Comparison of wavefront aberrations in the object and image spaces using wide-field individual eye models

Yongji Liu; Yongji Liu; Xiaolan Li; Lin Zhang; Lin Zhang; Xianglong Yi; Yuwei Xing; Kunqi Li; Yan Wang; Yan Wang

doi:10.1364/BOE.464781

1. Introduction

Myopia has become a global health concern due to its high prevalence and early onset, which leads to an increase in ocular pathology associated with myopia [1,2]. Optical interventions, such as Orthokeratology and multifocal contact lenses, have proved to be an effective way to control myopia progression [3–5]. The changes in peripheral refraction or higher-order aberrations introduced by the optical interventions are considered to be the explanation for its myopia control function [6,7]. Thus, accurately evaluating the wavefront aberrations especially peripheral aberrations in the image space is critical for myopia research. At present, peripheral wavefront aberrations of the eye are most commonly evaluated by Hartmann-Shack sensor(HS) based aberrometers [8,9], which measure the wavefront going out of the eye, i.e. the wavefront aberrations in the object space. The wavefront aberrations out of the eye are suitable for retinal imaging such as Optical Coherence Tomography (OCT), whereas wavefront aberrations into the eye are of significance in evaluating peripheral refractions of the eye, the optical outcomes of refractive surgery and Orthokeratology, the interaction between chromatic and higher-order aberrations of the eye and the optical performance of contact lenses or intraocular lenses. Thus, one question that needs to be answered is whether the wavefront aberrations, especially peripheral aberrations, in the object space are a proper representation of the wavefront aberrations in the image space.

The best way to answer this question is to compare the measured aberrations in the object and image spaces. One way to measure the wavefront aberrations in the image space is by the Laser Ray Tracing (LRT) aberrometers [10]. Both LRT and HS methods involve a double-pass process due to the difficulty of making the single-pass measurements. Moreno-Barriuso and Navarro studied the on-axis wavefront aberrations given by HS and LRT in both the artificial eye and real eyes [10]. They found that the HS and LRT gave the same results. In addition, the double-pass and single-pass measurements also give the same results. Other clinical studies comparing on-axis aberrations from the HS-based and the LRT-based aberrometers demonstrated no difference [11]. One thing worth pointing out is that the wavefront aberrations in Moreno-Barriuso and Navarro’s study and other clinical studies were on-axis wavefront aberrations, not including peripheral aberrations. Navarro et al. [12] first measured the wavefront aberrations across the horizontal meridian using LRT. Atchison et al. [13], using the HS, measured the wavefront aberrations in the horizontal meridian. When comparing the higher-order aberrations from these two studies with HS and LRT, differences in third-order and higher-order aberrations were found. However, these differences can not solely be explained as a difference in aberrations between the object and image spaces, because the subjects, pupil diameters, and measurement methods differed between the studies. Additionally, defocus and astigmatism, which seem to play a critical role in myopia control, were not compared in the above experimental studies probably because different subjects were involved.

Another way to compare the aberrations in object and image spaces is to do raytracing using eye models. This method holds the advantage that it can directly compare aberrations in the object and image spaces, avoiding the difficulties presented by the experimental measurements. Thibos et al. [14], using a reduced schematic eye model, pointed out that there is a difference in the transverse spherical aberration between the object and image spaces. Atchison and Charman [15], using into-the-eye and out-of-the-eye ray tracing, compared the longitudinal spherical aberration in the object and image spaces, finding that the difference in longitudinal spherical aberrations between both spaces was rather small, though they did notice an intendancy of increasing difference with the amount of ametropia. What the above studies focused on is the on-axis aberrations, not including the comparison in peripheral aberrations. Escudero-Sanz and Navarro [16] compared different types of aberrations in the object and image spaces over large visual fields as an application of their proposed wide-angle eye model. They found significant biases between the results of experimental measurements (aberrations in the object space) and ocular aberrations obtained directly from the model (aberrations in the image space) for certain specific aberrations. Generical eye models, in which the model’s aberrations represent the average of a large sample group, were implemented to conduct the comparison in the above studies.

With the development of technology, more detailed information about the eye, such as corneal topography and wavefront aberrations of the human eye, is available, which makes it possible to construct individualized eye models, also named personalized eye models [17] or customized eye models [18]. The individualized eye models have the advantages that they can reflect the optical properties of the individual eyes and are an effective computation tool for understanding the optics of the human eye [19]. Therefore, in this paper, we proposed to obtain the wavefront aberrations in the object and image spaces using individual eye models, thus comparing the aberrations, especially defocus and astigmatism, over large visual fields. This study aims to find out whether peripheral aberrations in object space are a proper representation of those in the image space. The answer to this question is of significance because of the aberrations’ possible role in myopia control.

