The statistics of refractive error maps : managing wavefront aberration analysis without Zernike polynomials

The refractive error of a human eye varies across the pupil and therefore may be treated as a random variable. The probability distribution of this random variable provides a means for assessing the main refractive properties of the eye without the necessity of traditional functional representation of wavefront aberrations. To demonstrate this approach, the statistical properties of refractive error maps are investigated. Closed-form expressions are derived for the probability density function (PDF) and its statistical moments for the general case of rotationally-symmetric aberrations. A closed-form expression for a PDF for a general non-rotationally symmetric wavefront aberration is difficult to derive. However, for specific cases, such as astigmatism, a closed-form expression of the PDF can be obtained. Further, interpretation of the distribution of the refractive error map as well as its moments is provided for a range of wavefront aberrations measured in real eyes. These are evaluated using a kernel density and sample moments estimators. It is concluded that the refractive error domain allows non-functional analysis of wavefront aberrations based on simple statistics in the form of its sample moments. Clinicians may find this approach to wavefront analysis easier to interpret due to the clinical familiarity and intuitive appeal of refractive error maps.

The effect of simulated astigmatic refractive error on reading performance

Background: Few studies have specifically investigated the functional effects of uncorrected astigmatism on measures of reading fluency. This information is important to provide evidence for the development of clinical guidelines for the correction of astigmatism. Methods: Participants included 30 visually normal, young adults (mean age 21.7 ± 3.4 years). Distance and near visual acuity and reading fluency were assessed with optimal spectacle correction (baseline) and for two levels of astigmatism, 1.00DC and 2.00DC, at two axes (90° and 180°) to induce both against-the-rule (ATR) and with-the-rule (WTR) astigmatism. Reading and eye movement fluency were assessed using standardized clinical measures including the test of Discrete Reading Rate (DRR), the Developmental Eye Movement (DEM) test and by recording eye movement patterns with the Visagraph (III) during reading for comprehension. Results: Both distance and near acuity were significantly decreased compared to baseline for all of the astigmatic lens conditions (p < 0.001). Reading speed with the DRR for N16 print size was significantly reduced for the 2.00DC ATR condition (a reduction of 10%), while for smaller text sizes reading speed was reduced by up to 24% for the 1.00DC ATR and 2.00DC condition in both axis directions (p<0.05). For the DEM, sub-test completion speeds were significantly impaired, with the 2.00DC condition affecting both vertical and horizontal times and the 1.00DC ATR condition affecting only horizontal times (p<0.05). Visagraph reading eye movements were not significantly affected by the induced astigmatism. Conclusions: Induced astigmatism impaired performance on selected tests of reading fluency, with ATR astigmatism having significantly greater effects on performance than did WTR, even for relatively small amounts of astigmatic blur of 1.00DC. These findings have implications for the minimal prescribing criteria for astigmatic refractive errors.

Prevalence and predictors of refractive error in a genetically isolated population : the Norfolk Island Eye Study

BACKGROUND: We aimed to determine the prevalence and associations of refractive error on Norfolk Island. DESIGN: Population-based study on Norfolk Island, South Pacific. PARTICIPANTS: All permanent residents on Norfolk Island aged ≥ 15 years were invited to participate. METHODS: Patients underwent non-cycloplegic autorefraction, slit-lamp biomicroscope examination and biometry assessment. Only phakic eyes were analysed. MAIN OUTCOME MEASURES: Prevalence and multivariate associations of refractive error and myopia. RESULTS: There were 677 people (645 right phakic eyes, 648 left phakic eyes) aged ≥ 15 years were included in this study. Mean age of participants was 51.1 (standard deviation 15.7; range 15-81). Three hundred and seventy-six people (55.5%) were female. Adjusted to the 2006 Norfolk Island population, prevalence estimates of refractive error were as follows: myopia (mean spherical equivalent ≥ -1.0 D) 10.1%, hypermetropia (mean spherical equivalent ≥ 1.0 D) 36.6%, and astigmatism 17.7%. Significant independent predictors of myopia in the multivariate model were lower age (P < 0.001), longer axial length (P < 0.001), shallower anterior chamber depth (P = 0.031) and increased corneal curvature (P < 0.001). Significant independent predictors of refractive error were increasing age (P < 0.001), male gender (P = 0.009), Pitcairn ancestry (P = 0.041), cataract (P < 0.001), longer axial length (P < 0.001) and decreased corneal curvature (P < 0.001). CONCLUSIONS: The prevalence of myopia on Norfolk Island is lower than on mainland Australia, and the Norfolk Island population demonstrates ethnic differences in the prevalence estimates. Given the significant associations between refractive error and several ocular biometry characteristics, Norfolk Island may be a useful population in which to find the genetic basis of refractive error.

