## Abstract

The human eye has ethnic difference, the existing typical eye models are based on western eyes. A generic eye model based on Chinese population is presented for the first time. The statistical analyzed ocular parameters based on measured data are used for the initial generic eye model, and the wavefront aberration data obtained at two different pupil diameters are used for reproduction by optimizing the initial generic eye model. The differences and similarities between Chinese generic eye model and western eye models are given. The Chinese generic eye model provides a suitable model for the related further researches and applications on Chinese eye.

© 2009 Optical Society of America

## 1. Introduction

From the point of view of optics, the human eye is an optical instrument [1] consisting of several refractive optical elements, such as cornea, pupil, lens, retina and so on. The building of a schematic eye model based on biometric measurement of the eye and optical properties of ocular refractive surfaces is helpful in teaching aids in optics, optometry, ophthalmology, psychology (vision and visual perception) and visual ergonomics [2]. Moreover, this can be used to understand and research the optical performances of eye in refractive surgical procedures, such as photorefractive keratectomy (PRK), laser-assisted in situ keratomileusis (LASIK) [3–5] and intraocular lens implants [6–8].

Since the first paraxial schematic eye was proposed by Listing in 1851 [2], many typical generic eye models [2, 9–15] have been proposed, developed and improved. For example, at the beginning of 20th century, Gullstrand [2] presented three eye models with different refractive surfaces. W. Lotmar [9] published a theoretical eye model with aspherics in 1971. J. W. Blaker [10] presented an adaptive model of the human eye in 1980. In 1997 Hwey-Lan Liou & Nobel A. Brennan [13] proposed a finite model eye, based on anatomical data of eyes. In 1999 Isabel Escudero-Sanz & R. Navarro [14] developed the eye model proposed by R. Navarro et al [12] in 1985 with wide-angle.

In the field of human eye models, the related researches or reports in China are not as many as those in the western countries. There are only a few reports about the eye models analysis until recent years. For example, in 2002 Zhao et al [16] analyzed the effects of aspheric surfaces and gradient-index on optical image of the eye based on the eye model of Gullstrand-Le Grand. In 2005 Liu et al presented a new eye model based on the eye models of Liou and Gullstrand-Le Grand with tear film [17] and with a shell-structure lens [18], respectively. Zhu et al [19] analyzed the influence of different factors on diopter accommodation of accommodative intraocular lens based on Liou model eye in 2007. These above researches in China are all based on western eye models, but the human eye has ethnic difference, and as far as we know, there is no report on the generic eye model based on Chinese population.

## 2. Method

Cooperating with the ophthalmologists in Jiangsu Provincial Hospital of Traditional Chinese Medicine (Nanjing University of Traditional Chinese Medicine Affiliated Hospital) and Shanghai EENT Hospital of Fudan University, we obtained the related data of the human eyes in a large population. Based on the statistical analyzed results of these measured data, a Chinese generic eye model is built with optical design software Zemax-EE (Zemax Development Corp. San Diego, Calif.).

In general, the image quality of an optical system with known parameters is evaluated by using an optical design program. However, the lens parameters of human eye are not known exactly because of the difficulties of measurement and the complexity of its structure. In the process of building the eye optical model, the image quality of the eye optical system is measured with Shack-Hartmann wavefront sensor, and with the measured known parameters such as corneal surface radii, intraocular distances and so on, the unknown lens parameters in the eye model are optimized by Zemax. Here the measured eye wavefront aberration is used to represent the image quality, and the lens parameters can be estimated by reproducing the measured eye aberration, so we refer to this method as reverse building. The flow diagram of reverse building is shown as Fig. 1.