2. Methods

A total of 5 myopes, with an average age of 17 ± 3 years and subjective average refraction of -5.67 ± 1.69 D, participated in this study. Wavefront aberrations and corneal topography of the corneal surfaces of 5 left eyes were collected at Tianjin Eye Hospital. One drop of 1% cyclopentolate was instilled 30 minutes before the wavefront measurements. The room illumination was reduced to ensure the measurements were done under a pupil diameter larger than 5.0 mm. The research followed the tenets of the Declaration of Helsinki and was approved by the Medical Human Research Ethics Committee of Tianjin Eye Hospital, with informed consent obtained from participants.

A commercial Hartmann-Shack sensor-based aberrometer (WaveScan WaveFront System, AMO Wavefront Sciences, Albuquerque, NM, USA) was modified to measure the monochromatic aberrations on-axis as well as off-axis. As shown in Fig. 1, a beam splitter was added between the aberrometer and the human eye to reflect the rays from the LED targets on the wall. The virtual images of the LED targets served as the fixation points in different visual fields. When the eye was viewing the target at a certain visual field, such as +15°, the light from the aberrometer will fall on the peripheral retinal area. Thus, the aberrations obtained from the aberrometer are the off-axis wavefront aberrations.

Fig. 1. The modified Hartmann-Shack aberrometer. For clarity, the distance between the aberrometer and the human eye was exaggerated. A beam splitter was added between the aberrometer and the human eye. The blue circles are the LED light source. The virtual images (green circles) of the LED targets created by the beam splitter were located in front of the left eye for fixation.

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The wavefront measurements were done out to the 15°eccentricity in the temporal, nasal, superior and inferior visual fields in steps of 5°while the participants viewed the virtual images of the LED targets, which were 2 meters away from the eye. The 0°LED target was aligned with the instrument’s internal fixation target. The pupil center was aligned with the instrument’s measurement axis. Three measurements were taken at each visual field, and their wavefront aberration coefficients were averaged. The small circle (SC) aperture method [20] (inscribed circle of 5 mm diameter) within the bigger elliptical pupil was used for fitting the on-axis and off-axis wavefront to the Zernike coefficients. The wavefront measurements were done at 0.78 µm, but the Zernike coefficients given by the aberrometer are at 0.55 µm due to the chromatic correction by the software.

Corneal topography of each subject was collected using a Pentacam corneal topographer (HR Premium, Oculus Optikgeräte GmbH, Wetzlar, Germany). The elevation data of the corneal anterior and posterior surfaces were exported for fitting with Matlab (R2016b, Mathworks, Natick, MA, USA) according to the following equation:

(1)$$Z = \frac{{{X^2} + {Y^2}}}{{R + \sqrt {{R^2} - (1 + K)({X^2} + {Y^2})} }} + \sum\limits_{\textrm{i} = 1}^{30} {{C_i}Zernik{e_i}(x,y)}$$

where X and Y represent corneal coordinates; Z represents corneal elevation, R represents the curvature radius and K represents the conic constant. $Zernik{e_i}(x,y)$ represents the ${i_{th}}$ Zernike polynomial and ${C_i}$ represents corresponding Zernike coefficients. Parameters such as R, K and ${C_i}$ were obtained after the fitting process. The corneal elevation data are referenced to the videokeratometric (VK) axis which does not coincide with the line of sight (LOS), the reference axis for the wavefront aberration data [21]. To compensate for this, the anterior corneal surface was decentered and tilted in the ray tracing in Zemax Optic Studio (version 2016, Washington Zemax LLC, USA) according to the following equations [22]:

(2)$$\left\{ \begin{array}{l} {\theta_x} = {\tan^{ - 1}}(TCP{C_x}/(WD + EP))\\ x = WD\tan ({\theta_x}) \end{array} \right.$$

(3)$$\left\{ \begin{array}{l} {\theta_y} = {\tan^{ - 1}}(TCP{C_y}/(WD + EP));\\ y = WD\tan ({\theta_y}) \end{array} \right.$$

where $EP$ is the entrance pupil position which was determined from the eye model, $WD$ is the working distance of the Pentacam corneal topographer (80 mm) [23], $TCP{C_x}$ and $TCP{C_y}$ are measured horizontal and vertical decentration components of the topographic center relative to the pupil center, ${\theta _x}$ and ${\theta _y}$ are horizontal and vertical angles between the LOS and the VK axis, x and $y$ are the horizontal and vertical decentration components in Zemax Optic Studio of the anterior corneal vertex from the pupil center.

