Models of Glaucomatous Visual Field Loss

Abstract

Purpose.: To evaluate and compare the ability of pointwise linear, exponential, and logistic functions, and combinations of functions, to model the longitudinal behavior of visual field (VF) series and predict future VF loss in patients with glaucoma.

Methods.: Visual field series from 782 eyes (572 patients) with open-angle glaucoma had greater than 6 years of follow-up and 12 VFs performed. Threshold sensitivities from the first 5 years at each location were regressed with linear, exponential, and logistic functions to estimate model parameters. A multiple-model approach applied the model with the lowest root mean square error (RMSE) at each location as the preferred model for future predictions. Predictions for each model were compared at 1, 2, 3, and 5 years after the last VF used to determine model parameters.

Results.: There were no clinically important differences between any of the models tested for fit; however, the logistic function had the lowest average RMSE (P < 0.001). For predictions, the exponential model consistently had the lowest average prediction RMSE for all time intervals (P < 0.001); the multiple-model approach did not perform better than the exponential model (P < 0.001).

Conclusions.: While the logistic model best fit glaucomatous VF behavior over a long time period, the exponential model provided the best average predictions. A multiple-model approach for VF predictions was associated with a greater prediction error than with the best-performing single-model approach. A model's goodness of fit is not indicative of its predictive ability for measurements of glaucomatous VFs.

Introduction

Glaucoma is a progressive optic neuropathy characterized by changes in the appearance of the optic nerve head and abnormalities of the visual field (VF). Detection of VF progression is of paramount importance in managing patients with glaucoma, and standard automated perimetry is currently the gold standard for evaluating functional change.¹ Event-based techniques as well as rule-based criteria from major clinical trials have been used to measure VF deterioration.^2–5 However, these approaches do not provide robust information about perimetric rates of change. Knowledge of the rate at which glaucomatous damage is worsening allows for appropriate and timely intervention in patients with faster rates of change to minimize visual loss and to optimally use treatment resources.^6,7 Pointwise regression of individual VF test locations has the advantage of providing spatial information about localized rates of change and increases the detection sensitivity of VF worsening compared to the use of global indices.⁸ 

Linear and exponential models have been used to fit and predict VF behavior.^9–12 The linear model assumes a constant additive rate of VF deterioration, while the exponential model assumes a constant multiplicative rate of change, that is, the change at any point in time is a constant percentage of the threshold sensitivity at that point. These assumptions may not necessarily apply to all locations of the VF, especially those that are initially normal.⁹ For example, VF locations may have normal threshold sensitivities for a period of time before beginning to deteriorate. The VF plotted over time for these locations would have a plateau appearance before the decline in sensitivity measurements. For these locations, neither the linear nor the exponential regression models would be able to account for the transition between “normal” measurements and deterioration. The logistic function is a nonlinear model with an inverse sigmoidal shape, which may better depict the course of glaucomatous damage from normal vision to perimetric blindness as measured with clinical perimetry because it can model asymptotes at the beginning and end of data series. Just as not all locations of the VF will progress at the same rate, all VF locations will not necessarily follow the same pattern of deterioration. We hypothesized that a combination of different models may better account for the perimetric behavior at various test locations across the VF. 

The purpose of this study was to evaluate the ability of the pointwise linear, exponential, and logistic functions, as well as a combination of these models, to fit longitudinal VF series and to predict future VF behavior in patients with chronic open-angle glaucoma. 

Methods

Patients

Patient VF data from the Jules Stein Eye Institute database and the Advanced Glaucoma Intervention Study (AGIS) with greater than 6 years of follow-up and 12 VFs were examined. Reliable VFs defined as <30% fixation losses, <30% false-positive rates, and <30% false-negative rates were included. If a location's initial three measured sensitivities were all 0 dB, the location was excluded from the analysis. All tests were performed with the Humphrey Field Analyzer (Carl Zeiss Meditec, Dublin, CA, USA) with 24-2 test pattern, size III white stimulus, and analysis was performed on sensitivity output in decibel values. The UCLA Human Research Protection Program approved the study, and all procedures adhered to the tenets of the Declaration of Helsinki and to the Health Insurance Portability and Accountability Act. 

Regression Modeling

Visual field modeling was performed on threshold sensitivities (dB) at each location, excluding the two test locations corresponding to the physiologic blind spot, with three models: linear, exponential, and logistic. Specifically, the exponential model is a logarithmic-transformed linear model, whereas the logistic model is a nonlinear function with an inverse sigmoidal curve.¹⁰ Figures 1A through 1C show examples of each model. The models are mathematically defined as follows: 

Figure 1

 

Examples of models fit to the same series of threshold sensitivities at a single test location. (A) Linear. (B) Exponential. (C) Logistic.

