Pointwise Rates of Visual Field Progression Cluster according to Retinal Nerve Fiber Layer Bundles

Abstract

Purpose.: To explore whether pointwise rates of visual field progression group together in patterns consistent with retinal nerve fiber layer (RNFL) bundles.

Methods.: Three hundred eighty-nine eyes of 309 patients from the Advanced Glaucoma Intervention Study with ≥6 years of follow-up and ≥12 reliable visual field exams were selected. Linear and exponential regression models were used to estimate pointwise rates of change over time. Clustering of pointwise rates of progression was investigated with hierarchical cluster analysis using Pearson's correlation coefficients as distance measure and an average linkage scheme for building the hierarchy with cutoff value of r > 0.7.

Results.: The average mean deviation (±SD) was −10.9 (±5.4). The average (±SD) follow-up time and number of visual field exams were 8.1 (±1.1) years and 15.7 (±3.0), respectively. Pointwise rates of progression across the visual field grouped into clusters consistent with anatomic patterns of RNFL bundles with both linear (10 clusters) and exponential (six clusters) regression models. One hundred forty-four (37%) eyes progressed according to the two-omitting pointwise linear regression model.

Conclusions.: Pointwise rates of change in glaucoma patients cluster into regions consistent with RNFL bundle patterns. This finding validates the clinical significance of such pointwise rates. The correlations among pointwise rates of change can be used for spatial filtering purposes, facilitating detection or prediction of glaucoma progression.

Introduction

Ever since the advent of automated perimetry, it has been well recognized that the threshold estimates of test locations across the visual field are highly correlated. ¹ Such correlations are a function of test location eccentricities, especially in normal visual fields, or the relationship of test locations with respect to specific nerve fiber bundle patterns, which is more likely in fields with abnormal test locations. The correlation among test locations tends to follow the pattern of the retinal nerve fiber layer (RNFL) bundles. It has been shown that clustering of abnormal locations in an arcuate pattern can better discriminate between glaucoma patients and normal subjects compared with when the abnormal points are not grouped together. ² The cross-sectional correlation of visual field test locations is also the basis for the Glaucoma Hemifield Test (GHT) originally described by Asman and Heijl. ³ Other visual field clustering schemes have been explored over time to define groups of related points across the visual field. ^1,4,5 In some definitions of early visual field loss, clustering of test locations is required to establish presence of early glaucomatous damage. ⁵  

There is renewed interest in the measurement and description of rates of damage in glaucoma. However, there is high variability in rates of change across the visual field. ⁶ In addition, determining the statistical significance of pointwise rates of progression is complicated by the high number of regression analyses carried out on individual test locations in a field series. Linear regression analysis of average threshold sensitivity in visual field clusters has therefore been explored to address this issue. ⁷ Determination of the statistical significance of pointwise rates of progression is further complicated by the correlated nature of sequential data. One assumption of linear regression is that the error term is independent from visit to visit, which may not be necessarily correct. There is no hard evidence in the literature regarding the strengths and topography of correlations of pointwise rates of progression. Current progression criteria such as pointwise linear regression (PLR) or guided progression analysis (GPA) do not require change to occur in points belonging to the same cluster. It has been shown that requiring progressing points to belong to the same GHT cluster increases PLR's specificity. ⁸  

The goal of the current study was to explore whether longitudinal rates of change at individual test locations across the visual field are correlated and, if so, whether the pattern is consistent with that of RNFL bundles. Such correlations would validate the biological significance of rates of progression. They could also be used as weighting schemes, so that information from related test locations could be incorporated to facilitate detection or prediction of glaucoma progression.  

Methods

Study Sample

Patient data collected during the Advanced Glaucoma Intervention Study (AGIS) were used. The AGIS design and methods are described in detail elsewhere. ⁹ All patients gave written informed consent for participation in AGIS. The Institutional Review Board of the University of California at Los Angeles approved this study of the AGIS data, and all procedures followed the tenets set forth in the Declaration of Helsinki. 

The cohort of patients who participated in AGIS consisted of 776 eyes of 581 patients with primary open-angle glaucoma. For this study, patients with ≥6 years of follow-up and ≥12 reliable visual field exams (AGIS reliability score <2) were included. Perimetry was performed with the Humphrey Visual Field Analyzer I (HFA, Carl Zeiss Ophthalmic Systems Inc., Dublin, CA) with the 24-2 test pattern, size III white stimulus, and full threshold strategy. 

