June 2016
Volume 57, Issue 7
Open Access
Multidisciplinary Ophthalmic Imaging  |   June 2016
Evaluating Descriptive Metrics of the Human Cone Mosaic
Author Affiliations & Notes
  • Robert F. Cooper
    Biomedical Engineering, Marquette University, Milwaukee, Wisconsin, United States
  • Melissa A. Wilk
    Cell Biology, Neurobiology and Anatomy, Medical College of Wisconsin, Milwaukee, Wisconsin, United States
  • Sergey Tarima
    Biostatistics, Institute for Health and Society, Medical College of Wisconsin, Milwaukee, Wisconsin, United States
  • Joseph Carroll
    Biomedical Engineering, Marquette University, Milwaukee, Wisconsin, United States
    Cell Biology, Neurobiology and Anatomy, Medical College of Wisconsin, Milwaukee, Wisconsin, United States
    Biophysics, Medical College of Wisconsin, Milwaukee, Wisconsin, United States
    Ophthalmology, Medical College of Wisconsin, Milwaukee, Wisconsin, United States
  • Correspondence: Joseph Carroll, Eye Institute, Department of Ophthalmology, Medical College of Wisconsin, 925 N 87th Street, Milwaukee, WI 53226-0509, USA; jcarroll@mcw.edu
  • Footnotes
     RC's Current affiliation: Departments of Ophthalmology and Psychology, University of Pennsylvania, Philadelphia, Pennsylvania, United States.
Investigative Ophthalmology & Visual Science June 2016, Vol.57, 2992-3001. doi:10.1167/iovs.16-19072
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      Robert F. Cooper, Melissa A. Wilk, Sergey Tarima, Joseph Carroll; Evaluating Descriptive Metrics of the Human Cone Mosaic. Invest. Ophthalmol. Vis. Sci. 2016;57(7):2992-3001. doi: 10.1167/iovs.16-19072.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose: To evaluate how metrics used to describe the cone mosaic change in response to simulated photoreceptor undersampling (i.e., cell loss or misidentification).

Methods: Using an adaptive optics ophthalmoscope, we acquired images of the cone mosaic from the center of fixation to 10° along the temporal, superior, inferior, and nasal meridians in 20 healthy subjects. Regions of interest (n = 1780) were extracted at regular intervals along each meridian. Cone mosaic geometry was assessed using a variety of metrics − density, density recovery profile distance (DRPD), nearest neighbor distance (NND), intercell distance (ICD), farthest neighbor distance (FND), percentage of six-sided Voronoi cells, nearest neighbor regularity (NNR), number of neighbors regularity (NoNR), and Voronoi cell area regularity (VCAR). The “performance” of each metric was evaluated by determining the level of simulated loss necessary to obtain 80% statistical power.

Results: Of the metrics assessed, NND and DRPD were the least sensitive to undersampling, classifying mosaics that lost 50% of their coordinates as indistinguishable from normal. The NoNR was the most sensitive, detecting a significant deviation from normal with only a 10% cell loss.

Conclusions: The robustness of cone spacing metrics makes them unsuitable for reliably detecting small deviations from normal or for tracking small changes in the mosaic over time. In contrast, regularity metrics are more sensitive to diffuse loss and, therefore, better suited for detecting such changes, provided the fraction of misidentified cells is minimal. Combining metrics with a variety of sensitivities may provide a more complete picture of the integrity of the photoreceptor mosaic.

