Both spherical and astigmatic refractive errors are common in the human eye.1 Although myopia has a clear environmental component, astigmatism is largely inherited2 and corneal.3 The corneal origin can partly be attributed to the mechanical pressure of the eye lids on the cornea in the vertical pupil meridian (astigmatism with the rule). Astigmatism shows an age-dependent prevalence—high at very young age, less in the middle age, and a later increase at prepresbyopic age.4
Developmental Aspects of Astigmatism
Although the biological mechanisms of the development of astigmatism have been less extensively studied than those of spherical refractive errors, it appears clear that, different from spherical errors, imposed astigmatism is NOT compensated by directional changes in eye growth (in this case, of cornea or lens).5 Only Irving et al.6 proposed that astigmatism is feedback controlled in the growing chicken eye, but this was not confirmed by later studies by Schmid and Wildsoet.7 Studies in monkeys8 showed that random astigmatic refractive errors were induced when lenses were worn that imposed with-the-rule, against-the-rule, or oblique astigmatism. In particular, the axes of any induced astigmatism were not appropriate to compensate for the astigmatism imposed by astigmatic spectacle lenses. In summary, astigmatism is a common optical deficiency in vertebrate eyes, and it could be expected that neural processing in the visual system is optimized to extract maximal visual information from the retinal image also in the presence of astigmatism.
Previous Studies on the Effects of Defocus on Contrast Sensitivity and VA
To be able to compare effects of spherical and astigmatic defocus on spatial vision, it is important to normalize the testing conditions— same viewing targets (numbers, letters, Landolt C, or Tumbling E), same luminances and, in the case of astigmatic defocus, same axes. However, such a normalization has generally not been attempted, making comparisons difficult. For example, Woods et al.9 measured the effects of spherical and astigmatic defocus on contrast sensitivity for vertical sine wave gratings and found, not surprisingly, a prominent effect of the axis of astigmatism. Astigmatic defocus of +2.00 diopter (D) had no effect as long as the plus power meridian was parallel to a grating pattern. If the power meridian was perpendicular to the grating, the effect was similar to that seen with +2.00 D spherical defocus. With the power meridian at 45°, an intermediate effect was observed. Bradley et al.10 observed the most noticeable decline of Snellen acuity when the power meridian of imposed astigmatism was at 0 or 90°. With accommodation intact and the reading target between the near and the far point, positive and negative astigmatic defocus had similar effects. Comparing Landolt C and Tumbling E, Reich and Ekabutr11 found a similar reduction in visual acuity (VA) for both optotypes when spherical and astigmatic defocus of the same dioptric magnitude were imposed. Interestingly, in this study, spherical defocus reduced VA more than astigmatic defocus (but see results of the current study). Imposing various aberrations with an adaptive optics system, Atchison et al.12 compared blur detection thresholds for spherical, cross-cylinder astigmatic defocus, and trefoil. They found that the impact of astigmatism on VA varies with the orientation of its axis, when tested with high contrast letter charts with different angular letter sizes of 0.1, 0.35, and 0.6 logarithm of the minimum angle of resolution (logMAR).
Comparisons of the Effects of Simulated and Real Spherical Defocus on VA
Using schematic eye models, the image on the retina can be calculated for various optical aberrations and used to predict spatial visual performance of subjects. Greivenkamp et al.13 simulated retinal images of Snellen letters with spherical defocus (ranging from 0 to 5.00 D), using the Kooijman eye model.14 They found that the predicted VA showed good agreement with published data, which describe the decline of VA with increasing defocus. Doshi et al.15 compared the effects of defocus on VA in calculated images using three different eye models (Indiana eye,16 Indiana Eye with spherical aberration,17 and the Kooijman Eye14). Under all testing conditions [pupil size of 2, 4, 6, and 8 mm, low contrast (10%), and full contrast Bailey-Lovie charts], good correlations were observed between VA as predicted by the eye models and the measured values in the subjects.
However, to the best of our knowledge, it has not yet been examined whether the decline of VA in the presence of astigmatism can also be predicted by letter charts that were convolved with astigmatic point spread functions (PSF). If this would be true, it could be concluded that the decline in VA with astigmatism is explained just by low pass filters, applied in different pupil meridians.
Nine subjects were enrolled in the study with an average age of 27.2 ± 1.8 years (range, 24 to 29 years). Eight of them were naïve to the experimental procedure. Permission was obtained from the Ethics Commission of the Medical Faculty of the University of Tuebingen. The research followed the tenets of the Declaration of Helsinki and in addition, informed consent was obtained from all subjects after explanation of the nature and possible consequences of the study. All subjects received a thorough subjective refraction of the OD by a certified optometrist (AO), using a letter chart at 6 m distance and trial lenses. VA was logMAR −0.1 in the OD, with optical correction if necessary. Three subjects were myopic (spherical equivalents, −2.00 D, −1.00 D, and −2.50 D) and six were emmetropic (average spherical equivalent, −0.08 ± 0.12 D). Astigmatic refractive errors were <0.25 D in all subjects.
