A comparative analysis of the influence of refractive error on image acuity using three eye models

To analyse and compare image acuity for different refractive errors generated by either altering axial length or corneal curvature and using three human eye models with two pupil sizes.

peripheral refraction, 10 accommodation or modulation transfer function. 1,7he variety of available models warrants a need for comparative analysis to determine how models may vary in simulating image acuity and how parameters used in generating refractive error may influence image acuity.While the variety of existing eye models does not currently cover the diversity of ocular biometry, the results of these studies can contribute to the further development of more biologically relevant eye models that recognise individual variations 11 as well as taking into account environmental effects such as luminance.The size of the pupil varies with luminance and accommodation and is also affected by a range of other external factors that influence the autonomic nervous system.
Refractive errors are caused by a mismatch between ocular components and can be quantified as axial, refractive or secondary myopia. 12Axial myopia is caused by an axial length (AL) that is excessively long; refractive myopia is attributed to changes in the structure or location of the image-forming structures of the eye (most generally the cornea) and secondary myopia is a refractive state for which a single, specific cause (e.g., drug, corneal disease or systematic clinical syndrome) can be identified and that cause is not a recognised population risk factor for myopia development. 12Secondary myopia will not be considered in this investigation.
This study aimed to investigate the influence of different refractive errors and the major causal parameters (AL and corneal curvature) and to compare those results for three human eye models and two different pupil sizes.[16]

METHODS
Ansys Zemax OpticStudio 22.3 (ansys.com) was used for the numerical calculations.5][16] These models were chosen because they are among the most commonly used and represent an older, more simplified model (Liou-Brennan 13 ) as well as more recent and advanced models (Goncharov 9 and Navarro [14][15][16] ).The parameters of the emmetropic eye for each model are shown in Tables 1-3.The Liou-Brennan eye model is comprised of six surfaces, representing the two corneal surfaces, three lens surfaces and the retina (Table 1).The Goncharov eye model has five refractive surfaces with only two surfaces representing the lens (Table 2), while the Navarro eye model has eight refractive surfaces with four surfaces representing the lens 14 (Table 3).
Simulations were made for real pupil diameters of 3 and 6 mm (corresponding to optical entrance pupils of 3.47 and 6.94 mm, respectively.This is in biological range for photopic and mesopic conditions [17][18][19] ) for a monochromatic beam ( = 550 nm ) along the optical axis (without incident light angle), for eye models with refractive errors ranging from −2 to +2 D in 0.25 D increments, as well as for an emmetropic eye.Refractive errors were simulated in two ways: by changing the AL or power of the cornea in relation to their values in the emmetropic eye.AL changes were simulated by altering the depth of the posterior chamber, while corneal refractive power was altered by changing the radii of curvature of the anterior and posterior corneal surfaces while maintaining the ratio between them. 20efractive errors created by varying the CRC simulations were created with and without changing the conic constant 21 in order to ascertain the influence of an aspherical cornea.
Every AL simulation commenced from the emmetropic eye with the object placed at infinity (Figure 1a) to check the spot diagram root mean square (RMS) values and compare these to values in the literature. 2,6,14The RMS value corresponded to the spot size on the retina and was correlated with the image acuity, such that the smaller the RMS, the better the image acuity.After comparison, the object plane was changed to the specific finite length that represented the far point for every respective refractive error length 20 (Figure 1b) based on the refractive error formula: The image plane was then moved by changing the posterior chamber depth (Figure 1c) to find the smallest spot diagram RMS value.After altering the AL, the position of the object plane was changed back to infinity (Figure 1d) to create the optical mismatch of the selected refractive error.Similar simulations to those described above were used to create models with refractive error induced by altering corneal curvature (Figure 2).This was performed for a cornea with and without a conic constant.
All simulations were optimised for the smallest RMS value.This optimisation goal is defined as the least aberrated optical system.
There was no alteration of the lens shape or thickness; therefore, the aberrations present in the original models 9,13,14 remain unaltered.All of the results are for an unaccommodated lens.The negative values for hyperopic eyes and positive values for myopic eyes correspond to the simulation method used in this article.The values pertain to the power of the eye.

