectable. In one of these
images an acceptable fit was obtained by manually choosing initial
parameters. In the other image, a fit was obtained by constraining
z m = 0 and setting
x 0 and
y 0 equal to the estimated center of
the cup, close to the middle of the image. Radius and slope values were
set equal to the means of the rest of the normal group for the purposes
of calculating the other derived parameters in this image.
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 =
y 0, calculated
using the averages of the parameters for the normal (solid line) and
glaucoma (dashed line) groups.
The model generally provided a good description of the images,
especially in the normal group, in which the value of the fit (the root
mean square of the difference between the model and real image)
averaged 0.076 mm. This is also evident from the visual comparisons
(Fig. 2) . That the fit in the glaucoma group, which averaged 0.092 mm,
was less good suggests a correlation between glaucoma and irregularity
in cup shape and means that the value of the fit can be used to
discriminate between normal and glaucomatous images.
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 (z m) and
radius (r 0) 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 f p, 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 f p differed significantly between the two groups and were found to
increase the accuracy with which images could be classified. The value
of f p 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 f p (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
(
g r ) one for the nasal half of
the region (
g r N), and one for the
temporal half (
g r T).
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:
f R 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 (z m) 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 (z 500) with d′ =
0.71, differed more between the two groups than did z m, 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%.
We performed a jackknife (leave-one-out) cross-validation procedure in
which data from the case classified were not used in the calculation of
group means and covariance values. This gives a better estimate of the
method’s ability to generalize to new cases—that is, cases not used
to derive the classification function. With this procedure, six
additional wrong classifications were identified, and the overall
accuracy of the method was reduced to 86%. However, of the new
mistakes, three had P values that fell in the 0.4 to 0.6
range, two had P values in the 0.2 to 0.4 range, and only
one had P > 0.7. Overall, the accuracy of the
confidently classified cases was reduced to 88%.
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 mm
2, and this was
significantly larger (
P < 0.001) than the area in the
correctly classified normal subjects, which was 2.335 ± 0.578
mm
2.
Comparison with Standard HRT Parameters and Visual Field Indices
We compared the accuracy of the present method with that which
could be obtained from the standard set of shape parameters calculated
by the HRT operating software. The DFA formula of Mikelberg et
al.
21 which is incorporated into the software (ver. 2.01)
gave a sensitivity of 49% and a specificity of 98%. Adjusting the
classification threshold to give more equal values yielded a
sensitivity of 77% and a specificity of 77%. A more fair comparison
is to subject the data to a new DFA. Thirteen of the parameters
(
ag [disc area] was excluded because it is derived from
mr [mean radius]) were entered into a forward-stepping
DFA, using an F-to-enter of 4.0 and an F-to-remove of 3.0. The overall
accuracy was 84%, and the cross-validated accuracy was 83.5%. The six
parameters selected by the analysis were
abr (area below
reference),
mhc (mean height of contour),
mr (mean radius),
var (volume above reference),
vas (volume above surface), and
vbr (volume below reference).
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 (
f p). However,
there was also a strong correlation (
r = 0.52) with the
model’s measure of cup radius (
r 0)
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 (
z 500) and the
corresponding HRT measure (
mdg). The fit of the parabolic
surface (
f p) 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
z 500 and
r 0 showed weak correlations with HRT
parameters
hvc and
var, on which the shape of the
cup would be expected to have little effect.
The HRT parameter csm (cup shape measure) has been shown by
several studies to give good discrimination between groups, although
its structural interpretation seems obscure. The model parameter that
correlated most strongly with csm was temporal gradient
measure, g r T (r =
0.40), however the correlation with cup diameter
(r 0) was nearly as strong
(r = 0.37).
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.
Discussion
Our results show that it is possible to analyze and classify
images of the ONH with a degree of reliability that is comparable to
that of existing methods, without the need for prior manual outlining
of the optic disc. The method that we propose takes only a few seconds
to perform on a computer and requires no user intervention at any stage
in the numerical analysis. As such, it is reliable and repeatable and
is not dependent on subjective estimates of the position of the borders
of the optic disc. These estimates may vary within and between
operators and complicate the comparison of studies conducted in
different centers. Although the details of the method itself are
somewhat technical, the structural interpretation of the parameters
extracted by it is less complex. The measures identified by the model
as particularly useful in indicating glaucomatous damage to the disc
include the horizontal and vertical components of image curvature
(i.e., the amount by which the nerve fiber layer surrounding the cup
bulges upward into the vitreous) and the steepness of the cup walls.
