Modeling Human Macular Cone Photoreceptor Spatial Distribution

Abstract

Purpose: The purpose of this study was to investigate the spatial distribution of human cone photoreceptors and examine cone density differences between the retinal meridians and quadrants.

Method: Using adaptive optics scanning laser ophthalmoscopy, the maculae were imaged in 17 eyes of 11 subjects with normal chorioretinal health aged 54 to 72 years. We measured cone density at 325 points within the central 10 degrees radius of the retina. Cone density spatial distributions along the primary retinal meridians and in four macular quadrants (superior-nasal, superior-temporal, inferior-temporal, and inferior-nasal) were analytically modeled using the polynomial function to assess the meridional and quadrantal difference.

Results: The mean and 95% confidence interval for the prediction of cone density along the primary retinal meridians was modeled with a 7-degree one-variable polynomial (R² = 0.9761, root mean squared error [RMSE] = 0.0585). In the 4 retinal quadrants, cone density distribution was described by a 2-variable polynomial with X degree 3 and Y degree 4 (R² = 0.9834, RMSE = 0.0377). The models suggest no statistically significant difference between medians and between quadrants. However, cone density difference at corresponding spatial locations in different areas can be up to 25.6%. The superior-nasal region has more areas with high cone density, followed by quadrants of inferior-nasal, inferior-temporal, and superior-temporal.

Conclusions: Analytical modeling provides comprehensive knowledge of cone distribution across the entire macula. Although modeling analysis suggests no statistically significant difference between medians and between quadrants, the remarkable cone density discrepancies in certain regions should be accounted for in applications requiring sensitive detection of cone variation.

Cone photoreceptors within the macula of the human eye are essential phototransduction cells ensuring high acuity vision and color perception.¹ The spatial distribution of these cells dictates the spatial information conveyed to higher stages of visual processing.² In a seminal work, Curcio and co-workers systematically quantified the two dimensional (2D) distribution of the human photoreceptor (including cones and rods) using the whole mounts of the fixed donor retinas, establishing the basic knowledge of the human cone topography.³ 

Recent advance in adaptive optics (AO) retinal imaging has enabled in vivo imaging of cone photoreceptors in the human eye routinely.⁴ The fundamental spatial properties of cone density have been studied in eyes with normal chorioretinal health, including density variation with retinal eccentricity,^5–8 the influence of the eye length,^5,9,10 refractive error, gender, race/ethnicity and ocular dominance,⁵ the aging effect,^5,11,12 the variability of foveal cone density,¹³ the interocular symmetry of the foveal cone mosaic,^13–15 cone density distribution differences between the fovea and the periphery,¹⁶ and the foveal cone packing mosaic pattern.¹⁷ These studies replicated the cone spatial distribution characteristics obtained by histology, such as a peak of density at the foveal center, a sharp density drop in the fovea and parafovea, and then a mild decline with increasing eccentricity, wide variability of foveal cone density, isodensity contours elongated along the horizontal meridian, and higher density in the nasal than the temporal periphery.^{5,7,9–11,13,15–24} In vivo cone density measured in the normal eyes by AO imaging not only enriched our knowledge of visual sampling and limit, but also provided a reference for assessing cone loss in various retinal diseases.^4,25–28 

However, in the published work, cone density has been mainly examined along the primary retinal medians or in the fovea. The 2D distribution across the whole macula is inadequately studied, leaving fundamental questions regarding cone density differences within individuals in medians and quadrants incompletely understood or clarified. In practical application, cone density was often modeled as a function of retinal eccentricity,^{11,16,20,23,28,29} and hinted a statistically isotropic distribution in the macula, despite the histologic and in vivo finding of the anisotropic cone density along different meridians.^{5–11,13,15–24} The incomplete characterization of the cone density distribution and variability within individual eyes can obscure the study of cone density variation between subjects. 