In Fig. 1, the generic eye model is built based on the measured data and fitting simulations mainly. The initial eye model is from the measured data, the unmeasured lens gradient-index (GRIN) distribution and chromatic dispersion of ocular media are simulated as initial parameters with the data from related references [10,12–15,20,21]. These two parts will be introduced hereinafter. In the process of reverse building, the eye wavefront aberrations, described by Zernike polynomials, are measured by eye aberrometers which are based on the Shack-Hartmann sensor principle [22]. The measured eye aberrations of the outward going wavefront scattered by the retina have an opposite sign to the inward going one. In order to reproduce the eye image quality with high fidelity, the measured eye wavefront aberrations with two different pupil diameters are introduced simultaneously by phase plates at the entrance pupil of eye model. The system of the initial eye model with the phase plates will be optimized to minimize the RMS wavefront aberration. In this way, after optimization and when the phase plates are removed, the eye model alone will replicate the measured wavefront aberrations. The method of introducing the measured wavefront aberrations to eye models had been proven to be effective [23, 24].

## 3. Statistical analyzed of the measured data

Currently, myopia accounts for a large proportion of the normal population in China, and an emmetropic selection is necessary in many subjects. After the selection with a large amount of data, 50 emmetropic eyes of no eye diseases with uncorrected visual acuity better than 20/25, spherical power ranged from +0.75~-1.75DS, cylinder refraction <0.75D are chosen. The subjects are adults with average age of (26.64±5.22) years. The measured parameters include anterior and posterior radii, central thickness of the cornea, intraocular distances (anterior chamber depth, lens central thickness, and the overall axial length; vitreous depth is obtained by subtracting the other distances from the overall axial length), and eye aberrations. Every parameter is obtained with at least three measurements. In the two hospitals, the measurement apparatuses used are different (Table 1). The measurement of corneal anterior surface topography and wavefront aberrations were taken with CRS-Master Twinline system (Carl Zeiss, Jena, Germany), Orbscan ⎕ corneal topographer and Zywave wavefront aberrometer (Baush & Lomb, USA). The intraocular distances were measured with Pentacam anterior segment analysis system (Oculus, Wetzlar, Germany), IOL-Master optical biometer (Carl Zeiss, Jena, Germany) and A/B ultrasonic diagnostic instrument.

#### 3.1 The corneal parameters and intraocular distances

Generally speaking, if a quantity is composed of many small independent random factors affecting the results, it can be considered show a normal distribution (see Central Limit Theorem). The measured corneal parameters and intraocular distances are analyzed with normality test with statistical analysis software SPSS 13.0 (SPSS Inc., Chicago). The mean values of measured data are adopted in the original generic eye model, the 95% confidence interval of these data are used to be as the variable range during optimization.

The graphic method of normality test in SPSS 13.0 by Probability-probability Plot (P-P plot) and Quantile-quantile Plot (Q-Q plot) can be used [25]. In the P-P plot, using the cumulative frequency of samples as abscissa and the corresponding cumulative probability calculated in accordance with the normal distribution as ordinate, the sample values are the scattered points in the Cartesian coordinate system. If the sample is subject to normal distribution, then these scattered points will be around the diagonal of the first quadrant distribution. Likewise, in the Q-Q plot, using sample quantile as abscissa and the corresponding quantile calculated in accordance with the normal distribution as ordinate, the sample values are the scattered points in the Cartesian coordinate system. Similar to P-P plot, if the sample is subject to normal distribution, then these scattered points will show a diagonal line of the first quadrant distribution in the Q-Q plot. The method of Q-Q plot is more efficient [25], so we use this graphic method to deal with the measured corneal parameters and intraocular distances.

Take the anterior corneal surface curvature radii for example. According to the corneal toric surface properties with orthogonal steep meridian (the smallest radius of curvature) and flat meridian (the largest radius of curvature) [26], the steep curvature radius R_{x} and the flat curvature radius R_{y} are measured. The Q-Q plots of these data are shown as Fig. 2 and Fig. 3 respectively.

From Fig. 2 and Fig. 3, the data points are around the diagonal of the first quadrant distribution in (a) Normal Q-Q Plots. It indicates that the anterior corneal surface curvature radii appear normal distribution. In (b) Deviation from Normal Q-Q Plots, the residuals distribute around Y=0 mostly, and the absolute values of most residuals are less than 0.12 and 0.2 respectively, it indicates that the normality properties of the anterior corneal surface curvature radii are nice.

