Optical performance of progressive addition lenses (PALs) with astigmatic prescription

The progressive addition lens (PAL) is a spectacle lens design with progressive refractive power changes across the lens surface to provide sharp vision at different viewing distances for patients with reduced accommodative strength. It has gained in popularity not just for presbyopic patients, but also patients with occupational (office, driving, or digital device) and therapeutic (e.g., myopia control) needs. However, despite the increasing prevalence of astigmatism in adults > 40 years old who rely on PAL correction, no metric is available to reflect the optical variation in PALs with astigmatic prescriptions. Based on recent studies, four novel optical metrics sensitive to variation of refractive power across the lens surface of PALs have been developed. These metrics were used to compare the optical performance of PALs of various prescriptions, designs, and manufacturers. For each lens, the refractive power profile was first measured with a Moire-deflectometry-based instrument.The data was then exported and analyzed using a two-dimensional error map for each of the four metrics. The results revealed significant impacts of astigmatic prescription, providing evidence for the usefulness of these metrics in quantifying the optical performance of PALs for patients with astigmatic prescriptions.


Scientific Reports
| (2021) 11:2984 | https://doi.org/10.1038/s41598-021-82697-0 www.nature.com/scientificreports/ patients' visual performance. Given the complex spatial distribution of aberrations across the lens surface of PAL, a comprehensive assessment of optical quality would require a global metric that can reflect the optical quality of the whole lens surface area. In this respect, Sheedy 17,18 proposed to evaluate the power distribution of PALs by estimating the width of clear "zones" designated for distance, intermediate, and near vision, based on the departure from the target prescription. This method provides a quantitative description of the area that provides clear vision, but the method has not been tested in PALs with astigmatic prescriptions 19 .
The primary aim of this study was to develop quantitative global metrics to examine the influence of astigmatism correction on the optical performance of PALs. The hypothesis is that the optical performance of PALs is quantifiable with metrics sensitive to astigmatic prescription. To test this hypothesis, four metrics varying in complexity were developed and used to compare the optical performance of PALs of different prescriptions, designs, and manufacturers. The results indicate that metrics sensitive to astigmatic prescription can quantitatively differentiate the optical performance of PALs.

Methods PAL samples.
To compare the influence of astigmatic axis on lens optics, different designs of PALs from three lens manufacturers (Essilor, Hoya, and Zeiss) with identical addition powers (+ 2.00D), but with cylindrical axis of orthogonal direction (i.e., 90 vs. 180), were tested. Because myopia (or short-/near-sightedness) is increasing in prevalence worldwide, especially in East Asian population 20 ), distance prescriptions of PALs were based on the characteristics of refractive error reported in a recent study on the Hong Kong Chinese clinical population 21 : − 4.00 DS, − 4.00 DS/− 2.00 DC × 90, and − 4.00 DS/− 2.00 DC × 180. For each lens manufacturer, two popular PAL models with two different progressive corridors were included (see Table 1). A total of 12 lenses from each lens manufacturer (3 prescriptions × 2 models × 2 corridor lengths) were provided in-kind for measurement. All lenses had a refractive index of 1.6. Figure 1A-C shows the instruments and steps involved in measuring the refractive power profile of PALs. Based on the Moire deflectometry principle, the Rotlex Free Form Verifier (Rotlex, Omer, Israel) used a diverging light to illuminate the PAL lens surface. Light refracted from the PAL passed through two gratings creating a Moire interference pattern that was converted into arrays of local wavefront properties. These   www.nature.com/scientificreports/ arrays were used to calculate two-dimensional refractive maps of spherical and cylindrical powers. The software required the input of the following lens parameters for the calculation of the refractive power map: lens refractive index (provided by the lens manufacturer), center thickness (measured by a lens gauge), and front surface radius (measured by a lens clock). The locations of the two standard laser-engraved symbols of the PAL were identified with standard procedures and marked by a fine-tip marker. The lens front was placed face down according to the  Since all prescriptions have identical spherical equivalent, they lie on the same astigmatic plane. All prescriptions lie on the big blue circle, representing a common cylinder magnitude of J . The symbol �J α stands for the unwanted errors after subtracting the target (with axis α = 0 and 90) from the actual prescriptions. Note that in this example, the first metric (JCC error map) predicts the same optical performance for the two prescriptions ( �J α = 0 ). The second metric (vectorial error map) J 45 components also could not differentiate between the two prescriptions when compared to the target prescription, but the different J 0 components for different axes ( α = 0 and 90) predicted different optical performance (see text for details). www.nature.com/scientificreports/ user's manual. The alignment of the tested lens was checked by positioning the two ink marks (laser-engraved symbols) within the green rings on the screen (Fig. 1B). Once all the parameters were set, five repeated measurements were acquired for each lens and averaged.