Three different experiments were done in which defocus was either introduced by trial lenses or calculated in the letter chart. Experiment (1), myopic spherical defocus was imposed. Experiment (2), positive or negative astigmatic power was added at 0, 45, or 90°. Experiment (3), a cross-cylinder with zero spherical equivalent was introduced at 0, 45, or 90°. In all three experiments, the amount of defocus was determined that was necessary to reduce VA by 20, 50, or 75%. The subjects had to read the characters and the detection threshold was set to 50% correct answers. VA was expressed in logarithmic scale: for instance, logMAR (0.79) = −0.0969 or approximately logMAR −0.1. A 20% reduction to a VA of 1.00 is equivalent to a logMAR of about 0.0. A reduction by 50% is equivalent to a VA of 0.63 (logMAR 0.2), and a reduction by 75% equivalent to a VA of 0.32 (logMAR 0.5). The advantage of a logarithmic scale is that the numbers are linearly related to VA. A headrest, used to minimize head movements, was adjusted so that the tested eye was centered to the computer screen.
Conventional acuity charts were used with the character sizes equivalent to a VA of logMAR 0.0 (VA 1.00), logMAR 0.2 (VA 0.63), and logMAR 0.5 (VA 0.32). The sizes of the letters were calculated for the test distance of 4 m and the characters (numbers and letters) were 5% smaller than the equivalent size of a Landolt C.18 The highest spatial resolution that was tested was logMAR 0.0 (smallest separable angle 1′ or 0.016° at 4 m). One pixel at 4 m was equivalent to 0.005° so that 1′ was equivalent to about three pixels. To reduce the risk that letters were learned during repeated presentations, eight different charts (four with letters and four with numbers) were used in random sequence. Because letter sizes varied for the different acuity levels, five characters were presented for logMAR 0.5 and six for logMAR 0.2 as well as for logMAR 0.0. To minimize the effects of crowding, there was a minimum distance between the characters of 20′.19
Defocused retinal images were simulated using ZEMAX 8.0 (ZEMAX Development Corporation, Bellevue, WA), and the Liou-Brennan Eye Model,20 which uses aspheric surfaces to simulate the optical properties of the cornea as well as a gradient index crystalline lens model. The parameters of the model (curvature, asphericity, thickness, refractive indices, etc.) were entered into the ZEMAX lens editor. A so called “XY paraxial” surface (which acts as an ideal thin lens) was used to simulate an astigmatic lens in front of the eye, positioned 10 mm in front of the corneal vertex. This surface allowed changing the optical power in the X and Y meridian without introducing further higher order aberrations. The acuity chart axial position was placed at 4 m in front of the eye and ZEMAX calculated a PSF of all the optical system (the object placed at 4 m, the astigmatic surface and the Liou-Brennan eye model), by tracing rays on the system and calculating the optical path difference of any ray with respect to the chief ray at the plane of the exit pupil of the eye. Later, the PSF could be convolved with any object. Monochromatic light of 550 nm was assumed for the simulation. In correspondence with the artificial pupil used in the experiments, the entrance pupil diameter of the eye was set to 3 mm. Astigmatic defocus was introduced in steps of 0.25 D (an example of a calculated astigmatically defocused reading chart is shown in Fig. 1). Because the highest amount of astigmatic defocus that was used in this study was ±4.50 D, magnification changes of the retinal image, relative to the “no defocus condition” were <5% and were ignored.
Retinal image defocus was introduced with trial lenses, placed in a trial frame. Corneal vertex distance was set to 10 mm. Astigmatic defocus was imposed in steps of 0.25 D in experiments (1) and (2) and in steps of 0.50 D in experiment (3). An artificial pupil with a diameter of 3 mm was placed into a trial frame and was carefully centered to the pupil of the OD for each subject, using a video technique. Because the subjects were young, none of them had a pupil size smaller than the artificial pupil.
Acuity charts were generated in Microsoft PowerPoint (Microsoft, Washington) and were presented at 4 m distance on a conventional computer monitor (size: 17 inch, angular subtense of 5° × 4°, EIZO FlexScan T 68, Model No. MA-1991) with a luminance of 96 cd/m2. The luminance response function of the computer monitor for different pixel gray levels (from 0 to 255, in steps of 5 pixel gray levels) was determined. Between pixel gray levels between 100 and 220, the response curve was almost perfectly linear, indicating that each step in pixel brightness increased the screen luminance by the same amount. No signs of saturation in the luminance response were observed for pixel values outside the linear range. All tests were done in polychromatic white light, produced by four fluorescent light tubes on the ceiling. Their spectrum matched roughly the sun spectrum (although discrete emission peaks were present) with prominent emission around 550 nm where the spectral sensitivity of the eye is maximal.21 Ambient illuminance in the test room was measured at the height of the chinrest with the photocell directed toward the reading chard (the computer screen) and was determined around 320 lux.