Figure 4 and Fig. 5 depict the Normal distribution histogram of the anterior corneal surface steep curvature radius R_{x} and the anterior corneal surface flat curvature radius R_{y}, respectively. It is calculated that the average values are R_{x}=(7.77±0.21)mm, R_{y}=(7.93±0.23)mm, and the 95% confidence intervals of R_{x} and R_{y} are (7.71, 7.83)mm and (7.86, 8.00)mm respectively. The data of other parameters are dealt with the same method as the anterior corneal surface curvature radii. The mean values ± standard deviation (SD) and 95% confidence intervals of these measured parameters are listed in Table 2.

With the values of ocular parameters in Table 2, we can make a comparison with the experimental mean values of western eyes [27–31]. For the anterior and posterior corneal surfaces, by comparing with the data presented by Dubbelman et al [27], it shows the similar values in Chinese eyes. The lens thickness in the eye models of Navarro and Escudero & Navarro was from the study by Brown [28], which employed in vivo Scheimpflug imaging [29]. The lens thickness in Liou & Brennan model eye was based on experimental data measured by various techniques in 100 normal emmetropic human subjects age 18–70 yr [30]. On the other hand, the dispute on the in vivo measurement data between Dubbelman et al [31] and Koretz et al [29] indicates that the accuracy of the measurement data has not been determinated till now. So, we can make the comparison of lens thickness between eye models [9–14]. For the lens central thickness, Chinese eyes have smaller lens central thickness. Moreover, comparing with the values in western eye models, it shows Chinese eyes have thicker anterior chamber depth and shorter overall axial length. These mean values will be adopted as the original Chinese generic eye model, and the corresponding 95% confidence intervals will be the variable ranges to achieve the purpose of optimization process always with the actual measurement data.

#### 3.2 The eye wavefront aberrations

Zernike polynomials are widely used to describe the human eye aberration. According to that the human eye pupil diameter size adjusts with the environmental changes, the eye wavefront aberrations described by 5-order Zernike polynomials coefficients at two different pupil diameters (3mm and 6mm) were measured. The coefficients are analyzed with SPSS 13.0, too. The average values of Zernike polynomials coefficients are depicted as Fig. 6 (3mm pupil diameter) and Fig. 7 (6mm pupil diameter) respectively.

From Fig. 6 and Fig. 7, we can know that defocus Z^{0}
_{2} is the dominant eye aberration no matter the size of pupil diameter. When the pupil diameter is 3mm, the other ones are primary spherical Z^{0}
_{4}, astigmatism x Z_{2}
^{2}, coma x Z^{1}
_{3}, trefoil y Z^{-3}
_{3}, coma y Z^{-1}
_{3} and trefoil x Z^{3}
_{3} orderly. When the pupil diameter is 6mm, the other ones are primary spherical Z^{0}
_{4}, coma x Z^{1}
_{3}, astigmatism y Z^{-2}
_{2} and trefoil x Z^{3}
_{3} orderly.

## 4. The key optical models of eye models

#### 4.1 The corneal optical models with bi-conic surface

The cornea, which is at the forepart of human eye optical system, is one of the main refractive optical elements. The common way of describing the corneal aspheric surface is to use a conicoid in the form

where the origin is at the surface apex, *x* and *y* are the surface coordinate, *z* is the axis of revolution, *R* is the radius at the apex, and *k* is the conic coefficient, is less than -1 for hyperboloids, -1 for paraboloids, between -1 and 0 for ellipses, 0 for spheres, and greater than 0 for oblate ellipsoids.

