Validation of Point-Wise Exponential Regression to Measure the Decay Rates of Glaucomatous Visual Fields

Abstract

Purpose.: This study was conducted to validate a recently described technique for measuring the rates of visual field (VF) decay in glaucoma.

Methods.: A pointwise exponential regression (PER) model was used to calculate average rates of faster and slower deteriorating VF components, and that of the entire VF. Rapid progressors had a faster component rate of >25%/year. Mean deviation (MD) and visual field index (VFI) forecasts were calculated by (1) extrapolation of linear regression of MD and VFI, and (2) calculation de novo from the PER-predicted final thresholds.

Results.: The mean (± SD) years of follow-up and number of VFs were 9.2 (± 2.7) and 13.7 (± 5.8), respectively. The median rates of the decay were −0.1 and 3.6 (%/year) for the slower and the faster components, respectively. The “rapid progressors” (32% of eyes) had a mean decay rate of 52.2%/year. In comparison with actual values, the average absolute difference and the mean squared error for MD forecasts with linear extrapolation of indices were 3.58 dB and 31.91 dB², and with the de novo recalculation from PER predictions were 2.95 dB and 17.49 dB², respectively. Similar results were obtained for VFI forecasts. Comparisons of the prediction errors for both the MD and VFI favored the PER forecasts (P < 0.001).

Conclusions.: PER for measuring rates of VF decay is a robust indicator of rates across a wide range of disease severity and can predict future global indices accurately. The identification of “rapid progressors” identifies high-risk patients for appropriate treatment.

Introduction

The ability to estimate rates of functional loss from glaucoma is essential if visual disability is to be reduced optimally with treatment. ¹ By identifying patients with faster rates of visual field decay and with appropriately early intervention, long-term visual function can be preserved and limited resources applied appropriately to optimize patient outcomes. 

The systematic analysis of a single visual field (VF) is well established, ² but a clinically useful assessment of the rate of worsening remains a challenge. Given the discrepancy among clinicians in the interpretation of VFs, ^3,4 a great deal of interest has been generated in the “perfect tool” for an objective assessment of VF decay. Current methods either are event-based or trend-based, and involve analysis of serial VFs. Unfortunately, no available gold standards are devoid of limitations, either in terms of accounting for confounding factors, or having a limited range of operation. ⁵  

In addressing the shortcomings of the current tools for assessment of glaucomatous VF deterioration, we recently described a method based on point-wise exponential regression (PER) analysis of threshold sensitivities in a group of patients with advanced VF loss. ⁶ We identified patients at potentially highest risk of deterioration by analyzing single locations in serial VFs and assessed their individual rates of decay. As such, individual locations were assigned as having a fast or slow rate of decay, while preserving the spatial distribution in the analysis. In our study, we used PER to validate the technique in a group of patients with less severe glaucomatous VF loss compared to the previously described group of patients, ⁶ and to detect the technique's accuracy in predicting future values of the mean deviation (MD) and VF index (VFI) by using individually regressed thresholds compared to linear extrapolation of the indices. 

Methods

Patient and Visual Field Data

Two cohorts of patients with reliable VFs were used to represent a wide range of glaucoma severity. Reliability was defined as <30% fixation losses, <30% false positive, and <30% false negative rates. The first group included University of California at Los Angeles (UCLA) patients diagnosed with open-angle glaucoma with 6 or more years of follow-up and who underwent 8 or more whole VF examinations. The tests were performed with the Humphrey visual field analyzer (Carl Zeiss Ophthalmic Systems Inc., Dublin, CA) with the 24-2 test pattern, size III white stimulus, and with either full threshold, SITA Standard, or SITA fast strategies. The results from this group then were compared and combined with a second cohort of patients from whom data were collected during the conduct of the Advanced Glaucoma Intervention Study (AGIS), and who had 6 or more years of follow-up and underwent 12 or more VFs. The results of the latter group have been reported previously. ⁶ In accordance with the tenets set forth in the Declaration of Helsinki, the individual Institutional Review Board of UCLA approved the study. 

Rates of VF Decay and Partitioning into Faster and Slower Components

The rates of VF decay were calculated by PER analysis of threshold sensitivities of the 54 test locations, in decibels (dB), excluding the two test locations at the physiologic blind spot. ⁶ Each VF location was regressed exponentially against time and a regression coefficient (equivalent to rate) was obtained. The mathematical model used in these calculations was $y=e^{a+bx}$. The 54 locations were ranked according to the rate of decay and partitioned into two groups (faster and slower). For each partitioning, a t-test was performed and the corresponding P values were adjusted for multiple testing. The Benjamini-Hochberg correction was used to find the optimal P value to maximize the difference between the faster and slower groups by minimizing the P value between component rates. ⁷ A minimum of 5 test locations was required in either the slow or fast groups. Global rates of decay then were obtained from the mean of all the test locations, and the mean fast and slow components. The fast component then was analyzed for any potential subdivisions within the group. The rates of change in these components then were compared to the rates of change in the MD. 

