BUEHREN, TOBIAS PhD; ISKANDER, D ROBERT PhD; COLLINS, MICHAEL J. PhD; DAVIS, BRETT BAppSc
The link between nearwork and refractive error development is well established.1 One mechanism that has been suggested to play a role in this relationship is known as functional hyperopia caused by under accommodation for near targets.2 The associated retinal blur is thought to act as a cue to eye growth, in a similar way to that shown in animal models. However, attempts at slowing human myopia progression based on the model of functional hyperopia during nearwork have shown only limited success,3,4 indicating that there might be more complex optical mechanisms involved. The question arises of how the retina decodes blur information and what kind of cue is needed to initiate a response in eye growth.5
Schaeffel et al.6 were first to show that eye growth of chicks can be manipulated with defocusing lenses. Subsequently, many animal models have been developed and it is now well established that emmetropization is a vision dependent phenomenon.7 The mechanism has been described as retinal image-mediated ocular growth.8 Although there is evidence from animal research that accommodation is playing a role in this mechanism,9,10 a number of studies have shown that accommodation is not essential for the retina to trigger a response in eye growth.11–14
In humans, the significant increase in myopia prevalence in recent decades points toward environmental factors playing an important role. At the same time, there is also clear evidence for the role of genetics in myopia development.15 Similarly genetics appear to play some role in the development of astigmatism. However, the relative significance of genetic versus environmental factors is not clear.16–20 Although astigmatism appears to play some role in emmetropization of spherical equivalent refractive errors during early ocular development,21–23 the time course of resolution of astigmatism and the ocular components involved indicate a different underlying mechanism.24 Yet later in life increasing levels of myopia are associated with an increased prevalence of with-the-rule astigmatism25,26 and it is not clear whether this relationship is associated with environmental or genetic factors.
The etiology of astigmatism has long been linked to mechanical forces of the eyelids along the vertical meridian of the cornea. Grosvenor27 suggested that ocular rigidity interacts with the tightness of the eyelids to produce corneal astigmatism, and suggested that variation in ocular rigidity is why astigmatism does not always develop. This hypothesis, which is different to that of retinal image-mediated ocular growth, could account for the mirror symmetry of the axes of corneal astigmatism observed between right and left eyes28 and may also explain changes in astigmatism found later in life.29–32 On the other hand, the reduction in infant astigmatism observed early in life could provide evidence for vision-dependent emmetropization of astigmatism.33,34 Kee et al.35,36 recently showed that rhesus monkeys show a corneal response to astigmatic defocus suggesting that visual experience can alter corneal shape. Even though the response in rhesus monkeys is not coordinated (i.e., does not compensate for the imposed astigmatic deprivation) this finding could have significant implications for the etiology of astigmatism and also myopia in humans. If the cause of refractive astigmatism in human eyes was due to a corneal response to visual experience, then the retinal image creating myopic astigmatism in humans may contain both spherical and cylindrical cues to eye growth.
Several studies have suggested that higher-order aberrations could play a role in myopia development by contributing to retinal blur.37–40 Changes in higher-order aberrations with accommodation have consistently shown a shift of spherical aberration in the negative power direction.37,41–43 Because spherical aberration can affect the best focal plane44–47 and also the accommodation stimulus/response function,48,49 it could also create directional cues for eye growth during nearwork. Other significant higher-order aberration changes associated with nearwork have been found in Zernike vertical coma and trefoil along 30° (trefoil30) terms.50,51 These changes were associated with a wave-like distortion of the cornea because of eyelid forces during reading in downward gaze,50,51 and have been found to primarily affect the retinal image along the vertical meridian.40
The recent developments in retinal image quality metrics such as the visual Strehl ratio based on the modulation transfer function (VSMTF)52 have shown good correlation with subjective visual acuity.47,53 These techniques have enabled detailed investigations of the interaction between lower and higher-order aberrations and their relationship to retinal image quality.40,48,49 Another application of these tools is the potential to find characteristics in the retinal image, which could act as cues to eye growth.
An issue that we have not addressed in this study is peripheral aberrations.54–56 There is an increasing interest of myopia researchers in the field of peripheral vision and eye shape.57 Incorporation of neural limits, refraction, and wavefront aberrations of peripheral vision to the overall visual performance of the eye, may prove to be an important component of human refractive error development. The role of nearwork induced wavefront aberrations in peripheral vision might be one aspect of this issue. During nearwork, the concept of a three-dimensional measure of defocus within the peripheral visual field adds further complexity to the issue.58 In this paper we have restricted our modeling to potential higher-order aberration cues in central vision.