R ESULTS
The RMS values for AL-induced refractive errors for three different models are plotted against refractive error in Figure 3.For the 3-mm pupil, all models show similar results, with the Navarro model giving slightly higher RMS values than the other two models around the emmetropic range.For the 6-mm pupil, the simulations show greater diversity.The Goncharov and Liou-Brennan models indicate similar results for myopic eyes, while the Navarro eye model had an RMS value that was greater than the other two models.For hyperopic eyes (≥1.00 D), the RMS  The RMS values for corneal curvature-induced refractive errors with a conic constant are plotted against refractive error in Figure 5. Similar results were seen with the 3-mm pupil for all models.Lower RMS values were found compared with previous simulations (Figures 3 and 4), as shown in Figure 5 for an emmetropic eye with a 6-mm pupil.The Liou-Brennan and Goncharov eye models have a higher RMS value for hyperopic eyes (≥0.75 D) than the Navarro eye model.Both models showed similar results for the 6-mm pupil.
The minimal RMS values are shown in Table 4. Values below the diffraction limit are indicated.The mean Airy disc value for these models was around 3 μm.For the 3-mm pupil, the results are relatively close for the Liou-Brennan and Goncharov eye models.However, higher RMS values were seen for the Navarro eye model than the other two models.This was also found for the 6-mm pupil size: a greater degree of similarity is found in the RMS values from the Liou-Brennan and Goncharov eye models, whereas the minima RMS values for the Navarro eye model were higher (Table 4).
The relationship between refractive errors induced either by altering AL or changing CRC is shown in Figure 6.The horizontal points correspond to the corneal curvature simulation  6).
The ratio between the two variable parameters (AL/ CRC) was calculated and is shown in Table 5 with

DISCUSSION
There are fewer models that can be applied to ametropic eyes than to the emmetropic eye.A previous study considered models for myopia from clinical measurements using a centred optical model to predict spherical aberration and a model with tilt and decentration to predict changes in peripheral refraction. 10The centred model predicted increasing

