For each of the wavefronts, we first calculated the ocular refractive power map which represents the distribution of the refractive power across the pupil. This can be done analytically using the recently developed concept of refractive Zernike power polynomials.59 The refractive power map simply represents wavefront aberrations in dioptres rather than micrometers.
In the next stage, the refractive power was then used to estimate the best sphero-cylinder60 which, in turn, was transformed to a sphero-cylindrical wavefront. In the following, the resulting wavefront was subtracted from the original wavefront producing a wavefront with a zero sphero-cylinder. Finally, to each of the derived wavefronts we added numerous sphero-cylindrical combinations ranging from −1.50 to +1.50 D in 0.125 D steps of spherical power (resulting in 25 combinations), 0.00 to −1.25 D in 0.125 D step of cylinder power (11 combinations) and axes between 0 and 157.5° in 22.5° steps (8 combinations). This gave a total of 2025 combinations. The sphero-cylindrical combinations were added to the wavefronts by transforming them from the power domain to the wavefront domain. Thus for each of the 25 original wavefronts, we calculated 2025 different combinations of sphero-cylinder with a fixed amount of higher-order aberration. It is crucial to note that the original amounts of higher-order aberrations (in terms of the Zernike coefficients values) do not change as a result of the procedure outlined (i.e., only the sphero-cylinder was varied). The resulting wavefronts with various sphero-cylinder and higher-order aberration combinations were then used to calculate retinal image plane metrics. All data processing was performed using routines developed from first principles in MATLAB.
The Strehl ratio is defined as the ratio of the observed peak intensity of an aberrated optical system when compared with the theoretical maximum peak intensity of the diffraction limited optical system. The visual Strehl ratios (VSMTF, VSOTF) take into account the contrast sensitivity function (CSF) of the human eye as described by Campbell and Green.61 In visual Strehl ratios, the optical or modulation transfer functions are weighted by the CSF in the spatial frequency domain before integration. In this way, the spatial frequencies at around 4 to 6 cycles per degree are given the greatest emphasis. The visual Strehl ratio based on the optical transfer function (VSOTF) has been reported to show good correlation with high contrast visual acuity,47,52,53 but has recently been identified to have some limitations in the way it is calculated.62 To analyze retinal image metrics and refractive power maps in this study, we used the visual Strehl ratio based on the modulation transfer function,47,52,53 which has shown to perform similarly to the VSOTF.47
The wavefronts and relevant refractive power maps that showed the highest retinal image plane metric (i.e., peak of VSMTF) were used to calculate retinal images and refractive power histograms.40 Retinal images and refractive power histograms were also calculated for the wavefronts and refractive power maps that contained no sphero-cylindrical components (i.e., higher-order aberrations only). The recently proposed augmented visual Strehl ratio (VSOTF_A)62 was also used for the analysis and found to perform very similarly to the VSMTF. Characteristically there were no differences found between the two metrics, however the sphere and cylinder magnitude that would correspond to their peaks varied in the area of ±0.125 D. We therefore decided not to include the VSOTF_A data in Table 1.
The combination of coma, trefoil, and spherical aberration requires various sphero-cylindrical combinations to improve retinal image quality. The sphere and cylinder magnitude as well as the axis direction are dependent on both the sign and magnitude of third- and fourth-order aberrations.
An example of the effect of higher-order aberrations on sphere, cylinder, and axis is presented in Fig. 2. The VSMTF results of positive trefoil30, negative vertical coma, and negative spherical aberration (+Tr −Co −SA) for various HO-RMS levels are shown for each of the 2025 sphero-cylinder combinations. Each data point in the sub-plots in Fig. 2 shows a particular combination of the sphero-cylinder added to the wavefront. For the 0.1 μm HO-RMS level (top row), the VSMTF is high and peaks are at 0 D of sphere and cylinder power (VSMTF peak = 0.69). However, with increasing HO-RMS error (second to bottom row), VSMTF peak decreases and shifts toward hyperopia for the spherical component and toward against-the-rule astigmatism for the cylindrical component, as shown by the sphere, cylinder, and axis results, respectively. It is worth noting that the levels surrounding the one with the best image metric are almost as high, making the peak of the VSMTF less and less distinctive with increasing HO-RMS levels. This is analogous to an increased depth of focus.
