Mathematical Analysis of the Normal Anatomy of the Aging Fovea

Abstract

Purpose.: To mathematically analyze anatomical changes that occur in the normal fovea during aging.

Methods.: A total of 2912 spectral-domain optical coherence tomography (SD-OCT) normal foveal scans were analyzed. Subjects were healthy individuals, aged 13 to 97 years, with visual acuity ≥20/40 and without evidence of foveal pathology. Using automated symbolic regression software Eureqa (version 0.98), foveal thickness maps of 390 eyes were analyzed using several measurements: parafoveal retinal thickness at 50 μm consecutive intervals, parafoveal maximum retinal thickness at two points lateral to central foveal depression, distance between two points of maximum retinal thickness, maximal foveal slope at two intervals lateral to central foveal depression, and central length of foveal depression. A unique mathematical equation representing the mathematical analog of foveal anatomy was derived for every decade, between 10 and 100 years.

Results.: The mathematical regression function for normal fovea followed first order sine curve of level 10 complexity for the second decade of life. The mathematical regression function became more complex with normal aging, up to level 43 complexity (0.085 fit; P < 0.05). Young foveas had higher symmetry (0.92 ± 0.10) along midline, whereas aged foveas had significantly less symmetry (0.76 ± 0.27, P < 0.01) along midline and steeper maximal slopes (29 ± 32°, P < 0.01).

Conclusions.: Normal foveal anatomical configuration changes with age. Normal aged foveas are less symmetric along midline with steeper slopes. Differentiating between normal aging and pathologic changes using SD-OCT scans may allow early diagnosis, follow-up, and better management of the aging population.

Introduction

The fovea, derived from the Latin word for “small pit,” is the area of retina responsible for the highest visual acuity. This critical area is characterized by an avascular zone consisting of densely packed cones with elongated outer segments, and is surrounded by outwardly displaced inner retinal layers, which forms the foveal pit. ¹ The development of the fovea has been described from midgestation (approximately fetal weeks 20–22), when outward migration of the inner retinal layers begins to form the foveal depression, to approximately 13 years, when final cone density is reached. ² The anatomy of this specialized area of retina has long been under investigation. 

Initially, the development and structure of the fovea was described histologically. ^2,3 With the introduction of optical coherence tomography (OCT), high-resolution cross-sectional images of the fovea were made possible, enabling in vivo study of foveal structure. Subsequently, the prevalence of OCT use in the management of various retinal disorders has increased significantly over the past decade. 

Imaging with OCT is used for diagnosing and monitoring vitreoretinal disorders involving the fovea such as age-related macular degeneration, diabetic macular edema, macular holes, and vitreomacular traction. ^4–7 There is an increasing reliance on OCT to determine treatment plans for patients. As a result, it is imperative to differentiate normal aging process from abnormal pathological conditions. 

There is still an incomplete understanding, however, of how the normal fovea changes with age. In the literature, studies have looked at foveal thickness measurements in healthy eyes on OCT, ^8–10 as well as the use of mathematical models to describe foveal morphology. ^11,12 This is the first study to mathematically analyze the anatomical changes in the healthy fovea that occur with normal aging from years 10 to 100. 

Materials and Methods

This study was reviewed and approved by the Institutional Review Board at the University of Louisville. We analyzed 2912 spectral-domain optical coherence tomography (SD-OCT, Spectralis; Heidelberg Engineering, Heidelberg, Germany) normal foveal scans were analyzed; OCT data was collected as a compilation of all central B-scans, including central foveal thickness, and central foveal configuration. This was reviewed manually to ensure the central B-scan section was indeed selected for every eye included in the study. A retrospective chart review was performed, and data collected included age and sex of the patient, visual acuity, and significant medical and ocular history. Subjects included in the study were healthy individuals between ages 13 and 97 years, with visual acuity of 20/40 or better and without clinical evidence of any ocular pathology. Patients with an ophthalmic diagnosis of, but not limited to, diabetic retinopathy, age-related macular degeneration, epiretinal membrane, and history of retinal tear/detachment were excluded. Eyes (n = 390) were subsequently divided into nine age groups representing each decade of life (10–19, 20–29, 30–39, 40–49, 50–59, 60–69, 70–79, 80–89, and 90–99 years). 

Foveal thickness maps were analyzed using several measurements: retinal thickness around the fovea at 50-μm consecutive intervals, parafoveal maximum retinal thickness at two points lateral to the central foveal depression, the distance between two points of maximum retinal thickness, maximal foveal slope at two intervals lateral to the central foveal depression, and central length of foveal depression. 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. Symmetry of the foveal pit was established by dividing the numerical values of maximal slopes on each side along the foveal midline. The mathematical analog of the foveal configuration was analyzed using the automated symbolic regression software (Eureqa beta version 0.98, in the public domain at http://www.nutonian.com; Nutonian, Inc., Somerville, MA, USA), which uses a breakthrough machine learning technique called symbolic regression to unravel the intrinsic relationships in data and explain them as simple math. ¹³ Using symbolic regression, Eureqa can create incredibly accurate equations. Eureqa allows the user to choose the level of accuracy in which the function fits the gathered data. The fit for the current study was chosen to be 0.085, which provided a close fit of the curve to the data in each group. A unique mathematical equation representing the mathematical analog of the foveal anatomy was derived for every decade of life, between 10 and 100 years. The behavior of each curve was subsequently studied and analyzed. 