Cup size and other measures of surface irregularity in the region of
the cup are also informative. Before discussing the interpretation of
these shape changes in more detail, we will consider first the
limitations of the mathematical modeling method we have used and,
secondly, the limitations imposed by the study design.
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° × 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° × 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
f p. 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
f p < 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 glaucoma
29 40 ; however, cup
area and disc area are strongly correlated in normal
eyes
37 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
nets
23 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 (
g r ;
Fig. 3 ) showed a
statistically large difference between groups
(Table 5) . We defined
g r 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
r 0 and
z m 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
z m differed
less between the groups than did the alternative measure
(
z 500), 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
f R (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 others
36 and in the
present results, and the values of
f R 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
eye
18 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.
Appendix 1
The equation used to describe the surface topography of the ONH
was:
\[z(x,\ y){=}\ \frac{z_{\mathrm{m}}}{1{+}e^{(r-r_{0})/s}}{+}a(x-x_{0}){+}b(y-y_{0}){+}c(x-x_{0})^{2}{+}d(y-y_{0})^{2}{+}z_{0}\]
where
\[r{=}\sqrt{(x-x_{0})^{2}{+}(y-y_{0})^{2}}\]
This defines the depth of the surface (
z) as a function
of position (
x,
y) on the surface. Movement in
the positive
x direction is identified with movement in the
nasal direction, away from the fovea, and movement in the positive
y direction corresponds to movement toward the superior
retina. The first term on the righthand side of the equation represents
a circularly symmetric cup, centered on position
(
x 0,
y 0), with a depth
z m, a radius
r 0, and walls with a slope inversely
proportional to
s (the smaller the value of
s,
the steeper the slope). The following four terms describe a surface
with variable slant in the
x and
y directions
(parameters
a and
b, respectively) and with
variable curvature, assumed to be parabolic, in the
x and
y directions (parameters
c and
d,
respectively). The radius (
z 0) is a
constant offset in the
z direction.
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
\[f{=}\left\{\frac{1}{N}\ {{\sum}_{i,j}^{all\ pixels}}\ {[}{\gamma}I(i,\ j)-z({\alpha}i,\ {\beta}j){]}^{2}\right\}\ ^{1/2}\]
where
I(
i,
j) is the value of the
image at pixel (
i,
j) and
N is the
total number of pixels in the image. The terms α and β, which are
normally equal, scale pixel indices in the
i and
j directions to millimeters in the
x and
y directions, respectively, and γ 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
(
x 0,
y 0)
and depth (
z m), 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
z 0.
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° × 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 (α, β, and γ) 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 × 256-pixel images were reduced in size
by averaging over blocks of 4 × 4 pixels, to give typically
60 × 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
(x 0, y 0) and a radius = r 0 +
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 × 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
\[f_{R}{=}\left\{\frac{1}{N}\ {{\sum}_{i,j{\subset}R}}\ {[}{\gamma}I(i,\ j)-z({\alpha}i,\ {\beta}j){]}^{2}\right\}\ ^{1/2}\]
where
N is the number of points contained in
R. The larger the value of
f R ,
the worse is the fit of the model to the image in the region of the
cup. As described above, this value was found to be significantly
larger in eyes with glaucoma.
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
\[G_{x}(i,\ j){=}\ \frac{{\gamma}{[}I(i{+}1,\ j)-I(i,\ j){]}}{{\alpha}};G_{y}(i,\ j){=}\ \frac{{\gamma}{[}I(i,\ j{+}1)-I(i,\ j){]}}{{\beta}}.\]
The radial component
G r is given by
\[G_{r}{=}\ \frac{(x-x_{0})G_{x}{+}(y-y_{0})G_{y}}{{\vert}r{\vert}}.\]
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
g r , is given by
\[g_{r}{=}\mathrm{log}_{e}\ \left\{{{\sum}_{i,j{\subset}R}}\ thr(\mathrm{-}G_{r},\ 1)\right\}\]
where
thr(
x,1) equals
x, if
x > 1 and equals 0 if
x < 1. The
quantity
g r is therefore the log of the
sum of all those radial gradient values within region
R that
are more negative than (i.e., steeper than), a gradient of −1 mm/mm.
