Mathematical Analysis of Specific Anatomic Foveal Configurations Predisposing to the Formation of Macular Holes

Abstract

Purpose.: To mathematically analyze and to clinically describe specific anatomic foveal configurations predisposing to the formation of macular holes in comparison with normal foveal anatomy.

Methods.: In a retrospective observational case-control series, a total of 3882 optical coherence tomography (OCT) foveal thickness maps were analyzed; 96 foveal maps were identified before the formation of macular holes. Maps were analyzed using several anatomic measurements including: retinal thickness, foveal slope, and length of foveal depression. The mathematical analog of the foveal configuration was analyzed using automated symbolic regression software and the equation to describe the mathematical relationship in a 0.083 fit was derived for premacular hole foveas compared with normal age-matched foveas.

Results.: Premacular hole anatomic configuration was found to be significantly different from normal foveal anatomy for maximal slope (P < 0.05) and for central length of foveal depression (P < 0.05). The mathematical regression function followed a first-order cosine curve of level 12 complexity for normal fovea compared with a complex sine curve of level 30 complexity function for premacular hole fovea. Normal foveas had higher symmetry (0.86 ± 0.1, P = 0.03) along the midline, whereas premacular hole foveas had steeper maximal slopes (40 ± 18°, P = 0.01); 75% of these patients had similar foveal configuration in the fellow eye and 50% developed bilateral macular holes.

Conclusions.: Premacular hole foveal configurations are significantly different from normal foveal configurations. Suspicious macular configurations are easy to recognize on OCT scans and may allow early diagnosis, follow-up, and better management of macular hole–prone patients.

Optical coherence tomography (OCT), introduced in the 1990s, greatly enhanced our understanding of the pathogenesis and anatomic changes underlying macular hole formation. ^{1 –3} Abnormal tight hyaloid adhesions and vitreomacular tangential and anteroposterior traction were considered to play a role in the formation of idiopathic macular holes for many years before they could actually be anatomically demonstrated using OCT. ^{4 –6}  

The morphologic changes in the macula during idiopathic macular hole formation were first systematically suggested and staged by Gass, ⁵ based on slit-lamp clinical biomicroscopy observation. ⁴ Modern OCT imaging provides researchers with a high-resolution image and with better understanding of the intimate relationship between the posterior cortical vitreous and the macula. 

The cross-sectional imaging technique implicated incomplete posterior vitreous detachment, with residual oblique and tangential traction on the macula as a possible cause of intraretinal splitting and consecutive macular hole formation. ^{1 –3,7 –9} Although there were several studies emphasizing the importance of a vitreoretinal relationship for the formation of macular holes, ^{10 –12} no specific foveal OCT configuration predisposing patients to develop macular holes has been identified. 

An automated symbolic regression software tool may be used to detect equations and hidden mathematical relationships in scientific data. In this study, a regression software (Eureqa; Cornell Creative Machines Lab) was used to identify the mathematical equations describing the curve analog of the foveal anatomic configuration of two groups (the premacular hole formation group and the control group). The software is completely automated, but the fit in which the curves can fit the data can be selected by the user. In this study, this automated technique was used for the first time for collecting data from OCT measurements of patients with macular holes in comparison with patients with normal foveal configurations to describe specific anatomic foveal configurations that predispose to the formation of macular holes. 

This is the first attempt to mathematically describe an analog of the foveal anatomy to be used to distinguish pathologic from nonpathologic processes. 

Materials and Methods

This study was reviewed and approved by the Institutional Review Board of the University of Louisville and included patients that were examined at the retina clinic at the University of Louisville, Louisville, Kentucky. Two groups of patients were studied: the premacular hole formation group included patients who later developed a macular hole and the control group included age-matched patients who did not develop a macular hole during the 69-month follow-up period between January 2005 and September 2010. 

In the premacular hole formation group, 3882 OCT (Stratus OCT 3000, Carl Zeiss) foveal thickness maps of 647 patients with a diagnosis of macular hole were analyzed in an attempt to identify the patients who had, for any reason, an early OCT before the development of macular hole: 96 foveal maps of 16 patients were identified to be taken before subsequent formation of macular holes composed of 6 meridians of the earliest OCT map available for every patient. These patients were asymptomatic at the time their first OCT was obtained. The control group included 96 maps of 16 age-matched patients who were randomly selected from our OCT database. These patients did not have macular pathology and did not develop a macular hole throughout the follow-up period. 