MD and VFI Forecasts

The data from the two subset of patients (AGIS and UCLA) were combined in this portion of the study to provide a wide range of disease severity. The “actual” MD and VFI values were taken as the average of the last two examinations during the final follow-up period. Two techniques were used to perform MD and VFI predictions: (1) Linear extrapolation (according to the method used by Statpac) was used on both global indices obtained from the first four years of the data or half of the follow-up period if the patients were followed for more than 8 years, to estimate the value of the index at the end of follow-up. (2) PER of individual test locations was performed, and the MD and VFI were recalculated de novo from the individually predicted threshold values at the end of the follow-up period. 

MD and VFI Calculations

While details of the age-matched normals within the Humphrey VF analyzer are proprietary, the formulae for the calculation of MD and VFI have been described previously. ^8,9 and VFI are in the public domain.To estimate the normal age-matched threshold for each location, the difference between the observed thresholds and total deviation values for each decile age group was used. This was repeated for at least 10 patients and an average of “normal” was obtained for each location for every decile of age. The SDs and the weighting system used in the MD calculation for each 10-year age group were obtained from a previous publication by Heijl. ¹⁰ The estimated normal values for individual locations for each decile age group together with the SDs then were inserted into the known formulae. When compared to actual known values, there was a high level of correlation with a random sample of our estimates using the above methods (MD r² = 0.996, VFI r² = 0.991) 

Comparisons of the MD and VFI Predictions

The predicted MD and VFI, obtained for the mean time interval between the last two follow-up visits, with the Statpac linear regression and with PER of individual test locations then were compared to the mean of the last two actual values at those visits. Frequency distribution curves of predicted differences from actual values for MD and VFI were used to compare the predictive accuracy of the Statpac and PER model. The areas under the curves (AUC) were calculated and a Wilcoxon signed rank test was used to compare the values. ⁸  

Results

Patient Data

A total of 409 eyes of 279 open-angle glaucoma patients from the UCLA database was included in this study. The results were compared and then combined with 389 eyes (309 patients) from the AGIS database, the results of which have been reported previously. ⁶ The VF test strategies for the UCLA group included SITA standard (88%), full threshold (12%), and SITA fast (3%). The characteristics and demographic data for the two groups are given in Table 1. The UCLA group consisted of patients with less severe glaucoma (initial MD −5.5 ± 5.5, final MD −6.7 ± 7.3 dB) compared to the AGIS group (initial MD −10.9 ± 5.4, final MD −12.9 ± 6.9 dB), as shown as frequency distributions in Figure 1. 

Figure 1. 

 

The distribution of MD for the VF of the UCLA (409 eyes) and AGIS (389 eyes) groups. Gray bars: initial MD. Black bars: final MD. The number of eyes (frequency) is shown on the y axis.

Table 1. 

 

Characteristics of the Study Groups

	UCLA	AGIS
Eyes, n	409	389
Patients, n	279	309
Age, y	75 ± 11.0	64.7 ± 9.6
Follow-up, y	9.2 ± 2.7	8.1 ± 1.1
Baseline IOP, mm Hg	15.3 ± 5.1	15.3 ± 5.0
Baseline number medications	2.0 ± 1.0	2.8 ± 0.9
Eye, n (%)
Right	297 (72.6)	186 (47.8)
Left	112 (27.4)	203 (52.2)
Number of VFs	13.7 ± 5.8	15.7 ± 3.0
Initial MD* (dB)	−5.5 ± 5.5	−10.9 ± 5.4
Final MD *(dB)	- 6.7 ± 7.3	−12.9 ± 6.9

 Data are given as the mean ± SD, unless otherwise stated.

*  VF MD.

Rates of VF Decay, and Partitioning into Faster and Slower Components

The global rates of VF decay in (%/year), which were determined from the mean of all individual decay rates (54 locations), are shown in Figure 2. The distribution of rates for the AGIS group is shifted to the right compared to the UCLA group, indicating a faster rate of deterioration in the more advanced glaucoma group. In Figures 3 and 4, the rates of progression for the faster and slower components of the VF are shown for the two groups separately. The median rates of the decay were −0.1 and 3.6 (%/year) for the slower and the faster components, respectively. The faster component has a bimodal distribution (P < 0.0001), with the division at the 25%/year decay rate. It consists of a group with a mean decay of 3.3%/year (SD ±5.4%), and a second more rapidly deteriorating group with a mean decay of 52.2%/year (SD ±13.8%) as shown in Figure 3B. The latter group consisted of 32% of the eyes. The rate of the decay in the faster component was independent of the MD rate of decay, whereas there was a linear relationship between the slower component and the MD (Fig. 4). 

