Association Between Cup-to-Disc Ratio and Structural and Functional Damage Parameters in Glaucoma: Insights From Multiparametric Modeling

Abstract

Purpose: To model the association between vertical cup-to-disc ratio (CDR) and clinically relevant glaucomatous outcomes.

Methods: From a tertiary glaucoma clinic, clinician-estimated CDR, CDR derived from peripapillary optical coherence tomography scans, circumpapillary retinal nerve layer thickness, optic disc diameter, visual field (VF) mean deviation (MD), glaucoma hemifield test (GHT), and VF index were used to develop sigmoidal regression models of CDR, analyzed for floor/ceiling effects of functional or structural damage by model comparisons based on logarithms of Bayes factors (logBF, with logBF > 5 indicating very strong evidence).

Results: We selected 17,509 eyes of 10,420 patients. For all outcomes, there was strong evidence for sigmoidal regression compared with linear regression (all logBF > 650). Model fits were characterized by plateauing for CDR toward 0, with ceilings for functional outcomes below the values denoting normal VFs (MD = 0 and VF index = 100). The clinician-estimated CDR was stronger associated with functional outcomes (all logBFs > 347) and optical coherence tomography–measured CDR with retinal nerve layer thickness (logBF = 243). Areas under the receiver operator characteristic curve for abnormal MD and GHT vs. CDR were 0.626 and 0.653, respectively. Optic disc diameter only marginally improved predictability (areas under the receiver operator characteristic curve increase for abnormal MD/GHT: 0.001/0.006).

Conclusions: CDR is subject to ceiling effects for glaucoma-related outcomes and poor at discriminating early glaucomatous damage. CDR values should be interpreted with care, particularly in screening settings.

Translational Relevance: This interpretable nonlinear model helps to better explain specific impacts on and limitations of CDR, one of the most frequently recorded parameters for glaucoma diagnostics.

Introduction

Glaucoma is an optic neuropathy that causes characteristic damage to the optic nerve that can result in apoptotic retinal ganglionic cell death and eventual loss of the optic nerve axons.^1,2 Progressive retinal ganglion cell axon loss is reflected in the optic nerve head by larger cupping, which can be quantified by the ratio between the optic disc border and the cup border, that is, the cup-to-disc ratio (CDR). This parameter and its relationship to glaucoma has been studied systematically since the 1960s (e.g., Armaly 1967³ and 1969⁴). It has been shown that the vertical CDR is a better predictor for the development of glaucoma than the horizontal CDR.⁵ Although the diagnostic performance of the CDR has been criticized and alternative fundus inspection–based measures have been proposed and shown to be superior with respect to glaucoma diagnostics,⁶ there continues to be a requirement by American Medical Billing Association, which was established in 1997, to have physicians report on a patients’ optic nerve health via CDR during every comprehensive eye examination.^7,8 As a consequence, the specification of the CDR is required by common electronic medical record systems and is therefore the arguably most frequently documented quantitative parameter in optic nerve head assessments. 

Given the widespread use of the CDR, detailed knowledge of its relationship to functional vision is of particular interest. Frequently applied linear models, such as correlations, risk oversimplifying this relationship and ignore possible plateauing. Apart from glaucoma severity, the CDR has also been associated with demographics, such as race,⁹ and with other parameters related to retinal structure, such as optic disc size.¹⁰ Detailed, interpretable parameters, such as initial or final plateauing, inflection points, and corresponding slopes, would also help to better understand potential moderating effects of these parameters on the structure–function relationship. It would be of great interest to know, for instance, if the CDR is naturally higher for certain races and, if so, whether this factor has a protective effect on visual function or not. An interpretable multiparametric rather than a simple linear model would be of great benefit for this purpose. We did not, however, find a single publication that quantitatively modeled the CDR and published the numerical fitted parameters—not even for linear regression. 

In clinical practice, the CDR is typically estimated by visual inspection. Some optical coherence tomography (OCT) devices, such as Cirrus HD-OCT (Carl Zeiss Meditec, Jena, Germany), aim to determine the CDR in a more objective way based on peripapillary volume scans. Although substantial agreement between OCT-based and clinician estimates has been reported,¹¹ it remains unclear if the supposedly more accurate OCT CDR measurements are more closely associated with functional vision loss or other relevant glaucomatous parameters. 