Interocular corneal symmetry in refractive error groups

Purpose: To examine between eye differences in corneal higher order aberrations and topographical characteristics in a range of refractive error groups. Methods: One hundred and seventy subjects were recruited including; 50 emmetropic isometropes, 48 myopic isometropes (spherical equivalent anisometropia ≤ 0.75 D), 50 myopic anisometropes (spherical equivalent anisometropia ≥ 1.00 D) and 22 keratoconics. The corneal topography of each eye was captured using the E300 videokeratoscope (Medmont, Victoria, Australia) and analyzed using custom written software. All left eye data were rotated about the vertical midline to account for enantiomorphism. Corneal height data were used to calculate the corneal wavefront error using a ray tracing procedure and fit with Zernike polynomials (up to and including the eighth radial order). The wavefront was centred on the line of sight by using the pupil offset value from the pupil detection function in the videokeratoscope. Refractive power maps were analysed to assess corneal sphero-cylindrical power vectors. Differences between the more myopic (or more advanced eye for keratoconics) and the less myopic (advanced) eye were examined. Results: Over a 6 mm diameter, the cornea of the more myopic eye was significantly steeper (refractive power vector M) compared to the fellow eye in both anisometropes (0.10 ± 0.27 D steeper, p = 0.01) and keratoconics (2.54 ± 2.32 D steeper, p < 0.001) while no significant interocular difference was observed for isometropic emmetropes (-0.03 ± 0.32 D) or isometropic myopes (0.02 ± 0.30 D) (both p > 0.05). In keratoconic eyes, the between eye difference in corneal refractive power was greatest inferiorly (associated with cone location). Similarly, in myopic anisometropes, the more myopic eye displayed a central region of significant inferior corneal steepening (0.15 ± 0.42 D steeper) relative to the fellow eye (p = 0.01). Significant interocular differences in higher order aberrations were only observed in the keratoconic group for; vertical trefoil C(3,-3), horizontal coma C(3,1) secondary astigmatism along 45 C(4, -2) (p < 0.05) and vertical coma C(3,-1) (p < 0.001). The interocular difference in vertical pupil decentration (relative to the corneal vertex normal) increased with between eye asymmetry in refraction (isometropia 0.00 ± 0.09, anisometropia 0.03 ± 0.15 and keratoconus 0.08 ± 0.16 mm) as did the interocular difference in corneal vertical coma C (3,-1) (isometropia -0.006 ± 0.142, anisometropia -0.037 ± 0.195 and keratoconus -1.243 ± 0.936 μm) but only reached statistical significance for pair-wise comparisons between the isometropic and keratoconic groups. Conclusions: There is a high degree of corneal symmetry between the fellow eyes of myopic and emmetropic isometropes. Interocular differences in corneal topography and higher order aberrations are more apparent in myopic anisometropes and keratoconics due to regional (primarily inferior) differences in topography and between eye differences in vertical pupil decentration relative to the corneal vertex normal. Interocular asymmetries in corneal optics appear to be associated with anisometropic refractive development.

Effect of age and refractive error on the melanopsin mediated Post-Illumination Pupil Response (PIPR)

Melanopsin containing intrinsically photosensitive Retinal Ganglion cells (ipRGCs) mediate the pupil light reflex (PLR) during light onset and at light offset (the post-illumination pupil response, PIPR). Recent evidence shows that the PLR and PIPR can provide non-invasive, objective markers of age-related retinal and optic nerve disease, however there is no consensus on the effects of healthy ageing or refractive error on the ipRGC mediated pupil function. Here we isolated melanopsin contributions to the pupil control pathway in 59 human participants with no ocular pathology across a range of ages and refractive errors. We show that there is no effect of age or refractive error on ipRGC inputs to the human pupil control pathway. The stability of the ipRGC mediated pupil response across the human lifespan provides a functional correlate of their robustness observed during ageing in rodent models.