The individual eye models were built in Zemax Optic Studio. The starting point of the eye model used the parameters in the wide-field eye model proposed by Escudero-Sanz and Navarro [16]. Abbe number for each refractive medium was set the same as those in the other eye model proposed by Navarro [24]. The anterior and posterior cornea surfaces in the starting eye model were replaced by the parameters obtained from the fitting in Matlab. The corneal central thickness and anterior chamber depth in the starting eye model were replaced by the corresponding measured data from Pentacam. Axial out-going eye models, where the light source is located on the retina and rays propagate to the cornea, were constructed firstly because this orientation most accurately simulates the HS wavefront aberrometer’s measurements. The following optimizations were done under a wavelength of 555 nm.

The entrance pupil of the out-going eye model was set in a way to make the exit pupil size the same as the measured pupil diameter from the wavefront aberrometer. Afocal and Real Ray Aiming in Zemax Optic Studio were set on. The on-axis wavefront aberrations were set as the targets for optimization. The anterior and posterior surfaces of the crystalline lens were set as a Standard Sag surface type. The radii of curvature, conic coefficients, Zernike polynomial coefficients of the anterior and posterior surfaces of the lens, the lens thickness and vitreous depth were set as variables and optimized. Boundaries were set for radii of the lens surfaces, central thickness and vitreous thickness to make them fall into physiological boundaries of the anatomical values [25]. When the optimization was completed, the on-axis outgoing eye model was obtained. Based on the on-axis outgoing eye model, the wavefront aberrations of the peripheral visual fields were set as the optimization targets using multiconfiguration provided by the Zemax Optic Studio. The fields of view were set the same as the visual fields in the wavefront aberrations measurement. In each configuration, the exit pupil size was set as the same as the measured value from the aberrometer. Weights were assigned to each visual field but with reduced values as the visual fields increase. Boundaries on the central thickness, edge thickness and radii of the crystalline lens were set to meet the physiological boundaries of the anatomical values. The parameters of the crystalline lens were optimized again. Subsequently, the retina was set as a biconic surface. Coordinate break surfaces were added to the crystalline to simulate the decentration and tilt of the real crystalline lens. The optimization was conducted again. In the end, the individualized outgoing wide-field eye model was obtained for each eye.

The out-going wide-field eye models were reversed to obtain the in-going eye models, in which the rays propagate from the cornea to the retina. For the outgoing wide-field eye models, the objects’ heights on the retina were defined in a way to form the corresponding visual fields of the in-going eye models. The wavefront aberrations obtained from the out-going eye model are the wavefront aberrations in the object space while the wavefront aberrations from the in-going eye model are the wavefront aberrations in the image space. The difference in wavefront aberrations between the object and image spaces is defined as the wavefront aberrations in the image space minus those in the object space.

The Zernike coefficients given by the Zemax Optic Studio were rearranged to follow the OSA standard [26]. The refractions of the eyes are often used in myopia research, thus, the wavefront aberrations were converted to the mean sphere M, regular astigmatism ${J_{180}}$ and oblique astigmatism ${J_{45}}$ components using the following equations [27]:

(4)$$M = \frac{{ - \left[ \begin{array}{l} (2\sqrt 3 C_2^0 - 6\sqrt 5 C_4^0 + 12\sqrt 7 C_6^0)(1 + {\cos^2}\phi )\\ + (\sqrt 6 C_2^{ - 2} - 3\sqrt {10} C_4^{ - 2} + 6\sqrt {14} C_6^{ - 2})\sin 2\alpha {\sin^2}\phi \\ + (\sqrt 6 C_2^2 - 3\sqrt {10} C_4^2 + 6\sqrt {14} C_6^2)\cos 2\alpha {\sin^2}\phi \end{array} \right]}}{{{r^2}{{\cos }^2}\phi }}$$

(5)$$\begin{array}{l} {J_{45}} = \frac{{ - \left[ \begin{array}{l} (2\sqrt 3 C_2^0 - 6\sqrt 5 C_4^0 + 12\sqrt 7 C_6^0)\sin 2\alpha {\sin^2}\phi \\ + (\sqrt 6 C_2^{ - 2} - 3\sqrt {10} C_4^{ - 2} + 6\sqrt {14} C_6^{ - 2})[2{\cos^2}2\alpha \cos \phi + {\sin^2}2\alpha (1 + {\cos^2}\phi )]\\ + (\sqrt 6 C_2^2 - 3\sqrt {10} C_4^2 + 6\sqrt {14} C_6^2)\cos 2\alpha \sin 2\alpha {(1 - \cos \phi )^2} \end{array} \right]}}{{{r^2}{{\cos }^2}\phi }}\\ \end{array}$$