In this study, we analyze and systematically compare vertical CDR estimated by clinicians and CDR measured by Cirrus HD-OCT to glaucomatous visual field (VF) loss (Humphrey Field Analyzer, Carl Zeiss Meditec), as well as circumpapillary retinal nerve fiber layer thickness (RNFLT) measured by Cirrus HD-OCT. We investigate possible plateauing (floor or ceiling) effects of severity of functional or structural loss for the CDR approaching 0 or 1. Specifically, of particular clinical interest, we study whether or how the CDR is discriminative of early functional vision loss. 

Methods

From all routine patient care measurements of the Massachusetts Eye and Ear (MEE) glaucoma service between 2011 and 2021, all available clinician estimates of vertical CDR were selected and extracted from electronic medical records if they had accompanying (≤100 days time difference) reliable (false-positive rate ≤15%, fixation loss rate ≤20%) Humphrey SITA Standard or SITA Fast 24-2 VFs and reliable (signal strength ≤6) peripapillary Cirrus HD-OCT volume scans (standard optic nerve head centered protocol; i.e., 200 horizontal B-scans with 200 A-scans each). All patients were from the MEE glaucoma service. Not all of them were glaucoma patients, although they would at least be suspects of glaucoma (for various reasons), if they were tested at the glaucoma service. We included all patients tested at this service, regardless of a glaucoma diagnosis. The MEE glaucoma service team consists of nine clinicians specialty trained in glaucoma, covering all levels of seniority (see https://www.masseyeandear.org/specialties/glaucoma for details). If more than one measurement per eye fulfilled these criteria, only the most recent measurement of the eye was selected. If both eyes of a patient fulfilled all aforementioned criteria, both eyes were included. 

From the VFs, the mean deviation (MD), VF index (VFI), and glaucoma hemifield test (GHT) outcomes were extracted, and it was checked if the MD was flagged as abnormal (≤5% probability) by the machine (which was the case for MD of ≤−2.1 dB). For the GHT-specific subanalysis, only eyes with test results within normal limits, borderline, or outside normal limits were included. From the XML export of the OCT volume scans, the vertical CDR and the global average RNFLT were extracted. 

All data analyses were performed with R (version 3.6.3). For the sigmoidal regression analyses, the R library DRC¹² (version 3.0-1) was used. Receiver operating characteristic curves were obtained with the help of the R library pROC¹³ (version 1.18.0). 

For data analyses with binary outcomes, we performed logistic regression with the CDR as the regressor and optic disc diameter as a possible covariate and calculated the area areas under the receiver operator characteristic curve. To associate the CDR with continuous structural parameters (mean RNFLT) and functional parameters (MD, VFI), we checked for possible asymptotes (ceiling or floor effects) for a CDR toward 0 or 1. In other words, we investigated whether structural or functional severity outcomes reach a ceiling or floor for low or high CDR values. This was implemented by fitting a sigmoidal regression curve, specifically a four-parametric logistic curve model, as described in Figure 1, with parameter s denoting the steepness of the slope, parameter i denoting the location of the inflection point, and parameters u and l representing upper and lower asymptotes, respectively. To specifically check whether asymptotes for decreasing CDR were below the level of functionally healthy vision (MD = 0 dB or VFI = 100), we additionally calculated models where we fixed upper asymptote u to MD = 0 dB, respectively, and VFI = 100 then compared it with the original four-parametric model, where l was a free parameter. 

Figure 1.

 

Schematic illustration of the four-parametric sigmoidal regression curve to model the relationship between CDR and structural and functional outcomes.

To quantitatively compare different models, we performed a Bayesian model comparison by Bayes factors (BFs), mathematically approximated by the Bayesian information criterion (BIC)¹⁴ as log(BF(model1,model2)) = (BIC(model1) − BIC(model2))/2. BFs, unlike P values, quantitatively represent the evidence for one model vs. the other. To interpret BFs, we follow a widely used recommendation¹⁵ and consider 1 < logarithm of the BF (logBF) ≤ 3 as positive, 3 < logBF ≤ 5 as strong, and logBF > 5 as very strong evidence in favor of model 1 vs. model 2. Analogous negative values represent evidence in favor of model 2 compared with model 1. 