To fully understand the fundamental spatial structure of macular cone photoreceptors in the human eye, it is essential to evaluate cone density in two dimensions across the entire macula in eyes with normal chorioretinal health. The result can provide a reference to assess the impact of aging and chorioretinal disease on cone photoreceptors, thereby leveraging the technical advantages offered by AO high-resolution retinal imaging. Thus, this study analytically examined the spatial distribution of macular cone density in healthy human eyes. It addressed fundamental questions about directional and regional cone density differences within the same eye. We evaluated the cone density distribution in the primary retinal meridians and four retinal quadrants (Q1 = superior-nasal; Q2 = superior-temporal; Q3 = inferior-temporal; and Q4 = inferior-nasal) using the data acquired by a research grade AO scanning laser ophthalmoscope (AOSLO)^30,31 in a group of older participants. 

Methods

This study followed the tenets of the Declaration of Helsinki and was approved by the Institutional Review Boards at the University of Alabama at Birmingham and the University of California – Los Angeles. Written informed consent was obtained from participants after the nature and possible consequences of the study were explained. The research complied with the Health Insurance Portability and Accountability Act of 1996. 

Cone Density Data

Cone photoreceptor images were collected from the normal subjects in a project studying age-related macular degeneration using AO imaging.³² The subject enrollment, image acquisition and process, and cone density measurement methods have been reported elsewhere.^12,32 For the readers’ convenience, we summarize the procedures here. 

The participants were recruited from the clinical registry of the Department of Ophthalmology and Visual Sciences of the University of Alabama at Birmingham. The inclusion criteria for healthy subjects were that the participant was older than 50 years of age, with a best-corrected visual acuity (BCVA) of 20/25 or better, refractive error within ±6 diopters (D) spherical and ±3 D cylinder, no history of ocular and systemic disease or corneal surgery for refraction correction (e.g. LASIK), and no significant cataract. However, participants with intraocular lenses implanted after standard cataract surgery were included. The subjects’ chorioretinal health was evaluated by multimodal ophthalmic imaging, including color fundus photography, near-infrared reflectance (λ = 830 nm), blue reflectance (λ = 488 nm), fundus autofluorescence (excitation, 488 nm; emission >600 nm), and spectral domain optical coherence tomography (SD-OCT). An experienced masked grader graded the color fundus photographs using the Age-Related Eye Diseases Study 2 (AREDS2) severity scale.³³ Subjects deemed with healthy macular health were rated as having AREDS2 grade 1 in both eyes. 

The macular cones were imaged by a research AOSLO using a low coherent near-infrared superluminescent diode (λ = 840 nm, Δλ = 50 nm)^30,32,34,35 through a dilated pupil (with 1.0% tropicamide and 2.5% phenylephrine hydrochloride) in all participants. The AOSLO images were continuously recorded across the macula. Post-processing corrected the nonlinear distortion caused by the resonant scanner, registered successive frames to enhance the signal-to-noise ratio, and montaged all individual images with cell-to-cell precision using custom and commercial software (Photoshop, Adobe Systems Inc., Mountain View, CA). 

The AOSLO montage was registered to the SD-OCT en face image using custom software. A coordinate system was set up using the fovea center as the origin point and with the abscissa and ordinate along the horizontal and vertical retinal primary medians in the fundus of a right eye in the standard perimetric projection. The foveal center determined by the peak cone density in eyes with foveal cones could be unambiguously identified on AOSLO and verified with the OCT B-scans. In the eyes whose foveal cones could not be imaged by AOSLO, the deepest point of the foveal pit in the OCT B-scan that features the central light reflex and the absence of inner retinal landmarks (the inner plexiform and inner nuclear layers overlying the outer nuclear layer) was considered the foveal center.³⁶ A sampling grid of 20 degrees × 20 degrees divided into a lattice of 1 degree × 1 degree was overlaid on the AOSLO image. Cone density was measured at 325 mesh points, including 317 grid nodes in 10 degrees radius macula (Fig. 1, left panel) and 8 points at 0.2 degrees and 0.5 degrees meridian eccentricity (zoomed in the white box on the top left corner of Fig. 1), using the “find Maxima” function of ImageJ (version 1.53c; http://imagej.nih.gov/ij). Cone density was expressed in cells/degree². 