Adaptive optics (AO) enhanced ophthalmoscopes permit noninvasive visualization of the human retina with cellular resolution. Imaging of the cone,15 rod,68 and retinal pigment epithelium (RPE)913 mosaics has been demonstrated in healthy and diseased eyes. While pathology can often be quite striking when imaged with single-cell resolution, the ability to use these images to detect subtle changes relies on the ability to extract quantitative information about the mosaic of interest. This process often involves assessing metrics derived from the cell locations within an image. Metrics such as density,1424 spacing,12,14,15,23,2531 and regularity19,3234 are frequently used to characterize the cone mosaic. Despite their broad use, there has been minimal evaluation of the ability of these metrics to detect disruptions of the photoreceptor mosaic. Such testing is needed to objectively assess the strengths and weaknesses of these metrics in evaluating retinal mosaics, especially with the growing demand to image the photoreceptor mosaic over time (either following therapeutic intervention or to monitor disease progression). 
One of the more significant factors known to affect metrics used to describe the cone mosaic is undersampling. Undersampling can come from two sources: cell misidentification or cell loss.35,36 First, algorithms used to automatically or semiautomatically identify cells in retinal mosaics have some nonnegligible errors that can vary substantially with image quality.14,15,34 As most metrics rely on cell identification rather than the retinal image itself (though Cooper et al.37 uses a Fourier transform-derived spacing extracted directly from the image), the error introduced by this undersampling is an inherent feature of most current AO analyses. How this source of undersampling affects a given metric provides a direct measure of its “robustness.” Second, various retinal diseases result in the actual loss of cells from the mosaic.21,22,25,2933,3842 How a metric changes in response to known amounts of cell loss defines its “sensitivity.” As there is a wide range of metrics used to assess retinal mosaics, it is critical to characterize how each metric is affected by undersampling: an ideal metric should be sensitive enough to detect cell loss, but robust enough to not be affected by small errors in cell identification. 
Due in part to the optical waveguiding properties of photoreceptors, the cone mosaic can be imaged with particular ease. In fact, the cone mosaic can be resolved in some individuals even without using AO.4346 Moreover, cone photoreceptors drive the majority of our visual function and are affected in a variety of retinal diseases. Thus, there is continued interest in the development and validation of metrics for detecting disruptions or changes in the cone mosaic. Following the approach developed by Cook,35 in which he compared versions of the same mosaic that had different amounts of undersampling, we examined the performance of a number of metrics by applying known amounts of diffuse cell loss (i.e., undersampling) to photoreceptor mosaic coordinates derived from images of the human cone mosaic. This pattern of cone mosaic disruption has been observed in conditions such as retinitis pigmentosa,25,41 cone-rod dystrophy,25 red-green color vision deficiency,38 and acute macular neuroretinopathy.22 In addition, this type of undersampling approximates the expected pattern that might occur as a result of errors in manual or automated cell detection. The data presented here provide a useful framework for understanding the strengths and limitations of these metrics, and highlight the important “philosophical” issue of whether the insensitivity (or robustness) of a metric to diffuse cell loss represents a strength or a weakness when trying to determine whether a given cone mosaic is normal or abnormal. 
Methods
Human Subjects
This research followed the tenets of the Declaration of Helsinki, and was approved by the institutional review boards at the Medical College of Wisconsin (Milwaukee, WI, USA) and Marquette University (Milwaukee, WI, USA). Twenty subjects with normal trichromatic vision were recruited for this study (median age: 23.5, range, 9–67 years; Supplementary Table S1). Subjects provided informed consent after the nature and possible consequences of the study were explained. Individuals with high myopia or hyperopia (>10 diopters [D]) were excluded from this study. Axial length measurements were obtained on all subjects using an IOL Master (Carl Zeiss Meditec, Dublin, CA, USA). To convert from image pixels to retinal distance (μm), we first acquired images of a Ronchi ruling positioned at the focal plane of a lens with a 19-mm focal length to determine the conversion between image pixels and degrees. An adjusted axial length method47 was then used to approximate the retinal magnification factor (in μm/degree) and convert to micrometer per pixel. 
Imaging the Human Photoreceptor Mosaic
The photoreceptor mosaic was imaged using an AO scanning light ophthalmoscope (AOSLO), where both confocal48 and nonconfocal split-detector1 imaging modalities were acquired simultaneously. Imaging was performed along the temporal, inferior, nasal, and superior meridians using a 790-nm superluminescent diode. Using a 1.0° field of view (FOV), each meridian was sampled every half degree from fixation out to 6°, and then every degree from 7° to 10°. Using a 1.5° FOV, each meridian was sampled every degree from fixation out to 10°. To correct for static intraframe distortion resulting from the sinusoidal motion of the resonant optical scanner, we estimated the distortion from images of a stationary Ronchi ruling and then resampled each frame over a grid of equally spaced pixels. Then, a reference frame was selected manually from within each image sequence for subsequent registration using custom software.49 Montages of overlapping split-detector and confocal images using both 1.0° and 1.5° FOVs were created semiautomatically using custom software. To simplify the process of montaging, custom software was created in MATLAB (Mathworks, Natick, MA, USA) that allows the user to rapidly screen which images should be included in a montage. After screening, the selected images were automatically placed in a corresponding Photoshop (Adobe, San Jose, CA, USA) file at a location extracted from the digitized image acquisition notes. Once the montage was “seeded” using this software, the user manually positioned the images within Photoshop to achieve a more accurate alignment. 
Analyzing the Cone Photoreceptor Mosaic
Because foveal cones could not be reliably resolved in all subjects, the location of peak foveal density was determined using a previously described method.50 First, cone coordinates were semiautomatically identified from a foveal montage using a previously described cell identification algorithm.15 Isodensity contour maps were generated from the resulting coordinates. Six contours (at 80%–93% of the peak cone density) were extracted from each map, and the center (x, y) position of each contour was averaged to provide an estimate of the location of peak foveal cone density within the foveal montage. 
Regions of interest (ROIs) were then extracted from each montage, relative to the location of peak foveal cone density, using custom software (Photoshop and MATLAB). The size of each ROI varied as a function of eccentricity, using published AOSLO-derived cone density data51 to estimate the area necessary to encompass approximately 100 cones at each ROI as described next. Using the minimum foveal cone density observed by Wilk et al.51 (84,000 cones/mm2), we set the area of ROIs at the location of peak foveal cone density to 37 × 37 μm. Due to the minimal change in cone density beyond 10°, we set the area of ROIs at and beyond 10° to 100 × 100 μm. We next fit an exponential function to these areas, establishing an eccentricity-to-ROI area relationship. We obtained ROIs at the foveal center, every 50 μm from 50- to 600-μm eccentricity, every 200 μm from 600- to 1600-μm eccentricity, and every 300 μm from 1600- to 3100-μm eccentricity. Within 500 μm of peak foveal cone density, ROIs were extracted from the confocal modality, while beyond 500 μm, ROIs were extracted from the split-detector modality due to superior cone contrast. When either blood vessels or seams between overlapping images occurred at a desired ROI sampling location, we adjusted the ROI's location to a nearby unobstructed area. To enable easier comparison, each ROI was binned based on the nearest sample location. Regions of interest within an eccentricity bin were then compared across all subjects. On average, the ROIs deviated from their bin location by 4.7 μm within 600 μm of the foveal center, and 67.4 μm beyond 600 μm from the foveal center. Cone coordinates were then semiautomatically identified within each confocal ROI,15 and manually identified within each split-detector ROI using custom software (Java 1.8; Oracle, Redwood City, CA, USA) by a single observer (RFC). 
Mitigation of Boundary Effects
All geometrical descriptors extracted from a discrete set of coordinates are subject to boundary effects at the ROI edges. The edge cells do not necessarily contribute all of their connected neighbors to a spacing measurement, or even all of the area that they encompass to a density measurement. To mitigate this effect, we used the Voronoi tessellation to establish which cell locations should be included for analysis. Only cones with their corresponding Voronoi cell fully contained within the ROI (i.e., “bound”) were considered for the metric calculations. 
Descriptive Metrics of the Cone Mosaic
The cone coordinates for each ROI were analyzed using the following spacing and regularity metrics (regularity metrics, as the name implies, capture the variation of a particular metric over an ROI): 
Density.
As mentioned above, Voronoi tessellation of the cone coordinates was used to define the bound Voronoi cells in a given ROI. Density was defined as the ratio of the number of bound Voronoi cells in an ROI to the summed area of the bound Voronoi cells. The shaded Voronoi polygons in Figure 1 represent bound Voronoi cells. 
Figure 1
 
