Mathematical Projection Model of Visual Loss Due to Fuchs Corneal Dystrophy

Abstract

Purpose.: To devise a mathematical disease classification model for eyes with primary guttata cornea, on the bases of endothelial loss trajectory and probability of advanced disease.

Methods.: A series of 1971 patients (3281 eyes), some with and some without guttata corneas, undergoing specular microscopy were retrospectively reviewed. The eyes were classified into four stages; stage 0, without guttae; 1, guttata cornea without edema; 2, mild Fuchs' corneal dystrophy (FCD); and 3, severe FCD, according to clinical records, and patient age and corneal endothelial cell density (ECD) were plotted. Nonparametric density smoothing was used to create a contour map, and a best-fit curve for ECD loss was calculated. The relation between ECD decrease rate and the stages were evaluated.

Results.: Endothelial decrease rate in stage 0 was 0.44%/year, which was compatible with that of normal eyes reported in previous studies. Decrease rates of stages 1, 2, and 3 were 0.81%, 2.65%, and 3.08%/ year, respectively. The age-ECD loss curves of 1.40%/year (ECO_1.4) and 2.00%/year (ECO_2.0) further divided stage 1 into three subgroups; stage 1a, asymptomatic guttata cornea; 1b, borderline guttata cornea; and 1c, pre-FCD. The ECO_2.0 cutoff line differentiated eyes with FCD from those without edema with a sensitivity and specificity of >90%. Stage 1c eyes were below ECO_2.0 and had a decrease rate as high as FCD.

Conclusions.: This mathematical model can be used to predict the prognosis of patients with primary guttata cornea.

Fuchs' corneal dystrophy (FCD) is a progressive, bilateral corneal dystrophy. ¹ There is a progressive loss of corneal endothelial cells with secretion of an abnormally thickened basement membrane, leading to corneal guttae formation. ¹ On specular microscopy, these corneal guttae are observed as dark areas. ^1,2 As endothelial function deteriorates, corneal edema increases and visual acuity declines, ² and FCD is a major indication for keratoplasty (corneal transplants) in the United States. ^{3 –5} Although FCD is recognized as a dominantly inherited disorder, females are predisposed to it and develop corneal guttae 2.5 times more frequently than do males, progressing to corneal edema 5.7 times more often than do males. ⁶ The prevalence of primary guttata cornea and FCD are lower in Japan than in the United States. ^7,8 This difference in prevalence is thought to be mainly attributable to the racial difference. ⁷  

Primary guttata cornea is believed to be a preliminary stage of FCD. Krachmer et al. ⁶ graded guttata cornea and FCD according to a spread of guttae and reported that there was a positive correlation between age and grade of guttae. However, the exact natural course of guttata cornea, or whether all cases of guttata cornea progress to FCD remains to be determined. A prospective study that follows the decline in endothelial cells density (ECD) with age would be ideal for predicting the natural course of guttata cornea; however, a very long follow-up would be required, and recruiting asymptomatic potential patients is practically impossible, especially in Japan. A retrospective study with a large database and an adequate mathematical model can be used in a similar way to predict the prognosis of patients with guttata cornea. In this report, we retrospectively reviewed age and ECD in a large group of hospital-based patients and evaluated the prevalence of guttae, male:female ratio, and distribution of age and ECD. In addition, we propose a new classification of guttata cornea based on a mathematical model that adequately predicts the prognosis of disease. 

Methods

Subjects

Clinical records of outpatients who underwent specular microscopy for corneal endothelial cell counts from January through December 2009 in six hospitals affiliated with the Fuchs' Corneal Dystrophy Study Group of Japan were retrospectively reviewed. The purpose of specular microscopy for those patients were routine examination before ocular surgery, follow-up for corneal diseases that were thought to have little effect on endothelium (such as keratoconus or lattice corneal dystrophy), or follow-up for diagnosed Fuchs' corneal dystrophy. Patients who had a history of trauma, corneal infection, intraocular inflammation, intraocular surgery, or laser iridotomy were excluded from the study. Endothelial photographs were taken at the center of the pupillary area with a noncontact specular microscope (Nonkon Robo F & A; Konan Medical, Nishinomiya, Japan, or EM-3000; Tomey, Nagoya, Japan), and analyses of the photographs were performed with an automatic cell analysis system attached to the microscope. Data concerning patient age, sex, presence of guttae, and ECD were recorded. The eyes were classified into four groups by slit lamp examination according to modified Stocker's classification ⁹ :