Figure 1.

 

Cone density sampling across the macula. Left panel: A montaged image of adaptive optics scanning laser ophthalmoscopy overlaid on the infrared reflectance image taken in the right eye of a normal human subject. The green grid represents a 1 degree interval. The green abscissa and ordinate indicate the horizontal and vertical retinal meridians, with the origin set at the fovea center. The yellow circle delimits the central 10 degrees radius, that is, the macula. Cone density was measured at the intersecting points of the green grid within the circle, a total of 317 points. Cone density was also measured at 0.2 degrees and 0.5 degrees eccentricity along two meridians (8 points), zoomed in the white box on the top left corner. The optical coherence tomography B-scan acquired along the horizontal retinal meridian crossing the fovea center is underneath the infrared retinal image. The white arrowhead points to the fovea center. The right panel is a summary plot of number of eyes included in the analysis, with a range of 1 to 17 eyes per location.

Cone Density Modeling and Evaluation

The spatial profile of cone density distribution, that is, peaking at the foveal center and declining with eccentricity and a sharp drop in the fovea and parafovea, intuitively suggests that cone density can be modeled using a (combination of) exponential, Gaussian, or Rational function. Meanwhile, it can be fitted to a polynomial or Fourier series with sufficient terms or orders. After an evaluation of the goodness and parsimony of models³⁷ with these functions along four primary retinal medians (superior, inferior, nasal, and temporal) and four macular quadrants (Q1 = superior-nasal; Q2 = superior-temporal; Q3 = inferior-temporal; and Q4 = inferior-nasal), we selected a 7-degree one-variable polynomial to model cone density along the primary retinal meridians and a two-variable polynomial with X degree 3 and Y degree 4 accurately for cone density in the 4 retinal quadrants (Supplementary Tables S1, S2). The modeling analysis was performed using Matlab Apps Curve fitter (Matlab R2023a; The MathWorks, Inc. Natick, MA, USA). 

Logarithmic transformation was applied to the cone density values to make the data conform to a normal distribution and reduce the variability.^38,39 The estimated mean and 95% confidence interval for prediction of the model coefficients were reported. 

Results

Subject Characteristics

Cone density was collected from 17 eyes of 11 subjects (7 men and 4 women) aged 60.0 ± 5.0 years (ranging from 54 to 72 years). Within this age range, cone density does not exhibit a significant age-related decline.¹² To avoid the correlation of the cone densities between the fellow eyes, the pointwise mean was calculated for modeling analysis if both eyes of a subject were imaged.^3,40 Visual acuity = 0.05 ± 0.12 (logarithm of the minimum angle of resolution, ranging from −0.1 to 0.3), and refraction error = 0.03 ± 1.38 D (ranging from −3.25 D to 1.75 D). 

Due to missed image acquisitions, shadows of blood vessels, or poor image quality in some regions, the cone density at each sampling point was determined using varying numbers of eyes that provided adequate data (see Fig. 1, right panel). 

Cone Density Models

Figure 2 illustrates the cone density and 95% predicted confidence intervals along the primary retinal medians using the 7-degree polynomials. The standard residuals randomly spread around the zero line, indicating a consistent variance. The Q-Q plot of the residuals forms a straight line with an approximately 45 degrees slope, suggesting that the residuals follow a normal distribution. Thus, the cone density spatial distributions are properly modeled. Table 1 lists the model coefficients and corresponding 95% predicated intervals. The modeling accuracies are shown in Table 2. 

Figure 2.

 

Modeling cone density along 4 primary retinal meridians with 7-degree polynomials. Left column: Cone density and the 95% confidence interval for prediction along the nasal, temporal, superior, and inferior meridian from the top to the bottom rows. Middle column: The standard residuals of the modeled cone density along the corresponding retinal meridians. Right column: The quantile to quantile (Q-Q) plots of the residuals of the modeled cone density along the corresponding retinal meridians.