A schematic of a hexagonally arranged patch of cones illustrating the relationship between the distance measurements used in this study. A single cone (red circle) and its six closest neighbors (open circles) are highlighted for clarity. The NND is defined as the distance from a given cone to its closest neighbor (orange dashed line). The FND is defined as the distance from each cone to its most distant neighbor (blue dashed line), and ICD is defined as the average distance between a cone and all of its neighbors (dashed lines). In order to mitigate boundary effects, only cones with bound Voronoi regions (shaded region) are included when calculating each metric. The regularity of each of these metrics (M) is defined as the mean (μM) of the metric for all cones with bounded Voronoi cells, divided by the metric's SD (σM).
Figure 1
 
A schematic of a hexagonally arranged patch of cones illustrating the relationship between the distance measurements used in this study. A single cone (red circle) and its six closest neighbors (open circles) are highlighted for clarity. The NND is defined as the distance from a given cone to its closest neighbor (orange dashed line). The FND is defined as the distance from each cone to its most distant neighbor (blue dashed line), and ICD is defined as the average distance between a cone and all of its neighbors (dashed lines). In order to mitigate boundary effects, only cones with bound Voronoi regions (shaded region) are included when calculating each metric. The regularity of each of these metrics (M) is defined as the mean (μM) of the metric for all cones with bounded Voronoi cells, divided by the metric's SD (σM).
Percent Six-Sided Voronoi Cells.
The number of sides of each bound Voronoi cell was determined, and the number of Voronoi cells with six sides was divided by the total number of bound Voronoi cells within an ROI. 
Density Recovery Profile Distance (DRPD).
The density recovery profile (DRP) is a method based on a two-dimensional autocorrelogram that is an expression of the spatial density of cells as a function of the distance of each cell from all other cells.26,36 To automatically determine spacing from the DRP, we first determined the width of each bin as defined by equation 16 in Rodieck et al.,36 assuming a reliability of two. After calculating the DRP, we interpolated between each bin using splines. We then found the first local maximum within the spline that was greater than the DRP density mean. The x-axis location (distance) of the maximum was taken as the DRPD. 
Nearest Neighbor Distance (NND).
The distance between a given cone and its closest neighbor, where the neighbors of a given cone are comprised of all cones with adjacent Voronoi cells. The NND reported for each ROI is the average NND for all of the cones with bound Voronoi cells in that ROI (Fig. 1, orange dashed line). 
Intercell Distance (ICD).
The average distance between a given cone and each of its neighbors, where the neighbors of a given cone are comprised of all cones with adjacent Voronoi cells. The ICD reported for each ROI is the average ICD for all of the cones with bound Voronoi cells in that ROI (Fig. 1, black dashed lines). 
Farthest Neighbor Distance (FND).
The distance between a given cone and its farthest neighbor, where the neighbors of a given cone are comprised of all cones with adjacent Voronoi cells. The FND reported for each ROI is the average FND for all of the cones with bound Voronoi cells in that ROI (Fig. 1, blue dashed line). 
Nearest Neighbor Regularity (NNR).
The mean nearest neighbor distance (NND) for all of the cones with bound Voronoi cells in an ROI divided by the standard deviation (SD) of the NND for all of the cones with bound Voronoi cells in that ROI. 
Number of Neighbors Regularity (NoNR).
The mean number of sides of all bound Voronoi cells in an ROI divided by the SD of the number of sides of all bound Voronoi cells in that ROI. 
Voronoi Cell Area Regularity (VCAR).
The mean area of the bound Voronoi cells in an ROI divided by the SD of the area of the bound Voronoi cells in that ROI. 
Examining the Sensitivity of Metrics to Undersampling
After calculating each metric from each normal ROI, we used a statistical classifier to determine the threshold at which a metric could sensitively detect diffuse loss. To create the classifier, both the eccentricity of each ROI and the normal metric values from each subject were transformed to conform to statistical assumptions for linear models. Each ROI's eccentricity was transformed as follows:  where Eμm is eccentricity in μm, and Et is the transformed eccentricity value. The metric values were transformed using the natural log. These transformed data were then fit to a polynomial (orders 1–4). The (1–4) polynomial coefficients from each fit were used to create a 95% prediction ellipsoid, which defined the plausible values for each coefficient.  
We used this classifier to assess the sensitivity of each metric to undersampling with the following process: First, we randomly selected a subject and removed between 5% and 80% of the cone coordinates from each of their ROI's (again, representing diffuse cell loss due to disease, or cells missed during the identification step). Cones were removed by first permuting the cone coordinate list according to a uniform random distribution using the randperm MATLAB function. After permuting the cone coordinate list, the number of coordinates defined by the percent loss was removed from the beginning of the list. Next, the remaining (now undersampled) cone coordinates were analyzed using the metrics described above. We then transformed the resultant metric and eccentricity data and performed a polynomial fit as described above on these undersampled mosaics. Finally, we determined if the set of fit coefficients were significantly different from normal by comparing them with the prediction ellipsoid using Hotelling's t-squared statistic with a 95% significance cutoff.53 This process was repeated 1000 times for each cone loss percentage to calculate an “abnormal mosaic detection rate.” 
Using this process, we assessed the detection rate of abnormalities (or statistical power) for each metric at different percent loss values. A metric was considered sensitive to loss at a given percent cone loss when it correctly identified abnormal mosaics in 80% of trials. Finally, at each eccentricity, we constructed 95% pointwise prediction intervals (PIs) for each of the above metrics to describe pointwise uncertainty. 
Results
We were able to obtain images from all 20 subjects across each eccentricity. The numerical results are summarized in the Table (for meridian-specific values, refer to Supplementary Table S2). Figure 2A illustrates the expected exponential decrease of cone density with eccentricity as reported in previous studies,5,19,20,51,52 and the 95% PI for our population. The PI appears larger near the foveal center due to the increased normal variability in foveal cone density. In contrast, the cone spacing metrics increased monotonically as a function of eccentricity (Fig. 3), with the 95% PI being smaller near the fovea (<500 μm). The three regularity metrics and percent six-sided cells followed previously observed patterns,19,3234 peaking at about 250 μm (Fig. 4). 
Table
 
