**Purpose**:
Retinal pigment epithelial (RPE) cells serve as a supporter for the metabolism and visual function of photoreceptors and a barrier for photoreceptor protection. Morphology dynamics, spatial organization, distribution density, and growth patterns of RPE cells are important for further research on these RPE main functions. To enable such investigations within the authentic eyeball structure, a new method for estimating the three-dimensional (3D) eyeball sphere from two-dimensional tissue flatmount microscopy images was investigated.

**Methods**:
An error-correction term was formulated to compensate for the reconstruction error as a result of tissue distortions. The effect of the tissue-distortion error was evaluated by excluding partial data points from the low- and high-latitude zones. The error-correction parameter was learned automatically using a set of samples with the ground truth eyeball diameters measured with noncontact light-emitting diode micrometry at submicron accuracy and precision.

**Results**:
The analysis showed that the error-correction term in the reconstruction model is a valid method for modeling tissue distortions in the tissue flatmount preparation steps. With the error-correction model, the average relative error of the estimated eyeball diameter was reduced from 14% to 5%, and the absolute error was reduced from 0.22 to 0.03 mm.

**Conclusions**:
A new method for enabling RPE morphometry analysis with respect to locations on an eyeball sphere was created, an important step in increasing RPE research and eye disease diagnosis.

**Translational Relevance**:
This method enables one to derive RPE cell information from the 3D eyeball surface and helps characterize eyeball volume growth patterns under diseased conditions.

^{1}In addition to providing nutrients to retinal visual cells, this cell layer performs numerous functions essential to the choroid and the photoreceptors,

^{2}including scattered light absorption, transepithelial transport, spatial buffering of ions, maintenance of the visual cycle, phagocytosis of photoreceptor outer segments, and secretion of growth factors and immunosuppressive factors. Without RPE cells, photoreceptor cells die, and the choroid degenerates. Therefore RPE cells play an important role in maintaining retinal homeostasis and photoreceptor survival.

^{3}

^{–}

^{8}AMD is the leading cause of untreatable and irreversible central vision loss and legal blindness in industrialized countries.

^{9}

^{–}

^{12}RPE cell morphological characteristics, such as shape, size, number, and geometric packing, provide informative features to determine RPE aging. For example, the size of an RPE cell in the fovea region is commonly smaller than that in peripheral areas,

^{4}

^{,}

^{13}and the average RPE cell size increases with age.

^{13}

^{,}

^{14}The RPE cell shape is closer to a regular hexagon in the central regions than in the outer zones.

^{15}

^{,}

^{16}It is also known that the number of RPE cells decreases with age.

^{14}

^{,}

^{17}

^{–}

^{19}RPE morphometries, such as cell density, eccentricity, form factor, and percent of hexagonal cells, also vary with aging and show significant topography-associated features.

^{19}

^{–}

^{21}Therefore accurate quantification of the RPE cell morphometrics is essential for research and clinical diagnosis of AMD.

^{22}microscopy has been used to quantitatively analyze RPE cell morphology.

^{4}

^{,}

^{6}

^{,}

^{18}

^{,}

^{19}

^{,}

^{21}

^{,}

^{23}However, the experimental procedure for generating such a flatmount image usually introduces significant RPE cell distortion and loss, especially on the segmented lobe borders.

^{22}

^{,}

^{23}This makes it difficult to measure RPE cell morphology on whole flatmount images accurately. In practice, regions not close to the lobe borders are selected for analysis.

^{21}

^{,}

^{23}However, the morphometry features of whole tissue flatmount images would provide more informative insights into RPE cell histology. To achieve this, numerous digital image processing methods

^{24}

^{–}

^{31}can be used to restore distorted or lost tissue regions in a RPE flatmount image. However, because of the considerable variations in RPE cell morphometry,

^{4}

^{,}

^{13}

^{,}

^{15}

^{,}

^{16}

^{,}

^{19}

^{,}

^{21}accurate determination of the RPE cell locations on the three-dimensional (3D) eyeball model is a prerequisite for digital image inpainting and restoration analysis. The lack of methods for accurately locating tissue regions on a 3D eyeball structure makes the quantitative RPE cell morphometry analysis short of the anatomic and geographic interpretation. The geographic delineation of the 3D eyeball in current research pivots around the eyeball physiological structures. The common terminologies used to depict the geographic regions on the 3D eyeball surface include fovea, optic nerve head, perifovea, parafovea, and the peripheral areas from the central zones defined by the fovea or optic nerve head. The definitions of these locations of interest are imprecise and insufficient to support further investigations of RPE cell quantitative morphometrics. Furthermore, the quantification of tissue location–specific dynamics of RPE cell morphometry features, alignment orientation, and spatial organization are important for research and clinical investigations of AMD. Ideally, such investigations ought to be conducted in reference to authentic 3D eyeball structures. The loss of 3D eyeball structure information in two-dimensional (2D) tissue flatmount microscopy images presents a challenge for associating RPE cell features with the 3D eyeball anatomic architecture.