Preoperative visual acuity of cataract patients. Repeatability of visual acuity and refractive error measurements in clinical settings

Visual acuities at the time of referral and on the day before surgery were compared in 124 patients operated on for cataract in Vaasa Central Hospital, Finland. Preoperative visual acuity and the occurrence of ocular and general disease were compared in samples of consecutive cataract extractions performed in 1982, 1985, 1990, 1995 and 2000 in two hospitals in the Vaasa region in Finland. The repeatability and standard deviation of random measurement error in visual acuity and refractive error determination in a clinical environment in cataractous, pseudophakic and healthy eyes were estimated by re-examining visual acuity and refractive error of patients referred to cataract surgery or consultation by ophthalmic professionals. Altogether 99 eyes of 99 persons (41 cataractous, 36 pseudophakic and 22 healthy eyes) with a visual acuity range of Snellen 0.3 to 1.3 (0.52 to -0.11 logMAR) were examined. During an average waiting time of 13 months, visual acuity in the study eye decreased from 0.68 logMAR to 0.96 logMAR (from 0.2 to 0.1 in Snellen decimal values). The average decrease in vision was 0.27 logMAR per year. In the fastest quartile, visual acuity change per year was 0.75 logMAR, and in the second fastest 0.29 logMAR, the third and fourth quartiles were virtually unaffected. From 1982 to 2000, the incidence of cataract surgery increased from 1.0 to 7.2 operations per 1000 inhabitants per year in the Vaasa region. The average preoperative visual acuity in the operated eye increased by 0.85 logMAR (in decimal values from 0.03to 0.2) and in the better eye 0.27 logMAR (in decimal values from 0.23 to 0.43) over this period. The proportion of patients profoundly visually handicapped (VA in the better eye <0.1) before the operation fell from 15% to 4%, and that of patients less profoundly visually handicapped (VA in the better eye 0.1 to <0.3) from 47% to 15%. The repeatability visual acuity measurement estimated as a coefficient of repeatability for all 99 eyes was ±0.18 logMAR, and the standard deviation of measurement error was 0.06 logMAR. Eyes with the lowest visual acuity (0.3-0.45) had the largest variability, the coefficient of repeatability values being ±0.24 logMAR and eyes with a visual acuity of 0.7 or better had the smallest, ±0.12 logMAR. The repeatability of refractive error measurement was studied in the same patient material as the repeatability of visual acuity. Differences between measurements 1 and 2 were calculated as three-dimensional vector values and spherical equivalents and expressed by coefficients of repeatability. Coefficients of repeatability for all eyes for vertical, torsional and horisontal vectors were ±0.74D, ±0.34D and ±0.93D, respectively, and for spherical equivalent for all eyes ±0.74D. Eyes with lower visual acuity (0.3-0.45) had larger variability in vector and spherical equivalent values (±1.14), but the difference between visual acuity groups was not statistically significant. The difference in the mean defocus equivalent between measurements 1 and 2 was, however, significantly greater in the lower visual acuity group. If a change of ±0.5D (measured in defocus equivalents) is accepted as a basis for change of spectacles for eyes with good vision, the basis for eyes in the visual acuity range of 0.3 - 0.65 would be ±1D. Differences in repeated visual acuity measurements are partly explained by errors in refractive error measurements.

Investigation of the effect of simulated lens yellowing, transparency loss and refractive error on in vivo resonance Raman spectroscopy

To separately investigate the impact of simulated age-related lens yellowing, transparency loss and refractive error on measurements of macular pigment (MP) using resonance Raman spectroscopy.

Prevalence of refractive error in Europe: the European Eye Epidemiology (E3) Consortium