•  
  Stage 1: Guttata cornea without the stroma or the epithelium being affected
•  
  Stage 2: Permeation of corneal stroma with fluid, edema of epithelium, and bullae formation
•  
  Stage 3: Late stages with subepithelial connective tissue formation, vascularization, and scar formation

Other eyes without corneal guttae were classified as stage 0. During the rest of the article, the term Fuchs' corneal dystrophy (FCD) represents stage 2 and 3, since eyes in these stages have symptoms related to corneal edema. The study complied with the Declaration of Helsinki. Approval was granted by the Committee for the Protection of Human Subjects of each hospital. 

Mathematical Model of Endothelial Cell Loss Rate

To construct a mathematical model of decrease in endothelial cells, we made the following two assumptions:

1.  
   The ECD at 5 years of age is 3600 cells/mm². This is common to all classes.
2.  
   From 5 years of age, the decrease rate (percent/year) of ECD is constant in each class, but different between classes.

Murphy et al. ¹⁰ reported that during first 2 years of life ECD decreased rapidly because of corneal growth, and after that the decrease rate slows down to 0.56%/year. The effect of corneal growth on ECD ends at 5 years of age or earlier. To simplify our mathematical model, we assumed that ECD at 5 years of age was common to all classes and regarded this point as the base point of age-ECD curve in our mathematical model. Because the onset of FCD is in adulthood, we believe that this assumption is acceptable. We substituted the mean ECD of normal 5-year-old children (3600 cells/mm²) in the report of Nucci et al. ¹¹ for the base point. We assumed that the (percentage) decrease rate is dependent on the class, and it is constant in each class from 5 years of age. Based on these assumptions, the following differential equation stands:  

 

where t is age 5 years; E _(t) is endothelial cell density at t years (in cells per square millimeter); and D is the decrease rate (percent). 

The solution to the differential equation is the following:  

 

Using this mathematical model, an age-ECD curve in each class can be drawn by the least-squares method. An age–ECD curve of optimal decrease rate can be drawn as well. 

Statistical Analysis

Scatterplotting, analysis of variance (ANOVA), nonparametric density smoothing, age-ECD curve, and other statistical analyses were calculated by or written in commercial software (Excel 2007; Microsoft, Redmond, WA, and JMP 8 software; SAS, Cary, NC). P < 0.05 was considered statistically significant. 

Results

Characteristics of Patients

Age, sex, and stage of reviewed patients and eyes are presented in Table 1. The prevalence of guttata cornea (stage 1+2+3) was 12.73%. The prevalence of stage 1 was 10.65%, and FCD (stage 2+3) was 2.08%. The male: female ratio in each stage was as follows; 1: 1.03 (stage 0), 1: 1.88 (stage 1), 1: 2.43 (stage 2), and 1: 4.67 (stage 3). Females were more predisposed to stage 1 or FCD than males, and the ratio increased in advanced stages. 

Table 1.

 

The Age, Sex, and Stages of Reviewed Patients and Eyes

 

Age-ECD Curve of 2.0% Differentiates Fuchs' Dystrophy

Figure 1, left shows the scatterplot between age and ECD for each stage. Nonparametric density smoothing was drawn on the scatterplot (Fig. 1, right), which represents the contour of plot density. The age-ECD curves based on our mathematical model were drawn by the least-squares method. Table 2 shows ECD with sample sizes at 5-year intervals for grades 0 to 3, which enables the mean ECD data of grade 0 to 3 to be compared at various ages. 

Figure 1.

 

Scatterplots (left) and contour maps of nonparametric density smoothing (right) of each stage. (A-1, A-2) Stage 0, (B-1, B-2) stage 1, (C-1, C-2) stage 2, and (D-1, D-2) stage 3. Red curves: age-ECD curves of each stage calculated by least-squares method. The decrease rates of each stage were 0.44% (stage 0), 0.81% (stage 1), 2.65% (stage 2), and 3.08% (stage 3). The contour maps showed that the age-ECD curve of 2.00% decrease rate (ECO_2.0, black curves) ran through a trough between peaks of all stages. Most of the peaks in stages 0 and 1 were located above ECO_2.0, whereas peaks of stages 2 and 3 were located below ECO_2.0.