(The log was taken because the resultant distribution of values more
closely approximate a normal distribution.) We similarly calculated
indices for gradients in the nasal and temporal halves of region
R, denoting these by parameters
g r N and
g r T, respectively.
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
I 500, we define the index as
\[z_{500}{=}{\gamma}I_{500}-z_{0}.\]
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
f p. 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 = (
x 1,
x 2,…
x D ), 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 function
35 :
\[P(\mathbf{x}){=}\ \frac{1}{(2{\pi})^{D/2}{\vert}\mathbf{C}{\vert}^{1/2}}\ \mathrm{exp}\left\{\mathrm{-}\ \frac{1}{2}\ (\mathbf{x}-\mathbf{u})^{\mathrm{T}}\mathbf{C}^{-1}(\mathbf{x}-\mathbf{u})\right\}\]
where
u is the mean of
x taken over the
group in question (normal or glaucomatous) and
C is the
within-group covariance matrix. When cross-validation was used, data
from the case being classified were excluded from the data used to
calculate the means and the covariance matrices. This gives a less
biased estimate of the ability of the classification method to
generalize to new data (data not used to derive the classification
method in the first place).
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
\[P(G{\vert}\mathbf{x}){=}\ \frac{P(\mathbf{x}{\vert}G)P(G)}{P(\mathbf{x}{\vert}N)P(N){+}P(\mathbf{x}{\vert}G)P(G)}\]
where
P(N) and
P(G) are the prior
probabilities that the subject in question is normal or has glaucoma.
In the present case the prior probabilities were 0.5 because the sample
sizes (
n = 100) were equal, but this would not be the
case with unequal sample sizes or if the normal population were
screened.
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, z 0 and b). Some sets of parameters were obviously closely
related (e.g., z m and z 500; f and f R; and g r , g r N , and g r T) 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, z 500, g r T , f p, f R, and r 0. 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.
Supported by the British Columbia Health Research Foundation and the Glaucoma Research Society of Canada.
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@interchange.ubc.ca
| Parameter Name | Symbol | Description |
| Nasotemporal slant | a | Overall component of tilt in the nasotemporal axis (mm/mm) |
| Vertical slant | b | Overall component of tilt along the vertical axis (mm/mm) |
| Horizontal image curvature | c | Overall curvature along the naso-temporal axis (mm2/m) |
| Vertical image curvature | d | Overall curvature in the vertical direction |
| Cup position | x0, y0 | Position of center of cup in image (mm) |
| Cup radius | r0 | Distance from center of cup to the cup wall at half-height (mm) |
| Cup slope | s | Slope of cup wall (mm) |
| Cup depth | zm | Depth of the cup (mm) |
| Vertical offset | z0 | Offset of the image in the vertical direction (mm) |
| Parameter Name | Symbol | Description |