The Intersubject Mean (±SD) of Each Metric as a Function of Eccentricity
Table
 
The Intersubject Mean (±SD) of Each Metric as a Function of Eccentricity
Figure 2
 
Assessing the sensitivity of cone density to varying amounts of cell loss. (A) The mean of all 20 subjects' density as a function of eccentricity (solid line) with the 95% prediction interval (dashed lines). (B) The statistical power curve for cone density (solid line) does not reach the 80% threshold (horizontal dashed line) until a 24% cone undersampling, implying that density cannot reliably detect an abnormal cone mosaic until greater than 24% of the cells have been lost.
Figure 2
 
Assessing the sensitivity of cone density to varying amounts of cell loss. (A) The mean of all 20 subjects' density as a function of eccentricity (solid line) with the 95% prediction interval (dashed lines). (B) The statistical power curve for cone density (solid line) does not reach the 80% threshold (horizontal dashed line) until a 24% cone undersampling, implying that density cannot reliably detect an abnormal cone mosaic until greater than 24% of the cells have been lost.
Figure 3
 
Mean population cone spacing measurements. Four different spacing measurements: (A) NND; (B) DRPD; (C) ICD; (D) FND are plotted as a function of eccentricity (solid lines) with their respective 95% prediction intervals (dashed lines). All four spacing metrics increased monotonically with eccentricity.
Figure 3
 
Mean population cone spacing measurements. Four different spacing measurements: (A) NND; (B) DRPD; (C) ICD; (D) FND are plotted as a function of eccentricity (solid lines) with their respective 95% prediction intervals (dashed lines). All four spacing metrics increased monotonically with eccentricity.
Figure 4
 
Mean population regularity measurements. Three different regularity measurements: (A) NNR; (B) VCAR; (C) Number of neighbors regularity and percent six-sided cells (D) are plotted as a function of eccentricity (solid lines) with their respective 95% prediction intervals (dashed lines). All three regularity metrics and percent six-sided cells increased in the parafoveal region and decreased near the foveal center.
Figure 4
 
Mean population regularity measurements. Three different regularity measurements: (A) NNR; (B) VCAR; (C) Number of neighbors regularity and percent six-sided cells (D) are plotted as a function of eccentricity (solid lines) with their respective 95% prediction intervals (dashed lines). All three regularity metrics and percent six-sided cells increased in the parafoveal region and decreased near the foveal center.
To characterize how each metric was affected by undersampling, we first applied undersampling to a single ROI that exhibited average metric values (JC_10145, 200-μm eccentricity). Figure 5 illustrates the effect of 40% and 80% undersampling on this particular ROI. Qualitatively, the mosaic appears less regular with fewer cells remaining. However, without a priori eccentricity information, the ROI could simply be from a location more distant to the fovea. The histograms of each type of spacing each appear different; NND remains tightly clustered about the mean, whereas the mean and spread of ICD and FND measurements dramatically change as increasing amounts of loss are applied. In the DRP, the mean only slightly changes; in fact, the estimated spacing decreased, though this is likely an artifact due to the bin size selection algorithm. All measurements of regularity and percent six-sided cells for this ROI decreased in response to undersampling (Fig. 6). In this single ROI, the percentage of six-sided cells decreased by a similar amount (by 39% between 0%–40% undersampling, by 40% between 40% and 80% undersampling) between each percent undersampling. Number of neighbors regularity decreased by 47% between 0% and 40% undersampling, and 32% between 40% and 80% undersampling. Interestingly, VCAR decreased by 75% between 0% and 40% undersampling, and roughly half that (31%) between 40% and 80% undersampling, implying that the metric changes more with lower amounts of loss. NNR was the opposite, decreasing only 27% between 0% and 40% undersampling, but substantially more (79%) between 40% and 80%. 
Figure 5
 