Statistical Analysis

Statistical analysis was carried out using statistical software (SPSS version 17.0; SPSS, Inc., Chicago, IL, USA). Multivariate analysis was performed using two-way ANOVA analysis. Values were considered significant if P < 0.05. 

Results

An equation was derived for each age group, representing the mathematical analog of the foveal anatomy (Fig. 1). Each equation was of good fit to the data, as indicated by the R ² goodness of fit value. The level of complexity of the curve, referring to the complexity of the mathematical equation describing the curve analog of the foveal anatomic configuration, was used to characterize the behavior of the regression function graphically. For the second and third decade of life, the mathematical regression equation followed first-order sine curve of complexity levels 10 and 8, respectively. For the fourth and fifth decades of life, the mathematical regression equation also followed a sine curve but have additional terms present with slightly increased complexity levels of 14 and 11, respectively. Beginning in the sixth decade of life, the mathematical analog deviated from a trigonometric pattern and adopted curves of increasing complexity. For example, in the sixth decade of life, the curve illustrates the behavior of a polynomial function of level 25 complexity. The regression functions of highest complexity occur in the seventh and ninth decades of life with exponential functions of level 43 complexity (Fig. 2). The foveal complexity increased in a sigmoidal manner according to the decade of life and the formula:  

 

 

Figure 1

 

The mathematical analog of foveal configuration for each decade of life is shown, with each graph depicting accuracy versus complexity. (A) 10–19 years. (B) 20–29 years. (C) 30–39 years. (D) 40–49 years. (E) 50–59 years. (F) 60–69 years. (G) 70–79 years. (H) 80–89 years. (I) 90–99 years. Size, level of complexity; solution, the mathematical regression function for each corresponding age group.

Figure 2

 

Foveal analog complexity, volume, and symmetry along the midline as a function of age are shown. Foveal complexity increased in a sigmoidal manner according to the decade of life. Foveal symmetry decreased quadratically with age. Foveal volume for each decade of life negatively correlated with age.

Demographic and foveal characteristics for each age group are shown in the Table. Young foveas had higher symmetry (0.92 ± 0.10) along the midline, whereas aged foveas had significantly less symmetry (0.76 ± 0.27, P < 0.01) along the midline and significantly steeper maximal slopes (29 ± 32°, P < 0.01). Foveal symmetry decreased quadratically with age according to the formula (Fig. 2):  

 

 

Table.

 

Demographic and Foveal Characteristics for Each Decade of Life

Age Groups	10–19	20–29	30–39	40–49	50–59	60–69	70–79	80–89	90–99	P Values
Mean age, y	17.16 ± 2.53	24.39 ± 2.95	33.03 ± 3.62	44.79 ± 2.73	54.98 ± 3.12	65.33 ± 2.77	74.37 ± 2.97	84.03 ± 2.97	91.44 ± 2.24	0.32
Volume, mm3	4.64 ± 1.37	4.55 ± 1.00	4.53 ± 1.72	4.39 ± 1.11	4.39 ± 1.00	3.87 ± 1.10	4.00 ± 1.33	3.70 ± 1.24	3.19 ± 0.89	0.21
Maximum slopes	21 ± 12°	23 ± 12°	25 ± 13°	26 ± 15°	28 ± 17°	29 ± 19°	32 ± 22°	28 ± 23°	29 ± 32°	<0.01*
Midline foveal symmetry	0.92 ± 0.10	0.90 ± 0.12	0.89 ± 0.13	0.87 ± 0.15	0.83 ± 0.23	0.83 ± 0.24	0.79 ± 0.24	0.78 ± 0.25	0.76 ± 0.27	<0.01*
Regression function complexity	10.00	8.00	14.00	11.00	25.00	43.00	16.00	43.00	21.00	<0.01*
Eyes, n	31	59	30	39	59	79	49	35	9	0.52
Males, %	41.9	74.6	53.3	41.0	33.9	35.4	49.0	37.1	44.4	0.53
Females, %	58.1	25.4	46.7	59.0	66.1	64.6	51.0	62.9	55.6	0.53
Caucasians, %	74.2	69.5	80.0	82.1	84.7	83.5	89.8	91.4	88.9	0.33
Asian, %	12.9	10.2	0.0	5.1	0.0	0.0	0.0	0.0	0.0	N/A†
Middle Eastern, %	12.9	6.8	6.7	0.0	0.0	0.0	0.0	0.0	0.0	N/A†
African Americans, %	0.0	6.8	6.7	12.8	15.3	16.5	10.2	8.6	11.1	N/A†
Hispanics, %	0.0	6.8	6.7	0.0	0.0	0.0	0.0	0.0	0.0	N/A†

*  Values of P < 0.05 were considered statistically significant.

