Journal of Refractive Surgery

Original Article Supplemental Data

Prediction of Visual Acuity and Contrast Sensitivity From Optical Simulations With Multifocal Intraocular Lenses

Joaquín Fernández, PhD; Manuel Rodríguez-Vallejo, PhD; Javier Martínez, OD; Noemi Burguera, MSc; David P. Piñero, PhD

Abstract

PURPOSE:

To evaluate whether the prediction of visual performance based on the modulation transfer function area (MTFa) calculated with optical simulations is better correlated with visual acuity or contrast sensitivity obtained from defocus curves in patients implanted with a trifocal intraocular lens.

METHODS:

Biometric eye data from 43 patients were used to create a mean eye model. A trifocal intraocular lens with a power obtained from the mean of the eyes implanted was incorporated into the model and the MTFa was calculated at the 11 defocus planes corresponding the 11 defocus locations measured in clinical practice. Simulations were conducted for pupil diameters of 2.5, 3, 3.5, and 4 mm. The MTFa correlation with visual acuity and contrast sensitivity was evaluated with the mean obtained after stratification of the clinical sample in four groups according to the previous pupil diameters.

RESULTS:

A linear model predicted the visual acuity and contrast sensitivity from MTFa with similar accuracy to nonlinear models, with R2 approximately 0.50 for visual acuity and approximately 0.42 for contrast sensitivity. A change of −0.01 logMAR and −0.02 logC was produced per unit of MTFa for visual acuity and contrast sensitivity, respectively. The mean difference between the visual acuity and contrast sensitivity obtained from the model and that measured in clinical practice was close to zero, but the bias varied depending on the defocus lens used, with higher deviation at −0.50 and −3.00 diopters of defocus.

CONCLUSIONS:

The MTFa obtained from optical simulations can be used to predict the mean visual acuity and contrast sensitivity consistently, with contrast sensitivity being more sensitive but with higher bias.

[J Refract Surg. 2019;35(12):789–795.]

Abstract

PURPOSE:

To evaluate whether the prediction of visual performance based on the modulation transfer function area (MTFa) calculated with optical simulations is better correlated with visual acuity or contrast sensitivity obtained from defocus curves in patients implanted with a trifocal intraocular lens.

METHODS:

Biometric eye data from 43 patients were used to create a mean eye model. A trifocal intraocular lens with a power obtained from the mean of the eyes implanted was incorporated into the model and the MTFa was calculated at the 11 defocus planes corresponding the 11 defocus locations measured in clinical practice. Simulations were conducted for pupil diameters of 2.5, 3, 3.5, and 4 mm. The MTFa correlation with visual acuity and contrast sensitivity was evaluated with the mean obtained after stratification of the clinical sample in four groups according to the previous pupil diameters.

RESULTS:

A linear model predicted the visual acuity and contrast sensitivity from MTFa with similar accuracy to nonlinear models, with R2 approximately 0.50 for visual acuity and approximately 0.42 for contrast sensitivity. A change of −0.01 logMAR and −0.02 logC was produced per unit of MTFa for visual acuity and contrast sensitivity, respectively. The mean difference between the visual acuity and contrast sensitivity obtained from the model and that measured in clinical practice was close to zero, but the bias varied depending on the defocus lens used, with higher deviation at −0.50 and −3.00 diopters of defocus.

CONCLUSIONS:

The MTFa obtained from optical simulations can be used to predict the mean visual acuity and contrast sensitivity consistently, with contrast sensitivity being more sensitive but with higher bias.

[J Refract Surg. 2019;35(12):789–795.]

Interest in the prediction of the postoperative visual performance before the implantation of a multifocal intraocular lens (IOL) has increased in recent years.1 For individual predictions, new devices have emerged with the aim of simulating how the patient would see after surgery with the implantation of a multifocal IOL.2–6 However, the main drawback of these technologies is that they would underestimate the postoperative visual performance with increasing age, especially after 50 years of age.7 Another approach is to predict the mean visual performance of the population depending on objective metrics resulting from measurements of the IOL in an optical bench or with optical simulations.8–13 The advances in the creation of models for predicting the visual performance have a great interest for manufacturers, which can estimate the visual performance of new designs before conducting a clinical trial, and surgeons, who can receive an estimation about the performance of an IOL in conditions such as with pupil diameter variation, something little described in the scientific literature due to the need for conducting well-controlled clinical studies with large sample sizes.