_{2}asphyxiation before dissection. After the enucleation process, the eyeball was fixed in zinc formalin fixative (Z-fix; Anatech Ltd., Battle Creek, MI) for 10 minutes followed by washing with phosphate-buffered saline three times. The fixed eyeball was trimmed to remove extraocular fat and muscles before size measurement was performed. The mouse eyeball size was measured with a noncontact light-emitting diode (LED) micrometer (model 7030M with an LS-7601 controller and LS-H1W software; Keyence America Corp, Itasca, IL) as previously described.

^{32}For five eyes from the control C57BL/6J mice, the eyeball size measurement was performed with a Keyence IM-6145 digital micrometer. The eyeball was placed in two different positions, that is, axial (anterior-posterior) and horizontal (equatorial), and the device automatically measured the diameter of each dimension. As we hypothesized that the eyeball is a sphere with a slight corneal bulge, the ground truth of the eyeball size was set with the equatorial diameter, that is, the average of the superior-inferior length and the nasal-temporal length. After the eyeball size measurement, the RPE flatmount was prepared with radial cuts from the center of the cornea back toward the optic nerve. The iris and the retina were removed with forceps, and additional incisions were made at the ciliary body and cornea margin to relieve tension from the sclera. Next, the RPE flatmounts were stained for rabbit anti-ZO1 antibody (1:100 dilution, catalog no. 61-7300; Invitrogen, Carlsbad, CA) and goat anti-rabbit immunoglobulin G (IgG) secondary antibody (catalog no. O11038; Invitrogen). The resulting flatmounts were imaged with a confocal imaging system (model C1; Nikon Inc., Melville, NY) with argon laser excitation at 488 nm. The confocal images were digitally merged (Adobe Photoshop CS2; Adobe Corp, San Jose, CA).

*R*, where

*R*is the radius of the sphere.

^{33}Many other examples of the assembly of 3D bulging objects from 2D flat panels exist: parachutes, hot air balloons, tents, domes, corners in heating, ventilation, and air conditioning (HVAC) ductwork, and sails.

*O*and

*N*are the center and the north pole of the sphere, respectively. In Figure 2, angle θ is formed by lines

*ON*and

*OA*where

*A*is an arbitrary point on the sphere. The south pole of this sphere model corresponds to the optic nerve head; the top dome of the sphere represents the cornea. In this coordinate system, the length of the shortest arc (i.e., the geodesic distance) from point

*A*to south pole

*S*on the sphere is

*L*=

*R*× (π − θ), where

*R*is the sphere radius. To clarify terminology, a line of latitude for point

*A*is “latitude

*A*” and its corresponding concentric circle after projection on the south pole plane is “circle

*A*”. The change in arc length during the flatmount process is assumed to be negligible. Therefore arc length

*L*is identical to the radius of “circle

*A*” on the 2D flatmount plane. With Figure 2, the length of “latitude

*A*” can be computed as

*R*× sin θ × 2π. The perimeter of “circle

*A*” on the south pole plane is

*L*× 2π. If the sphere is evenly cut into four lobes unfolded on the south pole plane, some gaps would occur between the neighboring lobes. This is illustrated in Figure 3(a). The sum of the lengths of all the gap arcs (

*G*) on “circle

*A*” is

*G*=

*L*× 2π −

*R*× sin θ × 2π. From the previous discussion,

*L*=

*R*× (π − θ). Therefore the following equation governs the relationship across the gap arc length (

*G*) in the 2D south pole plane, the geodesic distance from point

*A*to the south pole on the sphere, and the sphere radius:

*G*and

*L*in Equation (1) can be measured from the flatmount images through image processing. Thus sphere radius

*R*can be estimated with such measures through least-squares curve fitting.