In an effort to identify potential anatomic characteristics predisposing to idiopathic macular hole formation, the earliest baseline OCT images available before macular hole development were systematically reviewed and analyzed using the Retinal Thickness analysis function on the OCT. The fellow eye OCT scans of patients in the premacular hole formation group were also analyzed in an attempt to find predisposing anatomy for future macular hole formation. Systematic measurements of macular OCT geometrical characteristics were obtained and compared with the control group. These measurements included foveal volume (FV), two points of parafoveal maximum thickness (PMT), distance between the two PMT points (dPMT), and central macular thickness (CMT). The maximal slope for each side of the fovea was calculated as the maximum difference in thickness divided by the distance at 50-μm consecutive intervals around the fovea and normalized to an aged-matched group. Symmetry of the foveal pit was established by dividing the numerical values of the maximal slopes on each side along the foveal midline, as seen in Figure 1. The mathematical analog of the foveal configuration was analyzed using the automated symbolic regression software (Eureqa, version 0.82 beta), which enables the user to choose the level of accuracy in which the function fits the gathered data. The fit for our study for both groups was chosen to be 0.083, which proved to provide a close fit of the curve to the data in both groups. Other levels of fit may be alternatively chosen, provided that the same level of fit is used for both the control and the study groups. OCT images of patients' fellow eyes were systematically studied and analyzed. 

Figure 1.

 

OCT image: premacular hole formation (A) and control (B). Maximum retinal thickness: premacular hole formation (a) and control (a′). Maximum slope: premacular hole formation (θ) and control (θ′).

Medical records were reviewed for age and sex of the patient, visual acuity, lens status at presentation, and the presence of posterior vitreous detachment. All eyes demonstrating obvious gross structural macular abnormalities, abnormal vitreomacular traction distorting the foveal configuration, and/or the retinal architecture were excluded from the present study. Also all patients with other macular pathology such as diabetic maculopathy, prior macular surgery, macular scars, age-related macular degeneration, traumatic or myopic macular holes, macular holes associated with a rhegmatogenous retinal detachment, or with any clinical evidence or prior medical history of maculopathy were excluded from the present study. 

Statistical Analysis

Measurements were compared between the control group and the premacular hole group using a two-tailed paired t-test and ANOVA. P < 0.05 was considered statistically significant. 

Results

Premacular hole anatomic configuration was found to be significantly different from normal foveal anatomy, as shown in Table 1. The premacular hole foveal maps had steeper slopes (40 ± 18°) in comparison with a subtle slope (21 ± 12°) of the control maps (P = 0.01), as demonstrated in Figure 1. A statistically significant difference was found between the symmetry along the foveal midline of the control group (0.86 ± 0.1) compared with the premacular hole formation group (0.74 ± 0.14, P = 0.03). In the premacular hole group 43% of patients had already developed a posterior vitreous detachment documented by clinical observation and by OCT imaging. A statistically significant difference between premacular hole foveal maps and controls was found only in the maximal slopes and in the amount of symmetry along the foveal midline. There was no difference in foveal contour between the different meridians. Furthermore, there was no difference between the slopes in eyes that had posterior vitreous detachment compared with the eyes that did not have vitreous detachment. No statistically significant difference was found in all other measured parameters, including FV, PMT, dPMT, or CMT between the groups. 

Table 1.

 

Foveal Characteristics of “Premacular Hole Formation” and Control Groups

Parameter	Control	Premacular Hole	P *
Number of patients	16	16
Age, y	72.0 ± 7.7	71.9 ± 7.9	P = 0.53
Months to develop macular hole	—	13.5 ± 10.7
Maximum retinal Thickness, μm	289.6 ± 20.1	312.9 ± 37.5	P = 0.12
Central foveal Thickness, μm	173.4 ± 24.2	200.8 ± 35.2	P = 0.25
Maximum slope	21 ± 12°	40 ± 18°	P = 0.01*
Foveal symmetry along the midline	0.86 ± 0.1	0.74 ± 0.14	P = 0.03*

*  Values of P < 0.05 are considered statistically significant.