(6)$${J_0} = \frac{{ - \left[ \begin{array}{l} (2\sqrt 3 C_2^0 - 6\sqrt 5 C_4^0 + 12\sqrt 7 C_6^0)\cos 2\alpha {\sin^2}\phi \\ + (\sqrt 6 C_2^{ - 2} - 3\sqrt {10} C_4^{ - 2} + 6\sqrt {14} C_6^{ - 2})\cos 2\alpha \sin 2\alpha \cos \phi {(1 - \cos \phi )^2}\\ + (\sqrt 6 C_2^2 - 3\sqrt {10} C_4^2 + 6\sqrt {14} C_6^2)[2{\sin^2}2\alpha \cos \phi + {\cos^2}2\alpha (1 + {\cos^2}\phi )] \end{array} \right]}}{{{r^2}{{\cos }^2}\phi }}$$

where r is the major semi-diameter of the pupil and $\phi$ is the visual field angle. $\alpha$ is the angle between the horizontal axis and the line connecting the field point to the origin of the coordinates. $\alpha = {0^ \circ }$ for the horizontal visual field whereas $\alpha = {90^ \circ }$ for the vertical visual fields.

3. Results

For the data presented in this section, a positive visual field represents the nasal side along the horizontal visual field and the superior side along the vertical visual field. A negative visual field represents the temporal side along the horizontal visual field and the inferior side along the vertical visual field.

The accuracy of the optimized out-going individual eye models was evaluated by the fitting error (FE). The FE is calculated by Eq. (7):

(7)$$FE = \frac{{|RM{S_{Model}} - RM{S_{Measure}}|}}{{RM{S_{Measure}}}} \times 100\%$$

where $RM{S_{Model}}$ and $RM{S_{Measure}}$ are the RMS of the model eye and that from the measurement. Figure 2 shows the FE of the eye models as a function of the visual field. For the 0°visual field, the FE was about 1.47%. The FE increased with the visual fields, reaching about 2.94% for the temporal field and about 2.10% for the nasal field. Along the vertical meridian, the largest FE was about 1.69% and 2.37% for the inferior and the superior visual field respectively. Figure 2 also shows a relatively large inter-subject difference in the FE as shown by the error bars.

Fig. 2. Fitting error vs. the visual field. The error bars represent the standard deviation.

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The first-order characters of the individual in-going eye models were evaluated at 555 nm wavelength and the results were listed in Table 1. The optical power of the eye models is between 59.63 D and 61.56 D with an average of 60.42 D. The axial length is larger than 24.00 mm which could be explained by the fact that all the subjects are myopic. The average entrance pupil and exit positions are 3.29 mm and 3.94 mm away from the corneal vertex.

Table 1. Physiological parameters of the eye models after optimization^a

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Table 2 shows the parameters of the crystalline lens of the eye as well as the radii of the curvature of the retina after optimization. The average crystalline lens power is 21.68 D. The average anterior surface and posterior surface radii of the crystalline lens are 10.21 mm and -6.22 mm. Most anterior and posterior surface conic is negative. The average retina radius is 11.99 mm and 11.92 mm for horizontal and vertical meridians respectively. All the above parameters fall in the anatomical statistical range [25,28].

Table 2. Parameters of the crystalline lens and the retina of the eye models after optimization

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The in-going and the out-going models of one eye were shown in Fig. 3. The in-going eye model represented by Fig. 3. (b) showed an on-axis defocus because the participants in our study are myopic.

Fig. 3. The schematic layout of the eye models. (a) Out-going wide-field eye model. The colored lines represent point sources that originated from various retinal eccentricities spanning a ±15° angle range in the pupil. (b) In-going wide-field eye model. The colored lines represent rays with different visual fields going into the eye model.

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In the following section, the aberrations in the image space were obtained from the in-going eye models whereas the aberrations in the object space were obtained from the out-going eye models. Though the individual eye models were constructed employing the horizontal and vertical wavefront aberrations over the ±15° visual fields, we extended our analysis out to the ±45°horizontal and vertical visual fields, the maximum visual fields at which peripheral aberrations were able to be obtained from all individual eye models. A validation of this extrapolation over the horizontal ±35°was given in the discussion section and Appendix A. Though we could not validate the results in the vertical visual fields beyond ±15° or horizontal visual fields beyond ±35°at present, the results in these visual fields were also illustrated in the following section, intending to investigate the possible role of the visual fields on the differences in aberrations between the object and image spaces.