Authors who have explored the relationship between optical metrics and visual acuity have focused their studies on the prediction of visual acuity even though modulation transfer function (MTF) is measured in terms of contrast.8–13 MTF can be described as the contrast that an optical system can transfer and the most similar visual performance metric is the contrast sensitivity. The aim of this study was to evaluate whether the prediction of visual performance based on MTF area (MTFa) calculated with eye models is better correlated with visual acuity or contrast sensitivity obtained from defocus curves in patients implanted with a trifocal IOL.

Patients and Methods

Patients and Procedures

The study was approved by the Ethics Committee of Research, Almería Center, Torrecardenas Hospital Complex, and performed in adherence with the tenets of the Declaration of Helsinki. Data from 43 patients implanted with the Liberty IOL (Medicontur Medical Engineering Ltd. Inc., Zsámbék, Hungary) at Qvision, Ophthalmology Department of Vithas Virgen del Mar Hospital were retrieved from our historical database. Exclusion criteria were any surgical complication reported in the clinical history that might affect the postoperative visual performance. Other exclusion criteria were those for which the implantation of a spherical multifocal IOL was not indicated according to the habitual clinical practice, such as previous or coexisting ocular pathology that could decrease visual acuity (eg, corneal diseases, uveitis, retinopathy, glaucoma, dry eye, or amblyopia), corneal regular astigmatism of 1.20 diopters (D) or greater or irregular astigmatism higher than 0.5 µm measured with Pentacam HR (Oculus Optikgeräte, Wetzlar, Germany) at 4 mm.14

The data retrieved included biometric eye parameters for building a mean eye model, visual acuity defocus curve (VADC) and contrast sensitivity defocus curve (CSDC) measured with the Multifocal Lens Analyzer (version 1.0.8; Test-eye; defocuscurve.com, Spain),15 and the spherical equivalent refractive error at the 3-month follow-up visit. The tomography obtained in the patients with the Pentacam HR was used for creation of the eye model, including anterior (rc1) and posterior (rc2) corneal radii, asphericity from both corneal surfaces (qc1 and qc2) at 6 mm, actual lens position measured from vertex normal at the anterior cornea to the anterior IOL surface, and pupil location from posterior cornea. Effective lens position was calculated by means of the actual lens position plus the location of the object principal plane of the IOL implanted in each patient. The characteristics of the IOL implanted in each patient were provided by the manufacturer, including thickness and anterior and posterior radii (these are not detailed in the results because they were required to be kept as confidential by the manufacturer). The axial length was retrieved from the IOLMaster 500 (Carl Zeiss Meditec AG, Jena, Germany) and the pupil diameters at photopic and mesopic conditions from the Keratograph 5M (Oculus Optikgeräte).

Prediction Models

An eye model was created in Zemax 13 Professional (Zemax Development Corporation, Bellevue, WA) (Figure AA, available in the online version of this article). The corneal fictitious index was selected according to a previous study in which we found that the measured residual refractive error can be better predicted by fitting this index with the axial length (nc = 0.007 * axial length + 1.17).16 Refractive indexes for the other media were selected according to the most used by other eye models.17 Radii of the surfaces and distance between surfaces were obtained with the mean of biometric parameters described in the previous section.

(A) Eye model diagram created from the mean sample. The diffractive pattern of the multifocal intraocular lens (MIOL) has been exaggerated for making it visible in the image. Although rays have been traced, the computation included controlled interference. (B) Two-dimensional modulation transfer function (MTF) at the retinal plane after placing a defocus lens in front of the eye model as in clinical practice. Eleven plots were conducted for the 11 defocus lenses. MTF area (MTFa) represents the integrated area below the MTF line up to 50 lp/mm.

Figure A.

