**Purpose**:
To investigate a novel approach for structure-function modeling in glaucoma to improve visual field testing in the macula.

**Methods**:
We acquired data from the macular region in 20 healthy eyes and 31 with central glaucomatous damage. Optical coherence tomography (OCT) scans were used to estimate the local macular ganglion cell density. Perimetry was performed with a fundus-tracking device using a 10-2 grid. OCT scans were matched to the retinal image from the fundus perimeter to accurately map the tested locations onto the structural damage. Binary responses from the subjects to all presented stimuli were used to calculate the structure-function model used to generate prior distributions for a ZEST (Zippy Estimation by Sequential Testing) Bayesian strategy. We used simulations based on structural and functional data acquired from an independent dataset of 20 glaucoma patients to compare the performance of this new strategy, structural macular ZEST (MacS-ZEST), with a standard ZEST.

**Results**:
Compared to the standard ZEST, MacS-ZEST reduced the number of presentations by 13% in reliable simulated subjects and 14% with higher rates (≥20%) of false positive or false negative errors. Reduction in mean absolute error was not present for reliable subjects but was gradually more important with unreliable responses (≥10% at 30% error rate).

**Conclusions**:
Binary responses can be modeled to incorporate detailed structural information from macular OCT into visual field testing, improving overall speed and accuracy in poor responders.

**Translational Relevance**:
Structural information can improve speed and reliability for macular testing in glaucoma practice.

^{1}and measurement variability abounds. Better examination procedures designed to reduce this measurement variability would have a clinical impact.

^{2}

^{1,3}Bayesian algorithms, such as Swedish Interactive Threshold Algorithm (SITA),

^{3,4}Quantile Estimation After Supervised Training (QUEST),

^{5}or Zippy Estimation by Sequential Testing (ZEST),

^{6}use subject's response to iteratively update a prior distribution of probable threshold values. The algorithm stops when the responses from the subject are deemed sufficient to reliably estimate the threshold, reaching a stopping criterion.

^{7}Such an approach has allowed fast estimation of sensitivity values with minimal to no loss in accuracy.

^{3}Most importantly, it makes it easier to introduce additional information as prior knowledge in the testing procedure, for example by modifying the starting threshold distribution.

^{8}described a method to include such information in the ZEST strategy and showed, through simulations, the effect of the accuracy of the structural predictions on the final results of the strategy. Later, Ganeshrao et al.

^{9}used a decision tree approach to inform the testing strategy with structural measurements from circumpapillary retinal nerve fiber layer (CP-RNFL) OCT scans and tested the improvements via simulations on a 24-2 grid. Despite this, the proposed model was mostly limited by the fact that CP-RNFL scans provide circular measurements of the RNFL thickness around the optic nerve head and do not quantify the loss of tissue at each tested location; thus, the damage could be mapped only to whole sectors of the visual field. This is especially limiting when it comes to testing the macular region, where much more detailed structural features can be measured.

^{10,11}Yet, structural guidance for perimetric testing could be greatly beneficial for patients who undergo macular assessment, especially when they suffer from advanced damage and poor fixation and are tested with denser grids. For example, in the Humphrey Visual Field Analyzer (Zeiss Meditec, Dublin, CA), a typical 10-2 grid tests 68 locations, as opposed to the 54 in the 24-2 grid.

^{11–13}Although the actual improvement in diagnostic ability of dense macular grids still needs to be clarified,

^{14}the importance of precise macular testing is being increasingly recognized, especially considering that a typical 24-2 grid does not allow for accurate detection of macular damage.

^{12,15}

^{11}showed how the combination of detailed structural and functional information in the macular region can help identify features of glaucoma damage, such as specific areas more vulnerable to damage, and interindividual variations in the anatomy that could affect the clinical evaluation. In this work, we aimed at using the detailed two-dimensional structural information provided by macular SD-OCT scans to build a structure-function model for the macula that could be easily employed to inform perimetric testing. To improve the accuracy of our model, we used a fundus perimeter employing an implementation of the ZEST as the standard testing strategy and equipped with scanning laser ophthalmoscopy (SLO) tracking to acquire measurements from healthy subjects and patients with glaucoma.