The RMS of two participants with the largest (KZW) and least (DYS) difference in aberrations under 3 mm and 6 mm pupil diameter between the object and image spaces were illustrated in Fig. 4. For simplicity, the Zernike coefficients in 0°, 15°, 35° and 45° nasal and superior fields were shown. The Zernike coefficients in the object and image spaces seem to coincide with each other even for visual fields up to 45° under 3 mm pupil diameter. Under 6 mm pupil diameter, slight differences in Z4 (defocus), Z5 (45°astigmatism) and Z8 between the object and image spaces were observed for KZW, whereas FM barely showed any differences.

Fig. 4. Zernike coefficients of the two participants showed the largest (KZW) and least (DYS) difference in aberrations under 3 mm and 6 mm pupil diameters between the object and image spaces for different visual fields. N and S stand for the nasal and superior fields respectively.

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The point spread function (PSF) along the horizontal visual field of the above two eyes were shown in Fig. 5. The PSFs in the object and image spaces looked the same for both eyes at each peripheral visual field.

Fig. 5. The PSFs along the horizontal visual field under 3 mm pupil diameter in object and image spaces for the two eyes with the largest and least difference in aberrations between the object and image spaces.

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The mean sphere, regular astigmatism, and oblique astigmatism in the object and image spaces under 3 mm and 6 mm pupil diameter were calculated and illustrated in Appendix B. To illustrate the difference between the object and image spaces clearly, the differences in the mean sphere, regular astigmatism, and oblique astigmatism between the object and image spaces as a function of the visual field under pupil diameters of 3 mm and 6 mm were shown in Fig. 6. Figure 6 illustrates the difference between the object and image spaces of each participant along with the average of five participants.

Fig. 6. The difference in the mean sphere, regular and oblique astigmatism between object and image space as a function of the visual field under a pupil diameter of 3 mm and 6 mm. The light blue area represents a refractive error below ±0.25D.

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Though Zernike coefficients did not exhibit much difference between the object and image spaces, the difference in the mean sphere and astigmatism between the object and image spaces is quite noticeable. In addition, inter-subject differences were observed. It was shown that the differences in the mean sphere of each individual along the horizontal visual field under pupil diameters of 3 mm and 6 mm are in the range of -0.25 D and +0.25 D, except for KZW in 45°visual field under 6 mm pupil diameter, the absolute value being slightly larger than 0.25 D. The differences in regular astigmatism showed the largest inter-subject difference in comparison with the differences in the mean sphere and oblique astigmatism. The differences in regular astigmatism of each individual along the horizontal visual fields were below 0.25 D for 4 eyes whereas KZW showed differences beyond -0.25 D for most of the horizontal visual fields, reaching a maximum of about 0.6 D and 0.75 in 35° and 45° visual fields. The difference in oblique astigmatism along the horizontal visual field was below 0.25 D for most of the eyes except for KZW for visual field beyond 35 ° under 6 mm pupil diameter. Along the vertical visual field, similar trends were observed.

The average differences in the mean sphere, regular astigmatism, and oblique astigmatism between the object and image spaces under pupil diameters of 3 mm and 6 mm as a function of the visual field are also shown in Fig. 6 (red thick curve). The average differences in the mean sphere, regular and oblique astigmatism along the horizontal and vertical visual fields ranging from -45°to +45°are below 0.25 D for pupil diameters of 3 mm and 6 mm. Thus, the difference in the mean sphere and astigmatism between the object and image spaces is of little clinical significance for the visual field ranging from -45°to +45°. Implementing a different method, Escudero-Sanz and Navarro [16] compared astigmatism between the object and image spaces. They found a good agreement between object and image space (called simulation I and II in their papers) for small and moderate visual fields up to about 40°. They did not give the exact value of the difference, however considering the scale of the vertical axis in Fig. 4 in their paper, it may be reasonable to believe that our results agree with their findings.