^{34}

^{35}

*L*in the southern hemisphere; that is,

*L*≤

*R*× π/2. On the northern hemisphere (i.e.,

*L*>

*R*× π/2), the tissue width starts to decrease with the increase in

*L*. By contrast, the gap arc length continues to increase. In higher-latitude zones of the northern hemisphere, the gap arc length between lobes is much larger than the tissue lobe widths. Ultimately, at the tips of the lobes, tissue vanishes, and the gaps close to become a continuous circle with radius

*R*× π. The cut lobes may have irregular shapes in any zone (or latitude) because of the difficulty of the dissection, as shown in Figure 5(a). However, the proposed model as depicted by Equation (1) is fully applicable regardless of latitude or hemisphere.

*E*is introduced to compensate for this tissue distortion effect:

*E*=

*k*× sin θ, where

*k*is an unknown tissue distortion coefficient that depends on how 3D eyeball tissues are flattened to the 2D flatmount. With this error term, the proposed model considering the tissue deformation effect becomes:

*G*,

*L*, and

*k*in Equation (3) must be known before sphere radius

*R*can be estimated. The length of the gaps between lobes

*G*and the radius of the concentric circle on the 2D flatmount plane

*L*can be directly derived from the flatmount image by counting the number of pixels with image processing methods. The tissue distortion coefficient

*k*can be determined with Equation (3) using training samples with known ground truth

*R*.

*M*

_{1},

*M*

_{2},

*M*

_{3}, and

*M*

_{4}. The middle axis length is the distance from the origin to a lobe-marker point. In this implementation, Aperio ImageScope V12.3.3

^{36}was used to mark these critical marker points manually on the flatmount images.

*L*∈ [1,

_{Arr}*M*], where \({M_{min}} = \mathop {\min }\limits_n \{ {{M_n}} \},{\rm{\;}}n = 1,2,3,4\). For each circle with the radius

_{min}*L*∈

_{i}*L*, the number of circle pixels in the background region was counted to measure the length in-between lobe gaps

_{Arr}*G*∈

_{i}*G*. The length of the arithmetic array was set to be 500 in the implementations. The maximum of the lobe middle axis length \({M_{max}} = \mathop {\max }\limits_n \{ {{M_n}} \},{\rm{\;}}n = 1,2,3,4\) was used to estimate corneal angle θ after sphere radius

_{Arr}*R*was estimated.

*k*was introduced in Equation (3) to correct the tissue distortion error, it was necessary to estimate

*k*before we could calculate sphere radius

*R*and corneal angle θ. The error-correction term can be learned from a number of samples with known ground truth

*R*by the least-squares algorithm in the model. As all tissues were prepared similarly, the resulting tissue distortion coefficients were similar. As one tissue distortion coefficient

*k*was estimated from each such sample with a known

*R*, the mean of the tissue distortion coefficients was taken as the learned

*k*for further

*R*estimation. To solve

*R*from Equation (3) with the least-squares method, the optimization process was prevented from converging to a trivial solution (i.e.,

*R*≈ 0) by setting the initial value for

*R*much larger than 0, (e.g., 100). With the estimated

*R*, the corneal angle θ can be readily found as:

*L*is the radius of the corresponding concentric circle measured with the 2D flatmount image.

*k*was estimated based on Equation (3) with data points excluding those from the low-latitude zones. In the evaluations, the last 10%, 20%, and 30% of data points (i.e., 50, 100, and 150 data points from the 500 data points) in close proximity to the low-latitude zones were removed from

*L*and

_{Arr}*G*, respectively. Similarly to test the error effect from the high-latitude zones, the tissue distortion coefficient

_{Arr}*k*was estimated based on Equation (3) with data points excluding partial data from the high-latitude zones. In the evaluations, the first 10%, 20%, and 30% of data points (i.e., 50, 100, and 150 data points from the 500 data points) in close proximity to the high-latitude zones were removed from

*L*and

_{Arr}*G*, respectively. The resulting estimates of the tissue distortion coefficient

_{Arr}*k*from 23 samples based on the full dataset and partial data points excluding some from the low- and high-latitude zones are shown in Table 1 and Figure 6(a-b), respectively. Additionally, partial data points were removed from the low- and high-latitude zones, and the resulting deviations in the estimated tissue distortion coefficient

*k*from the reference estimate with the full data point set are shown in Figure 6(c-d). The fitting curves associated with the tissue distortion coefficients estimated with distinct data point sets excluding some from the low- and high-latitude zone in one typical sample are shown in Figure 6(e–f).