The study, which adheres to the Declaration of Helsinki and all federal and state laws, was approved by the MEE Institutional Review Board. The Institutional Review Board waived the need for informed consent owing to the retrospective nature of the study. 

Results

We included 17,509 eyes of 10,420 patients (56.4% female; mean age, 62.3 ± 15.9 years; self-reported race: 8.0% Asian, 13.7% Black, 63.2% White, 0.4% other, and 14.7% declined or missing response) that fulfilled the selection criteria. Functional and structural measurements (mean ± standard deviation): clinician-estimated CDR, 0.58 ± 0.19; OCT CDR, 0.62 ± 0.15; RNFLT, 82.92 ± 14.5 µm; MD, −3.96 ± 6.29 dB; VFI, 90.59 ± 18.77. The MD was flagged as abnormal (<5% probability) for 8569 eyes (48.9%). For the GHT-specific subanalysis, the three included GHT categories were subject to the following frequency counts: within normal limits, 7783 eyes; borderline, 2401 eyes; and outside normal limits, 6828 eyes. 

The Pearson correlation between clinician-estimated and OCT-measured CDR was 0.78. A Bland–Altman analysis (Fig. 2) shows a trend of a lower clinically estimated CDR compared with the OCT measurement (mean difference, −0.036; 95% confidence interval, −0.27 to 0.20). As illustrated in Figure 2, there are significant outliers that approach the maximum value of 1.0. These are rare, and in all cases of such extreme outliers that we checked, this was due to disc-cup segmentation errors by the Cirrus software. 

Figure 2.

 

Bland–Altman plot of clinician-estimated vs. OCT-measured CDR. Note: The diagonal stripes are the result of the higher discretization (typically only one digit after the decimal point) of clinician-estimated CDRs vs. the nearly continuous machine estimates.

Figure 3 illustrates the clinician-estimated (left) and OCT-measured (right) CDRs compared with functional and structural measures. Clinician-estimated and OCT-measured CDRs are plotted against glaucoma related outcome measures (top, VF MD; middle, VFI; bottom, average circumpapillary RNFLT). The red circles are the local means at the respective CDR values. The stroked blue lines show traditional linear regressions. The red curves show the sigmoidal regressions (four-parametric logistic model). For MD and VFI, additional three-parametric models were fitted where the upper asymptote of the four-parametric model was fixed to the healthy level of the respective parameter (MD = 0 and VFI = 100, respectively). The cyan curves show these three-parametric model fits. The logarithms of the BFs (logBF) of the model comparisons for the traditional linear vs. four-parametric sigmoidal models are presented in Table 1. All logBF values are considerably >5, which indicates very strong evidence in favor of the four- parametric models compared with linear regression. 

Figure 3.

 

Clinician-estimated (left) and OCT-measured (right) CDR vs. glaucoma related outcome measures (top, VF MD; middle, VFI; bottom, average circumpapillary RNFLT). The red circles are the local means at the respective CDR values. The stroked blue lines show the linear regressions. The red curves show the sigmoidal regressions (four-parametric logistic model). For MD and VFI, additional three-parametric models were fitted where the upper asymptote of the four-parametric model was fixed to the healthy level of the respective parameter (MD = 0 and VFI = 100, respectively). The cyan curves show these three-parametric model fits.

Table 1.

 

LogBF of the Model Comparisons for the Traditional Linear vs. Four-parametric Sigmoidal Models

 

Table 2 contains the respective coefficients of the four parameters for each outcome. Note that the fitted upper asymptotes of the four-parametric models are below the values associated with healthy vision (MD = 0 and VFI = 100, respectively). We then investigated whether these differences from the respective levels of healthy subjects are within the range of random fluctuations in model parameter effects or are statistically relevant by comparing the four-parametric models, where the upper asymptote is a free parameter, with models where the upper asymptote was fixed to the level of normality for MD and VFI, which leaves only three free parameters. Table 3 shows the logBF values for these comparisons. All logBFs are >5, indicating very strong evidence for the four-parametric models. 