Table 2.

 

Mean ECD with Sample Sizes at 5-Year Intervals for Grades 0 to 3

	0–9 y		10–14 y		15–19 y		20–24 y		25–29 y
	Eyes	ECD	Eyes	ECD	Eyes	ECD	Eyes	ECD	Eyes	ECD
Stage 0	4	3073.3 ± 392.6	7	3020.4 ± 330.1	47	2769.2 ± 530.1	31	2837.4 ± 567.3	60	2853.1 ± 507.6
Stage 1	0	—	0	—	0	—	4	2765.0 ± 128.8	6	2954.5 ± 175.6
Stage 2	0	—	0	—	0	—	0	—	0	—
Stage 3	0	—	0	—	0	—	0	—	0	—

	30–34 y		35–39 y		40–44 y		45–49 y		50–54 y
	Eyes	ECD	Eyes	ECD	Eyes	ECD	Eyes	ECD	Eyes	ECD
Stage 0	58	2732.6 ± 511.3	54	2741.9 ± 324.7	80	2672.2 ± 462.5	99	2687.8 ± 507.8	128	2754.6 ± 370.5
Stage 1	0	—	4	2423.0 ± 474.1	7	2503.7 ± 541.9	7	1934.3 ± 763.9	14	1865.2 ± 703.0
Stage 2	0	—	0	—	0	—	2	881.0 ± 60.8	2	592.0 ± 120.2
Stage 3	0	—	0	—	1	461.0	1	622.0	0	—

	55–59 y		60–64 y		65–69 y		70–74 y		75–79 y
	Eyes	ECD	Eyes	ECD	Eyes	ECD	Eyes	ECD	Eyes	ECD
Stage 0	195	2701.2 ± 408.1	325	2671.9 ± 464.4	384	2677.7 ± 449.1	494	2698.4 ± 435.0	496	2691.2 ± 421.3
Stage 1	25	2105.2 ± 673.3	28	2219.4 ± 695.5	39	2124.8 ± 743.7	61	2242.5 ± 719.4	44	2159.0 ± 741.7
Stage 2	4	645.8 ± 224.3	2	797.5 ± 282.1	7	562.9 ± 329.5	7	730.7 ± 149.5	7	483.0 ± 183.7
Stage 3	2	284.5 ± 21.9	0	—	0	—	2	302.5 ± 3.5	7	524.0 ± 418.9

	80–84 y		85–89 y		≥90 y
	Eyes	ECD	Eyes	ECD	Eyes	ECD
Stage 0	309	2698.9 ± 440.4	116	2624.5 ± 457.3	22	2563.7 ± 299.3
Stage 1	47	2264.2 ± 556.2	17	2279.2 ± 597.9	5	2962.0 ± 597.1
Stage 2	7	680.6 ± 318.1	3	723.3 ± 155.7	0	—
Stage 3	5	302.4 ± 5.4	3	482.3 ± 97.1	2	352.5 ± 74.2

 Eye data are expressed as the number, and the ECD in cells per square millimeter.

The decreased rate curve of stage 1 age-ECD was 0.81%, which was closer to that of stage 0 (decrease rate, 0.44%) than that of stage 2 (2.65%) or stage 3 (3.08%). The decrease rate of stage 0 in our study was 0.44%, which is within the range of normal eyes reported in previous studies (Table 3). ^{10,12 –16} Contour maps show that most of the peaks in stage 0 and 1 were located above the age-ECD curve of the 2.00% decrease rate, whereas peaks of stage 2 and 3 were located below this curve. Table 4 shows binary classification based on the age-ECD curve of a 2.00% decrease rate, designated novel ECD cutoff 2 (ECO_2.0), dividing stages 0+1 and stages 2+3 (Table 4) or stage 1 and stages 2+3 (Table 4). The high sensitivity and specificity of these classifications suggested that ECO_2.0 is an adequate cutoff between eyes with corneal edema and those without edema. 

Table 3.