To estimate the prevalence of refractive error in adults across Europe. Refractive data (mean spherical equivalent) collected between 1990 and 2013 from fifteen population-based cohort and cross-sectional studies of the European Eye Epidemiology (E³) Consortium were combined in a random effects meta-analysis stratified by 5-year age intervals and gender. Participants were excluded if they were identified as having had cataract surgery, retinal detachment, refractive surgery or other factors that might influence refraction. Estimates of refractive error prevalence were obtained including the following classifications: myopia ≤−0.75 diopters (D), high myopia ≤−6D, hyperopia ≥1D and astigmatism ≥1D. Meta-analysis of refractive error was performed for 61,946 individuals from fifteen studies with median age ranging from 44 to 81 and minimal ethnic variation (98 % European ancestry). The age-standardised prevalences (using the 2010 European Standard Population, limited to those ≥25 and <90 years old) were: myopia 30.6 % [95 % confidence interval (CI) 30.4–30.9], high myopia 2.7 % (95 % CI 2.69–2.73), hyperopia 25.2 % (95 % CI 25.0–25.4) and astigmatism 23.9 % (95 % CI 23.7–24.1). Age-specific estimates revealed a high prevalence of myopia in younger participants [47.2 % (CI 41.8–52.5) in 25–29 years-olds]. Refractive error affects just over a half of European adults. The greatest burden of refractive error is due to myopia, with high prevalence rates in young adults. Using the 2010 European population estimates, we estimate there are 227.2 million people with myopia across Europe.

Two-year changes in refractive error and related biometric factors in an adult Chinese population

Importance: This article provides, to our knowledge, the first longitudinal population-based data on refractive error (RE) in Chinese persons.

Objective: To study cohort effects and changes associated with aging in REs among Chinese adults.

Design, Setting, and Participants: A 2-year, longitudinal population-based cohort study was conducted in southern China. Participants, identified using cluster random sampling, included residents of Yuexiu District, Guangzhou, China, aged 35 years or older who had undergone no previous eye surgery.

Methods: Participants underwent noncycloplegic automated refraction and keratometry in December 2008 and December 2010; in a random 50% sample of the participants, anterior segment ocular coherence tomography measurement of lens thickness, as well as measurement of axial length and anterior chamber depth by partial coherence laser interferometry, were performed.

Main Outcomes and Measures: Two-year change in spherical equivalent refraction (RE), lens thickness, axial length, and anterior chamber depth in the right eye.

Results: A total of 745 individuals underwent biometric testing in both 2008 and 2010 (2008 mean [SD] age, 52.2 [11.5] years; 53.7% women). Mean RE showed a 2-year hyperopic shift from −0.44 (2.21) to −0.31 (2.26) diopters (D) (difference, +0.13; 95% CI, 0.11 to 0.16). A consistent 2-year hyperopic shift of 0.09 to 0.22 D was observed among participants aged 35 to 64 years when stratifying by decade, suggesting that a substantial change in RE with aging may occur during this 30-year period. Cross-sectionally, RE increased only in the cohort younger than 50 years (0.11 D/y; 95% CI, 0.06 to 0.16). In the cross-sectional data, axial length decreased at −0.06 mm/y (95% CI, −0.09 to −0.04), although the 2-year change in axial length was positive and thus could not explain the cross-sectional difference. These latter results suggest a cohort effect, with greater myopia developing among younger persons.

Conclusions and Relevance: This first Chinese population-based longitudinal study of RE provides evidence for both important longitudinal aging changes and cohort effects, most notably greater myopia prevalence among younger persons.

Age, gender, biometry, refractive error, and the anterior chamber angle among Alaskan Eskimos.

BACKGROUND:

The prevalence of angle-closure glaucoma (ACG) is greater for Eskimos/Inuit than it is for any other ethnic group in the world. Although it has been suggested that this prevalence may be due to a population tendency toward shallower anterior chamber angles, available evidence for other populations such as Chinese with high rates of ACG has not consistently demonstrated such a tendency.

METHODS:

A reticule, slit-lamp, and standard Goldmann one-mirror goniolens were used to make measurements in the anterior chamber (AC) angle according to a previously reported protocol for biometric gonioscopy (BG) (Ophthalmology 1999;106:2161-7). Measurements were made in all four quadrants of one eye among 133 phakic Alaskan Eskimos aged 40 years and older. Automatic refraction, dilated examination of the anterior segment and optic nerve, and A-scan measurements of AC depth, lens thickness, and axial length were also carried out for all subjects.

RESULTS:

Both central and peripheral AC measurements for the Eskimo subjects were significantly lower than those previously reported by us for Chinese, blacks, and whites under the identical protocol. Eskimos also seemed to have somewhat more hyperopia. There were no differences in biometric measurements between men and women in this Eskimo population. Angle measurements by BG seemed to decline more rapidly over life among Eskimos and Chinese than blacks or whites. Although there was a significant apparent decrease in AC depth, increase in lens thickness, and increase in hyperopia with age among Eskimos, all of these trends seemed to reverse in the seventh decade and beyond.