Table 2.

 

Coefficients of the Four-Parametric Model Fits

 

Table 3.

 

LogBF of the Model Comparisons for Models Where the Upper Asymptote Was Fixed to the Level of Normal Functional Vision (MD = 0 and VFI = 100) Vs. the Four-Parametric Sigmoidal Models Where the Upper Asymptote was a Free Parameter

 

The relationship between the MD and optic disc diameter is detailed in Figure 4. The Pearson correlation for the entire severity range of MD is 0.283 (P < 0.001). When only looking at those eyes with normal MD (MD probability > 0.05), the Pearson correlation increases to 0.396 (P < 0.001). 

Figure 4.

 

Relationship between optic disc size and clinician-estimated CDR. (Top) Entire MD severity range. (Bottom) Only eyes with normal MD (MD probability > 0.05). The discrete CDR estimates by clinicians, typically only covering one digit after the decimal point, would result in many overlapping points. Therefore, for better visibility, the points were jittered for plotting. Points are colored by glaucoma severity, represented by VF MD. Regression lines are displayed in blue.

Figure 5 shows heat maps of logistic regression predictions of two outcomes: First, abnormal MD (probability of MD < 0.05; top) and second, GHT outside normal limits (GHT outer nuclear layer; bottom), with disc diameter and CDR as regressors. As illustrated by the respective plots, as the CDR increases in patients with low disc diameters, the probability of MD and GHT abnormalities increases faster and to higher values than in patients with high disc diameters. 

Figure 5.

 

Heat map of the logistic regression of patient's probability of MD of <0.05 and the probability of GHT outer nuclear layer given their disc diameter and CDR, respectively.

Figure 6 shows areas under the receiver operating characteristic curves (AUCs) for predicting abnormal MD (MD probability < 0.05) and GHT outside normal limits from (clinician-estimated) CDR alone vs. CDR and optic disc diameter together by logistic regression. Abnormal MD predicted from CDR alone vs CDR plus optic disc diameter were 0.653 and 0.659, respectively. The AUCs were not significantly different (DeLong's test, P = 0.73). The analogous AUCs with GHT outside normal limits as an outcome for CDR without and with disc diameter were 0.653 and 0.659, respectively. The AUCs were significantly different (DeLong's test, P < 0.001). These results were very similar for CDR determined by OCT (see Supplementary Fig. S1 for details). 

Figure 6.

 

Receiver operating characteristic (ROC) curves for models with abnormal VF MD (probability > 0.05, top) and GHT outside normal limits (bottom) as outcomes, respectively. The blue lines show the models for clinician-estimated CDR alone and the red lines for CDR with optic disc diameter as an additional covariate. The respective areas under the curve are annotated. An analogous figure for OCT determined CDR has been added as Supplementary Material.

Table 2 contains the parameter values of these superior four-parametric sigmoidal models (red curves in Figure 6). The last two columns of the table show the percentages of the two asymptotes at the minimum CDR (CDR = 0) and maximum CDR (CDR = 1), respectively. A value of 100% would mean that the respective asymptote was fully reached at minimum or maximum CDR. In other words, a value of or close to 100% indicates strong evidence for a plateau. For all conditions, at CDR = 0, the curve reached over 99% of its respective asymptote u, and the inflection points (parameter i), that is, the points of steepest decay, are at least at the level of CDR = 0.85, in many cases even beyond the highest possible CDR value of 1. This result indicates that both the structural and functional outcomes initially plateau close to their saturation level for small CDR values and then exponentially decay, with the steepest level of decay greater or equal to a CDR level of 0.85. For clinician-estimated CDRs, the lowest inflection point is even at a CDR as high as 0.94 (for outcome VFI), indicating a particularly long period of plateauing for lower CDR values. Of note, the upper asymptotes for the functional outcomes (MD and VFI) are lower than the values encoding normal functional vision: The MD asymptotes are −2.34 dB and −2.46 dB for clinician-estimated and OCT-measured CDR, respectively, whereas a normal MD is defined to be zero, and the VFI asymptotes are 94.8 and 95.5 for clinician-estimated and OCT-measured CDR, respectively, whereas a normal VFI is defined to be 100. To investigate whether these deviations from normal vision at CDR = 0 are random fluctuations of our model fits or indeed statistically substantial, we performed model comparisons of these four-parametric models with models where we fixed the upper asymptotes to CDR = 0 and VFI = 100, respectively (cyan curves in Figure 3). Table 3 shows the respective quantitative model comparison between the three-parametric (upper asymptote–fixed) models and the four-parametric models with the upper asymptote as a free parameter. All logBF values are >250 and thereby substantially >5, indicating strong evidence for the four-parametric models. 