An illustration of the effect of cone undersampling on histograms of cell distances (NND, ICD, FND) and the DRPD from a single subject (JC_10145, image acquired 200 μm from the fovea). In each plot, the blue dashed line is the mean of the histogram from the complete mosaic, while the orange dashed line is the mean of the histograms from the 40% (middle row) and the 80% undersampled mosaics (bottom row). On all plots, the y-axis is the number of cells within each histogram bin. The NND histogram is only marginally affected (indicated by the similarity in the blue and orange dashed lines), even with an 80% loss. Similarly, the DRPD is largely unaffected by cell loss; its estimated spacing is only affected when the bin size increases (bottom right) due to a decrease in density. In contrast, the mean (indicated by further separation of the blue and orange dashed lines) and spread of both ICD and FND increase substantially with cell loss.
Figure 5
 
An illustration of the effect of cone undersampling on histograms of cell distances (NND, ICD, FND) and the DRPD from a single subject (JC_10145, image acquired 200 μm from the fovea). In each plot, the blue dashed line is the mean of the histogram from the complete mosaic, while the orange dashed line is the mean of the histograms from the 40% (middle row) and the 80% undersampled mosaics (bottom row). On all plots, the y-axis is the number of cells within each histogram bin. The NND histogram is only marginally affected (indicated by the similarity in the blue and orange dashed lines), even with an 80% loss. Similarly, the DRPD is largely unaffected by cell loss; its estimated spacing is only affected when the bin size increases (bottom right) due to a decrease in density. In contrast, the mean (indicated by further separation of the blue and orange dashed lines) and spread of both ICD and FND increase substantially with cell loss.
Figure 6
 
The effect of cone undersampling on measurements of regularity and percent six-sided cells in the same ROI shown in Figure 5. The measured value (normal) is represented by white bars. Undersampling the mosaic by 40% (gray bars) and 80% (black bars) results in a reduction in all four metrics, though each metric decreases at a different rate. Note: Percent six-sided cells has been divided by 10 to fit the scale.
Figure 6
 
The effect of cone undersampling on measurements of regularity and percent six-sided cells in the same ROI shown in Figure 5. The measured value (normal) is represented by white bars. Undersampling the mosaic by 40% (gray bars) and 80% (black bars) results in a reduction in all four metrics, though each metric decreases at a different rate. Note: Percent six-sided cells has been divided by 10 to fit the scale.
We then used the prediction ellipse method described above to examine each metric's ability to detect undersampling in simulations from all 20 subjects. Density did not reliably detect an abnormality until 24% of the cones had been removed across all eccentricities (Fig. 2B). The NND and DRPD were remarkably insensitive to undersampling; an abnormal mosaic was unable to be detected for either metric until 53% and 55% of cone coordinates were removed, respectively (Fig. 7). In contrast, ICD and FND were able to detect an abnormal mosaic at 29% and 23% undersampling, respectively (Fig. 7). Of the regularity metrics, NNR was the least sensitive, and detected abnormality with above 35% undersampling (Fig. 7). The VCAR and percentage of six-sided cells and were similarly sensitive and were able to consistently detect a deviation from normal beyond 17% and 14% undersampling, respectively (Fig. 7). Of the regularity metrics, NoNR was the most sensitive, and was able to detect an abnormal mosaic after only 10% of the cone coordinates had been removed (Fig. 7). 
Figure 7
 
Statistical power curves (solid lines) for each metric as a function of the amount of undersampling (i.e., cell loss). The sensitivity of a given measurement was defined as the point at which the statistical power curve crossed 80% power (horizontal dashed line). Mean NND and DRPD are insensitive to cone undersampling and are only able to reliably detect cell loss when 53% and 55% of cone coordinates were removed, respectively. Mean ICD can reliably detect that the cone mosaic is abnormal after 29% of the cones were removed. Mean FND is the most sensitive of the four spacing metrics and is able to detect an abnormal mosaic after 23% of the cones were removed. Mean NNR was relatively insensitive and was unable to detect an abnormal mosaic until 40% of cones were removed. Percent six-sided cells and VCAR were able to detect abnormal mosaics with 17% and 14% undersampling, respectively. Mean NoNR was the most sensitive, detecting abnormal mosaics at 10% undersampling.
Figure 7
 