However, this rotationally symmetric representation of the corneal surface is inadequate, because that there are orthogonal steep meridian (the smallest radius of curvature) and flat meridian (the largest radius of curvature) on the corneal surface [26]. From the perspective of the optical structure, the curvature radius on meridian plane and sagittal plane is not equal, and that is the reason of the existence of corneal astigmatism. Therefore, the surface type of bi-conic which is more suitable for the cornea is used. In Zemax [32], the sag of a bi-conic expression is given by

where the origin is at the surface apex, the sagittal plane is on *xoz* plane, the meridian plane is on *yoz* plane, *c _{x}* and

*c*are the curvatures at the apex on sagittal plane and meridian plane respectively, i.e. ${c}_{x}=\genfrac{}{}{0.1ex}{}{1}{{R}_{x}},{c}_{y}=\genfrac{}{}{0.1ex}{}{1}{{R}_{y}}$

_{y}*R*and

_{x}*R*are sagittal radius and meridian radius respectively,

_{y}*k*and

_{x}*k*are conic coefficients on sagittal plane and meridian plane respectively.

_{y}For above corneal data analysis and below eye modeling, we use the same biconic surface model. The detailed analysis of the corneal biconic surface models with 8mm diameter were described in our previous paper [33].

#### 4.2 The lens GRIN distribution models

There are two methods of expressing the crystalline lens gradient-index (GRIN) distribution. One is that the refractive index changes stepwise with lens shell-structure, i.e. non-continuous. The other is that the refractive index is described with a continuous gradient index equation. However, the non-continuous structure of lens produces multiple foci [34], and when the number of shells increases to make the model consist with the lens fibre layer structure, the refractive index distribution will approach a continuous change. Therefore, the second method is used here.

The continuous gradient index equations of the GRIN lenses in the schematic eye models of Gullstrand [20], Blaker [10], Liou & Brennan [13] and A. V. Goncharov & C. Dainty [15] all derived from the general form of GRIN distribution

where *n*
_{0}(*z*)=*n*
_{00}+*n*
_{01}
*z*+*n*
_{02}
*z*
^{2}+⋯,*n*
_{1}(*z*)=*n*
_{10}+*n*
_{11}
*z*+*n*
_{12}
*z*
^{2}+⋯,*n*
_{2}(*z*)=*n*
_{20}+*n*
_{21}
*z*+*n*
_{22}
*z*
^{2}+⋯,*n*
_{3}(*z*)=*n*
_{30}+*n*
_{31}
*z*+*n*
_{32}
*z*
^{2}+⋯,*z* is the coordinate along the optical axis, *r* is the radial coordinate. The above-mentioned four continuous equations [20, 10, 13, 15] are

$$-\left[0.00052375+0.00005735\left(z-1.7\right)+0.00027875{\left(z-1.7\right)}^{2}\right]{r}^{2}-0.000066716{r}^{4}$$

$$+0.049467z-0.015958{z}^{2}+0.0001715{z}^{3}+0.000141{z}^{4}$$

Eq. (4)~(7) are proposed by Gullstrand, Blaker, Liou & Brennan and A. V. Goncharov & C. Dainty respectively. Among them, the lens was divided into anterior and posterior part to express the refractive index distribution in Liou model [13]. The posterior part, i.e. the second expression in Eq. (6) is modified from the original one with “z” replaced by “z-1.59” because only in this way the anterior and posterior refractive indexes are equal at z=1.59.

The integrated approach with these different expressions is used to achieve the averages of the lens refractive index in axial and radial direction. Then the new coefficients and GRIN expression are obtained by fitting these refractive index values with a single equation

Curves comparing the fitting new continuous GRIN expression with other ones in axial direction and radial direction are shown as Fig. 8 and Fig. 9 respectively. These comparison curves indicate that the new single equation has the consistent distribution with other research results and a compact form. The lens GRIN model as the initial parameters programmed in C language with Eq. (8) is loaded in Zemax in the purpose of finding the Chinese generic eye model. Although Liou & Brennan model eye has some drawbacks since the GRIN lens parameters used overestimate the dioptric power, the analysis of lens refractive index based on the GRIN distribution in Liou & Brennan model is used widely [15, 23]. For better comparability with other results, the data from this model are used here. Moreover, the fitting GRIN expression Eq. (8) obtained by averaging is only the initial lens refractive index distribution; it will be changed to reproduce the measured aberration by later optimization.

