**Purpose:**:
To analyze static visual field sensitivity with topographic models of the hill of vision (HOV), and to characterize several visual function indices derived from the HOV volume.

**Methods:**:
A software application, Visual Field Modeling and Analysis (VFMA), was developed for static perimetry data visualization and analysis. Three-dimensional HOV models were generated for 16 healthy subjects and 82 retinitis pigmentosa patients. Volumetric visual function indices, which are measures of quantity and comparable regardless of perimeter test pattern, were investigated. Cross-validation, reliability, and cross-sectional analyses were performed to assess this methodology and compare the volumetric indices to conventional mean sensitivity and mean deviation. Floor effects were evaluated by computer simulation.

**Results:**:
Cross-validation yielded an overall *R*^{2} of 0.68 and index of agreement of 0.89, which were consistent among subject groups, indicating good accuracy. Volumetric and conventional indices were comparable in terms of test–retest variability and discriminability among subject groups. Simulated floor effects did not negatively impact the repeatability of any index, but large floor changes altered the discriminability for regional volumetric indices.

**Conclusions:**:
VFMA is an effective tool for clinical and research analyses of static perimetry data. Topographic models of the HOV aid the visualization of field defects, and topographically derived indices quantify the magnitude and extent of visual field sensitivity.

**Translational Relevance:**:
VFMA assists with the interpretation of visual field data from any perimetric device and any test location pattern. Topographic models and volumetric indices are suitable for diagnosis, monitoring of field loss, patient counseling, and endpoints in therapeutic trials.

^{1}and retinal degenerations, such as retinitis pigmentosa (RP)

^{2–4}for screening, disease detection, and classification, monitoring for progression, correlation with activities of daily life, and, more recently, structure-function studies. Because the earliest and most disabling features of many forms of inherited retinal degeneration involve visual field loss, most often in the periphery, testing the entire visual field is crucial for the evaluation and monitoring of these patients. The most common means of full-field testing is kinetic perimetry, which can efficiently locate the borders of seeing areas. In comparison, static perimetry can better define small sensitivity levels, and detect subtle sensitivity gradients and emerging scotomas. In the absence of a fast, full-thresholding algorithm, previous full-field static perimetry studies have used the standard full-threshold staircase strategy and limited the number of test points to avoid long exam durations.

^{5,6}A new, fast full-threshold algorithm, GATE-i

^{7}has enabled full-field static perimetry with more test locations and practical test durations (Weleber RG, et al.

*IOVS*2009;50:ARVO E-Abstract 3813; Schiefer U, et al.

*IOVS*2009;50:ARVO E-Abstract 5354).

^{8}Other mathematical models have been developed to study the effects on the visual field of neurological disease, trauma, retinal degenerations,

^{9,10}and glaucoma.

^{11,12}

^{13}which are based on the average sensitivity value. These indices are appropriate for rectilinear grids with uniform spacing; however, for grids with radial patterns, central condensation, and unequal spacing, these indices become weighted averages. The weighting biases the indices to the regions of higher sampling density, which alters their interpretation and limits their comparison among grids with different sampling patterns. Furthermore, these indices are global measures and can be insensitive to local spatial or regional behavior.

^{14}In this study, we perform topographic modeling, interactive visualization, and data distillation of visual field sensitivity data using a custom software application we developed called VFMA. VFMA renders 3-D surface models of the hill of vision (HOV)

^{15}and its defects, and also provides quantitative functional measures. We focus on several visual function indices that are derived from the HOV volume, or the volume beneath the sensitivity surface. These indices capture the visual field magnitude and extent at all states of disease without weighting bias, and are more meaningful when comparing exams acquired with different grids than indices based on simple averaging, such as MS and MD. The volumetric indices are conceptually similar to the kinetic visual field global volume,

^{16}but VFMA allows measurements from the entire visual field as well as specific regions of interest. Furthermore, VFMA provides contour analysis and comparisons with normative data, supports perimeter test grids of any size and arrangement, and performs peripheral field modeling with minimal cartographic distortion. In this study, we used VFMA to demonstrate the clinical use of HOV volume analysis in patients with retinitis pigmentosa.