Statistical power curves (solid lines) for each metric as a function of the amount of undersampling (i.e., cell loss). The sensitivity of a given measurement was defined as the point at which the statistical power curve crossed 80% power (horizontal dashed line). Mean NND and DRPD are insensitive to cone undersampling and are only able to reliably detect cell loss when 53% and 55% of cone coordinates were removed, respectively. Mean ICD can reliably detect that the cone mosaic is abnormal after 29% of the cones were removed. Mean FND is the most sensitive of the four spacing metrics and is able to detect an abnormal mosaic after 23% of the cones were removed. Mean NNR was relatively insensitive and was unable to detect an abnormal mosaic until 40% of cones were removed. Percent six-sided cells and VCAR were able to detect abnormal mosaics with 17% and 14% undersampling, respectively. Mean NoNR was the most sensitive, detecting abnormal mosaics at 10% undersampling.
Discussion
We characterized the normal cone mosaic as a function of eccentricity using both new and previously described geometrical metrics. The metrics examined here had different 95% PI widths, suggesting each metric had different variance. While we examined a wide variety of metrics describing the cone mosaic, this is not an exhaustive list; new metrics may be derived as other retinal cell types are imaged, or as disease processes are better understood. Additionally, metrics can be derived directly from the retinal image; approaches based on analysis of the Fourier spectrum of the image (“Yellot's Ring”) are already in use,12,25,26,29,30,37,42,54 and others have been published to assess beam direction in the lamina cribrosa.55 Nevertheless, different metrics respond more sensitively to undersampling than others. NND, DRPD, and NNR were the least sensitive to cone undersampling, whereas percentage of six-sided Voronoi cells, VCAR, and NoNR were the most sensitive. Intuitively, one might think that the most sensitive metrics should always be used; however, there are some important points that should be reviewed to provide context to these results. 
The pointwise PIs constructed here represent the range that individual metric values will fall, within 95% likelihood. The PIs are constructed from 20 subjects; assuming our 20 healthy subjects are representative of the variance in the population, our estimate of the PI is more conservative than it would be had we included a larger population. For each metric's PI, we aggregated the results from all meridians to construct the PI. In our population, metrics measured along each meridian (temporal, inferior, nasal, and superior) behaved similarly, which may not always hold.52,5658 In contrast, the classifier tools were constructed for each metric to classify all ROIs from a single subject as either abnormal or normal. However, these classifiers were based on multiple regressions of our data. While density and spacing metrics fit lower order polynomial models closely (R2 goodness-of-fit > 0.95), the unusual shape of the regularity metrics required higher-order (fourth) polynomials to fit well (R2 > 0.8). While on average regularity metrics had R2 goodness-of-fit values above 0.8, without a closer fit (R2 > 0.95), our classifier may underestimate the true amount of variability in regularity metrics across all subjects. 
In addition to affecting the size of the PI, the sample size can cause artifacts when constructing the statistical power curves. The prediction ellipse-based classifier used to generate the power curves is constructed from the normal data with no loss; thus, the classifier should correctly identify normal mosaics at a rate similar to the significance level of 95%, or statistical power of 5%. 
A different issue relates to the type of cone loss that was adopted for these analyses. Photoreceptor loss is a dynamic process; when cones or rods die, their neighbors can move and fill the gaps, albeit to varying degrees.22,39,59,60 This is seen in part in the image in Figure 8, which is from a subject with significant cone mosaic disruption (evidenced by the interleaved dark regions throughout the image).38 The image has density and ICD values that correspond to only 48% of the normal mean at that eccentricity. However, the NND and DRPD values are consistent with a greater than 70% cell loss, and VCAR, NoNR, and percent six-sided are consistent with only a 20% cell loss for this retinal location. Given that each of these metrics describe a different aspect of the mosaic, and that there is such a large disparity between the actual value of each metric and the value predicted by simulated undersampling, the inconsistency of these metrics is likely indicative of an alternative type of loss (such as photoreceptor remodeling). Regardless, exploring the relationship between different metrics and examining how each responds to both simulated and real loss could enable a more quantitative description of the type of cone loss in different retinal degenerations/loss types. 
Figure 8
 
Evaluating metric performance in a real mosaic with cell loss. The images shown here are taken from a patient with a deutan color vision deficiency caused by the presence of a unique opsin haplotype encoded by their OPN1MW gene.38 As can be seen in the image (acquired at 200 μm from the fovea), the cone mosaic has a diffuse loss of normal waveguiding cones (example region outlined in dotted cyan) that mimicks the simulated undersampling performed in the present study. Density and ICD were consistent with a simulated 48% undersampling. However, NND and density recovery profile distance were consistent with a greater than 70% simulated cell loss, and VCAR, number of neighbors regularity, and percent six-sided were consistent with only a 20% simulated cell loss at the same location. These findings illustrate that there has likely been reorganization of the mosaic in addition to discrete loss of waveguiding cones.
Figure 8
 