| Cup gradient measure | gr | Overall steepness of cup walls |
| Cup gradient measure temporal | gr T | Overall steepness of cup walls on the temporal side |
| Cup gradient measure nasal | gr N | Overall steepness of cup walls on the nasal side |
| Fit in central region | fR | Dissimilarity between the model and image in the cup region |
| Fit of parabolic function | fp | Dissimilarity between the image and a smooth parabolic surface without a cup |
| Maximum cup depth | z500 | Average of the 500 largest depth values in the cup |
| Normal | Glaucoma |
| Total number | 100 | 100 |
| Age range (mean± SD) | 25–87 y (53 ± 14 y) | 27–81 y (61 ± 13 y) |
| Gender (M/F) | 43/57 | 51/49 |
| Eye (R/L) | 48/52 | 51/49 |
| Race | | |
| White | 94 | 87 |
| Asian | 6 | 12 |
| Black | 0 | 1 |
| Mean deviation (dB) | 0.3 ± 1.6 | −4.9 ± 2.7 |
| SD of HRT scans (mm) | 0.024 ± 0.008 mm | 0.029 ± 0.010 mm |
| Refraction (D) | −0.48 ± 2.1 D | −0.61 ± 2.4 D |
| Symbol | Description | Normal | Glaucoma | d′ | r (age) |
| c | Horizontal image curvature | 0.193 ± 0.091 | 0.051 ± 0.053 | −1.97* | 0.16 |
| d | Vertical image curvature | 0.045 ± 0.075 | −0.049 ± 0.046 | −1.55* | 0.23, † |
| a | Nasotemporal slant (mm/mm) | −0.093 ± 0.130 | 0.018 ± 0.097 | 0.98* | 0.002 |
| f | Fit of the model to the image (mm) | 0.076 ± 0.021 | 0.092 ± 0.024 | 0.71* | −0.061 |
| r0 | Cup radius (mm) | 0.444 ± 0.142 | 0.547 ± 0.154 | 0.70* | −0.064 |
| zm | Cup depth (mm) | 0.621 ± 0.249 | 0.739 ± 0.222 | 0.50* | −0.372* |
| b | Vertical slant (mm/mm) | 0.008 ± 0.072 | −0.01 ± 0.068 | −0.26 | 0.005 |
| s | Cup slope | 0.116 ± 0.059 | 0.107 ± 0.055 | −0.16 | −0.163 |
| z0 | Vertical offset (mm) | 0.982 ± 0.312 | 1.028 ± 0.339 | 0.14 | 0.209, † |
Table 5. Secondary Morphologic Indices
Table 5. Secondary Morphologic Indices
| Symbol | Description | Normal | Glaucoma | d′ | r (age) |
| gr T | Cup gradient measure temporal | 8.39 ± 0.58 | 9.06 ± 0.47 | 1.28* | −0.15 |
| gr | Cup gradient measure | 9.35 ± 0.49 | 9.90 ± 0.40 | 1.23* | −0.18 |
| gr N | Cup gradient measure nasal | 8.85 ± 0.48 | 9.31 ± 0.38 | 1.06* | −0.17 |
| fR | Fit in central region (mm) | 0.107 ± 0.038 | 0.147 ± 0.048 | 0.93* | −0.20, † |
| fp | Fit of parabolic function (mm) | 0.137 ± 0.047 | 0.178 ± 0.054 | 0.81* | −0.29, † |
| z500 | Maximum cup depth (mm) | 0.702 ± 0.247 | 0.878 ± 0.250 | 0.71* | −0.31, † |
Table 6. Classification Statistics
Table 6. Classification Statistics
| According to Range of P | | | | | |
| 0.4 < P < 0.6 | 0.6 < P < 0.7 0.3 < P < 0.4 | 0.7 < P < 0.8 0.2 < P < 0.3 | 0.8 < P < 0.9 0.1 < P < 0.2 | 0.9 < P < 1.0 0.0 < P < 0.1 |
| Number of cases | 6 | 21 | 25 | 25 | 123 |
| Number of mistakes | 2 | 8 | 7 | 1 | 5 |
| Correct (%) | 67 | 62 | 72 | 96 | 96 |
| According to Cumulative P | | | | | |
| 0.0 < P < 1.0 | 0.6 < P < 1.0 0.0 < P < 0.4 | 0.7 < P < 1.0 0.0 < P < 0.3 | 0.8 < P < 1.0 0.0 < P < 0.2 | 0.9 < P < 1.0 0.0 < P < 0.1 |
| Total cases (%) | 100 | 97 | 87 | 74 | 62 |