F I G U R E 6
Plots of axial length against corneal radius of curvature for models with the lowest root mean square (RMS) retinal spot size shown by refractive status for 3-mm (left) and 6-mm (right) pupil diameters for the Goncharov (green), Liou-Brennan (yellow) and Navarro (red) eye models.
spherical aberration with increased myopia, whereas the decentred model was limited in predicting changes in peripheral refraction. 10These models did not differentiate between the different biometric parameters that lead to myopia (AL, CRC).Li et al. developed the first eye models for children with refractive error, ranging from −5 to +5 D using paraxial optics for use in studies of myopia control, 22 while Rozema et al. 23 introduced a statistical approach to generate refractive errors, which resulted in the creation of a tool for generating synthetic biometric data that closely resembles clinically measured data.Scott and Grosvenor tested a structural model for emmetropic and myopic eyes and found that corneal curvature and posterior chamber depth had the greatest contribution to refractive state. 24his study investigated simulations of refractive error caused by altering different biometric parameters (AL or CRC) for three models.While it is recognised that the optical system of the eye includes the lens, this is a dynamic element before the sixth decade of life, and a contribution of the lens to refractive error has been found in eyes from this decade and older. 259][30] Although strong correlations between refractive error and corneal curvature have not been found in all studies, 31 the role of the cornea in emmetropisation and myopia development must also be recognised. 32The findings indicate that image acuity, as determined by RMS, can vary depending on the model used and pupil size applied.For the 3-mm pupil, all models gave similar results, with the Navarro model having slightly higher RMS values for the emmetropic eye (Table 4).In all three models for a 3-mm pupil, the rate of increase in RMS with refractive error was similar for hyperopic and myopic simulations, with a slightly higher RMS value (mean value for all simulations 2.01 ± 1.51 μm) for models representing myopia compared with those representing hyperopia (Figures 3-5).
For the 6-mm pupil, the Liou-Brennan and Goncharov eye models gave similar results, with RMS values lower than for the Navarro model (Figures 3-5).The Navarro eye model diverges from the other two, particularly for myopic refractive errors induced by variation in AL (Figure 3) and CRC variation without a conic constant (Figure 4).In the corneal curvature (with conic constant) induced refractive errors (Figure 5), the RMS values of low refractive errors (from −0.50 to +0.50 D) were lower than for the other two models of inducing refractive error (Figures 3 and 4).The RMS values of the Liou-Brennan and Goncharov eye models seen in Figure 5 are diffraction limited for the emmetropic eye.The rate of increase in RMS with refractive error was similar for hyperopia and myopia for both Liou-Brennan and Goncharov eye models for all types of refractive error simulations (Figures 3-5), whereas for the Navarro model, this symmetry was only seen in the corneal curvature (with conic constant) induced simulation method (Figure 5).
The differences between the Navarro and the other two models are a result of the complexity of the models and the tilts and decentration of optical components.The Goncharov model has rotational symmetry, 9 while the Liou-Brennan model has a slightly decentred pupil (0.5 mm nasal direction). 13Conversely, the Navarro eye model has tilted anterior and posterior corneal surfaces and a tilted pupil plane. 14he findings indicate how these variations can influence the modelling of refractive errors and the determination of image acuity.The tilt influences not only the RMS values of the image acuity plot but also its profile (Figures 3-5).
The RMS minima values are shown in Table 4.The optimal RMS value in optical models occurs when the RMS value is smaller than the Airy disc.The highest RMS value is visible in the AL-induced refractive errors.The corneal curvature-induced refractive errors give smaller RMS values than AL-induced errors, and the smallest RMS values are found in corneal curvature-induced refractive errors with a conic constant.With the latter method, the Goncharov and Liou-Brennan models give RMS values that are diffraction limited for both 3-and 6-mm pupils.For a narrow pupil, the model could be expected to be diffraction limited (or nearly diffraction limited). 33With a wider pupil, there is an increase in higher-order aberrations 34 and the requisite effect on visual acuity. 35The findings for the Goncharov and Liou-Brennan eye models are not consistent with these observations, that is, they indicate that even with a 6-mm pupil, the eye model is diffraction limited.This could be a limitation of the models.In the biological eye, light levels vary, and illumination is not monochromatic.Further modelling using different wavelengths and incorporating varying illumination levels is required.The relationship between AL and corneal radius values (Figure 6) indicates a wide spread of results for myopic, hyperopic and emmetropic eyes, similar to those seen in the Singapore Indian Eye Study. 36There are multiple outcomes that give the same refractive error, even within a single-eye model.The AL/CRC ratio (Table 5) shows a higher ratio for myopes than hyperopes for every model, which concurs with the findings of the Nigerian population study. 37Similar values can be seen in the spherical equivalent refractive index in Grosvenor and Scott, 38 where hyperopic eyes from 0.00 to 0.99 D fell in the AL/ CRC value of 2.96 ± 0.08, while an AL/CRC value of 2.87 was found for a range of refractive errors from 1.00 to 1.99 D. Myopic eyes from ≥1.00 D were reported to have a mean AL/CRC ratio of 3.02 ± 0.07 while those between −1.01 and −2.00 D had a mean ratio of 3.07 ± 0.06. 38Only the results from the Goncharov eye model lay within the previously reported AL/CRC range for myopic and hyperopic eyes. 38he emmetropic eye was found to have an AL/CRC ratio of around 3.0. 39Again, only the Goncharov eye model provided values that concurred with these findings. 398][39] The discrepancy between the findings of the models in this study and those from previous investigations [37][38][39] is likely to be attributed to parameters of the lens within these models: lens thickness and variations in the refractive index gradient.The low values of AL for myopic eyes, seen particularly in the Liou-Brennan models and to a lesser extent in the Goncharov models (Figure 6), while lower than may be expected for myopia are close to reported AL values in adult eyes with low levels of myopia. 40he refractive error simulations based on Liou-Brennan, Goncharov and Navarro eye models take no account of the variation of the ocular parameters with age or ethnicity.Differences in the ocular components are visible in early childhood; for example, Native American children appeared to have flatter corneas than Caucasians, while children from Asian ethnicities have moderate values compared with the other two ethnic groups. 41These differences are still visible in adulthood based on the ocular biometry studies. 36,37,42ven within the different ethnic groups, the values of AL and CRC can differ from one person to another. 36Biometric ethnic differences that start in childhood 43 can increase further with age and after cataract surgery. 44Most notably, current eye models are based on Caucasian populations. 9,13,14,45he effects of age have been investigated in eye models 9,14,45 but these start in young adulthood (around 20 years).There is a paucity of models for the developing eye that need to consider the changes in the biometry of the eye and the AL/CRC ratio as the eye grows. 41,46,47Furthermore, the growing eye is exposed to different mismatches (of AL or corneal radius) that could cause refractive errors.Developmental models could provide insight into the process of emmetropisation and the development of myopia.The variation of findings from eye models needs to be borne in mind so that appropriate models can be selected depending on the application.