Figure 2. 

 

Frequency distribution curve of the global rates of VF decay for the UCLA (n = 409) and AGIS (n = 389) groups. The number of eyes (frequency) and the average rate of decay for each eye is shown.

Figure 3. 

 

(A) Frequency distributions (n eyes) of the slower (median 0.1%) and faster (median 3.6%) components of VF decay for the UCLA group. Histograms show the distribution of the average rates for the faster and slower components. (B) Frequency distribution curve for the faster component of the VF decay for the UCLA group showing a bimodal distribution. If a critical cut off rate loss of 25% is taken to be the separation point of the two data sets, the first mode has a mean of 3.3% ± 5.4% and has a statistically significant distribution (P < 0.0001) from second mode, which has a mean of 52.2% ± 13.8%. The latter group, which consists of 32% of eyes, is considered to be “rapid progressors.”

Figure 4. 

 

The average of decay of the faster and slower components for the UCLA group is plotted against the rate of change for MD. The reference lines termed diffuse and focal are drawn. The horizontal line termed focal components represents focal areas of decay within the VF where there is a complete dissociation from changes in the MD values. The faster component is associated closely with this line and, therefore, represents test locations affected by true glaucomatous damage. The diagonal line termed diffuse is a reference line with slope equal to unity, and represents complete agreement between the component rate and MD change. The slower component closely follows this line and represents general diffuse damage in VF, which are derived mainly from non-glaucomatous deterioration.

MD and VFI Forecasts

The eyes from both subsets of patients were combined to give a total n = 798 for the purposes of making MD and VFI forecasts. Figure 5 shows the relationship between the predicted MD and VFI with the Statpac and the PER model on a frequency distribution (binary classifier) curve. Of the two curves on each graph, the one to the upper left represents the technique with the better predictive potential, that is the smallest difference from the actual value. In both cases of MD and the VFI forecasts, the de novo calculation of the indices from PER regression of individual threshold values was closer to the actual values compared to the linear regression of the indices themselves. Comparisons between the two methods of MD and VFI predictions are summarized in Table 2. The de novo calculations with PER of individual thresholds are more consistent (smaller mean squared error) with smaller prediction errors (smaller average absolute differences), and when the areas under the frequency distribution prediction errors (using PER versus linear regression) are compared, they are significantly different (P < 0.001) 

Figure 5. 

 

Frequency distribution curves (n of eyes) of MD and VFI forecasts. Frequency on the y axis indicates the number of eyes. The x axis indicates the difference between the predicted and expected MD and VFI. Comparison of the AUCs shows the PER model to be significantly (P < 0.001) better in predicting the MD and VFI, than the linear regression of the indices.

Table 2. 

 

Predicted versus Actual Values Obtained with Linear Regression of Indices and Calculation of the Indices De Novo after PER

	MD			VFI
	AD (dB)	MSE (dB2)	AUC	AD (%)	MSE (%2)	AUC
LINEAR	3.58	31.90	0.75	8.67	0.019	0.69
PER	2.95	17.50	0.77	7.19	0.012	0.72

 MSE, mean squared error; AD, the average of absolute difference in dB for MD and % for VFI; AUC, area under the curve of the frequency distribution curves.

Discussion

In our study, we validated a recently described technique for the assessment of the rates of VF decay in a group of patients with less severe disease than studied previously. ⁶ With the PER technique in the UCLA patients, we demonstrated that the rates of VF decay for individual eyes can be divided into faster and slower components. The faster component appears to be divided into two groups with a subgroup demonstrating a more rapid VF loss and this subgroup is identified as “rapid progressors.” Furthermore the faster component appears to be independent of changes in the rate of MD in contrast to the slower component with which it has a strong linear relationship (Fig. 4). As with its relationship to the MD, the faster component also appears to be independent of VFI as shown by an example in Figure 6. This suggests that test locations can be identified where local glaucomatous deterioration is occurring, independent of diffuse VF loss, which are more likely to be caused by conditions, such as media opacities and aging. In this study, by combining groups of patients with both severe (AGIS data) and the less severe (UCLA patients) glaucoma groups, we also demonstrated that the PER of threshold sensitivities can be applied across a wide range of VF severity, allowing differentiation of the rates of decay into fast and slow components, and hence enabling us to make predictions on regional rates of decay while preserving the spatial information provided by the VF. 

Figure 6. 

 

An example of a VF decay rate and a prediction display that summarizes the behavior of the VF in a typical glaucomatous eye, and provides predictions of future behavior of the VFs. Top left: gray scale of the initial VF in the series. Top middle: gray scale of the final VF in the series. Top right: gray scale of the rate of decay (percentage per year) at each test location. Bottom left: spatial partition of the VF into slower (green) and faster (red) rates components. Bottom right: average rates of decay of the slower (green) component, faster (red) component, the MD (black-dashed), and the VFI (blue).

