Clustering Visual Field Test Points Based on Rates of Progression to Improve the Prediction of Future Damage

Abstract

Purpose.: To develop new visual field (VF) sectors based on pointwise rates of glaucomatous VF progression (“progression regions”) and to evaluate their usefulness for predicting future progression.

Methods.: A training dataset consisting of 10 VFs from each of 412 eyes in 412 open-angle glaucoma patients and a validation dataset consisting of 15 VFs from each of 71 eyes in 45 patients were investigated. First, using the training dataset, the VF was divided into small regions, according to the rates of progression of all 52 test points in the VF. Then, using the initial four VFs of the validation dataset, total deviation (TD) values in the 10th VF were predicted by applying linear regression analysis in derived regions and the absolute prediction error was calculated. The analysis was iterated, predicting TD values of the 10th VF, but each time including an additional VF in the regression (from five to nine VFs). Absolute prediction errors were then compared with conventional pointwise linear regression (PLR) and regression based on Nouri-Mahdavi (NM) sectors.

Results.: Twenty-three progression regions were derived. In general, absolute prediction errors were significantly smaller for regression based on these regions compared with PLR and NM sectors.

Conclusions.: Predictions of VF progression can be improved by dividing the VF into small regions based on clusters of test points with similar progression rates.

Introduction

Standard automated perimetry is the gold standard method of visual field (VF) testing for the diagnosis and management of glaucoma. It is important when making glaucoma treatment decisions to accurately predict the rate of VF progression. In the Guided Progression Analysis (GPA) software on the Humphrey Visual Field Analyzer (HFA; Carl Zeiss Meditec AG, Dublin, CA, USA), the rate of VF progression is predicted by applying linear regression to global indices such as mean deviation (MD). On the other hand, pointwise linear regression (PLR) analysis^1–10 is more sensitive to detect focal VF progression than analyses based on global indices.^2,5,6,11,12 However, it has been reported that test-retest reproducibility of pointwise VF sensitivity is poor,¹³ and as a result, the false-positive rate of PLR is considerably high.¹⁴ One possible approach to overcome this problem could be to reduce variability by dividing the VF into subsectors¹⁵ and predicting progression based on these clusters of test points. Indeed the Octopus 900 EyeSuite software (Haag-Streit, Inc., Köniz, Switzerland) conducts a cluster trend analysis. 

To date, several methods for clustering VFs, using cross-sectional correlations of VF test locations^12,16–18 or rates of progression,¹⁹ have been proposed. One of the problems with cluster analysis, in either hierarchical or nonhierarchical clustering, is the determination of the optimized number of clusters. The optimal number of clusters is often decided subjectively in hierarchical clustering and k-means nonhierarchical clustering, which can yield hugely different results. Recently, there has been a renewed interest in objectively determining the optimum number of clusters. The “silhouette index” is one method of objectively interpreting and validating the results of clustering²⁰ so that the optimum number and shape of clusters can be automatically decided by measuring the minimum mean/medium split silhouette (MSS) score. Supporting this method, research has suggested that using MSS is the most appropriate to identify fine structure in data.²¹ Indeed, we have recently defined VF clusters in both 30-2 and 10-2 HFA VFs using this approach.²² The purpose of the current study was to develop new VF clusters using this clustering method based on pointwise rates of VF progression, and to investigate the usefulness of these clusters to predict future VF progression compared with PLR and regression based on existing Nouri-Mahdavi sectors (NM sectors).¹⁹ 

Methods

The study was approved by the Research Ethics Committee of the Graduate School of Medicine and Faculty of Medicine at the University of Tokyo. Informed consent was obtained from all subjects. This study was performed according to the tenets of the Declaration of Helsinki. 