| Correct (%) | 89 | 89 | 92 | 96 | 96 |
Table 7. Correlations with Standard HRT Parameters, Visual Field MD, and ROC
Areas
Table 7. Correlations with Standard HRT Parameters, Visual Field MD, and ROC
Areas
| ag | abr | mhc | hvc | mdg | csm | var | vbr | r0 | z500 | c | d | gr T | fR | fp |
| abr | 0.68 | | | | | | | | | | | | | | |
| mhc | 0.22 | 0.34 | | | | | | | | | | | | | |
| hvc | −0.04 | −0.36 | −0.22 | | | | | | | | | | | | |
| mdg | 0.48 | 0.70 | 0.32 | 0.03 | | | | | | | | | | | |
| csm | 0.21 | 0.41 | 0.12 | −0.21 | 0.18 | | | | | | | | | | |
| var | 0.12 | −0.52 | −0.29 | 0.73 | −0.27 | −0.25 | | | | | | | | | |
| vbr | 0.51 | 0.82 | 0.24 | −0.19 | 0.66 | 0.31 | −0.38 | | | | | | | | |
| r0 | 0.52 | 0.58 | 0.20 | −0.12 | 0.31 | 0.37 | −0.10 | 0.43 | | | | | | | |
| z500 | 0.52 | 0.66 | 0.24 | 0.13 | 0.94 | 0.19 | −0.13 | 0.64 | 0.32 | | | | | | |
| c | −0.34 | −0.64 | −0.53 | 0.54 | −0.47 | −0.27 | 0.62 | −0.41 | −0.44 | −0.32 | | | | | |
| d | −0.30 | −0.60 | −0.53 | 0.38 | −0.58 | −0.22 | 0.57 | −0.42 | −0.28 | −0.39 | 0.84 | | | | |
| gr T | 0.51 | 0.78 | 0.31 | −0.16 | 0.75 | 0.40 | −0.40 | 0.61 | 0.51 | 0.78 | −0.55 | −0.55 | | | |
| fR | 0.37 | 0.60 | 0.21 | 0.09 | 0.85 | 0.18 | −0.20 | 0.59 | 0.33 | 0.82 | −0.34 | −0.44 | 0.68 | | |
| fp | 0.54 | 0.72 | 0.25 | 0.07 | 0.94 | 0.28 | −0.18 | 0.72 | 0.44 | 0.94 | −0.38 | −0.45 | 0.77 | 0.83 | |
| MD | −0.09 | −0.43 | −0.41 | 0.32 | −0.27 | −0.23 | 0.48 | −0.24 | −0.29 | −0.20 | 0.55 | 0.48 | −0.46 | −0.27 | −0.26 |
| ROC area | 0.60 | 0.85 | 0.91 | 0.74 | 0.73 | 0.77 | 0.85 | 0.84 | 0.69 | 0.69 | 0.93 | 0.86 | 0.82 | 0.76 | 0.73 |
The authors thank the subjects who generously volunteered for this
study; Phil Hetherington and Tim Blanche for useful suggestions; and
Gerhard Zinser of Heidelberg Engineering for his comments on the
manuscript and for kindly supplying the programs HRTCOMP and DBSCALES.
Pederson JE, Anderson DR. The mode of progressive disc cupping in ocular hypertension and glaucoma.
Arch Ophthalmol. 1980;98:490–495.
[CrossRef] [PubMed]Quigley HA, Addicks EM, Green WR. Optic nerve damage in human glaucoma, III: quantitative correlation of nerve fibre loss and visual field defect in glaucoma, ischemic neuropathy, papilledema, and toxic neuropathy.
Arch Ophthalmol. 1982;100:135–146.
[CrossRef] [PubMed]Sommer A, Pollack I, Maumenee AE. Optic disc parameters and onset of glaucomatous field loss.
Arch Ophthalmol. 1979;97:1444–1448.
[CrossRef] [PubMed]Sommer A, Katz J, Quigley HA, et al. Clinically detectable nerve fibre atrophy precedes the onset of glaucomatous field loss.
Arch Ophthalmol. 1991;109:77–83.
[CrossRef] [PubMed]Weinreb RN, Dreher AW, Bille J. Quantitative assessment of the optic nerve head with the laser tomographic scanner.
Int Ophthalmol. 1989;13:25–27.
[CrossRef] [PubMed]Kruse FE, Burk RO, Völcker HE, Zinser G, Harbarth U. Reproducibility of topographic measurements of the optic nerve head with laser tomographic scanning.
Ophthalmology. 1989;96:1320–1324.