We have shown that combinations of frequently occurring higher- order aberrations have the potential to provide directional cues for sphero-cylindrical refractive error development. Combinations of coma, trefoil, and spherical aberration create retinal images, which can be improved with lower-order aberration combinations of defocus and cylinder. The level of higher-order aberrations necessary to induce clinically significant sphero-cylinders (i.e., ≥0.25 D) was in the range between 0.2 and 0.3 μm of HO-RMS for a 5-mm pupil diameter.
The question arises, whether these aberrations could induce a response in eye growth? There are numerous potential cues the visual system could use to navigate eye growth toward emmetropia. Studies on animals have isolated and investigated several cues such as accommodation,12–14 chromatic aberrations,63,64 spatial frequency content, or image contrast on the retina,65,66 to name just some. The results show that the retina is able to reliably navigate a response in eye growth even with severely reduced cues and very poor images.
In humans, chromatic aberrations have been suggested to provide directional information for accommodation that specifies the vergence of light.67 Rucker and Kruger have recently found a relationship between long- and middle-wavelength cone sensitivity and refractive error suggesting a link between chromatic aberrations and ocular elongation.68 However, for pupils larger than 3 mm these effects are moderated by monochromatic aberrations of the eye.69
The potential of monochromatic aberrations to provide cues to refractive error development in humans appears not to be supported by measurements of higher-order aberrations as a function of refractive error.70,71 For a fixed pupil size, myopic and emmetropic groups have similar baseline aberration levels for the unaccommodated eye. However, a number of factors are not taken into account in such comparisons, such as the role of accommodation, nearwork, pupil size, and the sign of defocus within the higher-order aberrations.
Numerous studies have shown differences in accommodation stimulus/response between myopes and emmetropes (for a review see the work of Chen et al.72). More recently significant correlations have been found between accommodation accuracy and spherical aberration for natural pupil sizes.48,49 Few studies have investigated differences in aberration levels between refractive error groups during accommodation.37,73 These studies have reported differences in fourth-order aberrations37 and optimized Strehl ratio73 between myopic and emmetropic groups.
Another potentially misleading factor in refractive error group comparisons based on higher-order wavefront RMS alone is the limited information the HO-RMS provides with respect to retinal image quality and the sign of defocus.74 If eye growth is influenced by directional or defocus sign cues within the retinal image, then the sign of defocus rather than the magnitude of wavefront aberrations would be of primary importance. This line of reasoning is supported by our simulations, which have shown that different combinations of higher-order aberrations with the same total RMS value can influence meridional best image plane in both positive or negative sign directions (Table 1). An important aspect of refractive error group's comparisons could also be the combination of internal37,73 and corneal51 wavefront aberrations with natural pupils during reading. Analysis of wavefront aberrations at fixed pupil sizes makes statistical comparisons easy, but may not represent the natural status of the eyes optics.