CONCLUSIONS:

Eskimos do seem to have shallower ACs than do other racial groups. Measurements of the AC angle seem to decline more rapidly over life among Eskimos than among blacks or whites, a phenomenon also observed by us among Chinese, another group with high ACG prevalence. This apparent more rapid decline may be due to a cohort effect with higher prevalence of myopia and resulting wider angles among younger Eskimos and Chinese.

Heritability of refractive error and familial aggregation of myopia in an elderly American population.

PURPOSE: To determine the heritability of refractive error and the familial aggregation of myopia in an older population. METHODS: Seven hundred fifty-nine siblings (mean age, 73.4 years) in 241 families were recruited from the Salisbury Eye Evaluation (SEE) Study in eastern Maryland. Refractive error was determined by noncycloplegic subjective refraction (if presenting distance visual acuity was < or =20/40) or lensometry (if best corrected visual acuity was >20/40 with spectacles). Participants were considered plano (refractive error of zero) if uncorrected visual acuity was >20/40. Preoperative refraction from medical records was used for pseudophakic subjects. Heritability of refractive error was calculated with multivariate linear regression and was estimated as twice the residual between-sibling correlation after adjusting for age, gender, and race. Logistic regression models were used to estimate the odds ratio (OR) of myopia, given a myopic sibling relative to having a nonmyopic sibling. RESULTS: The estimated heritability of refractive error was 61% (95% confidence interval [CI]: 34%-88%) in this population. The age-, race-, and sex-adjusted ORs of myopia were 2.65 (95% CI: 1.67-4.19), 2.25 (95% CI: 1.31-3.87), 3.00 (95% CI: 1.56-5.79), and 2.98 (95% CI: 1.51-5.87) for myopia thresholds of -0.50, -1.00, -1.50, and -2.00 D, respectively. Neither race nor gender was significantly associated with an increased risk of myopia. CONCLUSIONS: Refractive error and myopia are highly heritable in this elderly population.

Distribution and visual impact of postoperative refractive error after cataract surgery in rural China: study of cataract outcomes and up-take of services report 4.

PURPOSE: To model the possible impact of using average-power intraocular lenses (IOLs) and evaluate the postoperative refractive error in patients having cataract surgery in rural China.SETTING: Rural Guangdong, China.METHODS: Patients having cataract surgery by local surgeons were examined and visual function was assessed 10 to 14 months after surgery. Subjective refraction at near and distance was performed bilaterally by an ophthalmologist. Patients had a target refraction of -0.50 diopter (D) based on ocular biometry.RESULTS: Of the 313 eligible patients, 242 (77%) could be contacted and 176 (74% of contacted patients, 56% overall) were examined. Examined patients had a mean age of 69.4 +/- 10.5 years. Of the 211 operated eyes, 73.2% were within +/-1.0 D of the target refraction after surgery. The best presenting distance vision was in patients within +/-1.0 D of plano and the best presenting near vision, in those with mild myopia (<-1.0 D to > or =2.0 D) (P= .005). However, patients with hyperopia (>+1.0 D) reported significantly better adjusted visual function than those with emmetropia or myopia (<-1.0 D). When the predicted use of an average-power IOL (median +21.5 D) was modeled, predicted visual acuity was significantly reduced (P= .001); however, predicted visual function was not significantly altered (P>.3).CONCLUSIONS: Accurate selection of postoperative refractive error was achieved by local surgeons in this rural area. Based on visual function results, aiming for mild postoperative myopia may not be suitable in this setting. Implanting average-power IOLs significantly reduced postoperative presenting vision, but not visual function.

Spectacle acceptance among secondary school students in rural China: the Xichang Pediatric Refractive Error Study (X-PRES)--report 5.