Statistical analyses were performed with the statistics software package JMP 4.0 (SAS Institute, Cary, NC). A two-way paired t-test was used to calculate significances of differences. The Tukey-Kramer HSD test was used to test significance levels when different groups where compared.
Experiment 1: Reducing VA by Imposing Simulated or Real Positive Spherical Defocus
The amount of simulated spherical defocus, necessary to reduce VA at a defined rate, is plotted against the amount of real defocus that is necessary to achieve the same effect, in Fig. 2. Both variables show a high correlation (r2 = 0.819) with a slope of the linear regression of 0.83. Apparently, less simulated spherical defocus was needed than real defocus, although this was significant only for one of the tested visual acuities (logMAR 0.0, p = 0.08; logMAR 0.2, p = 0.003; and logMAR 0.5, p = 0.049).
Experiment 2: Reducing VA by Imposing Positive and Negative Astigmatic Defocus
Two observations were made, first, in some cases, the decline of the VA was different when astigmatism was imposed at different axes, both for simulated and real astigmatic defocus (Fig. 3, power axis of astigmatism positive and Fig. 4 power axis of astigmatism negative). Second, the average astigmatic defocus that was necessary to reduce VA by a defined amount was much lower for simulated than for real defocus (by 40 to 60%). A difference between simulated and real astigmatism persisted in all tested visual acuities and cylinder axes. The signs of imposed astigmatic defocus had no effect except for one case, where more negative defocus had to be induced with lenses than positive defocus to reduce VA by 75% (logMAR 0.5; p < 0.01).
Experiment 3: Reducing VA with a Cross-Cylinder While the Spherical Equivalent was Unchanged
Different from experiment (2), cross-cylinders were used with 0 spherical equivalent power to reduce VA by 20, 50, and 75%. Cross-cylinder defocus was generated in two different ways. First, a negative spherical and a positive astigmatic lens were combined and, second, a positive spherical and a negative astigmatic lens. Optically, these two conditions produce the same amount of defocus on the retina, and this is also reflected in near-identical visual acuities that were found with both lens combinations. The axis of astigmatism had some effect on measured visual acuities; however, only when the cross-cylinder defocus was simulated (Fig. 5, cross-cylinder defocus with positive astigmatic lenses and Fig. 6, cross-cylinder defocus with negative astigmatic lenses). More importantly, similar to results of experiment 2 (above), simulated cross-cylinder defocus was much more powerful in reducing VA than real cross-cylinder defocus (by 40 to 60%).
It was found that VA was worse with simulated spherical [experiment (1)] and considerably worse with simulated astigmatic defocus [experiment (2)] than with real defocus of the same magnitude. Also in the case of cross-cylinders [experiment (3)], VA was worse with simulated than with real astigmatic defocus.
Potential Limitations When the Effects of Defocus are Predicted by Use of an Eye Model
In a recent abstract, Gracia et al.22 described a similar observation, namely that VA was up to 2.5 times higher when the subjects saw through their natural aberrations, compared with when the visual targets were convolved with their previously measured individual aberrations. These results are not intuitive and require further analyses to determine possible reasons.
Possible differences between simulated and optically defocused targets include (1) changes in the position of the entrance pupil when lenses are used, which are not introduced in the case of simulated defocus, (2) effects of diffraction, (3) differences in the contrast between simulated and optically degraded targets, which depend on the linearity and dynamic range of the computer display (on which the targets are simulated using Fraunhofer theory and Fast Fourier propagation algorithms), (4) possible effects of fluctuations of accommodation, which would not help in the case of simulated targets, (5) possible cues inherent to polychromatic stimuli while the simulations were done with one wavelength (550 nm), and (6) potential effects of higher order aberrations in the optics of the subjects eyes that might interact with the imposed defocus.
- To analyze point (1), exact ray tracing was performed with ZEMAX, a professional optical design and simulation program. To exclude rays that should have not reached the retina because of changes in position and shape of the entrance pupil (varying power of the defocusing lens can alter these variables), the “ray aiming feature” was enabled in ZEMAX. For a lens of 3 D, the entrance pupil location is shifted from 3.09 to 3.81 mm behind the first corneal surface and its diameter increases from 3.00 to 3.14 mm (∼5%). Compared with the magnitude of the effects on VA (40 to 60%), these changes appear small.