Figure 7 shows the chromatic focal shift in the object and image spaces as well as the difference in the chromatic focal shift between object and image spaces. The chromatic focal shift was calculated as the difference in the mean sphere under the red (671 nm) and blue (475 nm) wavelength. Figure 7 (a) shows that the chromatic focal shift patterns in vertical and horizontal meridians both in the object and image space agree with the results in the literature [25]. Besides, Fig. 7 (a) also illustrates a rather small difference between object and image spaces. The exact differences in chromatic focal shift shown in Fig. 7 (b) demonstrated that the differences increased with the visual field for the visual field between ±45°, but the absolutes values were rather small, only up to about 0.05 D which was of little practical effect. Escudero-Sanz and Navarro [16] compared the longitudinal chromatic aberrations, which is the counterpart of the chromatic focal shift in the present paper, between the object and image spaces to find an agreement between the object and image spaces if using a corrected simulation. The method they used is different from our methods, thus it is hard to directly make a comparison. However, it seems that our results agree with their findings though an emmetropic generical eye model was used in their study whereas myopic individual eye models were used in the present study.

Fig. 7. The chromatic focal shift in the image and the object spaces (a) and the difference in the chromatic focal shift between object and image spaces (b).

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4. Discussion

The present work aims to investigate whether the clinically measured wavefront aberrations, especially peripheral aberrations, in the object space are a proper representation of the wavefront aberrations in the image space. Though defocus seems to play a critical role in myopia research, the difference in defocus between the object and image spaces has not been studied in previous work. Thus, particular attention was paid to the mean sphere, regular and oblique astigmatism because they were commonly investigated in myopia research. To our knowledge, this is the first study to conduct the comparison of the aberrations between the object and image spaces with the wide-field individualized eye models.

The individual eye models were constructed with wavefront aberrations over ±15° horizontally and vertically. The analysis was made over ± 45°horizontal and vertical visual field based on these eye models. To verify whether the eye models can be used to analyze the aberrations beyond ±15°, two extra model eyes of Yongji Liu were constructed, one with wavefront aberrations over ±15°horizontal visual fields (Model 1), the other with the wavefront aberration over ±35°horizontal visual fields (Model 2). The refraction of the eye is -0.75D. This additional eye was used to validate the extrapolation simply because our setup to measure wavefront aberrations does not work well when measuring wavefront aberrations beyond ±15°visual field due to the limited space between the aberrometer and the eye. For this additional eye, the peripheral aberrations were measured by COAS-HD aberrometer (Wavefront Sciences Inc., Albuquerque, NM, USA). Details on the wavefront aberrations measurement were given in the paper of Jaisankar et al. [29]. Corneal topography of the anterior and posterior corneal surfaces was measured by the same Pentacam corneal topographer as the subjects involved in this study. The models were constructed in the same way as described in the methods section. The difference in the mean sphere, regular astigmatism, and oblique astigmatism between the object and image spaces of the two models were illustrated in Fig. 8 in Appendix A. The comparison shows that the results given by Model 1 and Model 2 are quite close. The absolute difference given by the two models is less than 0.05 D under 3 mm pupil diameter over all visual fields. Under 6 mm pupil diameter, the maximum difference in the difference in mean between the two models is about 0.08 D, while the maximum difference remains below 0.15 D for regular and oblique astigmatism. Though the vertical peripheral aberrations were not available, the results in the vertical visual fields from the two models were also illustrated for reference. The magnitude of the difference for both the horizontal and vertical visual fields was the same. Based on these results, it may be safe to say that the results given by the present paper up to the ±35° horizontal visual field are reasonable.

Not much difference in Zernike coefficients of on-axis aberrations was observed in the present study, which seems to agree with the findings of previous studies [10,15,16]. However, it was found in this study that the difference in the mean sphere, the regular and oblique astigmatism were relatively noticeable though the Zernike coefficients did not show much difference. This is probably because several Zernike coefficients as well as the pupil diameter were considered to calculate the mean sphere and astigmatism.

Atchison compared the higher order peripheral wavefront aberrations up to 40°horizontal visual field obtained by HS [13]with those from Navarro et al. by LRT [12] to conclude that their results were similar to Navarro’s. Though their results seem to agree with our findings, different subjects and measurement methods were used in their study. Our study directly compared the difference between the image and object spaces of the same eyes. Rather than higher order aberrations, we mainly focus on the differences in the mean sphere and astigmatism. Differences in the mean sphere and oblique astigmatism showed slight changes over visual fields. In contrast, differences in regular astigmatism over 6 mm pupil size demonstrated an apparent trend that they increased with the visual fields. This may be because astigmatism increases with visual field and pupil size, thus resulting in an increased difference in regular astigmatism.