*k*decreased as more data points from the low-latitude zones were removed as shown in Figure 6(a). Additionally, the estimated

*k*shown in Figure 6(b) remained at about the same value as more data points were removed from the high-latitude zones. The difference between the tissue distortion coefficient

*k*estimated with the full and distinct partial data point sets was further computed. The relative difference by percentage for each sample is plotted in Figure 6(c) and (d) in which partial data were removed from low- and high-latitude zones, respectively. It is noticeable that the tissue distortion coefficient dropped significantly when more data points were removed from the low-latitude zones for most samples shown in Figure 6(c), whereas the tissue distortion coefficient did not change significantly (<5%) when more data points were removed from the high-latitude zones for the vast majority of the samples shown in Figure 6(d). Furthermore, the same patterns were observed with the curve fitting results shown in Figure 6(e) and (f) . In Figure 6(e), the interlobe gap-fitting curves associated with distinct tissue distortion coefficients deviate by a larger decrease from the theoretic tissue interlobe gaps from Equation (1) as more data points were removed from the low-latitude zones. In contrast, all interlobe gap-fitting curves associated with distinct tissue distortion coefficients were close to each other when partial data points were removed from the high-latitude zones. The difference in the error compensation term

*k*/2π estimated between all and partial data points for each sample was computed. The resulting variances of the error compensation term difference associated with different partial datasets are presented in Table 2. The variances with partial data excluded from the low-latitude zones were consistently larger than those estimated with partial data excluded from the high-latitude zones by at least three orders of magnitude.

*E*due to tissue distortion in Equation (2) increases when the latitude decreases, suggesting the equatorial zones have larger gap errors than the polar zones. Based on this conclusion that the low-latitude zones have a larger impact on gap error than the high-latitude zones, data points from the low-latitude zones have a more influential impact on the gap-radius fitting results than the same number of data points from the high-latitude zones. Additionally, we can conclude that the interlobe gap error is more sensitive to the manual artifacts in low-latitude zones than in high-latitude zones during the tissue flatmount preparation procedure.

*R*and Corneal Angle θ

*k*in Equation (3). The mean of the estimated tissue distortion coefficients was computed for estimating the eyeball radius of the testing samples. Additionally, the estimates from the non-error-correction model described in Equation (1) were used as results from a control group.

*R*was estimated. The estimated corneal angle θ can help characterize tissue surface coverage on the 3D eyeball sphere, and this measurement is helpful in analyzing microphthalmia, profound myopia or hyperopia, and many other eye diseases. The estimated radius

*R*and corneal angle θ from 23 samples are shown in Table 6. Additionally, the estimates using the LOO-CV strategy for learning the tissue deformation coefficient are shown in Supplementary Figure S1.

^{36}save the annotation file in the image folder and invoke the software to process all images in a batch mode. To further promote software dissemination and usage, a graphic user interface (GUI) is provided in MATLAB (MathWorks Inc., Natick, MA). The MATLAB GUI integrates the manual labeling process with the computational estimation process and makes it easier to use the software. The user interfaces for estimation of the tissue distortion coefficient and the eyeball size are shown in Supplementary Figure S2. The interface on the top was developed to estimate the tissue distortion coefficient

*k*using sample flatmount images with known size. With the “Load” button, users can load an Excel file (Microsoft Corp., Redmond, WA) with the following information: image name, image path, ground truth eyeball size, and unit conversion ratio (i.e., microns per pixel). Within this interface, users can then mark the concentric circle origin with the “Set” button and lobe-marker points with the “New” button for each image. After all images are marked, the interface can be used to estimate the tissue distortion coefficient for each image and export results to an Excel file with the “Export” button. The graphic interface at the bottom is used to estimate the eyeball diameter for each flatmount image with the learned tissue distortion coefficients. It loads the learned tissue distortion coefficients in the previous step and computes the average tissue distortion coefficient. Additionally, it loads an Excel file that includes information about the image names, image path, and unit conversion ratio (i.e., microns per pixel). Users use this interface to mark the concentric circle origin and lobe-marker points and export another Excel file containing estimated diameters for eyeballs.

^{21}

^{,}

^{37}

^{–}

^{39}Despite extensive investigations on this topic, current quantitative RPE cell analyses with histology slides lack accurate eyeball geographic location information. Analysis of RPE cells without reference to their geographic locations is a bottleneck in understanding RPE cell organization. To address this challenge, a novel modeling method for recovering individual 3D eyeball size from a 2D tissue flatmount microscopy image was proposed in this study. Such a technical method enables RPE features extracted from a 2D flatmount image to be mapped to a recovered 3D eyeball structure, allowing better insights according to their 3D locations.