Measurement.
Analysis. ASCII data files from the Rotlex Free Form Verifier were imported into Excel and analyzed using a custom Matlab algorithm. The analysis was performed over a 64-mm lens diameter with a resolution of 0.3 mm. Using power vector analysis 22 , the spherical (S), cylindrical power (C) and axis (α) components in the conventional prescription notation ( S, +C × α + 90 ) were converted into rectangular E = (M, J 0 , J 45 ) and polar forms (M, J × α): where M is the spherical equivalent, J 0 , J 45 and J are the cylindrical components of a Jackson cross-cylinder (JCC) lens at axis 0°, 45°, and α , respectively. For instance, for a prescription written in the conventional negativecylinder notation (− 4.00/− 2.00 × 180), the power vector coordinate is (− 5.00, + 1.00, 0) in rectangular form and (− 5.00, − 1.00 × 90) in polar form. The formulae above were used to generate a set of optical metrics for comparing refractive power differences, including spherical defocus, astigmatism, and overall optical blurs. To determine the impacts on lens optics www.nature.com/scientificreports/ due to the varying refractive powers across the lens surface, an error map (�) of refraction was constructed by subtracting the target refraction from actual refraction of the lens prescription at each position on the PAL surface. The mean and standard deviation of the error maps across the four samples of the same prescription (i.e., 2 models × 2 corridors lengths) are reported for individual manufacturers. To determine the influence of the PAL corridor length and model type on the reported error map, the mean absolute difference of the error maps between PALs having a different corridor length and model, respectively, were also calculated. Small differences indicate that the averaged error map reported is representative of the performance of the selected PAL prescription, whereas large differences indicate that the corridor length or/and model affects the performance of prescription differently. The error maps generated are described below: • For unwanted spherical defocus, the error map M was calculated for three target (desired) viewing distances (i.e., distant, intermediate, and near) of PALs: where M actual and M target are the actual (i.e., measured) and target (i.e., intended) spherical powers, respectively. The M target is computed as: where M PALs were −5.00D and −4.00D for PALs with and without astigmatism correction, respectively; V represents the vergences of light (0.00 D, − 1.50 D, − 2.50 D) from three target viewing distances (distant = infinity; intermediate = 66 cm; near = 40 cm); and R x is the residual accommodative power. To account for the residual capacity of an average presbyope to accommodate for nearby objects, as recommended by Andre 23 , a residual accommodative power of R x = 0.50D at the intermediate and near distances was assumed.
• For unwanted astigmatism, two optical metrics, which measure the difference between the actual and target cylindrical profiles at each position on the lens surface, as depicted in Fig. 2, were developed. www.nature.com/scientificreports/ • The first metric concentrated on the error between the actual and target astigmatic magnitude of the PALs at each location on the lens surface. The error map associated with a single JCC component at axis α , here termed the JCC error map �J α , was generated as follows: Although this metric can quantify the difference in magnitudes of actual and target astigmatic blur, it ignores the potential optical effects of astigmatic axis. The JCC error map is equivalent to half of the astigmatic magnitude ("C ") in Sheedy's study 17 .