1. Linear:  
   Display Formula (1)
2. Exponential:  
   Display Formula (2)
3. Logistic:  
   Display Formula (3)

The response variable y represents the threshold sensitivities in decibels; x represents time; α, β, and ζ are model parameters to be estimated. The linear and exponential models were estimated with the least squares method, while the logistic model was estimated with the Newton-Raphson method.¹³ If the regression for the linear model resulted in a line that dropped below zero, the negative values were censored at zero, given the arbitrary “floor” of 0 dB due to the dynamic range of the testing device. Both the exponential and the logistic models do not create negative values so no such censoring was required for those two models. 

Model Evaluation

Each model was used to fit the entire follow-up for each location to evaluate the overall goodness of fit. The error was defined as the difference between the observed and corresponding estimated sensitivities. The averages of the root mean square errors (RMSEs) for the three models were compared to determine which model provided the best fit. The RMSEs of different models were compared by using pairwise t-test with a Holm-Bonferroni correction. The calculated initial sensitivity was also assessed for at each location by using estimated model parameters and then compared to the observed initial sensitivity. 

Predictions

For predicting future VF sensitivities, data from the first 5 years of follow-up were used to estimate model parameters. Locations were excluded if the first three sensitivity measurements were zero. Models fitted to the first 5 years of VF data were then used to predict VF measurements at 1, 2, 3, and 5 years after the last VF used to estimate the model parameters. Negative threshold predictions were censored at zero, while locations with improvements in predicted threshold sensitivities were censored with the average threshold sensitivity of the first four visits. For locations that improved, the average of the first four measured sensitivities was a compromise between allowing for improvement and filtering out noise (false positives). Visual field tests for a 6-month period before and after each time period of interest were aggregated to determine the observed sensitivity for each location at that time point. The prediction error was defined as the difference between the observed and the predicted sensitivities. The averages of prediction RMSEs were used to determine which model had the best prediction. Models were also compared by the number of times each had the closest prediction to the actual measurement. A secondary analysis using the previously described methodology was performed by using data from the first 7 years of follow-up to estimate model parameters and predict future VF sensitivities 3 years later. 

Multiple Models

For the multiple-model approach, a single model was chosen for each location on the basis of whichever one had the lowest RMSE as calculated from the regression of the VF data from the first 5 years of follow-up. The average of the prediction RMSEs (as defined above) was then calculated and used to compare the multiple-model prediction to single-model predictions. 

Results

In this study, 782 eyes of 572 patients were included. Average (±SD) age at the initial visit was 66.3 (±11.4) years, and average follow-up duration was 8.6 (±2.1) years. Average number of VF tests per patient was 15.3 (±4.9), and the total number of VF locations included was 38,363. Average initial mean deviation was −7.8 (±6.0). For the UCLA group, 85% were tested with the Swedish interactive threshold algorithm (SITA) standard protocol, 12% with the full threshold, and 3% with the SITA Fast. For the AGIS group, all eyes were tested with the full threshold strategy. On average, 9.1 (±2.5) VF tests were used to calculate model parameters. Figure 2 demonstrates the distribution of the initial and final threshold sensitivities for all test locations. 

Figure 2

 

Frequency distribution of initial and final threshold sensitivities for all the VF locations in 782 eyes of 572 patients.

Model Fit

When fitting the entire VF series, the logistic model had the lowest RMSE by a slim margin (P < 0.001; Table 1). The logistic model had the best fit at 50.1% of locations, the exponential model in 41.4%, and the linear model in 8.5%. Figure 3 shows the RMSE for each model stratified by the initial sensitivity. There were no clinically important differences in the goodness of fits as measured by the RMSE between any of the models; however, the logistic model consistently performed the best when stratified by the initial sensitivity. Calculated initial sensitivities differed from observed initial sensitivity by an average of 2.21 (±2.42), 2.16 (±2.37), and 2.08 (±2.34) for the linear, exponential, and logistic function, respectively. 

Figure 3

 

Average RMSE for fitting the three regression models to VF data for the entire follow-up binned by the average of the initial two sensitivities. Each bin was of equal size with the range indicated by the x-axis labels.

Table 1

 

Average RMSE for Fitting Visual Field Data of Each Regression Model

Model	Average RMSE	Count
Linear	3.057	3,420
Exponential	3.056	16,707
Logistic	2.925	20,214

 Count is a tally of the number of times a model had the lowest RMSE of the three models for a specific location. In the event of ties, which occurred 1701 times, all models that had the lowest RMSE were counted. The average RMSE for the logistic model was statistically less than other models by Tukey's test (P < 0.001).