Evaluating metric performance in a real mosaic with cell loss. The images shown here are taken from a patient with a deutan color vision deficiency caused by the presence of a unique opsin haplotype encoded by their OPN1MW gene.38 As can be seen in the image (acquired at 200 μm from the fovea), the cone mosaic has a diffuse loss of normal waveguiding cones (example region outlined in dotted cyan) that mimicks the simulated undersampling performed in the present study. Density and ICD were consistent with a simulated 48% undersampling. However, NND and density recovery profile distance were consistent with a greater than 70% simulated cell loss, and VCAR, number of neighbors regularity, and percent six-sided were consistent with only a 20% simulated cell loss at the same location. These findings illustrate that there has likely been reorganization of the mosaic in addition to discrete loss of waveguiding cones.
A major concern with the translation of AO imaging to the clinical arena (specifically clinical trials) is that image quality may not always be sufficient to visualize the entire photoreceptor mosaic. In addition to differences in hardware capabilities, pathologies such as AMD and RP are linked with poor image quality due to age or secondary effects of the disease (e.g., cataracts or cystoid macular edema).61,62 In these situations, the use of a metric that is insensitive (i.e., robust) to undersampling (DRPD, NND, NNR) should be used. However, as shown here, these same metrics would be poorly suited for use in longitudinal studies, due to this very same insensitivity. Thus, one has to be very explicit with what it is they are trying to measure when choosing which metric to use. In the end, the most sensitive metric cannot be assumed to be the “best” metric. 
Acknowledgments
The authors thank Alfredo Dubra, PhD, Mara Goldberg, Christopher Langlo, Erika Phillips, Moataz Razeen, MD, Phyllis Summerfelt, and Jonathon Young for their contributions. 
Supported in part by the National Center for Research Resources and the National Center for Advancing Translational Sciences of the National Institutes of Health under award number UL1TR001436 and by the National Eye Institute of the National Institutes of Health under award numbers P30EY001931, R01EY017607, and T32EY014537. Additional support was provided by Foundation Fighting Blindness (Columbia, MD, USA); The Edward N. & Della L. Thome Memorial Foundation, Bank of America, N.A. Trustee (Boston, MA, USA); the Gene & Ruth Posner Foundation (Milwaukee, WI, USA); and the Richard O. Schultz, MD/Ruth Works Professorship (Milwaukee, WI, USA). This investigation was conducted in part in a facility constructed with support from the Research Facilities Improvement Program; grant number C06RR016511 from the National Center for Research Resources, NIH. 
Disclosure: R.F. Cooper, None; M.A. Wilk, None; S. Tarima, None; J. Carroll, None 
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Figure 1
 
A schematic of a hexagonally arranged patch of cones illustrating the relationship between the distance measurements used in this study. A single cone (red circle) and its six closest neighbors (open circles) are highlighted for clarity. The NND is defined as the distance from a given cone to its closest neighbor (orange dashed line). The FND is defined as the distance from each cone to its most distant neighbor (blue dashed line), and ICD is defined as the average distance between a cone and all of its neighbors (dashed lines). In order to mitigate boundary effects, only cones with bound Voronoi regions (shaded region) are included when calculating each metric. The regularity of each of these metrics (M) is defined as the mean (μM) of the metric for all cones with bounded Voronoi cells, divided by the metric's SD (σM).
Figure 1
 
A schematic of a hexagonally arranged patch of cones illustrating the relationship between the distance measurements used in this study. A single cone (red circle) and its six closest neighbors (open circles) are highlighted for clarity. The NND is defined as the distance from a given cone to its closest neighbor (orange dashed line). The FND is defined as the distance from each cone to its most distant neighbor (blue dashed line), and ICD is defined as the average distance between a cone and all of its neighbors (dashed lines). In order to mitigate boundary effects, only cones with bound Voronoi regions (shaded region) are included when calculating each metric. The regularity of each of these metrics (M) is defined as the mean (μM) of the metric for all cones with bounded Voronoi cells, divided by the metric's SD (σM).
Figure 2
 
Assessing the sensitivity of cone density to varying amounts of cell loss. (A) The mean of all 20 subjects' density as a function of eccentricity (solid line) with the 95% prediction interval (dashed lines). (B) The statistical power curve for cone density (solid line) does not reach the 80% threshold (horizontal dashed line) until a 24% cone undersampling, implying that density cannot reliably detect an abnormal cone mosaic until greater than 24% of the cells have been lost.
Figure 2
 
Assessing the sensitivity of cone density to varying amounts of cell loss. (A) The mean of all 20 subjects' density as a function of eccentricity (solid line) with the 95% prediction interval (dashed lines). (B) The statistical power curve for cone density (solid line) does not reach the 80% threshold (horizontal dashed line) until a 24% cone undersampling, implying that density cannot reliably detect an abnormal cone mosaic until greater than 24% of the cells have been lost.
Figure 3
 
Mean population cone spacing measurements. Four different spacing measurements: (A) NND; (B) DRPD; (C) ICD; (D) FND are plotted as a function of eccentricity (solid lines) with their respective 95% prediction intervals (dashed lines). All four spacing metrics increased monotonically with eccentricity.
Figure 3
 
Mean population cone spacing measurements. Four different spacing measurements: (A) NND; (B) DRPD; (C) ICD; (D) FND are plotted as a function of eccentricity (solid lines) with their respective 95% prediction intervals (dashed lines). All four spacing metrics increased monotonically with eccentricity.
Figure 4
 
Mean population regularity measurements. Three different regularity measurements: (A) NNR; (B) VCAR; (C) Number of neighbors regularity and percent six-sided cells (D) are plotted as a function of eccentricity (solid lines) with their respective 95% prediction intervals (dashed lines). All three regularity metrics and percent six-sided cells increased in the parafoveal region and decreased near the foveal center.
Figure 4
 
Mean population regularity measurements. Three different regularity measurements: (A) NNR; (B) VCAR; (C) Number of neighbors regularity and percent six-sided cells (D) are plotted as a function of eccentricity (solid lines) with their respective 95% prediction intervals (dashed lines). All three regularity metrics and percent six-sided cells increased in the parafoveal region and decreased near the foveal center.
Figure 5
 
An illustration of the effect of cone undersampling on histograms of cell distances (NND, ICD, FND) and the DRPD from a single subject (JC_10145, image acquired 200 μm from the fovea). In each plot, the blue dashed line is the mean of the histogram from the complete mosaic, while the orange dashed line is the mean of the histograms from the 40% (middle row) and the 80% undersampled mosaics (bottom row). On all plots, the y-axis is the number of cells within each histogram bin. The NND histogram is only marginally affected (indicated by the similarity in the blue and orange dashed lines), even with an 80% loss. Similarly, the DRPD is largely unaffected by cell loss; its estimated spacing is only affected when the bin size increases (bottom right) due to a decrease in density. In contrast, the mean (indicated by further separation of the blue and orange dashed lines) and spread of both ICD and FND increase substantially with cell loss.
Figure 5
 