Finally, we compared the associations between clinician-estimated CDR vs. OCT-measured CDR and glaucoma-related outcomes. The model comparison results are shown in Table 4. The logBF values indicate that the associations of clinician-estimated CDR with functional parameters (MD and VFI) are considerably stronger compared with OCT-measured CDR (all logBFs >347 and thereby substantially >5). As for associations with the structural parameter, however, the opposite was the case: the RNFLT was considerably stronger associated with OCT CDR compared with clinician-estimated CDR, as reflected by logBF = −243, which is negative and substantially <−5. 

Table 4.

 

LogBF of the Model Comparisons for Models Associating Clinician-Estimated CDR vs. OCT-measured CDR With Functional and Structural Glaucoma-Related Outcomes

 

Discussion

The vertical CDR is one of the most frequently recorded diagnostic parameters for glaucoma. Clinicians are expected to report patients’ optic nerve health via CDR during every comprehensive eye examination according to a requirement by American Medical Billing Association.^7,8 The CDR can be estimated visually by clinicians without time-consuming or expensive functional or structural measurements, which might additionally contribute to the popularity of this measure, especially in the context of screenings for risk factors of ocular disease. 

Based on a large data set consisting of structural and functional measurements of 10,420 patients from the MEE glaucoma service, we studied how the CDR is related to functional severity measures of glaucomatous vision loss (VF MD and VFI), as well as with a widely established structural measure, namely, circumpapillary RNFLT-measured by OCT. We investigated these associations comparatively for clinician-estimated CDR, as well as for CDR measured by the Cirrus HD-OCT machine, with a special focus on early disease levels. As a first step and as a foundation for the aforementioned data analyses, however, we derived a novel and more appropriate statistical modeling approach for the association between the CDR and our investigated outcomes. 

A Novel, NonLinear, Four-Parametric Model

Frequently applied linear models, for instance, in linear regression or correlation studies, implicitly assume that the relationship with the outcome is identical over the whole range of the independent variable. When associating the CDR with other glaucoma-related outcomes, there are good reasons to question this assumption. First, there might be floor effects, that is, levels of CDR beyond which the outcome does not change anymore. For instance, patients might have total VF loss at a certain CDR, so that their functional vision cannot decrease measurably anymore with further CDR increases. Second, CDR might not be able to discriminate levels of early structural or functional loss in glaucoma, so that there might be ceiling effects, that is, the outcome saturates on the same level for the range of CDR between 0 and a certain level. For instance, it is imaginable that early retinal ganglion cell axon damage is captured by circumpapillary OCT measurements without being visibly reflected in CDR changes. 

In addition to these effects, there are further potential reasons for nonlinearities. To begin with, there is an effect of simple geometry, as illustrated on Figure 7: Most structure–function studies are based on the idea to relate a count of retinal ganglion cell axons to a measure of functional vision loss. A typical proxy for this count is the circumpapillary RNFLT, determined by retinal layer segmentations on a circular OCT B-scan around the optic disc. An equivalent proxy directly on the optic disc would be rim area. The CDR, however, is not based on area, but rather on diameter and, as illustrated on the figure, the relationship of CDR to the corresponding rim area nonlinear. If we assume a circular disc shape, for instance, a 10% increase of CDR from 0 to 0.1 would correspond with a 1% rim loss, whereas a 10% increase from 0.9 to 1.0 would correspond with as much as a 19% rim loss. The precise rim loss values for an eye would depend on the individual shape of the optic disc, but because the underlying reason for this nonlinearity is related to geometry, the eye-specific rim loss values would be similar to those of our circular disc model. To sum up, CDR is monotonically, but nonlinearly related to ganglion cell axon counts. 