(A) Eye model diagram created from the mean sample. The diffractive pattern of the multifocal intraocular lens (MIOL) has been exaggerated for making it visible in the image. Although rays have been traced, the computation included controlled interference. (B) Two-dimensional modulation transfer function (MTF) at the retinal plane after placing a defocus lens in front of the eye model as in clinical practice. Eleven plots were conducted for the 11 defocus lenses. MTF area (MTFa) represents the integrated area below the MTF line up to 50 lp/mm.

The multifocal IOL simulated was the Liberty IOL. The design consists of a diffractive element using elevated phase shift technology for the generation of the intermediate foci by controlled interference.18 The labelled addition power at the IOL plane was +3.50 D for near vision and +1.75 D for intermediate vision. The simulations were performed with a custom software using the user defined surface feature of the Zemax. The applied method is based on an implementation of zone decomposition where the far-field light distribution is calculated by scalar diffraction.19

The two-dimensional MTF was obtained at the retinal plane after correction of the mean residual refractive error of the model and displacing the focus with 11 defocus lenses from +1.00 to −4.00 D in 0.50-D steps (Figure AB). The defocus lens was simulated with a paraxial XY surface at 12 mm from the anterior cornea. The range of spatial frequencies from 0 to 50 lp/mm was used for computing the MTFa at the retinal plane for each of the 11 defocus lenses. The process was repeated for pupils from 2.5 to 4 mm in 0.5-mm steps.

CSDC and VADC obtained from the clinical practice were stratified in four groups depending on the average between photopic and mesopic measurement of the pupil of each eye. The average from 2.26 to 2.75 mm was arranged in the 2.5-mm group, from 2.76 to 3.25 mm in the 3-mm group, from 3.26 to 3.75 mm in the 3.5-mm group, and from 3.76 to 4.25 mm in the 4-mm group. The mean obtained in each group was confronted with the MTFa predicted by the model for each pupil and for the 11 defocus lenses to create the prediction models.

Statistical Analysis

Only one eye per patient, randomly selected with a personalized MATLAB function (The Mathworks Inc., Natick, MA), was included in the study. The normality of data distributions for the demographic variables was tested with the Shapiro–Wilk test. A one-way analysis of variance was used for testing differences among pupil groups for each of the included demographic variables. The trapz function included in MATLAB was used for computing the MTFa and the Curve Fitting Tool App of MATLAB for calculating the models and R2 values. The statistical analyses were performed using the IBM SPSS 24.0 software for Windows (SPSS, Inc., Chicago, IL). Mean ± standard deviation was used for reporting the centrality and spread of the data. The logarithm of contrast (logC) or the logarithm of contrast sensitivity (logCS) were used for calculations, taking into account that the values are the same but with opposite sign (logC = −logCS).

Results

The eye model used for evaluating the MTFa is detailed in Table A (available in the online version of this article). This model results from the mean demographic biometric characteristics of the eyes included in the study. The theoretical performance of the IOL was dependent on the pupil diameter. Figure 1A shows that the MTFa was higher than 20 for all tested pupil diameters at far distance (0.00 D of defocus), with the higher variations at intermediate distance (−1.50 D of defocus), which was above 10 for pupils of 3 mm or less, and at near distance (−2.50 D), ranging from 11.34 (pupil of 4 mm) to 21.01 (pupil of 2.5 mm).

Eye Model Based on the Real Measurements of the Sample, Mean [95% Confidence Interval]a

Table A:

Eye Model Based on the Real Measurements of the Sample, Mean [95% Confidence Interval]

(A) Modulation transfer function area of the intraocular lens calculated in the eye model for different pupil sizes and with 11 defocus lenses. (B) Mean visual acuity and (C) contrast sensitivity defocus curves obtained for the total sample and for the eyes arranged in each of the groups according to pupil size. D = diopters

Figure 1.