We use an analytical approach to specifically investigate the three higher-order aberrations that have shown systematic changes associated with accommodation and nearwork in downward gaze. Test wavefronts were created from the third-order primary vertical coma and trefoil30, as well as the fourth-order primary spherical aberration. For each of the higher-order Zernike combinations, numerous sphero-cylindrical combinations were added to the wavefronts to find the best retinal image, based on retinal image plane metrics.
Five types of test wavefronts were created using a 5-mm pupil diameter. One wavefront type was composed of primary spherical aberration only. Four types were composed of the same three higher- order Zernike polynomial terms (trefoil30, primary vertical coma, and primary spherical aberration) in various combinations. Differences between the wavefronts were created by the sign (i.e., positive or negative) of each coefficient (Table 1). For each of the five types, higher-order wavefront RMS errors between 0.1 and 0.5 μm in 0.1 of a micrometer steps were used to create a total of 25 wavefronts. For wavefronts that included third- and fourth-order aberrations, the third-order RMS was in the order of twice the magnitude of the fourth-order RMS. An example of a test wavefront which is combined of +0.195 μm of positive trefoil30, −0.195 μm of negative vertical coma, and −0.12 μm of negative spherical aberration (+Tr −Co −SA), to produce a HO-RMS of 0.3 μm is shown in Fig. 1.
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.
In Fig. 3 the positive trefoil30, negative vertical coma, and negative spherical aberration combination (+Tr −Co −SA) for a 0.3 μm HO-RMS is shown. The top row shows the refractive power maps and retinal image reconstructions for zero sphero-cylinder (top left) and the sphero-cylinder showing the best retinal image plane metric (top right). The reconstructed letter E corresponds to a 0.4 log MAR letter size. The sphero-cylinder that optimizes the retinal image according to the VSMTF is +0.375 × −0.375/90. This means that for this combination of higher-order aberrations, hyperopic defocus and against-the-rule astigmatism provides best retinal image quality. It can be seen from the refractive power maps and retinal images (top row) that the sphero-cylinder clearly changes the characteristic of the refractive power distribution leading to an improved retinal image and better image quality metrics as shown by the VSMTF. The refractive power histograms (bottom row) confirm the changes in refractive power distribution showing an overall shift and, more importantly, a marked peak close to 0 D for the refractive power map with sphero-cylinder added to optimize VSMTF (bottom right) compared to the refractive power map without sphero-cylinder (bottom left).
Table 1 is a summary of all 25 wavefronts for the VSMTF results. It shows the higher-order Zernike coefficients, third, fourth, and HO-RMS, also the sphere, cylinder, axis, and best-spherical-lens for the VSMTF peaks, as well as the corresponding dioptric power range. The first five wavefronts (−SA, i.e., spherical aberration only) show the well-known relationship between spherical aberration and the best focal plane (i.e., defocus shift). They are included as a reference to be compared with the other 20 wavefronts. The other four types generally show an increase in both sphere and cylinder with increasing HO-RMS. The axis of astigmatism depends on the combination (i.e., sign) of each of the higher-order coefficients and is linked to both the sign of third- and fourth-order aberrations (Table 1). An opposite sign for the two third-order coefficients, which is associated with a wave-like distortion,50 generally results in increased changes in cylinder power in either with-the-rule or against-the-rule direction depending on the sign of spherical aberration. As a consequence of the increased cylinder change, the best spherical lens also changes slightly more in this type of test wavefronts compared with the other types. Any other possible combination of trefoil, coma, and spherical aberrations in terms of the signs of coefficients (a total of eight combinations is possible) will be equivalent to one of the four combinations that are shown in Table 1.
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.
Animal models have shown the concept of compensatory blur driven eye growth,7 which may include a corneal response.35,36 However, in these cases, changes in lower-order aberrations had been induced by directional lower-order aberration cues. Therefore it does not necessarily follow that lower-order aberration changes can be induced with directional higher-order aberration cues. A number of studies have reported that corneal aberrations are partially compensated by the internal optics of the eye.51,100–103 It is uncertain however whether the compensation results form an active or passive mechanism of ocular wavefront minimization. Although an active compensation mechanism has been described as meta-emmetropization,104 there is limited evidence for its existence.100 To this end, it can only be assumed that the effects of higher-order aberrations on retinal image blur may contribute to compensatory blur driven eye growth. The interactions between lower- and higher-order aberrations shown in this study could lead to a better understanding of this complex process.
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.
Contact Lens and Visual Optics Laboratory
School of Optometry
Queensland University of Technology
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