Figure 7.

 

Schematic illustration of the relationship between the ratio of rim area and disc area vs. CDR. The number of lost retinal nerve fiber ganglion cells is equivalent to a shrinking of the rim area, but the CDR is not a measure based on area but rather based on diameters. For this simplified illustration, we assume an optic disc shape of a circle, for which the area ratio would be described by 1 – CDR.² Although the precise functional relationship is eye specific and depends on the exact shape of the optic disc, this relationship will always necessarily be quadratic, as always a ratio of lines (diameters) is related to a ratio of areas. Owing to this nonlinear relationship, for instance, a 10% increase of CDR from 0 to 0.1 would correspond with 1% rim loss, whereas a 10% increase from 0.9 to 1.0 would correspond with as much as 19% rim loss, as annotated on the figure.

Independent of the above, even if a more appropriate proxy for ganglion cell loss than CDR were applied, there is a further argument against a linear relationship between ganglion cell counts and VF loss. In perimetry, light sensitivity is typically obtained in decibel (dB), which approximates a perceptually linear scale: A difference between, say, 15 dB and 16 dB is perceived about equally strong as a difference between 25 dB and 26 dB. However, as perception follows a logarithmic scale with respect to luminance, there is a massive relative luminance difference between these two equally perceived intervals. Ganglion cell loss might not be linearly related to the perceptually linear dB scale, in which VF total deviations or MD is represented, but rather be linearly related to luminance, so that models with exponential terms would be necessary, such as our four-parametric model introduced here. In summary, there are numerous potential sources of nonlinearities when relating the CDR to perimetry or to RNFLT, which underlines the need of a flexible and ideally interpretable model. 

To take these considerations into account, we introduce a four-parametric statistical model that, to our best knowledge, has not yet been applied for this purpose yet, and that explicitly and independently considers upper and lower asymptotes (Fig. 1). The four-parametric model fits, illustrated in Figure 3, very strongly outperform linear regression in all conditions (Table 1). 

Insights From the Model Parameters

The coefficients of the fitted parameters, shown in Table 2, give insights relevant to ophthalmology. First, at CDR = 0, the upper asymptotes (parameter u) of all model fits are very close to their respective 100% level, varying between 99.98% and 100.00%, which indicates ceiling effects, that is, an asymptotic behavior toward low CDRs. Further implications of this, particularly for early glaucomatous damage, will be discussed below. 

Second, the inflection point (parameter i) provides insight whether there is an attenuation of the relationship for an outcome toward higher values of CDR. If the fitted inflection point is below a CDR value of 1, that is, inside the range of possible CDR values, such an attenuation does occur for the respective outcome. This is the case for VFI with clinician-estimated CDR (i = 0.94) and even more pronounced for both MD and VFI with OCT-determined CDR (0.85 and 0.89, respectively). For CDRs higher than i, the relationship between the CDR and the respective outcome weakens compared with CDRs lower than i, that is, the curve moves toward floor effects, where the increasing CDR leads to a weaker decrease of VFI or MD. For the structural parameter RNFLT, we do not observe such behavior: The theoretical inflection points are always above CDR = 1. One reason possibly contributing to this is the nonsigmoidal and strictly monotonically decreasing relationship between CDR and rim area (see Fig. 7), with the rim area arguably more closely related to the mean circumpapillary RNFLT than the CDR. 