However, our research results revealed that the differences in the mean sphere and astigmatism between the object and image spaces over the visual field ranging from -45°to +45° were generally less than 0.25D for pupil diameters of 3 mm and 6 mm. Thus, the difference in mean sphere and astigmatism are of little clinical significance, which means the wavefront aberrations in object space over the horizontal ±35°or ±15° vertical visual fields under 3 mm and 6 mm pupil diameter are a proper representation of these in the image space and then can be used to evaluate retinal imaging of the human eye. The difference in the mean sphere, the regular and oblique astigmatism over the visual fields beyond this range demonstrated the same trend. These findings have implications in other fields beyond the myopic research area.

The on-axis aberrations in the object space are usually used to conduct wave-front guided corneal ablation, customized intraocular lens [30,31], or contact lens design [32]. The present study showed that differences in aberrations between the object and image spaces may play little role in the post-surgery optical quality of the eye with wavefront-guided corneal ablation. Other factors, such as whether aberrations in the corneal plane or in exit pupil plane should be used [15], whether corneal aberrations [33] or aberrations of the eye should be used for the surgery, may play more important roles in improving the outcomes of the wave-front guided corneal ablation.

Wavefront aberrations are needed to construct individual eye models. To make the eye models reflect the measured wavefront aberrations to the maximum extent, individual eye models were usually constructed in a reversed way, from the retina to the cornea, because the wavefront aberrations obtained from HS are aberrations in the object space. According to the present study, it is safe to say that the in-going individual eye model can be directly constructed with the aberrations in the object pace if the visual fields are not larger than 45°. Or put it in another way, the previous individual in-going eyes [34] constructed with aberrations obtained from HS are reliable if the visual fields are not larger than 45°.

The findings of this study are based on myopic eyes. By comparison with the results from the previous study, it seems that states of the refraction (myopic/hyperopic/emmetropic refraction) play little role in the difference in the mean sphere. In addition, a rigid theoretical analysis, from an optical theoretical view, is necessary to verify what we have found, but this is beyond the scope of the present study.

The present study has some limitations. Only five eyes’ data were studied in this preliminary investigation and inter-subject differences were shown. Thus, the average difference obtained in this study is only restricted to these five eyes. Whether or not it can be extended depends on a further study of a large number of eyes. However, we believe that the difference in the refractions of the individual eyes holds value for application in clinical practices, like determining the peripheral defocus to evaluate the effectiveness of optical interventions to control myopia progression.

Building an individualized eye model typically involves a trade-off between anatomical similarity and functional equivalence. Thus, the crystalline lenses were modeled as a constant refractive index instead of a gradient index which the anatomical structure of the crystalline lens actually is, aiming to keep the eye models as simple as possible while maintaining wave-front features. Whether the gradient index of the crystalline lens plays a role in the aberration difference between the object and image spaces was not explored in this study.

5. Conclusion

The results show that the difference in the mean sphere and astigmatism was below 0.25 D between the object and image spaces over the horizontal ±35°or ±15° vertical visual fields under 3 mm and 6 mm pupil diameter. In a word, the difference in aberrations between the object and image spaces may be ignored when evaluating the peripheral refraction of the human eye over such visual fields, which means the wavefront aberrations in the object space are a proper representation of the aberrations in the image space aberrations for this visual range.

Appendix A. The validation of the extrapolation over the horizontal ±35°

Fig. 8. The mean sphere, regular astigmatism and oblique astigmatism aberration between object and image spaces as a function of the horizontal and vertical visual field for Yongji Liu based on Model 1 and Model 2 under pupil diameters of 3 mm and 6 mm. The light blue area represents a refractive error below ±0.25D.

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Appendix B. The mean sphere, regular astigmatism and oblique astigmatism aberration in the object and image spaces

To evaluate the refraction difference between the object and image spaces, the mean sphere, regular astigmatism, and oblique astigmatism in the object and image spaces under 3 mm and 6 mm pupil diameter were calculated respectively according to equation (4-6) and shown in Fig. 9. For simplicity, here we only show the results of KZW and DYS as these two eyes demonstrated the largest and least differences in aberrations between the object and image spaces. It is shown in Fig. 9 that relative peripheral hyperopia (which means central mean sphere minus peripheral mean sphere is negative) was observed for DYS across the horizontal visual fields and in the temporal visual field for DKW. Relative peripheral hyperopia is usually observed in myopic eyes in myopic research [35]. Difference in the mean sphere and oblique astigmatism between the object and image spaces under 3 mm and 6 mm pupil diameters was hardly noticed for both eyes. KZW showed a difference in regular astigmatism in the nasal and inferior fields whereas DYS did not show noticeable differences between the object and image spaces.