Estimation of Pointwise Rates of Progression

Both linear and exponential models have been used to fit or predict longitudinal behavior of threshold sensitivity at individual test locations over time. ¹⁰ We used both models to describe clustering of pointwise rates of progression. In the aforementioned study, an exponential model best described progression at individual test locations across the visual field. 

Linear Model

y = a + bx  

where, 

a = intercept (baseline threshold here), and 

b = regression coefficient expressing change as dB/year. 

First Order Exponential Model

y = e^{a +bx} , or equivalently, ln(y) = a + bx. 

where, 

e = base of the natural logarithm, 

ln = natural logarithm, 

a = intercept, and 

b = regression coefficient. 

The rate of change is represented by the coefficient b in both models. For an exponential model, b is the average annual rate of change in ln(y). The quantity (100b) can be interpreted as the percentage loss per year as in exponential decay. 

Cluster Analysis of Rates of Change

Since visual field data are inherently noisy, any statistical model that summarizes trends in visual field data should be evaluated for detecting known patterns (i.e., patterns consistent with the anatomy of the RNFL). For each visual field series, decay rates for all visual field test locations (excluding the foveal location) were determined as outlined above with both exponential and linear models. We performed an agglomerative hierarchical cluster analysis based on all 54 pointwise exponential or linear rates of change of all eyes, regardless of whether the eye progressed or not. ¹¹ An agglomerative hierarchical cluster analysis is a “bottom up” approach of cluster analysis, in which each test location starts in its own cluster. In order to decide which test locations should be combined into one cluster, the hierarchical clustering starts by finding the similarity between every pair of test locations. We used Pearson's correlation coefficient as the measure of similarity between every pair, and pairs of test locations with the highest similarity (correlation coefficient) were merged up one level as a single cluster. Next, the similarity between every pair of clusters was calculated, and pairs of clusters with the highest similarity were merged up another level as a new single cluster. The similarity between any two clusters was the mean correlation of all pairwise correlations between all elements within each cluster (also called average linkage clustering). This process was repeated until the similarity between remaining pairs of clusters was too far apart to be merged (distance criterion), defined as 0.7 or above for the mean correlation within all clusters. A tree-like diagram (dendrogram) illustrates the similarity of rates for test locations and was mapped with a color-coded scheme (Fig. 1). The agglomerative hierarchical cluster analysis was performed using MATLAB Version R2009b (MathWorks, Inc., Natick, MA; see Appendix). 

Figure 1.

 

Results of hierarchical cluster analysis on pointwise rates of change based on exponential regression analysis of the threshold sensitivity at individual test locations over time. All eyes regardless of whether they progressed or not were included. Left: six final clusters are shown on a color scheme of the 24–4 visual field. Right: the dendrogram or the tree demonstrates that the correlation coefficients decrease as smaller clusters (leaves of the tree) are merged to form larger clusters towards the root of the tree. For each test location, there were 389 exponential decay rates (equal to the number of available eyes). The similarity or pairwise correlations between rates at 54 locations were calculated. The highest pairwise correlations are indicated in the figure as doublets (black lines connecting pairs of test locations), which form the leaves or the lowest level of the tree; although not all locations paired as evidenced by points not connected to any other test location. Top right: the changing magnitude of the Pearson's correlation coefficients (r) as the clustering moves from the leaves of the dendrogram towards the root is shown on the scale. A cutoff point of r > 0.7 was used to determine the final clusters.

Detection of Progression

The two-omitting pointwise linear regression analysis was used to detect eyes progressing according to a well-established method. ¹² It is considered a conservative approach to PLR with high specificity. In summary, a linear regression analysis of threshold sensitivity at each test location is performed against time, once after excluding the last available threshold and a second time after excluding the threshold before the last available one. The foveal and blind spot test locations (i.e., locations 15 degrees temporal and 3 degrees above and below the horizontal midline) were excluded from analyses. Test locations demonstrating a significant change on both regression analyses were considered to be worsening (slope ≤ −1.0 dB/year along with P ≤ 0.01) or improving (slope ≥ 1.0 dB/year along with P ≤ 0.01). Eyes in which the number of worsening test locations exceeded the improving points by 3 or more were considered to be worsening at the end of follow-up. 