^{16–18}This allowed precise localization of the stimuli on the structural maps. Furthermore, instead of using the final thresholds from the tests, we built our model based on subjects' binary responses (seen or not seen) to each stimulus projected during the test. This allowed the estimation of probability of seeing (POS) curves based on the structural damage. Such curves were then used to calculate structurally based prior distributions for a modified structural ZEST (MacS-ZEST). We also took advantage of a standard method developed by Raza and Hood

^{10}to quantify the structural damage in terms of estimated loss of ganglion cells. Finally, we tested the improvements in accuracy and speed via simulations, comparing our novel strategy with a standard implementation of the ZEST.

^{19}

*P*< 0.05). For all patients, the presence of conditions other than glaucoma that could have caused central visual field defects (including retinal or neurologic disease, cataract, or significant media opacities) were evaluated and excluded.

^{10}This method employs the histologic ganglion cell density (GCD) map provided by Curcio and Allen

^{20}from normal subjects and a normative thickness map of the GCL. These two maps are combined to obtain a volumetric GCD map that can later be used to convert any given GCL thickness into an estimated GCC. The normative thickness profile was obtained by averaging both eyes of 35 normal subjects for an independent study published previously.

^{21}The transformation of thickness into GCD was meant to account for the normal decrease of GC with eccentricity, helping to reduce the floor effect. In fact, the same structural thickness at different eccentricities would correspond to different densities. Moreover, this was meant to make our structure-function model comparable with other approaches that have related the sensitivity to the GCD.

^{22}

^{16,23}The determination of the PRL consists of a 10-second fixation trial, during which time the device maps the part of the retina used by the subject for fixation. These subjects were tested with the standard ZEST strategy implemented in the device.

^{24}as implemented in MATLAB. Projective transformation can account for linear distortions needed to match the images from the two devices, but they may converge to local minima, giving incorrect solutions. Therefore, all results were visually inspected (GM and DA) to ensure correct alignment. This provided a spatial transformation that could be used to map the coordinates of the tested locations onto the structural SD-OCT thickness map and hence on the estimated GCC map.

^{25,26}was then used to compensate for lateral displacement of ganglion cells. As this model was applied to all vertices of the tile, the squares were radially displaced and distorted (Fig. 2). Furthermore, since the Drasdo model is based on anatomical features, we applied the deformation taking the actual fovea, determined by the OCT scans, as the center of the displacement, which might not coincide with the center of the testing grid, resulting in asymmetric displacements of the tiles (Fig. 2). The local GCD was estimated by calculating the total estimated GCC within the deformed tile and dividing this number by its area.

_{10}(GCD), and the eccentricity and the intensity of each projection. Interactions between the intensity of the projection, the log

_{10}(GCC), and the eccentricity allowed for a change in slope and intercept of the logistic model, as these three parameters varied. An example of the resulting curves at different GCD (without considering eccentricity) is depicted in Figure 3. These logistic curves can be interpreted as estimated structural POS curves. The change in location along the horizontal axis (stimulus intensity) and the slope of the curve model the expected probability of response and its variability, respectively. The structure-function model was built and fitted in the software R (R Project for Statistical Computing, Vienna, Austria).

^{19}Such a strategy was used for comparisons in the simulations. Briefly, the prior distribution for each tested location was a combination (in a proportion of 4:1) of an empirical distribution of normal and abnormal thresholds.The mean of the prior distribution is used as the intensity of the first projection. The prior distribution is then multiplied by a likelihood function centered on the intensity of the projection to produce a posterior distribution. The function could be increasing (for a seen) or decreasing (for a not seen). The posterior distribution is finally scaled and used as the prior distribution for the next iteration. In our work, we used the same starting prior distribution and likelihood function as in Turpin et al.,

^{19}varying the mode of the normal peak according to the age of the subject. The model for the macular sensitivities was estimated using a large collection of visual field tests from 444 healthy subjects recruited for the validation study for the Compass.

^{18}These subjects were examined with a grid that contained all the points of a 24-2 grid with 12 additional macular points. A mixed model with eccentricity and age as predictors was used to estimate the normal sensitivity for the tested locations.

^{27}to provide open access to our algorithm for testing in a reproducible environment.