(A) Modulation transfer function area of the intraocular lens calculated in the eye model for different pupil sizes and with 11 defocus lenses. (B) Mean visual acuity and (C) contrast sensitivity defocus curves obtained for the total sample and for the eyes arranged in each of the groups according to pupil size. D = diopters

Other demographic variables that can act as confounding factors adding noise to the predictions models are shown in Table 1. Patients in the 4-mm group were younger than those in the other groups (P > .05) and the irregular astigmatism at 4 mm was higher in the 3-mm group (P < .05). Figures 1B–1C show the mean VADC and CSDC, respectively, for the eyes with a pupil close to that selected for simulations. The MTFa values at the 11 defocus locations of Figure 1A were used for modelling the predictions of the visual acuity and contrast based on the corresponding 11 defocus locations of Figures 1B–1C, respectively. Contrast was used instead of contrast sensitivity for calculating the prediction model because the comparison between metrics was easier because visual acuity followed the same slope orientation as contrast and opposite to contrast sensitivity (logC = −logCS).

Demographic Characteristics of the Eyes Included in the Analysis (Mean ± SD)

Table 1:

Demographic Characteristics of the Eyes Included in the Analysis (Mean ± SD)

Figure 2 shows that the R2 value was similar between the linear model and the exponential decay model, either for visual acuity (Figure 2A) or contrast (Figure 2B), but more noise was obtained for contrast. The slope in the linear model for contrast nearly doubled the slope for visual acuity, which means that an increase of the MTFa that produced one line of visual acuity change (0.1 logMAR) corresponded to two lines of contrast change (−0.2 logC or 0.2 logCS).

Prediction models of (A) visual acuity and (B) contrast through the modulation transfer function area (MTFa). Exponential decay (dotted lines) and linear (continuous lines) functions were used for fitting the models.

Figure 2.

Prediction models of (A) visual acuity and (B) contrast through the modulation transfer function area (MTFa). Exponential decay (dotted lines) and linear (continuous lines) functions were used for fitting the models.

Contrast was correlated with visual acuity (r = 0.73, P < .0005) and the linear relationship between them was modelled by means of the following equation: contrast = 1.086 * visual acuity – 0.639 (F[1,471] = 535, P < .0005, R2 = 0.53). According to the previous model, the best agreement with a visual acuity of 0.3 logMAR was obtained for a contrast of −0.3 logC, and for achieving both values, MTFa of 3 and 6.5 were required, respectively.

Figures 3A–3B show the predicted defocus curves obtained from linear models of Figures 2A–2B, either for visual acuity or contrast sensitivity, respectively. Figures 3C–3D show the difference between the predicted visual acuity or contrast and the experimental mean achieved for each pupil group at each defocus location. Although the mean difference was close to zero for both metrics, a uniform bias was observed with a clinical overestimation in comparison to the predicted performance at −0.50 and −3.00 D of defocus locations. On the other hand, a clinical underestimation was found at intermediate locations (−1.50 and −2.00 D) and also in the extreme locations of the defocus curve, especially at +1.00 D. The limits of agreement were 0.15 logMAR for visual acuity and 0.27 logC for contrast sensitivity (Figures 3C–3D).

(A) Visual acuity and (B) contrast sensitivity defocus curves obtained from the modulation transfer areas through the linear equations of prediction. Mean differences between the (C) visual acuity or (D) contrast obtained from the predictions and the mean obtained with real eyes in clinical practice. D = diopters

Figure 3.

(A) Visual acuity and (B) contrast sensitivity defocus curves obtained from the modulation transfer areas through the linear equations of prediction. Mean differences between the (C) visual acuity or (D) contrast obtained from the predictions and the mean obtained with real eyes in clinical practice. D = diopters

Discussion

In this study, we evaluated the agreement between the optical simulations of a diffractive trifocal IOL and the clinical visual performance measured through VADC and CSDC. Several authors have studied the relationship between optical metrics and visual acuity.8–13 Felipe et al.8 reported a linear relationship (R2 = 0.91) between visual acuity at three distances and the average MTF for a spatial frequency between 0 and 100 line pairs per millimeter (lp/mm) in a study including only multifocal IOLs. However, studies that also include monofocal IOLs evaluate a wider range of optical qualities and in this case non-linear models predict the visual acuity better, following either a power function10 or an exponential decay function.12