CONCLUSIONS
The influence of refractive error on image acuity varies depending on the simulation method of refractive error and the model used.This study generated refractive errors by altering AL and CRC separately to ascertain the effect of each of these factors independently.The combined effects of AL and corneal curvature on refractive error and the influence that it has on the image acuity need further investigation.As models become more sophisticated, personalised and biologically relevant, they will better represent the image acuity of the eye for varying refractive errors, ethnicities, ages and pupil sizes.The development of such models will require a cohesive collation of large databases of ocular biometry findings that cover all populations, ages and refractive errors.

AC K N O W L E D G E M E N T S
This project was funded by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 956720.The authors are grateful to Professor Rafael Navarro for providing the lens gradient user-defined function.

CO N F L I C T O F I N T E R E S T S TAT E M E N T
Authors have no conflict of interest to declare.

F I G U R E 1
Axial length-induced refractive error simulation process.The initial model has an object plane at infinity (a), a change of the object plane to the specific finite length corresponding to the far point (b), an alteration of the axial length (c) and a change of the object plane to infinity (d).value was similar for all of the models with the 6 mm pupil size.Below −1.00 D the Liou-Brennan and Goncharov eye models had similar results, but the Navarro eye model diverged.The difference between this eye model and the other two increased further as the refractive power of the eye model increased.For a 6-mm pupil, a slight shift of the RMS minima can be seen for both the Goncharov and Navarro eye models (shift from 0 to −0.25 D).The RMS values for refractive errors induced by varying the CRC, without a conic constant, are plotted against refractive error in Figure4.The results are similar to ALinduced refractive errors (Figure3), with the 3-mm pupil simulations showing even greater similarity between RMS values for all models than in the AL-induced simulations.For the 6-mm pupil, the hyperopic eyes (≥1.50 D) for the Goncharov and Liou-Brennan eye models had higher RMS values than the Navarro eye model.The RMS minima shift is visible only with the Navarro eye model (−0.25 D).

F I G U R E 2
Corneal shape-induced refractive error simulation process.The initial model has an object plane at infinity (a), a change of the object plane to the specific finite length corresponding to the far point (b), an alteration of the corneal radius of curvature (c) and a change of the object plane to infinity (d).F I G U R E 3 Root mean square (RMS) retinal spot size plotted against refractive error for axial length-induced refractive errors.Negative and positive values refer to hyperopia and myopia, respectively.method (with and without changing the conic constant), and the vertical points correspond to the AL simulation method.The three points for the emmetropic eye correspond to the three simulation methods used.For the 3-mm pupil, all models had a similar spread of values.For the 6-mm pupil, the spread of values for the Navarro eye model was greater than for the other two models.The Liou-Brennan model had the smallest range of values for emmetropic eyes using either pupil size.For all simulations and pupils, the Navarro eye model had the longest AL and the Liou-Brennan model the shortest AL, while the Goncharov eye model had the highest CRC.It is notable that AL values for myopic eyes were in the range of 22.42-23.08mm for the Liou-Brennan model and 23.20-23.81mm in the Goncharov model (Figure

1 DF
I G U R E 4 Root mean square (RMS) retinal spot size plotted against refractive error for corneal curvature-induced refractive errors without a conic constant.Negative and positive values refer to hyperopia and myopia, respectively.F I G U R E 5 Root mean square (RMS) retinal spot size plotted against refractive error for corneal curvature-induced refractive errors, with conic constant.Negative and positive values refer to hyperopia and myopia, respectively.increments of refractive error for 3-and 6-mm pupil sizes.The Liou-Brennan model had the smallest AL/CRC ratio, while the Navarro model showed the highest AL/ CRC ratio for all refractive errors and both pupil sizes.Notably, hyperopic refractive errors created by the Navarro model had a higher AL/CRC ratio than myopic refractive errors created by both the Goncharov and Liou-Brennan eye models.
Parameters used in the Liou-Brennan eye model.Parameters used in the Goncharov eye model.
T A B L E 1 T A B L E 3 Parameters used in the Navarro eye model.a Values for the biconic surfaces.
T A B L E 4 a Values below diffraction limit.
The mean values of axial length and corneal radius of curvature ratio with standard deviations.