Using the automated symbolic regression software (Eureqa) with a 0.083 fit (Figs. 2, 3) the mathematical regression for the control group followed a first-order cosine curve of level 12 complexity, following this mathematical equation:  

 

The mathematical regression for the premacular hole formation group followed a complex sine curve of level 30 complexity, according to the following equation:  

 

Demonstrated by OCT, the time from premacular hole configuration to the development of a full thickness macular hole was found to be 13.5 ± 10.7 months. In all, 75% of premacular hole patients had a similar anatomic foveal configuration in the other eye, and 50% of these patients consequently developed bilateral macular holes. 

Figure 2.

 

Mathematical representation of accuracy versus complexity: premacular hole formation (A) and control (B).

Figure 3.

 

Mathematical representation of retinal thickness versus the mathematical analog of the foveal configuration, produced by automated symbolic regression software (Eureqa): premacular hole formation (A) and control (B).

To demonstrate the ease of recognition of this specific foveal configuration we performed a small survey among current ophthalmology residents at the University of Louisville. After a short explanation of the clinical configuration of premacular hole (steep slopes and asymmetry of the fovea along the midline), residents were asked to identify scans with premacular hole configuration among ten OCT scans (Fig. 4). 

Figure 4.

 

Ten different OCT retinal thickness maps were used to test ophthalmology residents' ability to correctly differentiate premacular hole configurations from age-matched controls. (b, e, h) Premacular hole configurations. (a, c, d, f, g, i, j) Normal retinal thickness maps from age-matched controls.

Ophthalmology residents were successful in identifying the premacular hole configuration on OCT with a sensitivity of 1, specificity of 0.91, a positive predictive value of 0.83, and a negative predictive value of 1. All premacular hole configurations were correctly identified, confirming that this configuration can easily be recognized by any ophthalmologist aware of its specific OCT characteristics. 

Discussion

Despite being a common clinical presentation at retina clinics around the globe, the etiology of macular hole formation is still unknown. Data gathered in recent years support the hypothesis that vitreoretinal abnormalities and vitreomacular traction play a major role in idiopathic macular hole formation. ^8,10,11 This hypothesis was primarily based on observing the fellow eyes of patients diagnosed with macular hole because the incidence of bilateral macular hole formation was found to be relatively high. ^8,10  

OCT is a modern prevalent useful tool to evaluate foveal anatomy in all patients, specifically in those who had macular hole in one eye, and are at risk of developing a macular hole in the fellow eye. This study is the first to demonstrate a preexisting abnormal foveal anatomic configuration in patients who consequently developed macular holes. Using mathematical analysis this abnormal configuration was found to be significantly different from the norm. This mathematical model would appear to be the most accurate predictor yet to foresee with accuracy the subgroup of patients that will more likely progress to full-thickness macular hole. The abnormal configuration includes steeper foveal slopes and less symmetry of the slopes around the foveal pit, despite the lack of apparent vitreomacular traction on OCT. A larger study investigating the normal variation of maximal foveal slopes is indicated to determine normal slope variations. Such abnormal configuration may represent early tangential traction on the fovea that results in the widening of the central foveal area. 

The study results are in agreement with studies performed by Michalewska et al. ¹⁰ on the fellow eyes of 131 patients with a history of macular hole. They performed four OCTs during a 6-month follow-up period and described seven foveal abnormalities and their changes over time; 13% of patients were classified as having irregular foveal contour with very steep foveal slopes and a flat wide fovea. There was no evidence of vitreomacular traction associated with this irregular foveal contour. 

The risk of developing a macular hole in the other eye of patients with unilateral macular hole has been previously reported to be in the 11% to 13% range ^8,9 ; 75% of premacular hole patients in this study had a similar anatomic foveal configuration in the fellow eye and, indeed, 50% of these patients consequently developed bilateral macular holes. This high incidence may indicate that this predisposing foveal configuration represents a subset of patients with high risk to develop bilateral macular holes. 