Subjects and VFs

Visual field data were retrospectively obtained from a total of 483 eyes in 457 patients with glaucoma. Patients were followed up in the glaucoma clinic at the University of Tokyo Hospital. Visual field measurements were performed using the HFA with either the 30-2 or 24-2 program and the Swedish Interactive Threshold Algorithm Standard. When VFs were obtained with the 30-2 test pattern, only the 52 test locations overlapping with the 24-2 test pattern were used in the analysis. Of the 457 patients, 412 eyes in 412 patients were used as a training dataset for the creation of the new VF regions (“progression regions”). If both eyes met the inclusion criteria described below, one eye was randomly chosen to be included. In addition, 71 eyes from 45 patients were used as a testing dataset. Patients' first VFs were excluded from the analysis. Other inclusion criteria in this study were best corrected visual acuity better than 6/12, refraction within ±6 diopter ametropia, no previous ocular surgery except for cataract extraction and intraocular lens implantation, and no other posterior segment of the eye disease. Reliability criteria for VFs were applied: fixation losses less than 20% and false-positive responses less than 15%; the false-negative rate was not applied as a reliability criteria based on a previous report.²³ A glaucomatous VF was defined if a VF met any of the following criteria²⁴: a cluster of three or more points in the pattern deviation plot in a single hemifield (superior/inferior) with P less than 0.05, one of which must have been P less than 0.01, excluding the outermost test point of HFA 30-2 program; glaucoma hemifield test (GHT) result outside of normal limits; or abnormal pattern standard deviation (PSD) with P less than 0.05. Patients who underwent intraocular surgical treatments during the observed period were excluded. The VF of a left eye was mirror-imaged to that of a right eye for statistical analyses. 

Clustering VFs

First, PLR was carried out using total deviation (TD) values of each of the 52 test points in the training dataset; each eye's VF series was 10 in length. Then, using the obtained pointwise progression rates, the VF was clustered using the Hierarchical Ordered Partitioning and Collapsing Hybrid–Partitioning Around Medoids (HOPACH-PAM) algorithm,²⁵ using the MSS criteria, which is particularly apt at identifying structures in a dataset.²⁶ The HOPACH-PAM algorithm is a hybrid between hierarchical ordered partitioning and collapsing; HOPACH builds a hierarchical tree of clusters by recursively partitioning the data, while ordering and possibly collapsing clusters at each level to identify finite structures in a dataset.^25,27,28 One of strengths of this approach is that the optimum number of clusters is automatically inferred²⁵; this is in contrast to many other clustering methods, such as k-means and hierarchical clustering methods in that the number of optimum clusters is arbitrarily decided, which can lead to inconsistent results. 

Investigation of the Usefulness of Progression Regions

Using the validation dataset, average TD values were calculated for each of the progression regions. Linear regression was then applied to each progression region for an eye's first four VFs in its series and the regional average of TD values in the eye's 10th VF was predicted. Absolute prediction errors were then calculated for every test point in the VF in which the predicted value for each point in a region was allocated its regional average predicted value. This analysis was iterated, but each time an additional VF was included in the regression (from five to nine VFs) to predict the 10th VF. Similarly, prediction errors were calculated using the same iterative analysis for PLR and further, regression based on NM sectors.¹⁷ 

Strouthidis et al.²⁹ proposed a physiologically derived VF filter to reduce measurement noise, considering the anatomic map of the retinal nerve fiber layer (RNFL) distribution and distance between each test points on the 24-2 VF. For comparison, we investigated this “filtering” method alongside our novel approach, to measure the relative improvement in prediction errors. 

All statistical analyses were carried out using the statistical programming language R (ver. 2.15.0; The R Foundation for Statistical Computing, Vienna, Austria) and the HOPACH package.²⁶ Absolute prediction errors were compared using the paired Wilcoxon test. Holm's method^30,31 was used to correct P values for the problem of multiple testing. 

Results

Characteristics of the study population are summarized in the Table. 