[CrossRef] [PubMed]Dreher AW, Tso PC, Weinreb RN. Reproducibility of topographic measurements of the normal and glaucomatous optic nerve head with the laser tomographic scanner.
Am J Ophthalmol. 1991;111:221–229.
[CrossRef] [PubMed]Cioffi GA, Robin AL, Eastman RD, et al. Confocal laser scanning ophthalmoscope: reproducibility of optic nerve head topographic measurements with the confocal scanning laser ophthalmoscope.
Ophthalmology. 1993;100:57–62.
[CrossRef] [PubMed]Mikelberg FS, Wijsman K, Schulzer M. Reproducibility of topographic parameters obtained with the Heidelberg Retina Tomograph.
J Glaucoma. 1993;2:101–103.
[PubMed]Lusky M, Bosem ME, Weinreb RN. Reproducibility of optic nerve head topography measurements in eyes with undilated pupils.
J Glaucoma. 1993;2:104–109.
[PubMed]Rohrschneider K, Burk RO, Völcker HE. Reproducibility of topographic data acquisition in normal and glaucomatous optic nerve heads with the laser tomographic scanner.
Graefes Arch Clin Exp Ophthalmol. 1993;231:457–464.
[CrossRef] [PubMed]Rohrschneider K, Burk RO, Kruse FE, Völcker HE. Reproducibility of the optic nerve head topography with a new laser tomographic scanning device.
Ophthalmology. 1994;101:1044–1049.
[CrossRef] [PubMed]Bartz–Schmidt KU, Weber J, Heimann K. Validity of two dimensional data obtained with the Heidelberg retina tomograph as verified by direct measurements in normal optic nerve heads.
Ger J Ophthalmol. 1994;3:400–405.
[PubMed]Chauhan BC, LeBlanc RP, McCormick TA, Rogers JB. Test–retest variability of topographic measurements with confocal scanning laser tomography in patients with glaucoma and control subjects.
Am J Ophthalmol. 1994;118:9–15.
[CrossRef] [PubMed]Janknecht P, Funk J. Optic nerve head analyzer and Heidelberg retina tomograph: accuracy and reproducibility of topographic measurements in a model eye and in volunteers.
Br J Ophthalmol. 1994;78:760–768.
[CrossRef] [PubMed]Orgül S, Cioffi GA, Bacon DR, Van Buskirk M. Sources of variability of topometric data with a scanning laser ophthalmoscope.
Arch Ophthalmol. 1996;114:161–164.
[CrossRef] [PubMed]Burk ROW, Rohrschneider K, Noak H, Völcker HE. Volumetrische Papillenanalyse mit Hilfe der Laser-Scanning Tomographie: Parameterdefinition und Vergleich von Glaukom und Kontrolpapillen.
Klin Monatsbl Augenheilkd. 1991;198:522–529.
[CrossRef] [PubMed]Brigatti L, Caprioli J. Correlation of visual field with scanning confocal laser optic disc measurements in glaucoma.
Arch Ophthalmol. 1995;113:1191–1194.
[CrossRef] [PubMed]Mikelberg FS, Parfitt CM, Swindale NV, Graham SL, Drance SM. Ability of the Heidelberg Retina Tomograph to detect early glaucomatous visual field loss.
J Glaucoma. 1995;4:242–247.
[PubMed]Weinreb RN, Shakiba S, Sample PA, et al. Association between quantitative nerve fibre layer measurement and visual field loss in glaucoma.
Am J Ophthalmol. 1995;120:732–738.
[CrossRef] [PubMed]Brigatti L, Hoffman D, Caprioli J. Neural networks to identify glaucoma with structural and functional measurements.
Am J Ophthalmol. 1996;121:511–521.
[CrossRef] [PubMed]Uchida H, Brigatti L, Caprioli J. Detection of structural damage from glaucoma with confocal laser image analysis.
Invest Ophthalmol Vis Sci. 1996;37:2393–2401.
[PubMed]Hatch WV, Flanagan JG, Etchells EE, Williams-Lyn DE, Trope GE. Laser scanning tomography of the optic nerve head in ocular hypertension and glaucoma.
Br J Ophthalmol. 1997;81:871–876.