Fig. 9. The mean sphere, regular astigmatism and oblique astigmatism aberration in the object and image spaces as a function of the horizontal and vertical visual field for KZW and DYS under pupil diameters of 3 mm and 6 mm.

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Funding

National Natural Science Foundation of China (No. 81873684); Natural Science Foundation of Tianjin City (No. 19JCZDJC36600).

Acknowledgments

The peripheral wavefront aberrations of Yongji Liu were obtained from Prof. David A. Atchison's lab. The authors would like to express our appreciation for the reviewers’ comments and suggestions to improve our manuscript.

Disclosures

The authors declare no conflict of interest.

Data availability

Data underlying the results from measurements and simulations presented in this paper are not publicly available at this time but may be obtained from the corresponding author upon reasonable request.

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Subject's name	Sphere (D)	Cylinder (D)	Power (D)	Axial length (mm)	Entrance pupil position (mm)	Exit pupil position (mm)	Distance between ENP and EXP (mm)
DYS	-4.09	-1.94	60.50	25.31	3.43	4.08	0.66
HZH	-4.14	-1.10	59.63	26.40	2.68	3.31	0.63
DK	-5.74	-0.47	59.89	26.56	3.29	3.95	0.67
KZW	-5.95	-1.36	60.51	25.96	3.70	4.37	0.67
FM	-8.85	-0.39	61.56	26.73	3.36	3.99	0.63
Average	-5.75 ± 1.94	-1.05 ± 0.64	60.42 ± 0.75	26.19 ± 0.57	3.29 ± 0.38	3.94 ± 0.39	0.65 ± 0.02

Subject's name	Lens Front Radius (mm)/ conic	Lens Back Radius(mm)/ conic	Lens power (D)	Retina Y Radius (mm)	Retina X Radius(mm)
DYS	10.00/-3.13	-6.32/-1.21	21.38	13.57	13.00
HZH	10.20/-3.20	-5.67/-1.67	22.92	11.59	11.00
DK	10.23/-2.71	-5.99/-1.21	22.09	10.97	11.35
KZW	10.20/0.25	-7.05/0.00	20.01	13.25	13.00
FM	10.20/-2.44	-6.04/-1.18	22.00	10.55	11.25
Average	10.21 ± 0.01/-2.25 ± 1.43	-6.22 ± 0.52/-1.06 ± 0.63	21.68 ± 1.08	11.99 ± 1.36	11.92 ± 0.99

Subject's name	Sphere (D)	Cylinder (D)	Power (D)	Axial length (mm)	Entrance pupil position (mm)	Exit pupil position (mm)	Distance between ENP and EXP (mm)
DYS	-4.09	-1.94	60.50	25.31	3.43	4.08	0.66
HZH	-4.14	-1.10	59.63	26.40	2.68	3.31	0.63
DK	-5.74	-0.47	59.89	26.56	3.29	3.95	0.67
KZW	-5.95	-1.36	60.51	25.96	3.70	4.37	0.67
FM	-8.85	-0.39	61.56	26.73	3.36	3.99	0.63
Average	-5.75 ± 1.94	-1.05 ± 0.64	60.42 ± 0.75	26.19 ± 0.57	3.29 ± 0.38	3.94 ± 0.39	0.65 ± 0.02

Subject's name	Lens Front Radius (mm)/ conic	Lens Back Radius(mm)/ conic	Lens power (D)	Retina Y Radius (mm)	Retina X Radius(mm)
DYS	10.00/-3.13	-6.32/-1.21	21.38	13.57	13.00
HZH	10.20/-3.20	-5.67/-1.67	22.92	11.59	11.00
DK	10.23/-2.71	-5.99/-1.21	22.09	10.97	11.35
KZW	10.20/0.25	-7.05/0.00	20.01	13.25	13.00
FM	10.20/-2.44	-6.04/-1.18	22.00	10.55	11.25
Average	10.21 ± 0.01/-2.25 ± 1.43	-6.22 ± 0.52/-1.06 ± 0.63	21.68 ± 1.08	11.99 ± 1.36	11.92 ± 0.99

Comparison of wavefront aberrations in the object and image spaces using wide-field individual eye models

Abstract

1. Introduction

2. Methods

3. Results

4. Discussion

5. Conclusion

Appendix A. The validation of the extrapolation over the horizontal ±35°

Appendix B. The mean sphere, regular astigmatism and oblique astigmatism aberration in the object and image spaces

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (2)

Equations (7)

Biomedical Optics Express