CDR and Optic Disc Size

There is evidence that large disc size is associated with larger CDRs independent of the effects of glaucoma. Several studies have warned of potential diagnostic biases if not taking this effect into account.^10,16–19 Indeed, we found a weak positive correlation (0.28) between the optic disc diameter and the CDR for the entire data set (Fig. 4, top). Because the entire data set contains all levels of glaucoma severity, the positive association between disc size and CDR could also be a result of a potential positive association between disc size and glaucoma severity itself. Therefore, we additionally investigated disc size only for eyes without abnormal VF loss (MD probability > 0.05). The positive correlation increased to 0.40 (Fig. 4, bottom). This indicates that larger discs have higher CDR in absence of glaucoma. The decrease of the correlation to 0.28 when including the entire MD range might suggest that the increase of CDR owing to glaucoma occurs independently of disc size. As the entire data set contains all levels of glaucoma severity, the positive association between disc size and CDR could also be a result of a potential positive association between disc size and glaucoma severity itself. Therefore, we additionally investigated disc size only for eyes without abnormal VF loss (MD probability > 0.05). The positive correlation increased to 0.40 (Fig. 4, bottom). This finding indicates that larger discs have a higher CDR in the absence of glaucoma. The decrease of the correlation to 0.28 when including the entire MD range might suggest that the increase of CDR owing to glaucoma occurs independent of disc size. This finding motivated us to investigate further the interrelationship between CDR, disc diameter, and functional outcomes of glaucoma. Figure 5 summarizes this interrelationship for MD abnormalities (top) and GHTs (bottom). As expected, we observed the greatest functional glaucoma severity at low disc sizes together with high CDR (yellowish colors in the heat plots) and lowest functional glaucoma severity at high disc sizes together with low CDR (bluish colors in the heat plots). The plot indicates a continuous interactive effect between these two parameters and functional glaucoma severity. 

These results motivated us to study whether disc size as a covariate in addition to CDR improves the association to functional glaucoma outcomes. Interestingly, however, Figure 6 shows that the improvement by adding disc size was marginal at best. Although we indeed reproduce a previously suggested relationship between disc size and CDR (Figs. 4 and 5), the AUCs of our logistic regressions with functional outcomes do not indicate the expected substantial improvements. 

CDR and Early Glaucomatous Damage

The availability of clinician-estimated CDR, even in absence of expensive and/or time-consuming ocular measurements and assessments like perimetry or OCT, makes this parameter attractive in screening settings even outside of specialized ophthalmic care centers. To support screening successfully, the CDR should be able to discriminate outcome levels particularly at early stages of disease. 

The sigmoidal model fits for functional outcomes (Table 3) show upper asymptotes of >2.3 dB below the normal level for MD (which is defined as MD = 0) and >4% (in case of clinician-estimated CDR even >5%) below the normal level for VFI (which is defined as VFI = 100). To rule out potential random fluctuations of the model fits and to confirm that these upper asymptotes really differ from the levels defining the normal VF, we calculated a Bayesian model comparison with models where the upper levels were fixed to their normal levels and found very strong evidence that the observed lower than normal upper asymptotes were real (Table 3). This means that both clinicians-estimated and OCT-measured CDR are not able to discriminate early glaucomatous functional vision loss. The upper asymptotes provide upper bounds for the levels of vision loss CDR is able to discriminate. 

For the VF MD, for instance, the arguably most frequently used severity measure of glaucomatous vision loss in clinical practice, the upper asymptote is −2.34 dB. Note that the perimeter used in this study, the Humphrey Field Analyzer, already marks any MD of <−2.1 dB as abnormal on the VF printout. This implies that, especially for the lowest clinically estimated CDRs (values toward CDR = 0), it is particularly difficult to decide whether functional vision is already impaired to a level marked as abnormal by the machine. Moreover, owing to the asymptotic relationship toward CDR = 0, the slope of the curve is the smaller the lower the CDR level, which means that the earlier the disease level as quantified by the respective outcome, the less the CDR is able to discriminate it. 

The structural parameter RNFLT is not normalized relative to a healthy control population like MD and VFI, which does not allow an analogous data analysis to determine how much (if any) early thinning is indiscriminable by CDR. However, for RNFLT, we observe a similar asymptotic relationship toward CDR = 0. as for the two functional parameters, which makes it likely that a certain level of early retinal ganglion cell loss is undetectable by the CDR as well. 