Table

 

Characteristics of the Patients

	Training Data	Testing Data
No. of eyes/patients	412/412	71/45
Type of glaucoma
Primary open angle glaucoma	140	15
Normal-tension glaucoma	206	49
Secondary glaucoma	41	3
Angle-closure glaucoma	15	0
Congenital glaucoma	10	4
Age, y, mean ± SD	62.1 ± 13.2	61.4 ± 13.0
Sex, male/female	219/192	24/21
Eye laterality, right/light	195/217	30/41
Baseline IOP, mm Hg, mean ± SD	18.6 ± 6.0	16.5 ± 4.0
No. of VFs	11	16
Follow-up, y, mean ± SD	6.2 ± 1.0	8.1 ± 1.2
Initial MD, dB, mean ± SD	−7.44 ± 5.93	−7.28 ± 5.25
Final MD, dB, mean ± SD	−9.09 ± 6.70	−10.45 ± 6.07

Twenty-three progression regions were identified; see Figure 1. In general, the distribution of clustered test points tended to follow the mapping of the RNFL. As shown in Figure 2, mean absolute prediction errors decreased for all three approaches as more measurements (n) were included in the regression. Errors were significantly smaller for regression based on progression regions compared with PLR and NM sectors when n = 3:7 (P < 0.01) and n = 3:9 (P < 0.01), respectively. In addition, prediction errors associated with the structural filtering method were significantly higher than those with the regression based on progression regions (Fig. 3). 

Figure 1

 

The progression regions identified by clustering. Regions are illustrated in the 24-2 VF test pattern of a right eye. The blind spot is filled in black.

Figure 2

 

Mean prediction errors for predicting the 10th VF using initial the initial three to nine VFs. Prediction errors for regression based on PLR, NM sectors,¹⁹ and progression regions. The “n” represents the number of VFs used in the prediction for 10th VF. The double asterisks (**) indicate statistically significant difference at P < 0.01.

Figure 3

 

Mean prediction errors associated with the structural filtering method and those with the regression based on progression regions. The structural filtering method was obtained following a previous article.²⁹ The “n” represents the number of VFs used in the prediction for 10th VF. The double asterisks (**) indicate statistically significant difference at P < 0.01.

Discussion

In this study, progression regions were identified from a training dataset; test points in the 24-2 VFs of these training data were clustered based on their progression rates from the entire series of VFs (n = 10), which spanned approximately 6 years. As a result, 23 VF regions were obtained using the HOPACH-PAM clustering algorithm. These progression regions were then used to predict pointwise VF sensitivity in a separate validation dataset. This novel approach resulted in significantly smaller prediction errors than PLR or regression based on NM clusters that have been reported previously.¹⁸ In addition, prediction errors associated with the progression regions approach were far smaller than those associated with the structural filtering method.²⁹ 

Over the years, many different VF clusters have been suggested.^{12,15,17,18,32–35} Many of these studies were based on the assumed projection of RNFL bundles^32–35 or simply the cross-sectional relationship between test points in the VF.^12,15,17,18 One recent report also used the rate of deterioration and agglomerative hierarchical cluster analysis to derive VF sectors¹⁹; however, one problem with this clustering technique is the number of clusters is chosen arbitrarily.³⁶ Hierarchical techniques do not produce clusters per se, rather they produce trees (dendrograms) and clusters are generated by cutting the tree at a specific level.³⁷ In the research by Nouri-Mahdavi et al.,¹⁹ the optimum number of clusters was decided with a somewhat arbitrary cutoff level (the similarity between every pair of clusters had to be greater than 0.7 based on Pearson's correlation). However, if a different cutoff was applied, a completely different set of VF clusters would be generated. 

In the current study, VF clusters were derived using an automated and objective method (using the silhouette width index) and, as a result, a robust set of clusters was obtained. Interestingly, the obtained regions approximately followed the distribution of the RNFL and no clusters crossed the horizontal meridian. In addition, areas in which typical glaucomatous damage usually occur, such as the nasal step and Bjerrum scotoma, were clustered together independently from other regions. Furthermore, central test points, such as regions 12 and 15 (see Fig. 1), were composed of only a single test point; this result is in agreement with previous reports that suggested that these areas tend to progress independently from other test locations,^38,39 probably because of the rich RNFL distribution. Supporting this, in our previous article,²² 142 VF test points from the 30-2 and 10-2 VFs were analyzed together to obtain 67 VF clusters; as a result, 38 sectors were located outside the 10-2 VF, whereas 29 sectors were located inside the 10-2 VF. 