†   P value could not be calculated due to small sample numbers.

Foveal volume for each decade of life negatively correlated with age (Fig. 2) according to the formula:  

 

 

There were no significant sex differences in foveal parameters (P = 0.53). Subjects included those from different ethnic backgrounds; however, due to limited sample size for these populations, no conclusions could be made as to any possible difference in foveal parameters according to race. An attempt to compare between the two eyes of each individual failed because many of the subjects were seen for another condition in the fellow eye. There was no correlation between different visual acuities within each decade of life and the mathematical analog of the foveal anatomy. 

Discussion

With increasing reliance on OCT imaging in diagnosing and treating various disorders involving the fovea, better knowledge of what constitutes “normal” foveal anatomy is essential. Furthermore, understanding the normal aging foveal structure is important in the treatment of the many vitreoretinal disorders that span the different decades of life. This is the first study to mathematically analyze the anatomical changes in the healthy fovea that occur with normal aging. 

Previous studies have looked at OCT macular thickness measurements in healthy eyes in order to establish normative data for different OCT instruments. ^14–16 However, generalized macular thickness measurements across a broad population of patients are not sufficient. The effect of other parameters on macular thickness has been studied, including age, race, axial length, high myopia, and sex. ^{8–10,17–20} Males and Caucasians have been shown to have greater macular thickness on OCT than females and African-Americans, respectively. ^9,10,19,21 Results have varied as to age, with some studies finding no change in macular thickness ^9,16 and others finding a decrease in macular thickness with age. ^8,19  

In addition to central macular thickness measurements, more detailed analysis of foveal morphology, including depth, diameter, and slope, adds further insight into normal foveal anatomy amongst different demographic groups. A mathematical model, using several preset measurements, can achieve reliable and reproducible results across a variety of different datasets. Several studies have used a mathematical description to try and better understand foveal anatomy. 

Dubis et al. ¹² used an automated algorithm to extract the slope, depth, and diameter of the foveal pit from OCT images of 65 patients, finding considerable variation in all parameters. Wagner-Schuman et al. ²¹ also used an algorithm to assess foveal pit morphology in 90 patients. In that study, there was no difference between males and females in foveal pit morphology and African Americans had significantly deeper and broader foveal pits than Caucasian. ²¹ In the current study, the focus was to investigate the change in normal foveal anatomical configuration with each decade of life. 

Mathematical models have also been used to distinguish pathologic foveal changes from nonpathologic changes. Our group has previously described using a previous version of Eureqa, the mathematical anatomic foveal configurations in patients who subsequently developed a macular hole. ²² We described a significant difference between normal foveal configurations and premacular hole foveal configurations, which allowed earlier diagnosis and follow-up of patients prone to the development of macular holes. ²² Mathematical modeling of the fovea is also being used in other diseases, such as Parkinson, ²³ in order to distinguish normal fovea from pathologic. With this increasing use of mathematical analysis of the fovea to distinguish normal from pathologic, the need for an understanding of normal foveal structure over the decades of life is of even more significance. 

The current study used a mathematical analog of foveal configuration to show the normal foveal anatomical changes with age. Normal aged foveas are less symmetric, with significantly steeper slopes, although there was no significant difference in foveal volume as compared to younger foveas. The mathematical regression function became significantly more complex with aging. Not only is this important as a baseline for distinguishing pathologic changes from nonpathologic changes, this may perhaps also explain why many elderly patients with no obvious retinal pathology have unexplained visual complaints of decreased vision, the asymmetric altered morphology of the aged fovea may be less efficient in creating a crisp-clear image. Of note, other foveal variables including reflectivity, texture, color, irregularities, as well as circular and global symmetry may contribute to foveal characterization. These were not accounted for in our study and may be included in future research. 

In conclusion, as technology continues to advance, the importance of imaging in managing patients with vitreoretinal disease involving the fovea will only continue to grow, therefore it is pertinent to define the norm, and differentiate normal aging from disease process. This study showed that the normal fovea exhibits an increasing asymmetry with steeper slopes and more complexity with age. This knowledge may be used to help differentiate between normal and pathologic foveal aging changes on SD-OCT scans, allowing for earlier diagnosis, follow-up, and better management of the aging population. 

Acknowledgements

Presented in part at the annual meeting of the Association for Research in Vision and Ophthalmology, Orlando, Florida, United States, May 2014. 

Supported in part by an unrestricted grant from Research to Prevent Blindness. The authors alone are responsible for the content and writing of the paper. 

Disclosure: B. Nesmith, None; A. Gupta, None; T. Strange, None; Y. Schaal, None; S. Schaal, None