Clinician-Estimated Vs. OCT-Measured CDR

The vertical CDR as it appears in medical records is typically based on visual estimates by clinicians instead of real measurements. These estimates are necessarily subjective and expected to differ from CDRs quantitatively measured on the true vertical axis of the optic disc. Here, we therefore additionally compared these clinician estimates with quantitative measurements based on a peripapillary OCT volume scan. Although these OCT measurements are prone to errors as well, for instance owing to incorrect retinal layer segmentations or other measurement artifacts, they are likely closer to the true CDRs compared with the clinician estimates. 

Unsurprisingly, clinician-estimated and OCT-measured CDR are positively correlated (Pearson's R = 0.78; P < 0.001), with clinician estimates by an average value of −0.036 below OCT estimates (see Fig. 2). A possible reason for the lower clinician estimates might be that CDR assessment, which requires dilation, did not occur at every patient visit. Instead, for some visits, the previously estimated CDR from an earlier visit might have been copied into the medical record. Because patients might already slightly have progressed in their glaucoma severity, the clinician-estimated CDR might therefore be subject to a slight underestimation, which would explain the slightly lower values we observed here. 

One might expect the OCT-measured values to be more closely associated with the glaucoma-related outcomes studied here. Surprisingly, however, as for the two functional outcomes, the clinician-estimated CDR values have a considerably stronger associated with glaucoma related outcomes compared with the OCT measurements (Bayesian model comparison; see Table 4). A possible explanation could be that clinicians might (at least implicitly) take the entire holistic appearance of the ONH into account when generating the single CDR number for the medical records, whereas the machine strictly aims to estimate a numerical value based on a single axis through the optic disc and, therefore, ignores any other features or characteristics present on the fundus. In other words, clinicians might be biased by additional information visible on the fundus when making their estimates, and this bias, although possibly increasing deviations from the true vertical CDR, could yield to numerical values that are better associated with functional vision. The structural parameter RNFLT, in contrast, indeed has a substantially stronger association with OCT-measured CDR compared with clinician estimates. A reason for this might be that both of these measurements have been performed by the same machine and are based on exactly the same peripapillary volume scan. 

Although the large sample size based on the entire population of a major glaucoma service is a major strength of this study, a potential shortcoming is the lack of longitudinal data. Cross-sectional measurements, even based on a large sample size, necessarily limit interpretations. Future studies that include and compare changes of CDR and glaucoma-related outcomes over time would be beneficial. A further limitation is the lack of external validation: Although our dataset is large and supposed to cover a variety of disease specific conditions as well as a substantial heterogeneity of glaucoma specialists, all data are only from one institution (MEE). 

In conclusion, based on a sample size of >10,000 patients from a large glaucoma service, we studied the relationship between CDR and clinically relevant structural and functional outcomes related to glaucoma. We statistically derived a nonlinear, sigmoidal regression model that proved to be considerably superior to linear regression and that contains interpretable parameters particularly for the investigation of early and late disease stages, but also for better understanding previously reported relationships between CDR and demographics (e.g., race) and retinal characteristics. We found for all studied outcomes that CDR was subject to a ceiling of the outcome effect toward a CDR = 0 and that this ceiling was considerably below those values of the functional outcomes that define a healthy VF. As a consequence, the CDR seems to be particularly problematic for detecting early glaucomatous damage, which is the major purpose of screening. 

Acknowledgments

The authors thank Gadi Wollstein for his helpful comments on the nonlinearities between perimetry and RNFLT and Lucy Shen for insights into how CDR is dealt with and recorded by clinicians in electronic medical records. 

Supported by the National Institutes of Health (R01 EY030575, R00 EY028631, and P30 EY003790), Research to Prevent Blindness International Research Collaborators Award, Alcon Young Investigator Grant, and Schmidt Science Fellows (H.R.). 

Disclosure: A. McCalla, None; M. Wang, Genentech (F); M. Eslami, None; S. Kazeminasab, None; Y. Luo, None; H. Rana, None; S. Saha, None; M. Shi, None; Y. Tian, None; N. Zebardast, None; T. Elze, Genentech (F)