Predictions

For all time points predicted, the exponential model resulted in smaller average RMSEs than those of other models, including the multiple-model approach (P < 0.001 for all comparisons; Fig. 4). The number of times each model was chosen for the multiple-model approach is depicted in Figure 5. The multiple-model approach did not perform better than the best performing single-model approach, the exponential model (P < 0.001 for all time points; Table 2). Figure 6 shows the 3-year prediction average RMSEs stratified by the initial sensitivity for each model. As the time interval between predictions and the last VF measurement increased, prediction RMSEs increased for all models. In the secondary analysis, which uses 7 years' worth of data to predict VF sensitivities 3 years later, the exponential model still resulted in the lowest prediction errors, although by a smaller margin (results not shown). 

Figure 4

 

Average RMSE for VF predictions of each regression model for different number of years predicted. Model parameters for the linear, exponential, and logistic models were estimated with the first 5 years of VF data, and average RMSEs for the predictions of each model were calculated at various time points after the last VF used to estimate model parameters. The multiple-model approach used the model with the lowest RMSE for the 5-year fit as the model for predictions. The exponential model resulted in the lowest RMSE (P < 0.001 for all comparisons).

Figure 5

 

Percentages of the number of times the logistic, exponential, and linear functions were chosen for the multiple-model approach for each VF location. A grayscale legend is provided on the right.

Figure 6

 

Average RMSE for 3-year predictions of each regression model binned by the average of the initial two sensitivities. Each bin was of equal size with the range indicated by the x-axis labels.

Table 2

 

Average RMSE of Visual Field Predictions by Regression Models

Years Predicted, y	Linear	Exponential	Logistic	Multiple Model
1	4.766	4.521	5.146	5.150
2	5.200	4.891	5.518	5.526
3	5.969	5.497	6.148	6.131
5	7.561	6.720	7.406	7.328
8	8.840	7.395	8.391	8.251

 Model parameters were estimated with the first 5 years of VF data, and average RMSEs for the predictions of each model were calculated at various time points after the last VF used to estimate model parameters. The multiple-model approach used the model with the lowest RMSE for the 5-year fit as the model for predictions. The exponential model resulted in the lowest RMSE (P < 0.001 for all comparisons).

Discussion

For our cohort of patients, the logistic regression model consistently outperformed both the linear and exponential regression models for most accurately fitting the VF data series measured in decibels, but differences between the models were small and not likely clinically significant. For predictions, the multiple-model approach that chooses which model to use based on fitting ability of prior VF data did not outperform the single-model predictions. The exponential model outperformed all other models when predicting future VF loss. 

Modeling VFs to better predict rates of change and future deterioration has garnered significant interest, and many approaches have been proposed.^{9–12,14–17} The linear model, which uses an ordinary least squares approach to minimize error, is an easy model to use, but applying it to VF data violates certain assumptions as described by Pathak et al.,⁹ who also have showed that a nonlinear model better fits VF series. 

The advantage of the nonlinear logistic model lies in its theoretical ability to model VF progression across the entire range of glaucomatous damage. The natural asymptotes of the model reflect both the period of normal, stable VFs early in the disease, as well as the endpoint of perimetric blindness where perimetric measurements can no longer detect changes in visual function (floor effect). To our knowledge, no other single model is currently able to reflect the entire range of perimetric glaucoma progression. However, the cohort of VFs in this study represent eyes that are receiving care for glaucoma over a limited period of time (9 years on average), so very few eyes demonstrated the full range of progression, especially the initial period of stable, normal sensitivities. 

The logistic model has some intrinsic limitations when fitting a VF series. The shape of the inverse sigmoid curve defined by the logistic function can be modified, but will always have rotational symmetry about an inflection point, defined as where the function's second derivative equals zero. Specifically, the rate at each time point will be equal to that of another time point equidistant from the inflection point. However, at each of these pairs of points, the acceleration of the deterioration rate for the point located before the inflection point will be equal to the deceleration in decline for the other point, which results in rotational symmetry about the inflection point. Clinically, this symmetry means that the model most accurately represents VFs when the acceleration of VF loss in early stages will be equal to the deceleration of VF loss at the end stage of the disease. Symmetry in true VF sensitivities does not likely exist. Changes in VF rates of decay are not likely symmetric. Lower sensitivities are associated with a larger amount of noise, which in turn may cause a VF series to appear to “taper off” toward lower sensitivities. Though it can be argued that the slowing of VF damage near perimetric blindness is not representative of true glaucomatous damage, the models used in this study aim to reflect the perimetric measurements of glaucoma progression. 

The logistic model's sigmoid shape stems from it having an additional parameter as compared to the exponential and linear models. However, the increased complexity of the logistic model compared to the other two works against this model when making predictions, because the additional parameter requires more data points to develop predictions. As previously discussed, having a model that fits series of VF data better does not necessarily correlate with that model providing better VF predictions.¹⁸ It is for this reason that the predictions from the multiple-model approach likely do not perform as well as those of single-model approaches. 