An illustration of the effect of cone undersampling on histograms of cell distances (NND, ICD, FND) and the DRPD from a single subject (JC_10145, image acquired 200 μm from the fovea). In each plot, the blue dashed line is the mean of the histogram from the complete mosaic, while the orange dashed line is the mean of the histograms from the 40% (middle row) and the 80% undersampled mosaics (bottom row). On all plots, the y-axis is the number of cells within each histogram bin. The NND histogram is only marginally affected (indicated by the similarity in the blue and orange dashed lines), even with an 80% loss. Similarly, the DRPD is largely unaffected by cell loss; its estimated spacing is only affected when the bin size increases (bottom right) due to a decrease in density. In contrast, the mean (indicated by further separation of the blue and orange dashed lines) and spread of both ICD and FND increase substantially with cell loss.
Figure 6
 
The effect of cone undersampling on measurements of regularity and percent six-sided cells in the same ROI shown in Figure 5. The measured value (normal) is represented by white bars. Undersampling the mosaic by 40% (gray bars) and 80% (black bars) results in a reduction in all four metrics, though each metric decreases at a different rate. Note: Percent six-sided cells has been divided by 10 to fit the scale.
Figure 6
 
The effect of cone undersampling on measurements of regularity and percent six-sided cells in the same ROI shown in Figure 5. The measured value (normal) is represented by white bars. Undersampling the mosaic by 40% (gray bars) and 80% (black bars) results in a reduction in all four metrics, though each metric decreases at a different rate. Note: Percent six-sided cells has been divided by 10 to fit the scale.
Figure 7
 
Statistical power curves (solid lines) for each metric as a function of the amount of undersampling (i.e., cell loss). The sensitivity of a given measurement was defined as the point at which the statistical power curve crossed 80% power (horizontal dashed line). Mean NND and DRPD are insensitive to cone undersampling and are only able to reliably detect cell loss when 53% and 55% of cone coordinates were removed, respectively. Mean ICD can reliably detect that the cone mosaic is abnormal after 29% of the cones were removed. Mean FND is the most sensitive of the four spacing metrics and is able to detect an abnormal mosaic after 23% of the cones were removed. Mean NNR was relatively insensitive and was unable to detect an abnormal mosaic until 40% of cones were removed. Percent six-sided cells and VCAR were able to detect abnormal mosaics with 17% and 14% undersampling, respectively. Mean NoNR was the most sensitive, detecting abnormal mosaics at 10% undersampling.
Figure 7
 
Statistical power curves (solid lines) for each metric as a function of the amount of undersampling (i.e., cell loss). The sensitivity of a given measurement was defined as the point at which the statistical power curve crossed 80% power (horizontal dashed line). Mean NND and DRPD are insensitive to cone undersampling and are only able to reliably detect cell loss when 53% and 55% of cone coordinates were removed, respectively. Mean ICD can reliably detect that the cone mosaic is abnormal after 29% of the cones were removed. Mean FND is the most sensitive of the four spacing metrics and is able to detect an abnormal mosaic after 23% of the cones were removed. Mean NNR was relatively insensitive and was unable to detect an abnormal mosaic until 40% of cones were removed. Percent six-sided cells and VCAR were able to detect abnormal mosaics with 17% and 14% undersampling, respectively. Mean NoNR was the most sensitive, detecting abnormal mosaics at 10% undersampling.
Figure 8
 
Evaluating metric performance in a real mosaic with cell loss. The images shown here are taken from a patient with a deutan color vision deficiency caused by the presence of a unique opsin haplotype encoded by their OPN1MW gene.38 As can be seen in the image (acquired at 200 μm from the fovea), the cone mosaic has a diffuse loss of normal waveguiding cones (example region outlined in dotted cyan) that mimicks the simulated undersampling performed in the present study. Density and ICD were consistent with a simulated 48% undersampling. However, NND and density recovery profile distance were consistent with a greater than 70% simulated cell loss, and VCAR, number of neighbors regularity, and percent six-sided were consistent with only a 20% simulated cell loss at the same location. These findings illustrate that there has likely been reorganization of the mosaic in addition to discrete loss of waveguiding cones.
Figure 8
 
Evaluating metric performance in a real mosaic with cell loss. The images shown here are taken from a patient with a deutan color vision deficiency caused by the presence of a unique opsin haplotype encoded by their OPN1MW gene.38 As can be seen in the image (acquired at 200 μm from the fovea), the cone mosaic has a diffuse loss of normal waveguiding cones (example region outlined in dotted cyan) that mimicks the simulated undersampling performed in the present study. Density and ICD were consistent with a simulated 48% undersampling. However, NND and density recovery profile distance were consistent with a greater than 70% simulated cell loss, and VCAR, number of neighbors regularity, and percent six-sided were consistent with only a 20% simulated cell loss at the same location. These findings illustrate that there has likely been reorganization of the mosaic in addition to discrete loss of waveguiding cones.
Table
 
The Intersubject Mean (±SD) of Each Metric as a Function of Eccentricity
Table
 
The Intersubject Mean (±SD) of Each Metric as a Function of Eccentricity
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