[CrossRef] [PubMed]Iester M, Mikelberg FS, Courtright P, Drance SM. Correlation between the visual field indices and Heidelberg retina tomograph parameters. J Glaucoma. 1997a;6:78–82.
Iester M, Mikelberg FS, Swindale NV, Drance SM. ROC analysis of Heidelberg retina tomograph optic disc shape measures in glaucoma. Can J Ophthalmol. 1997b;32:382–388.
Iester M, Mikelberg FS, Drance S. The effect of optic disc size on diagnostic precision with the Heidelberg Retina Tomograph.
Ophthalmology. 1997c;104:545–548.
[CrossRef] Iester M, Swindale NV, Mikelberg FS. Sector-based analysis of optic nerve head shape parameters and visual field indices in healthy and glaucomatous eyes. J Glaucoma. 1997d;6:371–376.
Anton A, Yamagishi N, Zangwill L, Sample PA, Weinreb RN. Mapping structural to functional damage in glaucoma with standard automated perimetry and confocal scanning laser ophthalmoscopy.
Am J Ophthalmol. 1998;125:436–446.
[CrossRef] [PubMed]Bathija R, Zangwill L, Berry CC, Sample PA, Weinreb RN. Detection of early glaucomatous structural damage with confocal scanning laser tomography.
J Glaucoma. 1998;7:121–127.
[PubMed]Wollstein G, Garway–Heath DF, Hitchings RA. Identification of early glaucoma cases with the scanning laser ophthalmoscope.
Ophthalmology. 1998;105:1557–1563.
[CrossRef] [PubMed]Orgül S, Cioffi GA, van Buskirk EM. Variability of contour line alignment on sequential images with the Heidelberg Retina Tomograph.
Graefes Arch Clin Exp Ophthalmol. 1997;235:82–86.
[CrossRef] [PubMed]Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing. 1994; 2nd ed. Cambridge University Press Cambridge, UK.
Bishop CM. Neural Networks for Pattern Recognition. 1995; Oxford University Press
Weinreb RN, Lusky M, Bartsch D, Morsman D. Effect of repetitive imaging on topographic measurements of the optic nerve head.
Arch Ophthalmol. 1993;111:636–638.
[CrossRef] [PubMed]Teal PK, Morin JD, McCulloch C. Assessment of the normal disc.
Trans Am Ophthalmol Soc. 1972;70:164–177.
[PubMed]Bengtsson B. The variation and covariation of cup and disc diameters. Acta Ophthalmol. 1976;54:804–818.
Britton RJ, Drance SM, Schulzer M, Douglas GR, Mawson DK. The area of the neuroretinal rim of the optic nerve in normal eyes.
Am J Ophthalmol. 1987;103:497–504.
[CrossRef] [PubMed]Jonas JB, Gusek GC, Naumann GOH. Optic disc morphometry in chronic primary open-angle glaucoma.
Graefes Arch Clin Exp Ophthalmol. 1988;226:531–538.
[CrossRef] [PubMed]Tabachnick BG, Fidell LS. Using Multivariate Statistics. 1989; 2nd ed. Harper Collins New York.
Parfitt CM, Mikelberg FS, Swindale NV, Green S, Graham SL, Drance SM. The use of artificial neural networks to identify optic nerve head degeneration due to glaucoma [ARVO Abstract]. Invest Ophthalmol Vis Sci. 1995;36(4)S628.
Fechtner RD, Weinreb RN. Mechanisms of optic nerve damage in primary open angle glaucoma.
Surv Ophthalmol. 1994;39:23–42.
[CrossRef] [PubMed]Caprioli J. Discrimination between normal and glaucomatous eyes.
Invest Ophthalmol Vis Sci. 1992;33:153–159.
[PubMed]Radius RL, Maumenee AE, Green WR. Pit-like changes of the optic nerve head in open-angle glaucoma.
Br J Ophthalmol. 1978;62:389–393.
[CrossRef] [PubMed]Dichtl A, Jonas JB, Mardin CY. Comparison between tomographic scanning evaluation and photographic measurement of the neuro-retinal rim.
Am J Ophthalmol. 1996;121:494–501.
[CrossRef] [PubMed]