Bryan et al.¹⁴ have explored the usefulness of a Tobit model (censored linear model) to better fit and predict VF data, given the floor effect imposed by the limitations of the measurements. They found that the Tobit model performed worse than an uncensored model. In our article, we used a linear model instead of a Tobit model, but imposed a floor for predictions below zero. Censoring the linear model in this fashion vastly improved the linear model's predictive ability. One speculation to explain the difference in the performance of these two models is that as a test location's sensitivity approaches perimetric blindness, the apparent rate of decline slows because of the increased noise at lower sensitivities as well as the arbitrary floor imposed by measurement. This in turn “slows” the rate of decline as measured by the machine, which allows for the censored linear model to outperform the Tobit model. Alternatively, the rate of decline may actually physiologically slow down as glaucomatous damage becomes severe. Whether or not the apparent slowing of VF decay toward perimetric blindness is due to measurement artifact or a physiologic phenomenon, an exponential or logistic function would better fit the data because of its asymptotic trend toward zero. 

Another difference between this study and that of Bryan et al.¹⁴ lies in the treatment of locations that were modeled as improving. The latter limited predicted sensitivities to a maximum of 40 dB. In our data series 42% of locations were found to have a positive slope or were modeled as improving. For these locations, the predictions were set at the average of the first four measured sensitivities, which was a compromise between allowing for improvement and filtering out noise (false positives). If the perimetric sensitivity in some eyes actually improves after treatment, a component of glaucomatous damage still persists, and the VF may likely continue to decay but from a higher sensitivity. In these cases, it is not unrealistic to assume that the VF progression would be best represented by a function with an increasing segment followed by a decreasing segment. These cases are a limitation of the monotonic functions used in this study and warrant further investigation. 

Though the average fit and prediction RMSE for both the exponential and censored linear models were similar in our study, the exponential model consistently outperformed the latter by a small margin across the entire range of initial sensitivities. The exponential model also performed better, though by a smaller margin, when using 7 years of data to define parameters; however, many patients will not have that much data for analysis. The fact that the exponential model has a larger advantage when using a shorter and more clinically relevant time frame of data to define parameters strengthens its utility over other models in this study. These results are in contrast to those reported by Bryan et al.¹⁴ who have found that the uncensored linear model predictions outperformed those of the exponential model. In their study, a higher percentage of locations had an initial sensitivity of zero dB, and different treatment of these points for regression analysis could affect the performance of the models tested. 

Similar to models previously proposed, the prediction error values for each model in this study are still relatively high, and their clinical use should proceed with caution.^10,14,18 None of the models used in this study address the issue of patients receiving treatment during the course of the VF series. Multimodal functions may depict the VF progression in patients undergoing treatments or procedures that may stop damage to, or even improve, visual function. Also, a low signal-to-noise ratio associated with standard automated perimetry, due to both human and equipment factors, often makes test results difficult to interpret, and certainly negatively affects the regression of VFs. Because all eyes in this study had a diagnosis of glaucoma, the results may be biased against a model that assumes an initial period of decline and make the logistic model appear to not perform as well as the exponential model. Lastly, calculations were performed on a logarithmic scale of VF data, since that is the format used clinically. The RMSE differences will be larger when compared to studies that model VF data on a linear scale. 

Currently, the pathophysiology of glaucoma is not well understood, and no model truly represents the biologic underpinnings of the disease. In its most basic form, glaucoma progression can be thought of as a period of time when the patient is healthy, then deteriorating, and then functionally impaired, if allowed enough time for the disease to progress to perimetric blindness. A simplified form of this progression could be conceived as two plateau periods of little VF change that sandwiches a period of deterioration—a logistic function. Other uses for modeling VFs include identifying patients with more aggressive deterioration to better understand risk factors for the disease. This would be achieved with a model that best fits the data, which was the logistic model in this study. Another use is to predict future change so that clinicians have additional tools to help decide when treatment or more aggressive treatment is appropriate. This would be achieved with models that can most accurately predict future VF changes, which was the exponential model in this study. 

In summary, we explored the utility of a pointwise logistic model for fitting longitudinal VF data and for predicting future trends and compared it to linear and exponential models. The logistic model provided the best mathematical fit for VF series, though all models performed similarly clinically. The exponential model provided the best predictions of future sensitivities. The multiple-model approach, which picks models for predictions by their ability to fit the data, had a larger prediction error than any single model; a model's fitting ability was not alone indicative of its performance to predict VF measurements. 

Acknowledgments

Supported by Research to Prevent Blindness and the Simms/Mann Foundation. The sponsor or funding organization had no role in the design or conduct of this research. The authors alone are responsible for the content and writing of the paper. 

Disclosure: A. Chen, None; K. Nouri-Mahdavi, None; F.J. Otarola, None; F. Yu, None; A.A. Afifi, None; J. Caprioli, None