Table 1.

 

Cone Density and 95% Predicted Confidence Intervals Along Primary Retinal Medians^*

 

Table 2.

 

Cone Density Modeling Accuracy Along Primary Retinal Medians

 

Notably, modeling cone density directly using the raw data led to wrong fitting results (Supplementary Fig. S1). With the same data and the same math function, although all model functions could produce a high R² (>0.95), the residuals exhibited a distinctive “fan in” pattern from left to right rather than a random spread around the zero line. Meanwhile, the Q-Q plots manifested significant nonlinearity aberrating from the 45 degrees line. The non-stable variances and the aberrating Q-Q plots indicate poor modeling, which can also be perceived by the negative cone density in the models beyond 6-degree eccentricity. In contrast, the logarithmic transformation of the raw data stabilized the variance (see Fig. 2, middle column) and corrected for the skewness (see Fig. 2, right column) of the measurements, making the residuals follow a normal distribution. 

Figure 3 shows the cone density and 95% predicted confidence intervals in the 4 macular quadrants modeled using a 2-variable polynomial with X degree 3 and Y degree 4 (Table 3). Table 4 lists the model accuracy. 

Figure 3.

 

Modeling cone density in 4 retinal quadrants using polynomials with X degree 3 and Y degree 4. The middle surfaces are the estimated means, whereas the upper and lower surfaces show the 95% confidence intervals for predictions.

Table 3.

 

Cone Density and 95% Predicted Confidence Intervals in Four Retinal Quadrants^*

 

Table 4.

 

Cone Density Modeling Accuracy in Four Retinal Quadrants

 

Compare the Cone Density Distribution in Different Medians

By plotting all the modeled cone density curves and their confidence interval lines of the four meridians, we picked up the lowest upper and the highest lower bound as the “common confidence intervals” (Fig. 4, left). It is clear that the model curves of cone density along the four meridians are all within the common confidence interval, suggesting that there is no statistical evidence of a difference in the cone density distribution along the four meridians.^41–44 

Figure 4.

 

Comparing cone density in four primary retinal meridians. Left panel: The modeled cone density in four retinal meridians and their common confidence interval. The upper bound common confidence interval is the lowest upper bound of the 95% confidence intervals of modeling in the 4 meridians, whereas the lower bound is the highest lower bound of the 95% confidence intervals of modeling in the 4 meridians. Right panel: The magnified area is indicated by the box in the left panel. Cone density exhibits the most prominent difference (25.6%) at 3.7 degrees eccentricity between nasal and superior (red arrow).

However, the cone density difference can be up to 25.6% (difference/mean) at the eccentricity of 3.7 degrees between the superior and the nasal loci (see Fig. 4, right panel), with the latter having a higher density. Meanwhile, the biggest differences between superior and temporal, superior and inferior, nasal and temporal, nasal and inferior, and temporal and inferior are 18.6%, 24.5%, 9.18%, 23.6%, and 17.6%, respectively. Overall, the cone density is lower in the vertical meridian than in the horizontal meridian. At most eccentricities, the cone density is the highest in the nasal, temporal, and inferior, and the lowest in the superior meridian (see Fig. 4). 

Compare the Cone Density Distribution in Different Macular Quadrants

In the same way, we obtained the common confidence interval of the cone density in four macular quadrants. The results show cone density within the common confidence interval at all locations within the 9 degrees radius. However, cone densities near the 10 degrees eccentricity at some edge locations exceed the common confidence interval (Fig. 5). 

Figure 5.