• To overcome the limitation of �J α of ignoring the astigmatic axis, the second metric, the vectorial cylindrical error map, J v was computed by taking into account the vectorial differences between actual and target astigmatic components of the power vector ( J 0 , J 45 ), as follows: where the cylinder error maps associated with each J 0 and J 45 component were as follows: To illustrate how the second metric, J v , can overcome the limitation of the first metric, �J α , Fig. 3 shows the vectorial differences that could only be measured by J v but not the �J α metric. For illustration purposes, the two metrics were compared by their ability to detect differences in astigmatic prescription. Three prescriptions with identical spherical equivalent and cylinder (J), one actual (blue square) and two target (red J 45 x, y = J 45,actual x, y − J 45,target .   is not shown here. Each of the three symbols occupies a locus on the big blue circle, representing the same magnitude of J. Since the first metric ( �J α ) only considers the magnitude of J, comparing an actual refractive prescription to either of these two target prescriptions with identical J would return a zero value ( �J α = 0) , i.e., the metric could not differentiate between the two prescriptions with different astigmatic axes. In contrast, J v takes into account the vectorial differences, returning two distinctly different quantities relative to the actual prescription ( �J α , represented by red lines), indicating a differential optical performance.
It is important to note that when actual and target axes are the same, the vectorial cylindrical error map J v is equivalent to �J α . This is because, for α actual = α target = α , the following relationships: become: �J 0 = J actual cos (2α actual ) − J target cos 2α target �J 45 = J actual sin (2α actual ) − J target sin 2α target Table 2. Summary of two-way ANOVA results for different metrics. This statistical summary indicates the level of significance obtained with various metrics for the different manufacturers and prescriptions. The color code of the p-value is as follows: dark blue, p > 0.05 (not significant); light blue, p ≤ 0.05; yellow, p ≤ 0.01; red, p ≤ 0.001. Note that the blank field corresponds to the conditions where the statistical test is not applicable (i.e., presence of interaction between prescription and manufacturer, or absence of a significant main effect). H Hoya, E Essilor, Z Zeiss. www.nature.com/scientificreports/ By substituting those expression in J , we get: • To provide a comprehensive estimation of the overall optical effects introduced by the actual refractive power variation across the lens surface, the length of the power vector 22,24 was analyzed, as follows: A previous study showed that the length of the power vector can explain more than 90% of the variance in uncorrected visual acuity found in the ametropic population with simple myopia, myopic astigmatism, and mixed astigmatism 15 . In addition, the length of the power vector was found to accurately predict the change in visual acuity associated with optically induced variations in spherical and cylindrical refractive errors, with a correlation coefficient R 2 ranging from 0.90 to 0.99 in groups with mixed astigmatism, simple myopic astigmatism. and hyperopic astigmatism 25 .
• The overall optical blurs arising from the mismatch between actual and target prescription was computed by vectorial analysis of the power vector according to the above rule of subtraction 26 : The length of this vector difference is a measure of the overall error map of refraction L: where M, J 0 , and J 45 are the error maps associated with the spherical and cylindrical components, respectively, and are described individually above.
In order to quantify the useful region providing visual clarity on the lens surface, the "area of clear vision" was defined as an area exhibiting dioptric errors below a certain threshold corresponding to the tolerance of optical blur. Here, the area of clear vision within ± 0.25D of the target prescription was reported, reflecting both the level of accuracy provided by lens prescriptions and the level of blur 17 . This area was used to compare the optical performance between different PALs.
Area of clear vision. �J α metric. Figure 8 compares the area of clear vision determined by the JCC astigmatic magnitude alone ( J α ) . Using this metric that does not take into account the influence of the astigmatic axis, significant main effects on the area of clear vision were found for both prescription and manufacturer (Two-way ANOVA: prescription effect, p < 0.001; manufacturer effect, p < 0.006). Although all PALs bear some amount of unwanted astigmatism (Fig. 8A, dark blue area), PALs with astigmatic prescription tend to have a smaller magnitude of errors. For instance, the area of clear vision is more than 60% for PALs with − 2.00 DC × 90 prescription (Fig. 8B). Tables 2 and 3 use colored boxes to indicate significant effects on the area of clear vision for different metrics, as revealed by two-way ANOVA and post-hoc tests, respectively. While an unfilled (blank) box indicates no www.nature.com/scientificreports/ statistical test is applicable (either because the presence of an interaction effect, or because there was no main effect), the different colored boxes represent different p-values (dark blue, p > 0.05, not significant; light blue, p ≤ 0.05; yellow, p ≤ 0.01; red, p ≤ 0.001). Note that because no interaction effect was found for J α (dark blue, Table 3), the corresponding column for J α in Table 3 has only blank boxes. The following sections cover the results for different metrics, further detailed statistical results are available in Supplementary Tables.