Results

A total of 389 eyes of 309 patients were included in this study. Table 1 describes the clinical and visual field characteristics of the study sample. The average (±SD) follow-up time and mean visual field exams were 8.1 (±1.1) years and 15.7 (±3.0), respectively. The average (±SD) mean deviation was −10.9 (±5.4). The median (interquartile range) number of progressing test locations was 10 (5–21). Hierarchical clustering was performed on all eyes regardless of whether or not they progressed. Figure 1 shows the results of the hierarchical clustering for pointwise rates of change according to the exponential model. With a cutoff point of 0.7 for the average correlation within the clusters, six distinct clusters were defined. Neither of the test locations representing the blind spot was grouped within the clusters, although they were included in the cluster analysis. The black lines connecting pairs of test locations (doublets) refer to the highest correlations observed at the lowest level of clustering (leaves of the tree), when clusters consisted of only pairs of test locations. Figure 2 demonstrates the results of a similar approach using linear rates of change at individual test locations across the visual field. The same cutoff point for correlation coefficients (r ≥ 0.7) was set with the linear model. Ten individual clusters were defined by the hierarchical clustering approach, when linear rates of change were used. Based on the two-omitting PLR method, 144 (37.0%) eyes progressed at the end of the follow-up period. 

Table 1.

 

Clinical and Perimetric Characteristics of the Study Sample

Characteristics
Number of eyes/patients	389/309
Age (years, mean ± SD)	64.7 ± 9.5
Follow-up (years, mean ± SD)	8.1 ± 1.1
Baseline IOP (mm Hg, mean ± SD)	15.3 ± 4.9
Baseline number medications (mean ± SD)	2.8 ± 0.9
Race
White	44.7%
Black	54.2%
Other	1%
Sex
Male	47.3%
Female	52.7%
Eye laterality
Right	47.8%
Left	52.2%
Cataract surgery
No	57.8%
Yes	42.2%
Number of visual fields (mean ± SD)	15.7 ± 3.0
Initial MD (dB, mean ± SD)	−10.9 ± 5.4
Final MD (dB, mean ± SD)	−12.9 ± 6.9

 IOP, intraocular pressure; MD, mean deviation.

Figure 2.

 

Results of hierarchical cluster analysis on pointwise rates of progression based on linear regression analysis of the threshold sensitivity at individual test locations over time. For simplicity, only the final visual field cluster scheme is demonstrated here. Please refer to the legend for Figure 1 for further explanations.

Discussion

In this subgroup of patients from AGIS more than one-third of the eyes (37%) demonstrated progression based on established PLR criteria. Although different methods would result in varying proportions of worsening eyes, the two-omitting PLR is considered a conservative approach with high specificity, and it is reasonable to assume that a sizable minority of the eyes truly progressed in our study sample. We found that longitudinal perimetric rates of change grouped into clusters that were consistent with RNFL bundle patterns. This was true for both linear and exponential models. Although the clusters derived from exponential and linear models were not expected to exactly match, they were quite similar. 

Over the years, different clustering schemes have been described in the literature. ^{1, 13–15} Wirtschafter et al. defined the boundary lines of the visual field clusters by overlaying illustrations of the primate RNFL onto a scaled visual field map. ¹³ Similarly, the Glaucoma Hemifield clusters were originally based on the superimposition of RNFL bundle patterns and the 74 test locations belonging to the 30-2 testing strategy of the HFA. ³ Garway-Heath and coworkers defined visual field clusters based on the structure–function correlations between RNFL bundle defects on fundus photographs and visual field defects observed in a group of normal-tension glaucoma patients. ¹⁵ Mandava et al. ¹ defined visual field clusters using the cross-sectional correlation of threshold sensitivities at individual test locations on Octopus visual fields in normal and stable glaucomatous eyes. The visual field clusters based on exponential rates of progression in this study closely resemble those reported by Garway-Heath et al. ¹⁵ The correlation of rates of progression and distribution of clusters across the visual field would be expected to vary as a function of glaucomatous damage at baseline. Despite a potentially high correlation between any pair of test locations, the correlation would be observed only if the baseline threshold sensitivity in either test location was high enough to allow detection of any change. However, as can be seen comparing the clusters derived from linear and exponential regressions, general patterns emerge that are consistent with the scheme of clusters derived from cross-sectional correlation of structure and function in glaucoma. It is reassuring to observe that rates of progression follow a pattern of correlation similar to the RNFL bundle trajectories. This suggests that the rates of progression as measured with regression analyses reflect a biological phenomenon. 