 

Cone density (CD) distribution in four retinal quadrants. Left panel: The 2D map of the modeled cone density in 4 quadrants. Middle panel: Cone density in four quadrants compared with their common confidence interval (CCI). The upper bound common confidence interval is the lowest upper bound of the 95% confidence intervals of modeling in the 4 quadrants, whereas the lower bound is the highest lower bound of the 95% confidence intervals of modeling in the 4 quadrants. Cone density has the biggest difference (12.7%) at the points (8.6 degrees, 2.7 degrees) and (−8.6 degrees, 2.7 degrees), with the former having a higher cone density. Right panel: The relative cone density to the lower bound of the common confidence interval. Q1 = superior-nasal quadrant; Q2 = superior-temporal quadrant; Q3 = inferior-temporal quadrant; and Q4 = inferior-nasal quadrant.

The pairwise comparison of cone density across the four quadrants reveals variations. Between Q1 and Q2, Q1 and Q3, and Q1 and Q4, the cone density differences can be up to 12.7%, 9.4%, and 11.1%, respectively. Similarly, between Q2 and Q3, and Q2 and Q4, the differences are up to 10.9% and 11.9%, respectively. The highest difference between Q3 and Q4 is 7.3%. Overall, Q1 has the highest cone density, followed by Q4, Q3, and Q2. The largest difference across the macula occurs at the coordinates (8.6 degrees, 2.7 degrees) and (−8.6 degrees, −2.7 degrees), with the latter having a lower cone density (see Fig. 5). 

Discussion

Precise and comprehensive knowledge of cone spatial distribution across the macula is vital for understanding the fundamental constraints on visual perception and measuring the effects of aging and disease on photoreceptors. By analytically modeling cone density acquired in subjects with tightly controlled age range and refraction status, we examined the two-dimensional spatial distribution of the cone photoreceptors across the entire macula of the human eye. We answered fundamental questions regarding directional and quadrantal differences in cone density within the subject, which is essential for other aspects of accurate macular cone morphology. 

Because cone density was measured at discrete retinal loci in practical applications, properly modeling the cone density variation against the retinal location is essential for understanding the fundamental spatial arrangement of these cells. An analytical model of the cone density and its predicted variance (the 95% confidence interval) allows us to assess cone density at any retinal point across the macula, thereby providing precise measures of the fundamental limits of visual sampling and the impact of aging and chorioretinal diseases.^4,25,26 Albeit the importance, cone density modeling was inexplicitly and inadequately studied. Duncan and associates used a quadratic function to model cone spacing with data measured at retinal locations from 0.5 degrees to 3.5 degrees eccentricity.²⁹ Song and co-workers adopted an exponential function to fit the cone density acquired in the primary retinal medians.¹¹ Zayit-Soudry et al.,²⁰ Qin et al.,²³ and Cooper et al.⁴⁵ described cone spacing in normal human subjects using an exponential function. In a recent publication, Duncan and colleagues modeled the mean spacing of the cones as a function of location using a double exponential function to account for the spatial position of the region of interest in the macula.²⁸ In these pioneering studies, cone density or spacing was modeled either in one direction and did not consider potential directional differences, or along the primary retinal meridians and did not account for possible two-dimensional inhomogeneity across the macula. Our study addressed these inadequacies by examining the cone density in the primary medians and in four macular quadrants. Furthermore, in a recent study, Elsner and colleagues found cone densities at 0.63 mm retinal eccentricity were uncorrelated to those at 2.07 mm. This fact rules out models with a constant or proportional relation of cone density to eccentricity,¹⁶ indicating that the spatial variation of the cone photoreceptors should be modeled with higher-order mathematical functions. We thus modeled cone density with high-order polynomials based on an evaluation of different mathematical functions, including exponential, Fourier, Gaussian, and rational functions (see Supplementary Table S1). 