- Diffraction could have been a problem because the diffraction-limited logMAR VA for a 3 mm pupil size is ∼0.012 deg or 0.7 min (logMAR −0.35), close to the highest VA that was tested (logMAR 0.0). However, VA was better with real defocus, rather than worse, and therefore this factor appears to operate in the opposite direction. In human eyes, the modulation transfer function reaches an optimum between 2 and 4 mm pupil size, before effects of aberrations take over for larger pupil sizes.23
- The screen luminance-response function was carefully mapped (gamma was set to 1 during the entire study). Almost perfect linearity was found for pixel values vs. luminances between 100 and 220—the range in which the defocused letter charts were presented. Therefore, this factor can be ruled out.
- Although accommodation could enhance VA in the presence of astigmatism by scanning sequentially through the interval of Sturm, accommodation could not help in the case of imposed positive astigmatic power, when the target was already at 4 m distance (0.25 D from infinity). Negative accommodation would have been necessary to clear the defocus in the meridians behind infinity and humans do not have this option. Therefore, it can be excluded that accommodation improved VA in the case of real positive astigmatic defocus. Furthermore, only minor differences were found between positive or negative astigmatic defocus and only in the case of a 75% reduction of VA. In the case of lens-induced cross-cylinder defocus with positive or negative astigmatic lenses, there were no differences with the sign on VA. Also Bradley et al.10 found that the sign of imposed astigmatism had little effect on VA. Their measurements were done using Snellen charts, presented “in a big classroom,” with functional accommodation.
- Monochromatic light with a wavelength of 550 nm was used to simulate defocused images in ZEMAX. Of course, using a single wavelength for the simulations but a broadband light for the real experiments potentially carries a risk that additional cues are introduced that could be used by the visual system to enhance VA. But expanding the simulations to different wavelengths would require that the relative weighting of the resultant different monochromatic images must be known to be able to make predictions about VA. This is demanding and beyond the scope of this study.
- Higher order aberrations in the optics of the eyes could also have increased the depth of focus. However, because the average root mean square wavefront error of human eyes is only 0.3 μm,24 (equivalent to a quarter of a diopter), this factor is too small to explain the observed differences, the amount of imposed astigmatism in this study was up to 10 times higher than the depth of focus of the human eye (∼0.25 to 0.4 D25). Furthermore, it was shown that correcting higher order aberrations improves VA in young human subjects only marginally.12
In summary, a number of factors could explain the differences in VA with simulated and real astigmatic defocus, but we believe that the major confounders were analyzed and appear too minor to explain the observation. Therefore, we propose that additional special “tricks” of retinal or cortical image processing may be used to improve VA in the presence of real astigmatism, although the underlying image processing algorithms are not known and no direct proof can be provided here that they really operate.
Effect of the Axis of Astigmatism on VA with Letter Charts
In this study, VA was similarly reduced in most cases when the axes of real astigmatic and cross-cylinder defocus were at 0, 45, and 90 deg. This finding matches observations by Remon26 who found little influence of the axis of a simple myopic astigmatism on the VA. In contrast, Atchison et al.12 used cross-cylinders and found that the axis of astigmatism matters when testing with high contrast letter acuity chards. Atchison et al.27 observed that reading is facilitated with astigmatism against-the-rule vs. with astigmatism with-the-rule. Miller et al.28 imposed astigmatic defocus for 2 d in emmetropic subjects. They found that the subjects reported discomfort with astigmatic errors of 0.50 D but the level of discomfort was dependent on the axis of astigmatism. Although 70% of the subjects considered astigmatism at 0° unacceptable, even 90% complained when the axis of induced astigmatic error was at 45 or 90° (although the statistics is not reported in this article, the difference appears highly significant based on the standard deviations). Apparently, daily vision is differently affected by the axes of astigmatism, different from a letter chart with single characters and a test distance of 4 m. We used a wide range of different letters and numbers and produced eight different charts to reduce the bias for certain axes of astigmatism. It is likely that the exact design of the reading target determines how important different axes of astigmatism are. A secondary finding of this study, which remains unexplained, was that the axes of simulated astigmatism did affect readability of the letter charts.
A surprisingly high tolerance of the subjects was observed against real astigmatic and cross-cylinder blur, induced by trial lenses, compared with simulated defocus (in case of astigmatic and cross-cylinder defocus, there was a difference of about a factor of two). Possible reasons might be limitations inherent to the simulations, an influence of diffraction or the eye's higher order aberrations, or fluctuations of accommodation. Because the possible effects of these factors are small and do not seem to explain the observation, another possibility is that yet unknown image processing “tricks” are used by the neuronal machinery to improve VA in the presence of astigmatism.
This study was supported by the Werner Reichardt Center of Integrative Neuroscience, Tuebingen.
The parts of the study were presented as a poster at the 13th International Myopia Conference, July 26–29, 2010; Tübingen, Germany.
Institute for Ophthalmic Research
Section of Neurobiology of the Eye
72076 Tübingen, Germany
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