PURPOSE: To assess determinants of spectacle acceptance and use among rural Chinese children. METHODS: Children with uncorrected acuity < or = 6/12 in either eye and whose presenting vision could be improved > or = 2 lines with refraction were identified from a school-based sample of 1892 students. Information on obtaining glasses and the benefits of spectacles was provided to children, families, and teachers. Purchase of new spectacles and reasons for nonpurchase were assessed by direct inspection and interview 3 months later. RESULTS: Among 674 (35.6%) children requiring spectacles (mean age, 14.7 +/- 0.8 years), 597 (88.6%) were followed up. Among 339 children with no glasses at baseline, 30.7% purchased spectacles, whereas 43.2% of 258 children with inaccurate glasses replaced them. Most (70%) subjects paid US$13 to $26. Among children with bilateral vision < or = 6/18, 45.6% bought glasses. In multivariate models, presenting vision < 6/12 (P < 0.009), refractive error < -2.0 D (P < 0.001), and amount willing to pay for glasses (P = 0.01) were predictors of purchase. Reasons for nonpurchase included satisfaction with current vision (78% of those with glasses at baseline, 49% of those without), concerns over price or parental refusal (18%), and fear glasses would weaken the eyes (13%). Only 26% of children stated that they usually wore their new glasses. CONCLUSIONS: Many families in rural China will pay for glasses, though spectacle acceptance was < 50%, even among children with poor vision. Acceptance could be improved by price reduction, education showing that glasses will not harm the eyes, and parent-focused interventions.

Corneal hysteresis and axial length among Chinese secondary school children: the Xichang Pediatric Refractive Error Study (X-PRES) report no. 4.

PURPOSE: To evaluate the association between corneal hysteresis and axial length/refractive error among rural Chinese secondary school children. DESIGN: Cross-sectional cohort study. METHODS: Refractive error (cycloplegic auto-refraction with subjective refinement), central corneal thickness (CCT) and axial length (ultrasonic measurement), intraocular pressure (IOP), and corneal hysteresis (Reichert Ocular Response Analyzer) were measured on a rural school-based cohort of children. RESULTS: Among 1,233 examined children, the mean age was 14.7 +/- 0.8 years and 699 (56.7%) were girls. The mean spherical equivalent (n = 1,232) was -2.2 +/- 1.6 diopters (D), axial length (n = 643) was 23.7 +/- 1.1 mm, corneal hysteresis (n = 1,153) was 10.7 +/- 1.6 mm Hg, IOP (n = 1,153) was 17.0 +/- 3.4 mm Hg, and CCT (n = 1,226) was 553 +/- 33 microns. In linear regression models, longer axial length was significantly (P < .001 for both) associated with lower corneal hysteresis and higher IOP. Hysteresis in this population was significantly (P < .001) lower than has previously been reported for normal White children (n = 42, 12.3 +/- 1.3 mm Hg), when adjusting for age and gender. This difference did not appear to depend on differences in axial length between the populations, as it persists when only Chinese children with normal uncorrected vision are included. CONCLUSIONS: Prospective studies will be needed to determine if low hysteresis places eyes at risk for axial elongation secondary or if primary elongation results in lower hysteresis.

Visual disability, visual function, and myopia among rural chinese secondary school children: the Xichang Pediatric Refractive Error Study (X-PRES)--report 1.

PURPOSE: To evaluate visual acuity, visual function, and prevalence of refractive error among Chinese secondary-school children in a cross-sectional school-based study. METHODS: Uncorrected, presenting, and best corrected visual acuity, cycloplegic autorefraction with refinement, and self-reported visual function were assessed in a random, cluster sample of rural secondary school students in Xichang, China. RESULTS: Among the 1892 subjects (97.3% of the consenting children, 84.7% of the total sample), mean age was 14.7 +/- 0.8 years, 51.2% were female, and 26.4% were wearing glasses. The proportion of children with uncorrected, presenting, and corrected visual disability (< or = 6/12 in the better eye) was 41.2%, 19.3%, and 0.5%, respectively. Myopia < -0.5, < -2.0, and < -6.0 D in both eyes was present in 62.3%, 31.1%, and 1.9% of the subjects, respectively. Among the children with visual disability when tested without correction, 98.7% was due to refractive error, while only 53.8% (414/770) of these children had appropriate correction. The girls had significantly (P < 0.001) more presenting visual disability and myopia < -2.0 D than did the boys. More myopic refractive error was associated with worse self-reported visual function (ANOVA trend test, P < 0.001). CONCLUSIONS: Visual disability in this population was common, highly correctable, and frequently uncorrected. The impact of refractive error on self-reported visual function was significant. Strategies and studies to understand and remove barriers to spectacle wear are needed.