Our study extended previous work by measuring cone density across the whole macula. Previous studies have mainly documented the cone density at discrete retinal loci in the fovea and parafovea or along the primary retinal medians. The analytical model provides an intuitive visualization of the cone density distribution across the macula. Overall, the superior-nasal quadrant (Q1) has more areas with high cone density, which are then followed by the inferior-nasal quadrant (Q4), the inferior-temporal quadrant (Q3), and the superior-temporal quadrant (Q2). The nasal side retina had more regions with high cone density than the temporal side. Along the primary retinal meridians, cone density is higher along the horizontal (temporal-nasal) than the vertical (superior-inferior). These characteristic properties have only been reported in histology³ and partially measured by in vivo AO imaging studies.^7,8,11,17 

The models disclose that the mean cone densities in the four retinal medians are within the common confidence interval (see Fig. 4), indicating no statistical evidence of a difference in cone density distribution in the four meridians. Likewise, the models reveal that the mean cone densities in most locations of the four retinal quadrants are within the common confidence interval (see Fig. 5), suggesting no statistical evidence of a difference in cone density distribution in the four quadrants. Thus, the macular cone density can be described by a function of a single variable, that is, the eccentricity, in light of the statistical significance. However, the substantial discrepancy in the cone density at specific retinal locations should be noted. For example, cone density at (3.7 degrees, 0 degrees) nasal meridian is higher than at (0 degrees, 3.7 degrees) superior meridian by 25.6% (see Fig. 4), and cone density at (8.6 degrees, 2.7 degrees) in the Q1 superior-nasal quadrant is higher than at (−8.6 degrees, 2.7 degrees) in the Q2 superior-temporal quadrant by 12.7% (see Fig. 5). Lumping data at the same eccentricity in all quadrants can thus yield a significant variance, reducing the sensitivity to detect cone loss in retinal diseases and treatment, which requires appropriately established normative data with narrow confidence limits. Thus, A 2D map of cone density and its variance is ideal for precisely assessing cone loss. 

The strength of this study lies in a carefully established retinal coordinate system with a precisely determined origin point at the foveal center,³ which defines the spatial locations of the sampling points. Histological³ and in vivo imaging studies^10,12,13 have found a dramatic density drop from the fovea. Thus, precisely determining the foveal center location is critical. We have identified this point by the peak cone density^3,9,10,12,13 and checked with the OCT B-scans of the fovea.³⁶ Further, we made a continuous montage of the cone photoreceptor images and registered it to the fundus photograph, thereby establishing the retinal coordinates system to model the spatial distribution of the cone photoreceptors with rigorous mathematical functions. Furthermore, we carefully assessed the modeling correctness by evaluating the residual error distribution and correcting for the data abnormality, thereby avoiding inappropriate modeling only judged by the R². It is also worth mentioning that our data were derived from an older population, providing valuable age-matched comparisons for the diseased populations. The study also identified limitations to be addressed in future investigations. We used data acquired in a small number of subjects only. The spatial locations do not have an equal number of measurements. There are fewer measured data at 10 degrees eccentricity, which may be the contributory factor resulting in cone density at some edge points outside the common confidence interval. Another weakness is that all measurements were only performed at the integer grid points (see Fig. 1). A more random sampling is preferable for proper modeling. Future measurement in more subjects incorporated with split-detection imaging is warranted. 

In conclusion, analytical modeling of cone density and its predicted variance allows for assessing cone density at any spatial retinal location and indicates no statistically significant difference between medians and quadrants. However, due to the significant discrepancies in cone density in certain retina regions, a 2D cone density map should be considered for precisely assessing cone variation and cone loss. 

Acknowledgments

The authors thank Christine A Curcio, PhD, Cynthia Owsley, PhD, C. Douglas Witherspoon, MD, Christopher A Girkin, MD, and Mark E Clark, BS, for helping with data acquisition. 

Supported by research grants from the National Institute of Health (R01EY024378 and R01EY034218), W. M. Keck Foundation, Carl Marshall Reeves & Mildred Almen Reeves Foundation, Research to Prevent Blindness/Dr. H. James and Carole Free Catalyst Award for Innovative Research Approaches for AMD. 

Disclosure: X. Wang, None; S. Hoshi, None; R. Liu, None; Y. Zhang, None