 

Decrease Rates of Stage 0 in the Present Study and Normal Unoperated Eyes Reported in the Previous Studies

Author	Decrease Rate (%/y)	Nation
Murphy et al. 10	0.56	United States
Cheng et al. 12	1.00	England
Ambrose et al. 13	0.60	England
Numa et al. 14	0.30	Japan
Bourne et al. 15	0.60	United States
Rao et al. 16	0.30	India
Present study	0.44	Japan

Table 4.

 

Binary Classification of Clinical Stage

Clinical Stage	Classification Based On ECO2.0		Total
	Below ECO2.0	Above ECO2.0
Total Eyes
Stage 2+3	60	4	64
Stage 0+1	122	3095	3217
Total	182	3099	3281
Sensitivity, %	93.75
Specificity, %	96.21
Eyes with Guttata Cornea
Stage 2+3	60	4	64
Stage 1	27	281	308
Total	87	285	372
Sensitivity, %	93.75
Specificity, %	91.23

 Data are based on the age-ECD curve of 2.00% decrease rate as a novel ECD-cut-off (ECO_2.0), sensitivity and specificity to detect stage 2+3 from total eyes or the eyes with guttata cornea based on the classification.

Age-ECD Curve of 1.4% and 2.0% Divides Stage 1 into Three Distinct Groups

The contour map of stage 1 consisted of several peaks. Figure 2 shows that the age-ECD curve of the 1.40% decrease rate, designated novel ECD-cutoff point 1 (ECO_1.4), divides these peaks into a high-density group (>ECO_1.4), and a low-density group (<ECO_1.4). ANOVA revealed that the age-ECD curves of each group predicted ECD according to age, with statistical significance: The F ratio and P value were 803.3 and <0.0001 in the high-density group and 945.7 and <0.0001 in the low-density group. The decrease rate of the age-ECD curve in the high-density group was 0.56%, which was very close to that of the stage 0 age-ECD curve. On the other hand, the decrease rate in the low-density group was 2.00%, which coincided with ECO_2.0. These results suggest that the decrease rate of the high-density group in stage 1 was nearly normal, whereas the low-density group in stage 1 was located on the border between eyes with and without corneal edema. We therefore classified stage 1 on the basis of ECO_1.4 and ECO_2.0, as follows (Fig. 3):

•  
  Stage 1a, asymptomatic guttata cornea (AGC): above ECO_1.4
•  
  Stage 1b, borderline guttata cornea (BGC): between ECO_1.4 and ECO_2.0
•  
  Stage 1c, preliminary stage of FCD (pre-FCD): below ECO_2.0

Figure 2.

 

(A) The contour map of nonparametric density smoothing in stage 1. Stage 1 consisted of several peaks, and the age-ECD curve of 1.40% decrease rate (ECO_1.4, green curve) ran through a trough between peaks of high ECD group (black asterisks) and low ECD group (red asterisks). (B) High-density group in stage 1 above ECO_1.4. The age-ECD curve of this group (red curve) was close to that of stage 0 (red dotted curve), and the calculated decrease rate was 0.56%. (C) Low-density group in stage 1 below ECO_1.4. The age-ECD curve of this group (red curve) coincided with ECO_2.0 (black dotted curve), with a decrease rate of 2.00%.

Figure 3.

 

Proposed classification of eyes in stage 1 based on ECO_1.4 and ECO_2.0. Eyes in stage 1a above ECO_1.4 were named AGC, which had a decrease rate as low as stage 0. Eyes in stage 1c below ECO_2.0 had a decrease rate as high as FCD (stages 2 and 3), and therefore, this stage was named pre-FCD. Stage 1b between ECO_1.4 and ECO_2.0 was named BGC. The table below the graph shows the coordinates of ECO_1.4 and ECO_2.0.

Discussion

To obtain a sufficient number of age-ECD data to compare FCD (stage 2+3), guttata cornea without edema (stage 1), and control group without guttata cornea (stage 0), we performed a retrospective, hospital-based review of total 1971 outpatients. In this study, we found a somewhat higher prevalence of guttata cornea than that found in previous reports in Japan. The prevalence of corneal guttae was reported to be 3.7% (1.5% in men, 5.5% in women) in Japan, ^17,18 whereas it ranges from approximately 7% up to a remarkable 70.4% in North America, Iceland, and Europe. ^1,8,19 In our study, the fact that subjects were hospital-based may have caused a higher prevalence. However, such bias does not have an effect on the validity of the mathematical model derived from the data. The following tendency of prevalence was apparent in our group of subjects: First, females were more predisposed to stages 1, 2, and 3 than were males, and the female ratio increased as stages progressed. Second, the prevalence of FCD was much smaller than stage 1. An increase in the female ratio in progressing stages suggested that sex may have some role not only in the onset but also the progression of the disease. Apparent difference of prevalence between FCD and stage 1 suggest the existence of a patient group in stage 1 that does not progress to corneal edema despite having guttata cornea. 