There is no consensus on the most appropriate method of assessment of VF decay in an individual patient. By defining empirical 95% confidence intervals for a change at test locations or global indices and consequent comparisons to the baseline VF, event-based analyses identify statistically significant locations of progression but give little information about the rates of VF decay. ⁵ Trend-based analyses, on the other hand, have the potential to provide estimates of the rates of VF decay. There is little consistency among various investigators on the criteria set for the detection of VF progression. ^9–13 Analysis of global indices has the advantage of providing a single figure and, therefore, a quick guide for the clinician when analyzing serial VFs of an individual patient, but at the expense of losing spatial information. Point-wise linear regression of individual threshold sensitivities (regional trend based analysis) provides an estimate of VF worsening, ^14,15 and this technique is available currently as software for clinical use (PROGRESSOR; Medisoft Ltd., Leeds, UK). Decay is said to be clinically significant when the rate of deterioration is at least −1.0 dB/year and the statistical significance of the linear slope is P < 0.01. With these criteria for progression, Nouri-Mahdavi et al. found PLR to be at least comparable to the AGIS criteria for detection of longitudinal VF progression. ¹⁶ By comparing several methods, Gardiner et al. found that a “three omitting” technique leads to an increase in the specificity of PLR. ¹⁷  

Bengtsson et al. described the use of linear regression of VFI to predict long-term VF decay. ¹⁸ Final VFI for 100 patients with an average follow-up of 8.2 years and 11 VF examinations was predicted by linear regression of 5 initial tests. They found the technique to be a reliable predictor of future VF loss. Of patients 70% had a predicted final VFI within ± 10% of the estimated final VFI (particularly in patients with abnormal initial VF tests), with a correlation coefficient of 0.84 between predicted and estimated VFI. While this technique is useful in most cases, it may not be applicable reliably across the entire range of VF due to the way the VFI is calculated. While the effect of confounders (e.g., cataracts) is reduced by the use of pattern deviation probability maps to calculate VFI, the authors acknowledge that in advanced cases where even the 85th percentile most sensitive test point has been affected, such maps no longer can be used, and the reliability of VFI is reduced beyond a cut off of −20dB, where the index is derived from the total deviation plots. ¹⁴ Artes et al. recently compared the properties of VFI and MD in a 204 eyes with a range of MD from −2 to −10dB, and found them to be closely related (r = 0.88, P < 0.001), and concluded that VFI may have reduced sensitivity to changes in early glaucoma damage due to the ceiling effect cause by pattern deviation maps. ¹⁹  

To address this gap in the extremes of VF severity (very early and advanced glaucoma damage), we used our database in which a wide range of VF severities was represented to compare the predictive ability of our technique to the method described by Bengtsson et al. for VFI. ¹⁸ We also used the technique to predict MD as a further validation measure, and found that in the case of VFI and MD, de novo recalculations from PER of individual threshold sensitivities were statistically significantly closer to actual values (P < 0.001) than simple linear regression of both of these global indices. 

There are some limitations to the technique. To avoid detecting falsely deteriorating clusters, we required a minimum of five test locations in the smaller cluster when identifying the faster and slower components. Therefore, it is not possible to comment on the technique's accuracy when more focal areas of decay (i.e., smaller number of locations) are occurring. It also is possible that the technique may not identify a small subgroup of patients who may have slow diffuse glaucomatous visual loss. Given that the rates of decay are expressed in percentiles, when applying the model, clinicians would need to make a note of starting thresholds. 

In summary, we presented a novel method, PER, for measuring rates of VF decay that seems reliable across a wide range of visual field sensitivities. The advantages of this technique include the ability to identify focal glaucomatous areas of VF deterioration through a stratification of decay rates into slower and faster components within which a subgroup of “rapid progressors” can be identified allowing a targeted treatment, as well as the facts that it can produce reasonable and clinically useful predictions of future visual field trends, the spatial information of VF in the measurements of rates are preserved, a normative patient database is not required, and it can be applied on a variety of test strategies for detection of visual field decay. Further work is required to test the ability of the model to measure the effects of treatment on the rates of VF decay. 

Footnotes

 Supported by unrestricted grants from Research to Prevent Blindness, Inc.

Footnotes

 Disclosure: P. Azarbod, None; D. Mock, None; E. Bitrian, None; A.A. Afifi, None; F. Yu, None; K. Nouri-Mahdavi, None; A.L. Coleman, None; J. Caprioli, Allergan (C)

Footnotes

 Presented in part at the annual meeting of the Association for Research in Vision and Ophthalmology, Fort Lauderdale, Florida, May 1–5, 2011.