The finding that 43% of the cases that did develop macular holes already had posterior vitreous detachment is intriguing. This clearly indicates that there is a subgroup of patients in whom the development of macular hole is not directly related to the presence of vitreomacular traction. There has been some discussion in the literature that suggests the vitreous might not be the prime perpetrator of macular hole formation. Schubert et al. ¹³ discussed the possibility that a weakened or dehisced central fovea and focal proliferation of Müller or glial cells might create secondary traction via abnormal vitreoretinal adherence, leading to a full-thickness macular hole. This configuration may be indistinguishable from a scenario of the primary role of vitreoretinal traction, despite the most accurate of imaging modalities. There have been several studies describing a subset of patients in which the cortical vitreous has separated or was absent after vitrectomy, and thus was unavailable to mediate tangential or anteroposterior traction. ^{14 –17} This suggests that a degenerative component may be contributing to macular hole pathogenesis. One possible explanation is an inherent pathology in the cone-shaped zone of Müller cells that composes the central and inner part of the fovea centralis. ¹⁸ The importance of these cells is maintaining the structural integrity of the macula. Müller cell invasion and proliferation within the prefoveolar vitreous cortex, as described by Gass, ¹⁸ may lead to stronger than normal vitroretinal adhesion, causing part of the posterior hyaloid to remain firmly attached to the macula even after posterior vitreous detachment formation. Early changes and contraction of this prefoveolar tissue may result in transfoveal traction, leading to the premacular hole foveal configuration described in this study. Progression of this process can lead to full-thickness retinal dehiscence and macular hole formation in its later stages. Another possible explanation for this unique anatomic configuration is the loss of tissue (operculum), enabling the process of macular hole formation to begin. This tissue is composed of Müller cells and cortical vitreous that may give rise to an epiretinal membrane formation because these cells have been shown to be present in the vitreous of macular hole patients. ^18,19  

In patients with no apparent posterior vitreous detachment who exhibit this specific foveal anatomic configuration, a possible explanation is that this configuration represents changes caused by early manifestation of a strong adhesion between the posterior hyaloid and the internal limiting membrane not demonstrable by current OCT resolution. Because patients with obvious vitreomacular traction were excluded from this study, we indicate that this prediction technique is not relevant to that subgroup of patients. 

We believe the abnormal foveal OCT configuration illustrated in this study represents, at least for a subset of patients, the earliest recognizable OCT changes that indicate possible progression to macular hole formation. These changes are apparent on OCT before loss of foveal contour (stage 1A) can be observed biomicroscopically. Patients with this specific configuration are at higher risk of developing a macular hole. Clinically, this configuration is unique and can easily be identified by clinicians once recognized and searched for. Our survey among ophthalmology residents demonstrates correct identification of this specific configuration in all cases. 

The observation of this configuration enables physicians to identify at-risk patients even without a history of macular hole in the contralateral eye, leading to closer follow-up and, if needed, earlier surgical intervention, which was shown to correlate with better visual acuity outcome. ²⁰ This new analysis raises the question of what to do if a risk group is identified with high accuracy. There is no evidence yet that preventive pars plana vitrectomy may be of use. However, this new way of analysis may lay the ground for a newer impending macular hole study, patterned after the one that was inconclusively completed in the early 1990s. ²¹  

Although uncommon, cataract extraction surgery is a recognized risk factor for the formation of macular hole. ²² This known relationship is likely due to forces transmitted to the posterior segment and the macula altering its architecture during the cataract surgery. Because OCT imaging is now readily accessible, we believe that a patient recognized with a premacular hole foveal configuration before cataract extraction surgery should be alerted to the possible increased risk of macular hole formation after surgery. 

A study that follows patients over a timeline is needed to determine when a patient becomes high risk to develop a macular hole. Future research calls for a large-scale prospective study to follow patients with premacular hole configuration to establish the risk of developing full-thickness macular holes. 

The automated symbolic regression software (Eureqa) used in this study is unique and is capable of calculating a curves equation given its coordinates. Previously used for research in many fields of science, to our knowledge this is the first time it has been used to distinguish pathologic from nonpathologic processes in the eye. ²³ Because the foveal representation on an OCT two-dimensional cut is a curve and because OCT imaging is so widely used these days, we believe this way of mathematically describing the fovea may be useful for ophthalmology research and needs to be further explored in the future. 

In conclusion, the mathematical analog of a premacular hole foveal anatomic configuration was first described in this study and was found to be significantly different from normal foveal configuration. The clinical ability to identify this distinct macular configuration composed of steep nonsymmetrical foveal slopes and a wide fovea on OCT scans may allow early diagnosis, close follow-up, and better management of macular hole–prone patients. 

Footnotes

 Presented at the annual meeting of the Association for Research in Vision and Ophthalmology, Fort Lauderdale, Florida, May 2011.

Footnotes

 Supported in part by an unrestricted grant from Research to Prevent Blindness, Inc. New York, NY.

Footnotes

 Disclosure: Y. Barak, None; M.P. Sherman, None; S. Schaal, None