^{17,18}We performed a post hoc analysis of floor effects and their impact on the visual function indices. Progressively larger MSVs were imposed on the perimetry data we collected to simulate the floor effects induced by perimeters with smaller maximum stimuli.

^{2}(31.5 apostilbs) background, the GATE-i strategy,

^{7}and Goldmann stimulus size V. Fixation was monitored by the technician during the entire testing session. Subjects were tested with the radially oriented, centrally condensed, binocularly symmetric grid pattern shown in Figure 1. This grid consisted of

*N*= 164 points spanning a solid angle, or angular footprint, of 3.69 steradians (sr). Measured threshold values from the Octopus 101 were exported and converted

_{t}^{19}to differential luminance sensitivity (DLS) in decibels (dB); for the Octopus 900, this conversion was performed automatically by the manufacturer's EyeSuite software. The quality of each subject's exam was assessed by a reliability factor (RF), defined as the percentage of total catch trials resulting in either a false-positive or false-negative. Any exam with an RF > 15% for normal subjects or 25% for patients was excluded; based on this criterion, three normal and nine patient exams were excluded.

^{20}of the perimetry data. The surface was constrained outside the subject's field-of-view by adding

*N*= 60 artificial points with zero sensitivity along a circle with radius 120°, as shown in Figure 1b. The set of points

_{z}*N*=

*N*+

_{t}*N*= 224, (

_{z}*x*,

_{i}*y*) is location of the

_{i}*i*

^{th}grid point in angular coordinates and

*z*is the corresponding DLS value in dB. The HOV surface model was defined at location (

_{i}*x*,

*y*) by where

*w*= [

*w*

_{x}w_{y}w_{0}

*w*

_{1}

*w*

_{2}…

*w*]

_{N}*and*

^{T}*d*= [

*x y*1 φ(‖

*x−x*‖)]

_{i}*are (*

^{T}*N*+3) × 1 vectors of weights and displacements, respectively.

^{21}Here, φ(

*r*) =

*r*

^{2}log(

*r*) is the infinitely differentiable TPS radial basis function, and

*x*= [

*x y*]

*and*

^{T}*x*= [

_{i}*x*]

_{i}y_{i}*are 2 × 1 coordinate vectors. The weight vector was found by solving the matrix equation*

^{T}*ij*

^{th}element of the

*N*×

*N*submatrix Φ is Φ

*= φ(‖*

_{ij}*x*−

_{i}*x*‖), and the

_{j}*i*

^{th}row of the

*N*× 3 submatrix

*C*is [1

*x*]. Once the weight vector

_{i}y_{i}*w*was calculated, the surface was interpolated at an arbitrary location (

*x*,

*y*) by updating

*d*with the location coordinates and evaluating Equation 1. In this study, we interpolated the data from each exam onto a dense 501 × 501 point rectilinear grid with 0.36° spacing along each dimension, which is outlined in Figure 1b.

*θ*is the co-latitude angle,

*ϕ*is the azimuth angle, and

*S*is the selection region defined by the user. Here,

*V*represents the volume of the solid defined by

*ẑ*(

*θ*,

*ϕ*) and

*ẑ*= 0 over the angular region

*S*. The VFMA calculates volumes by the midpoint integration rule wherein the finely interpolated surface is summed and scaled by the pixel extent.

^{22}

*S*is selected for investigation. When the selection region is the entire grid, the result is the total volume,

*V*

_{TOT}. In this study, we analyzed

*V*

_{TOT}and the central field volume,

*V*

_{30°}, defined by a setting the selection region

*S*to be a circle with a radius of 30° centered on the point of fixation. The footprints of these volumes are depicted in Figure 1. We also examined the normalized index

*V*

_{30°}/

*V*

_{TOT}, which approaches zero as central sensitivity is lost and approaches one as peripheral sensitivity is lost. The blind spot was not removed before surface fitting and calculation of these volumes.

*z*sensitivity values, an HOV model of the defect was created by replacing each

_{i}*z*value with the DLS difference between the subject and an age-adjusted normal.