^{28}as implemented in the OPI package

^{27}to simulate subjects' response. This formula combines responses from glaucoma and healthy subjects and varies the slope of the sampling frequency of seeing function of the response according to the input threshold (see details in the Appendix). It also allows for arbitrary rates of false positive (FP) errors, indicating the rate at which the simulated patient is responding when no stimulus is seen, and false negative (FN) errors, indicating the rate at which the simulated patient is not responding, even when a stimulus is perceived. First, we simulated responses from reliable subjects (FP 3%; FN 3%). We then simulated increasing rates of either FP or FN errors (10%, 20%, and 30%), while keeping the other parameter fixed at 3% (either FN or FP errors, respectively). The simulation was stopped when each location reached a standard deviation (SD) in the posterior distribution <1.5 dB, a widely used limit for dynamic termination.

^{8,29,30}For both strategies, the likelihood function was a Gaussian cumulative distribution function (SD = 1) centered on the tested threshold, whose maximum and minimum values were capped at 0.03 and −0.03, respectively. In order to have realistic inputs to the simulations accurately reflecting the true relationship between structure and function in a real patient, we used data from the 20 glaucoma subjects in group 2, tested three times with a 4-2 strategy at eight locations. Results from the three tests were averaged to obtain accurate estimates of the real threshold. These results also represented an independent dataset set for validation of the MacS-ZEST. The averaging process and the alignment for the three tests with the fundus tracking was meant to improve the reliability of the estimated threshold.

^{31}Percentage reductions in MAE and number of presentations were evaluated using generalized linear mixed models with a log link function. This modeling approach provided estimates and standard errors of percentage change in the linear scale, while the log link function accounted for the skewed distributions of positive values (MAE and counts for the number of presentations).

*P*values are not reported as hypothesis testing is meaningless in a context where arbitrary increase in the number of simulations could yield statistical significance at any given α value. Bias and variability are also reported. Bias was calculated as the mean error of each estimated threshold from the input threshold, while variability was calculated as the mean absolute deviation (MAD) from the mean estimated threshold. All statistical calculations were performed in R.

*P*= 0.89) in MD between the two groups of glaucoma patients, indicating a similar severity in visual field loss.

^{10,32}our estimates suffered from a significant upward bias at lower thresholds due to the floor effect in the structural measurement. Therefore, the effective range of accurate prediction of the model was limited to values above 20 dB.

*n*= 20; 500 simulations per subject on eight locations). Error plots in Figure 6 show a comparison at different input thresholds of the 95% and 99% error limits for reliable responses, 20% FP errors, and 20% FN errors. In general, MacS-ZEST offered a better control over extreme errors, especially on the 99% error limits, in unreliable patients. The MacS-ZEST showed a slight upward increase in the average error at 15 dB (1.3 dB on average). This sensitivity value was obtained from the average of tests at one location in a single patient whose average sensitivity was lower than predicted by the structural model. This location was at the edge of a structurally damaged region (see Supplementary Fig. S1).

^{10}This is clear in Figure 4, where the estimates for thresholds below 20 dB are clearly less accurate and positively biased. Although this was accounted for by using varying weights for the abnormal curve when building the prior distributions in order to reduce the risk of bias at damaged locations, the resulting structural prior distributions were weakly informative for low sensitivities. Nevertheless, structural information was able to confine the threshold estimates to lower values, reducing the 95% and 99% limits of errors with high rates of FP errors. On the other hand, convergence for healthier locations was faster due to the more precise estimate from structural measurements. For these locations, the MacS-ZEST was also protective against increased rates of FN errors.

^{8}proposed a theoretical method to modify ZEST prior distributions according to educated guesses from hypothetical structural measurements by changing the mode of the normal component of the general ZEST prior distribution. However, their work focused more on assessing the theoretical benefits on precision and number of presentations, as well as the bias of the determined sensitivities for increasing differences of the structural estimate from the real threshold. Hence, they did not use an explicit structure- function model and actual structural data in their analysis. Other work from the same group

^{9}used RNFL SD-OCT data and the structure-function model developed by Hood and Kardon

^{32}to predict the expected threshold. They then used a mix of decision trees and ZEST strategy to guide the initial projections using the structural information. They found an important reduction in the number of presentations but not a large reduction in the error with the structural strategy. This is in agreement with our results. On the other hand, the actual increase in precision is much less than what was predicted by Dennis et al.,

^{8}where the effective range of structural predictions was assumed to span the whole domain of tested sensitivities. However, this is largely prevented with real structure-function models (and in our case) by the bottom floor effect.