Studies reporting non-linear relationships have used other metrics beyond the average MTF. Cardona et al.11 used the MTFa up to 100 lp/mm, suggesting that the increase in visual acuity was not noticeable for areas beyond approximately 20. Felipe et al.8 pointed out that the average MTF is almost equivalent to the MTFa and that it is easier to obtain and understand, so they decided to use the average MTF instead of the MTFa. Alarcon et al.10 also compared several metrics including cross-correlation coefficient,9 MTFa up to 50 lp/mm, weighted MTF, and weighted optical transfer function (R2 > 0.89),10 concluding that metrics that consider multiple spatial frequencies should be used instead of a singular spatial frequency. Specifically for the MTFa up to 50 lp/mm, several studies agree that is a good metric to predict the mean visual acuity achieved after a multifocal IOL implantation.8–13 Our results are in agreement with these studies and extend the findings for the first time to the contrast sensitivity. We hypothesized that if MTFa is an optical quality metric that describes the capability of a system to transfer the contrast of an object, the contrast sensitivity might be a more appropriate clinical metric than the visual acuity to predict, from the MTFa, the mean postoperative visual performance. The main argument for this hypothesis is because the visual acuity is based on a resolution task and the contrast sensitivity is based on a detection task.

Our results showed a higher slope in the model prediction for contrast sensitivity than for visual acuity, which means that, as expected, contrast sensitivity is a more sensitive clinical metric to detect changes in optical quality based on MTFa than visual acuity.20 This is in agreement with clinical findings reported in previous studies with IOLs on which significant differences have been reported for contrast sensitivity but not for visual acuity, such as after correction of chromatic aberration,21 in mesopic vision after spherical aberration correction,22 or among different pupil sizes.23

On the other hand, we found a slightly lower R2 value for contrast sensitivity than for visual acuity in the models. This can be explained by the poorer repeatability of contrast sensitivity, which results in a higher bias in the model.15,24 The higher variability of contrast sensitivity also explains why some authors found significant differences for visual acuity but not for contrast sensitivity, such as in corrections of spherical aberration.25 It is important to note that the significance of a statistical test depends on the magnitude of the effect and on the variability; therefore for equal sample sizes, contrast sensitivity has lower statistical power and this can lead to a type II error, non-rejecting the null hypothesis when it is really true.

Furthermore, an equivalent R2 value was obtained for the exponential decay and the linear fitting functions. These results are not in agreement with those reported by Vega et al.12 or Alarcon et al.,10 who found non-linear relationships between MTFa and visual acuity, but they are in agreement with those reported by Felipe et al.8 The differences are explained by including10,12 or not including8 monofocal IOLs in the model creation; therefore in studies with monofocal IOLs with MTFa higher than 28.75 or lower than 4.08 at any defocus location, the non-linear function will be preferable to our model.10,12

A direct comparison of the MTFa achieved with the IOL of the study with those reported from other IOLs should be interpreted with caution. No theoretical optical studies have been published with commercial IOLs and data reported by other authors come from optical bench measurements of real lenses. A difference up to 30% between the theoretical and manufactured performance might be found due to the manufacturing precision.26

Vega et al.12 reported a MTFa at intermediate (−1.50 D) of 12.6 for the FineVision IOL and 13.7 for the AcrivaUD Reviol Tri-ED IOL (3-mm aperture); the Liberty IOL achieved intermediate values of 12.52 at 2.5 mm of pupil diameter and 10.25 at 3 mm. The performance at intermediate for a 3-mm pupil was lower for the Liberty IOL (10.25) than for the FineVision IOL (12.6) or the AcrivaUD Reviol Tri-ED IOL (13.7), but the optical quality with both lenses at far (< 22.65 at 0.00 D) and the extended depth of focus at near (< 10.16 at −3.00 D) were higher for the Liberty IOL.

The MTFa obtained from optical simulations can be used to predict the mean visual acuity because it has been previously proved with measurements taken from an optical bench. In addition, our study demonstrates for the first time that mean contrast sensitivity can also be predicted with the MTFa. The contrast sensitivity was more sensitive to changes in MTFa than visual acuity, but the model showed a higher variability, mainly because contrast sensitivity is less repeatable than visual acuity and also can be more affected by other factors beyond the MTFa, such as irregular corneal astigmatism and age. Clinical studies looking for small differences between IOLs can benefit from incorporating CSDC, but more restricted inclusion criteria will be required to minimize the effect of confounding factors and higher sample sizes for increasing the statistical power.