#### 4.3 The chromatic dispersion models of ocular media

There are several different chromatic dispersion equations and data of ocular media, for example, Le Grand proposed using the Cornu equation to describe the refractive index of ocular media changes with wavelength [21]; different dispersion equations were presented in above-mentioned typical eye models. From reference [21], analyzing and fitting data, it was proposed that Cauchy’s equation [Eq. (9)] was the most suitable for the chromatic dispersion models of ocular media. But Conrady equation [Eq. (10)] with very simple form is empirical and designed for most common optical materials, including those of the eye, in the visible region [35]. Furthermore, Conrady equation is more accurate to the fifth decimal place than Cauchy [36]. Therefore, the chromatic dispersion models of ocular media based on Conrady equation is discussed here.

With the various references and sources about chromatic dispersion models of ocular media [12–14, 21], we fit their different data with Conrady equation to know the fitting precision. Table 3 gives the fitting coefficients of Conrady equation, max fitting error and Abbe number. The max fitting errors with 10^{-4} magnitude are very small. The coefficients A and B of Conrady equations fitted to Liou & Brennan [13] are the same because the equations they used show that the differences between the different media do not change with wavelength, which means their equations are not theoretically sustainable [21]. Therefore, we discard the usage of this chromatic dispersion source especially of lens in the process of subsequent discussion.

After the determination of fitting precision with Conrady equation, the Conrady equations of ocular media are fitted based on the some averages of various sources [12–14, 21] in the visible spectrum ranged from about 0.4µm to 0.75µm (Table 4). The fitting coefficients of Conrady equation, max fitting error and Abbe number are shown in Table 5. The Conrady equations for cornea, aqueous, lens, vitreous are Eq. (11)~(14) respectively.

Cornea:

Aqueous:

Lens:

Vitreous:

Figure 10 depicts the contrast curves of the fitting Conrady equations for ocular media with various sources of chromatic dispersion data [12–14, 21] in the visible spectrum. The chromatic dispersions of cornea (a), aqueous (b), lens (c) and vitreous (d) based on Conrady equations are consistent with the sources [12–14, 21]. The chromatic dispersion models of ocular media represented by Conrady equations are built as a refractive index database to be called in Zemax.

## 5. Chinese generic eye model

Based on above-mentioned statistical results of the measured data and the stimulation of the key optical models of eye model, the building of Chinese generic eye model is implemented in three stages:

1. With Zemax, the original eye model is structured according to averages in Table 2. The stop of the optical system is set on the pupil of eye model. Similar to corneal surface, the lens surface type is expressed by bi-conic with initial structural parameters referring to the data in existing eye models [2, 9–15]. The lens GRIN model constructed by C language is loaded in Zemax as the initial parameters, the chromatic dispersion models of ocular media represented by Conrady equations with the form of a refractive index database is also called in Zemax.

2. The measured eye aberration phase plate is introduced at the entrance pupil. The measured aberrations described by 5-order Zernike polynomials coefficients are obtained at eye pupil diameters of 3mm and 6mm respectively. Thus two phase plates of Zernike Standard Phase surface type are introduced into two systems of different pupil diameters simultaneously by the multi-configuration function in Zemax.

3. The initial systems consisting of the original eye model and the measured eye aberration phase plates are optimized after setting variables and building merit function to meet the target of eye model aberration equal to the measured values. In the original eye model, the initial measured eye parameters are from the means in Table 2, the initial unknown lens surface parameters from averaging the related data in the existing models [9–15] and the initial lens index distribution is expressed by Eq. (8). The merit function about the wavefront aberration RMS is built to be minimized in the optimization which is based on the damped least squares algorithm. The measured eye parameters, unknown lens surface parameters and the coefficients in Eq. (8) are set as variables. In the merit function, the variable ranges of the measured parameters are 95% confidence intervals shown in Table 2. The variable ranges of unknown lens surface parameters are referenced by the related research reports [2, 9–15, 20, 37, 38]. The values of refractive index at the lens surfaces are also constrained to keep same in the merit function. The lens surfaces and refractive indexes are optimized simultaneously. Because the angle of eye visual axis and optical axis is about 5°, the eye model with 5° field of view is optimized.