Our model is based on the assumptions that the ECD at 5 years of age is common to all classes and that the decrease rate of ECD percentage per year) is constant but with a different value of each class. The use of these assumptions may be a debatable point when discussing the validity of our study. However, the results of our mathematical model show ECD decrease rates that are acceptable when compared with clinical observations. The decrease rate of 0.44% in stage 0 is within the range of values of normal unoperated eyes reported in the previous studies. ^{10,12 –16} Furthermore, since ECO_1.4 and ECO_2.0 runs through a clearly defined trough between peaks on the scatterplot, and ECO_2.0 divided stages 0+1 and stages 2+3 or stage 1 and stages 2+3 with high sensitivity and specificity, we believe our mathematical model for classifying patients with guttae based on ECD decrease rates is adequate for predicting the prognosis. 

The ECO_1.4 and ECO_2.0 curves based on our mathematical model divided stage 1 into three subgroups, stage 1a, 1b, and 1c. The ECD decrease rate of stage 1a was close to that of stage 0, that is, almost normal. Schnitzer and Krachmer reported on 44 relatives of 12 families with guttata cornea which appeared normal on slit-lamp examination and endothelial cell parameters. ²⁰ These eyes presumably belonged to stage 1a of our classification. In addition, because the distribution of patients of stage 1a was located above ECO_1.4, the risk of progressing to corneal edema may be as low as stage 0. If a patient was on the curve of a 1.4% decrease rate, the ECD would be 1095 cells/mm² even when he was 90 years old. Presumption of low risk of stage 1 is supported by analysis of variance, showing that age-ECD curves of each stage had significant predictability. 

It was surprising that the age-ECD curve of the low-density group of stage 1 (stages 1b and 1c) coincided completely with ECO_2.0. The former was calculated by the least-squares method of the low-density group of stage 1, whereas the latter was obtained from trough between peaks of stages 0 to 3 on scatterplots. This result suggests that the low-density group of stage 1 was located on the border between stage 0 and FCD. Eyes in stage 1c below ECO_2.0 have a decrease rate as high as FCD, suggesting that these eyes have a risk to progress to FCD, even if there was no corneal edema present. This was the rationale for referring to stage 1c as pre-FCD. Further prospective study of patients in stage 1b and 1c is needed to determine whether stage 1c is a preliminary stage of FCD. 

Recently, several pathogenic mechanisms, such as oxidative stress or unfolded protein response, have been reported as causes of FCD. ^21,22 The difference in resistance against such stress may cause the difference in decrease rates between stages. Previous reports suggested that ECD of some eyes with guttata cornea did not decrease significantly compared with normal eyes after cataract surgery, ^7,23 whereas some eyes in other reports showed a significantly higher decrease. ²⁴ When we adapted data from these reports to our classification, we found that most of the former eyes with no difference in ECD (18/21 eyes) were categorized as stage 1a, suggesting that our classification may be used to identify patients with a higher risk of endothelial damage due to external stress. Future studies on guttata corneas using our classification may help clarify the mechanism of FCD progression. 

In conclusion, we assessed distribution and endothelial loss rate of guttata cornea stages 0 to 3 and determined new cutoff curves ECO_1.4 and ECO_2.0 by using scatterplots. Our mathematical model is a simple method for predicting the prognosis of patients with guttata cornea. 

Footnotes

 Supported by a grant from the Ministry of Health, Labor and Welfare, Japan. The sponsor or funding organization had no role in the design or conduct of this research.

Footnotes

 Disclosure: S. Hatou, None; S. Shimmura, None; J. Shimazaki, None; T. Usui, None; S. Amano, None; H. Yokogawa, None; A. Kobayashi, None; X. Zheng, None; A. Shiraishi, None; Y. Ohashi, None; T. Inatomi, None; K. Tsubota, None