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Demographic Characteristics of the Eyes Included in the Analysis (Mean ± SD)

Characteristic2.5 mm3 mm3.5 mm4 mmPTotal
No. of eyes513121343
Age (y)70 ± 671 ± 670 ± 864 ± 10.13268 ± 8
Effective lens position (mm)5.60 ± 0.195.41 ± 0.255.38 ± 0.325.58 ± 0.21.1225.47 ± 0.27
Pupil diameter (mm)
  Photopic1.93 ± 0.132.36 ± 0.232.72 ± 0.183.08 ± 0.25< .00052.63 ± 0.44
  Mesopic3.00 ± 0.13.72 ± 0.204.26 ± 0.254.92 ± 0.16< .00054.15 ± 0.66
  Average2.47 ± 0.103.04 ± 0.153.49 ± 0.154.00 ± 0.18< .00053.39 ± 0.53
Regular astigmatism (D)0.83 ± 0.290.72 ± 0.320.57 ± 0.230.74 ± 0.24.2640.70 ± 0.28
Irregular astigmatism at 4 mm (µm)0.15 ± 0.040.29 ± 0.170.17 ± 0.050.11 ± 0.03.001, .03,a .0005b0.19 ± 0.12
Corneal spherical aberration for mesopic pupil size (µm)0.01 ± 0.010.06 ± 0.070.07 ± 0.040.09 ± 0.04.045, .03c0.07 ± 0.05
Corneal Z40 for 6 mm of corneal diameter0.35 ± 0.030.33 ± 0.090.30 ± 0.080.25 ± 0.06.027, .04b0.30 ± 0.08

Eye Model Based on the Real Measurements of the Sample, Mean [95% Confidence Interval]a

MediumRadius of Curvature (mm)AsphericityDistance to Next Surface (mm)Diameter (mm)Refractive Index
Anterior cornea7.79 [7.72 to 7.86]−0.15 [−0.18 to −0.11]0.54 [0.53 to 0.55]61.334b
Posterior cornea6.41 [6.34 to 6.48]−0.24 [−0.29 to −0.19]3.35 [3.27 to 3.43]61.3374
PupilInfinityInfinity0.52 [0.46 to 0.58]3.39 [3.23 to 3.55]1.4625
Anterior surface IOLcr1lq1l1.1676
Posterior surface IOLcr2lq2l17.8236
RetinaInfinity6
Authors

From the Department of Ophthalmology (Qvision), Vithas Virgen del Mar Hospital, Almería, Spain (JF, MR-V, JM, NB); the Department of Optics, Pharmacology and Anatomy, University of Alicante, Alicante, Spain (DPP); and the Department of Ophthalmology, Vithas Medimar International Hospital, Alicante, Spain (DPP).

Dr. Fernández is a consultant for Medicontur. Dr. Rodríguez-Vallejo has designed and programmed the Multifocal Lens Analyzer test used in the current study, which is distributed by the Apple Store. The remaining authors have no financial or proprietary interest in the materials presented herein.

The authors thank the research and development department of Medicontur for providing the data required for calculating the optical simulation.

AUTHOR CONTRIBUTIONS

Study concept and design (JF, MR-V, JM, NB); data collection (JF, MR-V, JM, NB); analysis and interpretation of data (JF, MR-V, DPP); writing the manuscript (MR-V, DPP); critical revision of the manuscript (JF, MR-V, JM, NB, DPP); statistical expertise (MR-V); administrative, technical, or material support (JF, JM); supervision (JF, MR-V, JM, DPP)

Correspondence: Manuel Rodríguez-Vallejo, PhD, Department of Ophthalmology (Qvision), Vithas Virgen del Mar Hospital, 04120 Almería, Spain. E-mail: manuelrodriguezid@qvision.es

Received: July 22, 2019
Accepted: October 23, 2019

10.3928/1081597X-20191024-01

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