_{i}^{23}The resulting interpolated surface depicts the visual field in defect space, as opposed to the native DLS space. The defect space surface models are three-dimensional, finely sampled analogues of total deviation plots, and are useful for quantifying patterns for field loss. For example, a scotoma appears as a recession in the DLS surface and as an elevated ridge in the defect surface. In defect space, the volume is where

*z*is the age-adjusted normative DLS. These volumes measure the net defect volume in the selection region. In this study, we analyzed the defect space volumes

_{n}*D*

_{TOT}and

*D*

_{30°}. For defect space measures, the blind spot was removed before volume calculation.

^{20,24}LOOCV measures the residual DLS error at each of the test locations by reinterpolating the surface using all data except that location, and then accumulates the errors from all test locations. The error for the

*k*

^{th}location is

*e*=

_{k}*ẑ*(

_{k}*x*,

_{k}*y*) –

_{k}*z*, where

_{k}*ẑ*

*(*

_{k}*x*,

_{k}*y*) is the HOV surface interpolated at location (

_{k}*x*,

_{k}*y*) using the set of points

_{k}*R*

^{2})

^{24}and the index of agreement (

*d*),

^{25,26}as given by and

*R*

^{2}and

*d*. The

*R*

^{2}, which specifies the proportion of the data variation captured by the topographic model, is commonly used as a goodness-of-fit metric. The

*d*is an alternative metric ranging from 0.0 (no agreement between the model and observation) and 1.0 (perfect agreement).

*V*

_{TOT},

*V*

_{30°},

*V*

_{30°}/

*V*

_{TOT},

*D*

_{TOT}, and

*D*

_{30°}from VFMA as well as the conventional indices MS and MD. Repeatability performance was assessed by the coefficient of variation (CV), where CV = σ/μ, σ is the within-subject deviation estimated via 1-way ANOVA and μ is the corresponding mean. We also estimated the repeatability coefficient (RC) =

*N*test locations in each perimetry exam in this study, and to all age-adjusted normative sensitivity values. With this approach, the dynamic range of the sensitivity data monotonically decreases as

_{t}*M*increases, although the DLS values retain the same reference point. We simulated MSVs of

*M*= 4, 6, 9, 12, 15, and 18 dB. For each simulated MSV, all visual function indices were recalculated. and the LOOCV and repeatability analyses were repeated. The cross-sectional analyses were repeated for

*M*= 4 and 18 dB.

*V*

_{TOT},

*V*

_{30°},

*D*

_{TOT},

*D*

_{30°}, and the ratio

*V*

_{30°}/

*V*

_{TOT}are presented in Figures 2 and 3, and a volumetric measurement of the ring scotoma is presented for the field in Figure 3. The 3-D topographic representations generated by VFMA show the HOV contours and enhance the subtle variations in the visual fields.

*d*values indicating high accuracy, and also good consistency with similar

*R*

^{2}values and similar

*d*values in each subject group. By comparison, the performance of the NN interpolator was lower and less consistent among groups. The

*R*

^{2}and

*d*values from the VFMA TPS interpolator were significantly larger than those from the NN interpolator, overall (

*P*< 0.001) and within each subject group (

*P*< 0.001 in each case). The standard deviations for

*R*

^{2}and

*d*were largest in the RP group, which is likely a reflection of the diversity of visual field patterns within this group.

*R*

^{2}and

*d*values changed only slightly, getting worse in the RP and PCRP groups and better in the normal group. The improvement in the normal group is due to the increased uniformity and spatial autocorrelation of the far peripheral fields where the floor effects are more likely to occur. The worsening in the patient groups is a result of the floor effects causing abruptly increased sensitivity values at isolated test locations, which reduces the local autocorrelation and the accuracy of any neighborhood-based interpolation scheme. At every simulated MSV, the

*R*

^{2}and

*d*values from VFMA remained significantly larger than the NN interpolator.