The measured aberrations with opposite signs at different pupil diameters are shown as Fig. 11 (a) & (b). After the optimization and removing the phase plates, the wavefront aberrations of obtained eye optical system are shown in Fig. 11 (c) & (d). Comparing with the Fig. 11 (a) & (b), the predicted wavefront aberrations of obtained generic eye model agree approximately, and the RMS errors are 0.07µm and 0.17µm respectively.

The structural parameters of Chinese generic eye model and its schematic plots are shown as Table 6 and Fig. 12. The eye optical system is consist of 6 surfaces, the first and second surface represent the anterior and posterior of cornea; the third one represents the pupil; the fourth and fifth one represent the anterior and posterior of lens; the sixth one represents retina. The pupil is not exactly centered with respect to the rest of the eye and is often displaced slightly nasally by ~0.5mm [13]. Therefore, the displacement of the third surface is introduced as shown in Fig. 12.

## 6. Discussion

The parameters in Table 6 are in the actual measured ranges (Table 2). The surface performance comparison of main refractive optical elements (the cornea and lens) in Chinese generic eye model with existing western eye models [9–14] are shown in Table 7.

From the comparison, we can find that,

1. The radii of anterior corneal surface are similar, elliptical surface with the similar conic coefficients of anterior corneal surface in Chinese model and in most western models. The posterior corneal surface in Chinese model is also elliptical surface with the conic coefficients -0.14 (meridional) and -0.10 (sagittal), but in most western models except Liou model (-0.60), the conic coefficient is 0 (spherical surface).

2. The lens exact shape is not known, the lens structure optimized based on our measured eye aberrations in Chinese model has a great different from the ones in western models. The radii of anterior lens surface in Chinese model are bigger than that in western models. The anterior lens surface in Chinese model is hyperboloid, too. But the conic coefficients of anterior lens surface in Chinese model are smaller than that in most western models. Furthermore, the radii of posterior lens surface are similar, although the conic coefficients of posterior lens surface in Chinese model indicate hyperboloidal surface, while in western models, the posterior lens surface is expressed by paraboloid (the conic coefficient is -1) or oblate ellipsoid (Liou model) or spherical surface.

3. There will be infinite combinations of the lens GRIN distributions and surfaces that give the targeted performance. The reproduce precision can be improved by using complex surface type to express lens [23]. This work will be confirmed in further research on Chinese personalized eye models which need higher fidelity than the generic eye model.

Expect for wavefront aberrations, the MTF of eye models are presented. Fig. 13 displays the monochromatic MTF of Chinese generic eye model compared with experimental result given by P. Artal and R. Navarro [39] and two western eye models [13,14] for the pupil diameter of 3mm (a) and 6mm (b). In Fig.13 (a), the MTF curve of Chinese generic eye model coincides with the one of Escudero-Sanz model roughly, and the agreement between Chinese generic eye model and experimental data is better than the other two models, especially when the spatial frequency greater than 40 cycles/degree. In Fig.13 (b), the agreement between Chinese generic eye model and experimental data is as good as Liou model and better than Escudero-Sanz model. In this case, it is probable that the imaging quality of human eye has ethnic difference. But there is no experimental report about MTF of Chinese eyes to make a comparison.

## 7. Conclusion

To the best of our knowledge, the new schematic eye model presented in this paper is the first generic eye model that based on Chinese population. With the statistical analyzed ocular parameters based on measured data and the simulation of the key models of eye model based on references, Chinese generic eye model is obtained by reverse building in Zemax. This Chinese generic eye model, presented some differences and some close performances compared with existing western eye models, shows the speciality of Chinese eyes, is suitable for our Chinese eye research and application. The related further research of the optical performance of Chinese generic eye model and Chinese personalized eye models will be implemented in future.

## Acknowledgment

This research is supported by the National Nature Science Foundation of China (Research Project, No. 60678048).

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