Current criteria used for pointwise trend analyses (such as PLR) and event analyses (such as GPA) do not require the test locations demonstrating progression to be spatially or functionally related. The high specificity of such methods is mainly derived from confirmation of change over time (a minimum of one or two confirmations depending on criteria). Taking into account longitudinal spatial correlations found in this study could result in a similar sensitivity and specificity with potentially less need for confirmation, if the pattern of presumed progression follows the expected clustering patterns found in the current study. Gaussian and non-Gaussian spatial filtering of threshold sensitivity at neighboring test locations has been used to improve performance of trend analyses. ^16–19 We speculate that the correlation coefficients derived from the current study might be better suited for such spatial filtering compared with the cross-sectional weighting schemes used in prior studies. Gaussian and non-Gaussian filters are mostly based on information from neighboring points, while the correlation of visual field test locations normally goes beyond immediately adjacent points. This is the focus of an ongoing investigation by the authors. 

Trend analyses have been previously reported based on cross-sectionally defined clusters ⁷ or according to Glaucoma Hemifield Test clusters, which are based on RNFL bundle anatomic patterns. ^20,21 Further study is needed to determine whether clusters derived from correlation of longitudinal rates of progression, as found in this study, would improve the sensitivity of such cluster-based trend analyses. One challenging and unsolved issue in pointwise regression analyses has been the inherent correlation among test locations across the visual field. Given the fact that the correlation among clusters is lower than the correlation among individual test locations, linear and nonlinear mixed models with random slopes for such longitudinally defined clusters might result in increased sensitivity and specificity for detection of glaucoma progression. Mandava and colleagues found that clusters performed better than global indices for detection of localized glaucomatous loss and that the long-term fluctuation was lower in clusters compared with individual test locations. ¹  

Because of the nature of our data, we cannot take into account the effect of media opacity or cataract surgery on detection of progression. If anything, both would have had a negative effect on the correlation of rates, and therefore neither is expected to have significantly influenced the results. As mentioned, the exact arrangement of clusters depends to some extent on the specific patient sample and on the model used for the trend analyses. The results, however, were consistent between the two models applied to visual field data in this study and support the notion of clustering of progression rates across the visual field. We are currently exploring ways to incorporate the correlations found in this study for measurement of rates of worsening and prediction of progression in glaucoma. Another issue is the fact that the bracketing strategy used in the HFA's full threshold algorithm uses information from adjacent test locations as a starting point. This could potentially affect the final threshold at neighboring test locations and potentially influence correlation of longitudinal rates of progression. 

In summary, we report that longitudinal rates of change at test locations across the visual field tend to cluster according to RNFL bundle patterns, which is consistent with observed patterns of clustering of cross-sectional visual field data. The correlation of test locations within a cluster can potentially be used as a weighting scheme for detection or prediction of glaucoma progression. 

Footnotes

 Supported in part by a grant from Research to Prevent Blindness and the Gerald Oppenheimer Family Foundation Center for the Prevention of Eye Disease.

Footnotes

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

Footnotes

 Presented in part at the 2010 ARVO Annual Meeting, Fort Lauderdale, Florida, May 2–6, 2010.

Appendix 

The agglomerative hierarchical cluster analysis was performed using the following commands in MATLAB R2009 Statistics Toolbox: 

Y = pdist (X, ‘correlation'); 

*This command calculates the pairwise Pearson correlation coefficients (Ys) between Xs (rate change of each visual field location) *** 

Z = linkage (Y, ‘average'); 

*This command defines the average of correlation coefficients (Ys) as the linkage criterion between clusters, and the final clusters are stored in Z *** 

[H,T] = dendrogram (Z,'colorthreshold','default'); 

*This command generates a dendrogram plot (H) of the hierarchical, binary cluster tree represented by Z. 

Set (H,'LineWidth',2); 

*This command shows the dendrogram plot H.