^{31}This modeling approach could have potential further use, for example, to build structure-function curves by tracing the 0.5 level curves in the multivariate logistic model. One limitation is that slope estimates are biased when estimated using presentations from sequential adaptive strategies,

^{33}but this is unlikely to make any substantial differences for the applications we are proposing in a clinical scenario. However, we did not use the information on the slope derived from the structure-function model in the testing procedure. Future work might expand the current methodology to include the uncertainty of the threshold estimate derived from the structural data.

^{34}In addition, shifts and instability in fixation have been shown to arise in subjects with macular damage of the ganglion cells, for example, due to optic neuritis

^{35}or in glaucoma.

^{36}This is particularly relevant in our analysis as we selected patients for whom the central 10° were damaged. Furthermore, we could match the fundus images from the perimeter and the SD-OCT device, also equipped with fundus tracking, increasing the precision in the localization of the tested location onto the structural maps and with respect to the anatomical fovea. This had important consequences for the correct application of the Drasdo model

^{25,26}to correct for radial displacement of ganglion cells.

^{37}rather than for individual locations.

^{9}

^{8}and were evident at one of the locations from one of the glaucoma patients tested with the small grid, resulting in an upward spike in error at 15 dB. This location was at the edge of a damaged region (Supplementary Fig. S1), and the structural estimate was higher than the real threshold. Several different factors could have influenced this phenomenon, ranging from difficulties in estimating sensitivity from local measurement on borderline locations to the fact that population estimates for RGC displacements might be inaccurate when high spatial precision is needed.

^{25}One potential practical solution to this issue would be to account for gradient in the structural damage in the area surrounding the tested location, for example, adapting the size of the area used to calculate to GCD. Moreover, customized calculations for RGC displacements might help reduce the error and improve the precision of the structural estimate.

^{25}

^{38–40}The main future development will be the practical implementation of the strategy to test real patients evaluating test-retest variability, testing time, and offset in sensitivities. This is likely to have an impact in the clinical practice, facilitating the more extensive use of accurate macular tests for early detection of glaucoma, especially when employed for precise monitoring of patients with ocular hypertension. Additionally, patients with central visual field damage could benefit from a faster test that might allow, by incorporating structural changes, the early detection of macular damage progression, prompting timely therapeutic interventions.

**G. Montesano**, None;

**L.M. Rossetti**, CenterVue (C);

**D. Allegrini**, None;

**M.R. Romano**, None;

**D.P. Crabb**, CenterVue (C).

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*age*is the age of the subject (years),

*int*is the intensity of the projection,

*ecc*is the eccentricity of the stimulated location, and GCD is the local ganglion cell density. Interaction terms allow varying slopes of the POS curve at different eccentricities and local GCD values. The eccentricity was kept as a separate predictor in the final model as it had significant effect. This indicates that the transformation of thickness values into GCD alone was not sufficient to fully compensate for the distance from the fovea. The random part of the model included a random factor grouping projections on the same location (

*γ*) nested within a random factor for the subject (

_{ij}*ν*). A random slope was also used for the intensity of the projection (

_{i}*θ*). FP and FN errors were not included in the model. However, only 3/20 healthy subjects and 5/31 subjects with glaucoma (group 1) had a FP rate >0 as determined by Compass (no FN errors were recorded), and these errors are not individually tracked in the test history. The coefficients for the fixed effects may be seen in Table A1.

_{ij}^{28}as implemented in the OPI R package.

^{27}The response to a stimulus is sampled from a frequency of seeing function with the following formula:

*th*is the sampled threshold,

*fpr*and

*fnr*indicate the FP and FN rate, respectively, and

*pnorm*indicates the cumulative distribution function of a Gaussian function with mean

*tt*(the true input threshold) and variance

*Var*, evaluated at the intensity

*x*.

*Var*was modified according to the input threshold with the formula

*cap*is the maximum allowed variability, set at 6 dB, and

*A*and

*B*were selected from Table 1 in Henson et al.

^{28}to provide a combined response of glaucoma and normal subjects. Specifically,

*A*= −0.081 and

*B*= 3.27.