Pupil size could play an important role in the relationship between higher-order aberration cues and refractive error development. Younger age groups, who are generally at higher risk of myopia development, tend to have larger pupil sizes when compared with adult populations.75,76 For our theoretical calculations we used a 5-mm pupil diameter. Although this may appear large for an adult population, especially at near focus, in younger populations this is probably a good estimate.75,76 This is supported by the fact that the pupillary near response is smaller in younger age groups. Schaeffel et al.77 found that in children below the age of 10, the pupillary near response is absent for moderate levels of accommodation (up to 4 D). Even up to the age of 20, pupil constriction is smaller (<10%) than that of populations older than 20 years.78
The combinations of Zernike coefficients used in this study were selected specifically because of their previously shown association with accommodation and nearwork in downward gaze. Although there are countless of other combinations of higher-order Zernike coefficients interacting with lower-order terms, the investigation of arbitrarily selected higher-order Zernike combinations was not the aim of this study. Similarly, the HO-RMS errors created by the combinations of Zernike coefficients in this study were chosen to cover the range of HO-RMS levels published in studies of large populations of normal eyes.79–81 Although for a 5-mm pupil diameter, HO-RMS errors in the range of about 0.15 to 0.25 μm have been reported,79–81 in myopes these errors have shown to increase immediately after periods of nearwork reaching average levels closer to about 0.3 μm (0.27 μm ± 0.18 SD).51 To demonstrate that in some cases reading induced HO-RMS errors can reach substantial levels, Fig. 4 presents data of two real eyes after a 2-h reading session. The refractive power maps of both examples are characterized by significant distortions reaching HO-RMS levels of almost 1 μm in the first example (Fig. 4, top panel). The baseline wavefront HO-RMS for this myopic eye with WTR-astigmatism (Rx −6.75 × −1.75/6) was 0.144 μm. Introduction of the sphero-cylinder increases both VSMTF and HO-RMS, while the retinal image is visibly improved. The sphero-cylinder required for the refractive power map of the second example (Fig. 4, bottom panel: patient with predominantly against-the-rule refractive astigmatism, Rx −0.50 × −2.00/88), does not substantially improve the VSMTF, however, retinal image reconstruction shows a better recognizable E-letter despite a dramatically increased total RMS value. Baseline prereading higher-order wavefront RMS level for this ATR-astigmatic eye was 0.145 μm.
For the test wavefronts, the relative contribution of the third- and fourth-order RMS to the total HO-RMS was chosen such that third-order RMS had a larger input to the total HO-RMS when compared with the fourth-order component.79,80 It is clear that any change in relative contribution of each of the higher-order coefficients would have an effect on the relative effect of the sphere and cylinder components. Ultimately, the purpose of the test wavefronts was to give some examples of the interactions between lower- and higher-order aberrations. For example, the effects of combinations of higher-order aberrations on astigmatic axis, or the larger impact of spherical aberration on the best focal plane when combined with wave-like third-order aberrations.
Our model, which is based on three higher-order aberration terms will be limited in describing interactions in real eyes in detail. However, the vast majority of higher-order wavefront RMS of the human normal eye is contained within third- and fourth-orders.80 With respect to third-order aberrations, it becomes evident from our data that astigmatic axes other than 0 or 90° would require components of both of the coma (i.e., Z31 and Z3−1) and trefoil (i.e., Z33 and Z3−3) terms. Spherical aberration is generally the largest contributor to the fourth-order aberrations of the eye and the change in spherical aberration with accommodation has been well documented.37,41–43 We therefore have included higher-order aberrations in our model that make up more than 80% of the total higher-order RMS of the eye.80
As the test wavefronts we used in this study were not meant to cover all potential interactions between higher-order and lower-order aberrations, the image quality metric that we chose (i.e., VSMTF) is not claimed to be the only metric appropriate to quantify these interactions. In terms of predicting visual acuity, image plane metrics seem to perform better than pupil plane metrics.47,52,53,82,83 Although slight variation between different image plane metrics may occur as we found with the VSOTF_A, we would not expect a characteristically different outcome in terms of our sphero-cylinder results for most image plane metrics. Furthermore, the objective of this study was not to compare different retinal image quality metrics but instead to investigate potential cues to refractive error development. Therefore our results should be viewed as an estimate of the interactions between lower- and higher-order aberrations.