J v metric. The metric J v , which takes into account the influence of the astigmatic axis, shows a reduction in the range of the clear vision area, as compared to the metric �J α (Fig. 9 vs. Fig. 8, 25-45% vs. 40-80% for J v and �J α metrics, respectively), for PALs with astigmatic correction. These differences were particularly obvious with the − 2.00 DC × 90 prescription, indicating the dominant role of the astigmatic axis in the optical performance. Unlike �J α , there was a significant interaction effect when using J v to compare the area of clear vision across PALs (Two-way ANOVA: manufacturer × prescription, p < 0.05). The difference in optical performance between manufacturers was particularly obvious for PALs with − 2.00 DC × 90 prescription ( Fig. 9; see Tables 2 and 3 for statistical details).
M metric. The differences in optical performance between PALs were not simply affected by astigmatic magnitude ( J α ) and axis ( J v ), but tended to vary strongly with viewing distances. Figure 10 shows

Discussion
The four optical metrics developed in this study revealed the impacts of prescription and manufacturer on the optical performance of PALs (Tables 2 and 3). Whilst J α metric showed a significant effect of prescription on the optical performance, J v metric shows a significant interaction effect of prescription and manufacturer on the area of clear vision (Table 2). In contrast, comparing the optical performance at different viewing distances using M vs. L also revealed subtle differences (Table 2): for distant viewing, the performance depended on the interaction between prescription and manufacturer with M, but depended only on manufacturer with L ( Table 2); for intermediate viewing, it depended on the manufacturer with both M and L ; for near viewing, it depended on both prescription and manufacturer for both M and L . These results highlight the influence of optical parameters on optimizing lens performance across surface area. The area of clear vision provided by PAL is limited by the axis of the astigmatic correction. Although a metric such as J α could reflect the variation of optical blur due to astigmatic magnitude (Fig. 8), but the metric per se has little clinical value to one whose vision is sensitive to blur associated with the astigmatic axis-in this case a metric such as J v provides a better evaluation of optical performance (e.g., compare area of clear vision in Fig. 8 with Fig. 9). Nevertheless, all four metrics including J α underscore the pivotal role of astigmatic prescription in determining the optical performance of PALs. Of particular importance is the finding that, as supported by both M and L metrics (Table 2, Figs. 10 and 11), the area of clear vision at near viewing distance shows a significant reduction in PALs with − 2.00 DC × 90 when compared to the other two prescriptions. To the best of our knowledge, this is the first study that shows a significant impact of astigmatic prescription in PALs using novel, quantifiable metrics.
Although these results demonstrate some obvious differences between PALs with different astigmatic correction and between PALs from different manufacturers (Figs. 8, 9, 10, Tables 2 and 3), caution should be employed when generalizing these results to lenses with different prescriptions, because only a small number of PALs were used. Some variations in the distribution of optical errors across PALs are expected with individual lens parameters. In addition to these factors, other non-optical factors can also influence the resultant optical performance. For examples, studies using wavefront analysis 5,16 showed that the optical interactions between the ocular aberrations and PAL can influence the retinal image quality at specific lens zones. While such optical effects may be overcome by a personalized PAL design taking into account the potential spectacle tilts as-worn (e.g., pantoscopic tilt 4,7 and wrap angle 11 ) and the preferred visual habits (e.g., head and eye movements 9,27 ), it remains unknown how the intersubject variability in ocular parameters and the way eye and lens optics interact in real view condition 28 , affect the visual performance of a conventional PAL (i.e. with standard values). Since these parameters may interact with the optics of PALs in a complex manner, prescribing conventional PALs with astigmatic correction for myopic children warrants careful consideration, because growing evidence in animal studies 29 has indicated the influence of astigmatic blur in regulating early eye growth. www.nature.com/scientificreports/ In conclusion, this study used novel quantifiable optical metrics to show the differences in optical performance of PALs attributable to prescription and manufacturer. Further work is needed to investigate the potential impacts of these optical metrics on visual performance in real life conditions. Received: 8 June 2020; Accepted: 18 January 2021