In this study, prediction errors based on regression using progression regions were smaller than those with PLR, especially when only a small number of VFs were used in the prediction. Pointwise VF test-retest variability will be considerably higher than sectorwise test-retest variability,¹⁵ in particular in areas in which glaucomatous deterioration exist.¹³ Reflecting this, prediction accuracy was poor with PLR unless a sufficient number (approximately seven) of VFs were used in the regression. This finding is in agreement with previous studies that suggested five⁴⁰ or eight VFs were necessary for good accuracy, but sometimes higher.^41,42 Thus, it is clinically useful to analyze VF results in small regions rather than at each point to improve prediction accuracy. Furthermore, prediction errors were smaller using progression regions compared with the larger NM sectors (10 or 6 sectors) reported previously¹⁹; this may be because early focal progression may be masked when the regional average is calculated using larger sectors. Prediction accuracy is therefore a balance between the variability of pointwise VF sensitivity and the masking effect of taking the average of large sectors.¹⁹ 

Garway-Heath et al.³⁵ reported a structure-function map by comparing fundus photography and VF tests whereby the corresponding angle on the optic disc was identified for each VF test point. In general, our progression regions roughly followed this structure-function map; however, the progression regions also appeared to be influenced by their distance from the blind spot. For example, test points (x coordinate = −9, y coordinate = −15) and (x coordinate = 3, y coordinate = −15) correspond to 81 and 80 degrees above the papillo-macular bundle line on the optic disc in Garway-Heath's mapping,³⁵ respectively, but these test points belonged to different regions in the current results. This is probably because RNFL damage is influenced not only by the angle portion on the optic disc, but also by the length of the RNFL. It is controversial whether long RNFL runs at a different anatomical depth in the retina⁴³ and penetrates the optic disc at different eccentricity from short RNFL.⁴⁴ Nonetheless, it is apparent that the deterioration of long and short RNFL bundles do not always occur at the same time. For example, the nasal step and Bjerrum scotoma often develop independently, to some extent at least, despite belonging to adjacent RNFL bundles. 

A possible caveat for the clinical use of the current results is that sectorwise averages and predictions are not readily available in the clinical setting. It would therefore be clinically beneficial to develop support tools/software to analyze VF progression, as introduced in this study, similarly to other tools already used in glaucoma management, such as the VF progression analysis tool, PROGRESSOR (Medisoft, Inc., London, UK).⁴⁵ Indeed, it has been shown that using an analysis tool such as PROGRESSOR improves clinicians' decisions regarding VF progression.⁴⁶ A further study should be carried out to compare the prediction accuracy with existing techniques, such as in.^47–49 Previous reports have suggested varying the definition of a “progressive VF” to improve the accuracy of progression detection, such as using two VFs for confirmation (where progression is confirmed in more than two test points),⁵⁰ or by applying the binomial test on PLR results.⁵¹ A future study should be carried out to compare the clinical usefulness of the current approach with these previously reported methods, such as in Gardiner and Crabb,⁵⁰ and Karakawa, et al.⁵¹ 

In conclusion, novel VF progression regions were developed based on the rate of VF deterioration. The VF clusters identified tended to follow the mapping of RNFL bundles. It was also suggested that prediction error can be reduced using these progression regions compared with PLR and regression based on NM sectors,¹⁹ particularly when the number of VFs used in the prediction was small. 

Acknowledgments

Supported in part by Japan Science and Technology Agency (JST) CREST and Grant 26462679 from the Ministry of Education, Culture, Sports, Science and Technology of Japan. 

Disclosure: K. Hirasawa, None; H. Murata, None; H. Hirasawa, None; C. Mayama, None; R. Asaoka, None