But just how sophisticated would the eye's emmetropization mechanisms have to be to coordinate a response in eye growth based on higher-order aberrations? Because the eye's optical characteristics change as a result of visual task, it is plausible that emmetropization adopts a state of refractive equilibrium for different visual tasks (i.e., no condition is optimal but overall performance is). Subjective visual acuity measurements at different levels of accommodation are consistent with such a model. Visual acuity appears to be highest for intermediate object distances and diminishes slightly at far and closer target conditions.49,84,85 In line with this, higher-order wavefront RMS86 and spherical aberration levels49 were also found to be smallest at mid range distances. The eye's emmetropization mechanisms could be influenced by the optical status of the eye during and after close visual tasks. When accommodating, the spherical aberration of the eye consistently shifts in the negative power direction.37,87 Transient changes also occur in corneal aberrations during and after nearwork, that can persist for up to several hours postreading depending on the duration of the previous reading task.88 Therefore, during intensive periods of nearwork activity, the optical characteristics of the eye can change and are likely to be different throughout most of the day when compared with periods where little or no nearwork is performed.88,89 During that time, emmetropization could be driven by the optical status of the eye during near-visual task conditions. Studies that have investigated temporal variations in childhood myopia progression during the school year, have reported an association between nearpoint activity and myopia progression rates.90–92
The concept that higher-order aberrations can influence the best focal plane has been previously reported.45–47,53 Wilson et al.45 showed that even-order aberrations provide odd-error cues to the direction of defocus. Similarly Cheng et al.47 showed that the level of astigmatism that produced best image performance is influenced by the level of secondary astigmatism. In this study we have investigated the three higher-order aberration terms that have shown a systematic change as a consequence of accommodation and nearwork in downward gaze.37,41–43,50,51
The systematic changes previously reported have been a negative shift of primary spherical aberration associated with accommodation and a positive shift of trefoil30 as well as a negative shift of primary vertical coma associated with lid force-induced changes on the cornea during reading.
The direction of change of these three aberrations combined as a test wavefront, required hyperopic defocus in combination with against-the-rule astigmatism to provide optimized retinal image quality (Table 1). In the context of emmetropization (or compensatory blur driven eye growth), this combination would provide potential cues for myopia and with-the-rule astigmatism development. It is interesting to note, that due to nearwork, significant changes in several wavefront aberrations occur, which when combined have the potential to provide cues for the most commonly observed refractive error (i.e., myopia and with-the-rule astigmatism) among the population.
One important aspect of the interactions between lower- and higher-order aberrations shown in this study relates to the development of astigmatism. In various animal species,35,36,93–95 the effects of astigmatic deprivation on eye growth are not consistent but generally suggest that astigmatism is not eliminated through a visual feedback mechanism. In humans, the cause of astigmatism has long been associated with mechanical forces applied by the eyelids primarily along the vertical meridian of the cornea,27 thereby also suggesting a nonvisual mechanism. This theory implies the existence of differences in eyelid tension and/or corneal rigidity between astigmatic and nonastigmatic populations. Consequently, such differences seem likely to contain an inheritance component. However, there is little support for either differences in eyelid tension96 or a strong genetic component of astigmatism.19,20,97 On the other hand, if astigmatism could be induced actively by visual experience, the etiology of refractive astigmatism and the link between myopia and astigmatism might be explained on the basis of an optical emmetropization mechanism. The individual effects of lid force induced higher-order aberrations on the retinal image, in combination with internal higher-order aberrations during accommodation, could result in sphero-cylindrical refractive errors that have developed to optimize retinal image quality as a result of extensive nearwork (Fig. 5).
Although an optical mechanism may provide an alternative explanation for the etiology of refractive astigmatism, the importance of mechanical forces of the eyelids along the vertical meridian of the cornea should not be ruled out. Read et al.98 recently reported a number of associations between eyelid morphology and normal corneal shape. A hypothesis that combines both optical and mechanical aspects would fit in well with known features of astigmatism in human eyes. Corneal with-the-rule astigmatism of populations with little or no total astigmatism is mostly compensated by internal optics of the eye.99 Kelly et al.100 suggested that this well-known compensation of corneal astigmatism by internal astigmatism is determined by an active fine-tuning process. The corneal astigmatism induced by mechanical forces might be compensated fully by internal optics because no cue to refractive astigmatism is present.
The theory of retinal-image mediated ocular growth is well supported by numerous studies from animal and human research.8 Our results show that the effects of a combination of higher-order aberrations that occur in association with accommodation and nearwork in downward gaze, could provide retinal image cues to promote eye growth. The interactions between higher- and lower-order aberrations could provide an explanation for the association between myopia progression and increased incidence of with-the-rule astigmatism.
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