*V*

_{TOT}and MS, and also for

*D*

_{TOT}and MD, although among normals the

*D*

_{TOT}coefficient was somewhat larger than that for MD. For the indices in DLS space, the variation was larger among RP patients than normal subjects. The CVs of the regional volumetric indices

*V*

_{30°}and

*D*

_{30°}were similar to their global counterparts

*V*

_{TOT}and

*D*

_{TOT}. The largest CVs were found in the defect space indices of normal subjects; because normal subjects have negligible field loss, the visual function indices in defect space had mean values near zero which elevated the CVs. Similarly, the normalized index

*V*

_{30°}/

*V*

_{TOT}in RP patients had a mean near zero and, consequently, a relatively large CV. For the defect space measures of RP subjects, however, the CVs remained small. Although the repeatability coefficients in Table 3 are not comparable across indices, they provide useful summary information for a given index regarding how similar two observations on the same individual are likely to be.

*V*

_{TOT},

*V*

_{30°}and MS grew with MSV while the defect–space indices

*D*

_{TOT},

*D*

_{30°}and MD and the normalized index

*V*

_{30°}/

*V*

_{TOT}correspondingly shrank (Supplementary Tables S9 and S10). These indices changed considerably more in the RP group due to the greater impact of severe floor effects in regions with visual field loss. The within-subject standard deviations (Supplementary Tables S11 and S12) tended to decrease because the dynamic range was being reduced as the simulated MSV became larger. Accordingly, the CVs (Supplementary Tables S7 and S8) generally improved for the indices in DLS space. The changes in defect-space indices were less predictable, but were quite small for the range of MSVs considered.

*V*

_{30°}/

*V*

_{TOT}values, indicating a loss of peripheral field. For the PCRP group,

*V*

_{30°}/

*V*

_{TOT}was smaller compared to the RP and normal groups, while

*V*

_{TOT}was larger than RP and smaller than normal. This corresponds to more relative central field loss and some peripheral field loss, although not as much as seen in the RP group.

*P*values comparing these groups tended to be larger. This impact is most apparent in comparisons of the regional volumetric measures

*V*

_{30°}and

*D*

_{30°}, since these are the indices for which the original

*P*values in Table 4 were substantially larger than 0.0001. When more severe floor effects are simulated at an MSV of 18 dB, as shown in Supplementary Table S16, the mean differences between groups decreased for all indices, which impacted the statistical comparisons between the subject groups. Again, this is most apparent when comparing

*V*

_{30°}and

*D*

_{30°}between the RP and PCRP groups.

*RP1*

^{8}and are in agreement with a prior report of the central visual field in RP being more robust and intact compared to that of the periphery.

^{4}The visual field loss with a pericentral distribution seen in Figure 3 is consistent with autosomal dominant RP associated with a heterozygous mutation of

*NR2E3*

^{27}and an autosomal dominant mutation of

*TOPORS*.

^{28,29}

*V*

_{TOT}is a function of the grid extent and, because DLS values are non-negative, is monotonically nondecreasing with grid solid angle. Regional indices like

*V*

_{30°}have a fixed topographic footprint and do not have this dependency. Depending on the grid extent, the field may be truncated, especially in normal subjects. Compared to stimulus size III, size V has been shown to have a larger dynamic range

^{30}and smaller variability

^{31}in glaucoma patients. In RP patients, use of stimulus size V yields more seeing locations whereas use of size III has been suggested for early disease to allow access to statistical analyses of progression that are not available for size V.

^{32}Based on the concept of spatial summation, we predict that stimulus size is positively correlated with volume. Studies are currently underway to characterize the sensitivity of the volumetric measures to these factors.

*V*

_{30°}and

*D*

_{30°}to discriminate between the RP and PCRP groups. This result is somewhat expected, since increasing the MSV will artificially improve the sensitivities in the central fields of typical RP and PCRP patients, making them appear more similar and more difficult to discriminate.

^{33}Demonstrating the effects of visual field loss on how one sees, monocularly and with binocular viewing, has become an integral part of the care of patients with retinitis pigmentosa and allied disorders at the Oregon Retinal Degeneration Center. Such visual presentations are very helpful when explaining how field loss creates impairment, the concept of compensation of seeing field between the two eyes, and in discussing issues surrounding driving and disability.

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