*k*estimated by distinct data points with some removed from the low-latitude zones, which is shown in Figure 6(c). The difference between the interlobe gaps measured from the flatmount image and the associated fitting curve for this gap indicated the variation of the tissue distortion coefficient

*k*estimated with distinct partial data points. Curve fitting plots associated with samples 4, 1, and 16, representing relatively large, medium, and small differences between measured and fitting interlobe gap curves, are shown in Figure 10, respectively. The variations in the tissue distortion coefficient estimated with distinct partial data points were found to be large, medium, and small, respectively, and are shown in Figure 6(a). However, this correlation did not change the underlying trend that exclusion of data points from low-latitude zones led to a decrease in the estimated tissue deformation coefficient. In reference to the ground truth eyeball radius measured with noncontact LED micrometry at submicron accuracy and precision, the proposed method, considering the effects of tissue deformation, achieved 0.03 mm and 5.27% for the average radius difference and the average relative radius difference percentage, respectively.

*k*/2π ranged from 0.05 mm to approximately 0.25 mm, with an average of approximately 0.13 mm. As shown in Table 6, the ground truth value for the eyeball radius ranged from 1.47 to 1.71 mm, with an average of approximately 1.58 mm. By numeric comparisons, the numeric value of

*k*/2π was less than

*R*by an order of magnitude. As the average

*k*/2π was used to correct the estimation error in

*R*in Equation (3), the impact of the sample variation in

*k*on the radius

*R*estimation was limited. In the experiments, the least-squares algorithm was used to estimate

*k*and

*R*simultaneously. However, this simultaneous approach resulted in inexact estimates of

*k*and

*R*. Therefore we conclude that it is essential to estimate

*k*before

*R*.

^{6}

^{,}

^{21}Notably, a large number of in situ noninvasive fundus imaging technologies have emerged recently, including adaptive optics optical coherence tomography,

^{40}

^{,}

^{41}adaptive optics scanning laser ophthalmoscopy,

^{42}

^{–}

^{44}and fluorescence adaptive optics scanning light ophthalmoscopy,

^{45}

^{,}

^{46}among others. With advances in these imaging technologies, RPE morphometry analyses become more important in eye disease research, diagnosis, and treatment. The method proposed in this study initiates the effort of constructing a 3D digital eyeball with 2D RPE tissue flatmount microscopy images and makes it possible to leverage image restoration approaches

^{24}

^{–}

^{31}for such efforts. As the RPE layer is naturally a 3D anatomy structure in a 3D space, it is clinically and biologically meaningful to derive more key information about RPE cell morphometry, orientation, and spatial organization on the 3D eyeball surface. Additionally, the proposed method can be used to facilitate quantitative biometric characterization of mice eyeball size that is related to abnormalities in refractive or ocular development.

^{32}Such traits as eyeball volume recovered by the proposed method can also help researchers and clinicians better understand eye growth patterns under certain disease conditions,

^{47}

^{–}

^{50}the mechanisms vision systems follow to maintain visual clarity,

^{51}

^{,}

^{52}and key genes affecting visual pathways.

^{53}

^{–}

^{58}

^{59}This can be enhanced by development of a better model for characterizing individual tissue deformation specific to a customized tissue flatmount preparation process. (3) The process for identifying the concentric circle center and lobe-marker points can be automated to enable high-throughput image analysis of a large number of flatmount microscopy images. (4) As the anteroposterior axis is slightly different (1%–2% longer) from the sagittal or horizontal axes in some 3D eyeballs,

^{60}the 3D geometric sphere for modeling a 3D eyeball may be further improved with a 3D ellipsoid. However, it would be challenging in such methodology development, as we need to simultaneously estimate more parameters in the 3D ellipsoid model, such as the lengths of the three major axes, the polar angle (i.e., the angle respect to the polar axis), and the azimuthal angle (i.e., the angle of rotation from the initial meridian plane). Because of the substantial increase in the number of model parameters for estimation, the resulting method would become much more computationally expensive. Furthermore, given the limited measuring information derived from flatmount images, this change to the ellipsoid model would challenge the estimation robustness. Finally, it is operationally challenging to satisfy the ellipsoid model requirement during the sample preparation procedure. Making the 3D ellipsoid computational method as simple as possible, it is ideal to place the eyeball in a perfectly upright position. However, this is challenging to achieve in practice as eyeballs are made of soft tissues.

**H. Li**, None;

**H. Yu**, None;

**Y.-K. Kim**, None;

**F. Wang**, None;

**G. Teodoro**, None;

**Y. Jiang**, None;

**J.M. Nickerson**, None;

**J. Kong**, None

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