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Polychromatic defocus could affect the optimal residual spherical aberration that could yield the best image quality for patients with intraocular lenses (IOLs). Modulation transfer functions (MTFs) were generated using a model that included polychromatic defocus. The maximum MTF volume occurred at + 0.05 μm of overall ocular spherical aberration. For 3 case studies, the optimal overall ocular spherical aberration was ~0.05 μm more positive with the contribution of polychromatic defocus than without it. Overall, the model indicated that image quality was usually best when IOLs allowed overall ocular spherical aberration that was slightly positive, rather than strongly positive, zero, or negative.
©2010 Optical Society of America
1. Introduction
A variety of monofocal intraocular lenses (IOLs) offer different levels of asphericity intended to compensate for all, some, or none of the positive spherical aberration of the cornea. The Tecnis IOL (AMO Inc.) has −0.27 µm of spherical aberration with a 6-mm pupil aperture [1]. That value was equal and opposite to the average corneal spherical aberration of the study population first implanted with that IOL [1], yielding an average of 0 µm of overall ocular spherical aberration, representing full correction of spherical aberration. In contrast, the SofPort AO IOL (Bausch & Lomb Inc.) has 0 µm of spherical aberration over a 6-mm zone [2], representing zero correction of spherical aberration. Between these full-correction and zero-correction options lies the AcrySof IQ IOL (Alcon Laboratories, Inc.), which intentionally allows a slightly positive residual spherical aberration to remain in the optical system of the eye. The AcrySof IQ IOL (model SN60WF) has −0.20 µm of spherical aberration over the 6-mm optic; in various studies, this yielded an average overall ocular spherical aberration of + 0.07 to + 0.11 μm [3–5].
A slightly positive residual value for spherical aberration may be desirable. Several studies have reported that populations of healthy youthful eyes with good vision had a positive ocular spherical aberration [6], with an average value at + 0.08 µm to + 0.13 µm [7,8]. A population of 70 eyes with supernormal vision (20/15 Snellen or better) had average spherical aberration of + 0.11 ± 0.08 µm [9]. When 3 subjects with good corrected vision were tested with an adaptive optics system, peak contrast sensitivity occurred at + 0.06 μm of spherical aberration for the subjects fitted as a group, or at an average of + 0.09 μm after subjects were fitted individually [10].
Although small positive spherical aberration values may be advantageous on average, an individual eye may have a unique corneal wavefront aberration pattern that would best be corrected by a specific amount of asphericity from an IOL in order to yield optimal image quality at the retina. Therefore, some researchers have advocated customized selection of IOL asphericity for each eye, based on total higher-order aberration [11,12]. Other researchers selected IOLs for patients based only on spherical aberration, not on total higher-order aberration, and found that the group selected to have ~0.1 µm of residual spherical aberration had better contrast sensitivity than the group selected to have ~0.0 µm of residual spherical aberration [13]. Similarly, when a man was implanted contralaterally with 2 different aspheric IOLs, the eye with overall ocular spherical aberration of + 0.11 µm had better mesopic contrast sensitivity than the contralateral eye with overall ocular spherical aberration of + 0.02 µm [14].
A residual overall ocular spherical aberration of 0 μm would minimize the wavefront if no other aberrations existed; however, the interaction with other aberrations complicates the selection of optimal spherical aberration. This interaction among aberrations may dictate that a positive residual spherical aberration may be beneficial for the majority of patients. One parameter in the complicated optimization process could be the effect of longitudinal chromatic aberration [15]. This effect, represented by the polychromatic defocus aberration coefficient, c2 0, was not included in some previous models [11,12]. Idealistic models may assume that any second-order aberrations (defocus or astigmatism) that remained after cataract surgery were fully corrected by some means, such as spectacles [11]. However, even spectacles do not correct longitudinal chromatic aberration [16,17], which is a component of defocus.
This study was designed to consider the contribution of the polychromatic defocus aberration coefficient when identifying the optimal target residual spherical aberration of an eye, in order to identify a spherical aberration value that could enhance image quality for pseudophakic eyes.
2. Methods
2.1. Mathematical modeling
Polychromatic point-spread functions were generated in MatLab (The MathWorks, Inc., Natick, MA) by using equations previously described by Dai [18]. The model included the Stiles-Crawford effect, and point-spread functions were weighted in the image plane based on the retinal spectral response function. The following wavelengths were used to represent the visible spectrum: 400 nm, 450 nm, 500 nm, 550 nm, 600 nm, 650 nm, and 700 nm. All calculations assumed a pupil with a 6-mm diameter. Modulation transfer functions were generated as the modulus of the Fourier transform of the point-spread functions. The volume of each modulation transfer function was calculated with the spatial frequency limit up to 50 line pairs per millimeter (lp/mm) to represent 20/40 Snellen vision, and was calculated at the spatial frequency limit up to 100 lp/mm to represent 20/20 vision.
The mathematical parameters discussed above are similar to a previous idealistic model [11]. In this study, longitudinal chromatic aberration was investigated in relation to the defocus aberration coefficient, c2 0, which weights the defocus Zernicke polynomial term, Z2 0, as shown in Eq. (1):
where r and θ designate the polar coordinates of the pupil, W is the wavefront aberration pattern, n is the order of the aberration and f is the meridional frequency of the aberration (n = 2, f = 0 for defocus). Longitudinal chromatic aberration, C(λ), was approximated per Dai [18] using Eq. (2):where λ was the wavelength (from 0.4 µm to 0.7 µm) and the resultant C(λ) was expressed in diopters. When zero object vergence is present, Eq. (3) relates c2 0 to longitudinal chromatic aberration, C(λ):where R is the radius of the pupil (set to 3000 µm for this model). The resultant c2 0 is the polychromatic defocus aberration coefficient.2.2. Selected typical cases
Representative case studies were selected from a previously described population of eyes [11,19]. Institutional review board approval was obtained for retrospective chart review. The subjects were refractive surgery candidates or cataract patients. The population had no previous ocular or corneal surgery and no corneal pathology. All eyes had complete set of data points within the central 7-mm zone on Humphrey Atlas corneal topographic maps (Carl Zeiss Meditec, Inc, Dublin, CA). An idealistic mathematical model was used to identify 3 cases that represented recommended optimal overall spherical aberration values that were very positive, very negative, and approximately neutral. Then, the corneal topographies of these 3 cases were used in conjunction with the new realistic mathematical model to identify the new ideal residual spherical aberration in each case. During realistic and idealistic mathematical modeling, the astigmatism term of each eye was not kept.
3. Results
3.1. Defocus aberration coefficient
The defocus aberration coefficient, c2 0, was plotted as a function of visible wavelengths, as shown in Fig. 1 . An idealistic curve would be symmetrically centered on the average/median wavelength of 550 nm. Instead, the curve was asymmetrically skewed toward myopic defocus at shorter wavelengths, and skewed toward hyperopic defocus at longer wavelengths. Each wavelength had a different defocus aberration coefficient and therefore a different optimal spherical aberration, but the average focusing wavelength was 575 nm. At the center wavelength of 550 nm, the defocus aberration coefficient was + 0.16 µm, which corresponded to 0.125 D of myopia. A larger proportion of the c2 0 function was positive (shaded blue in Fig. 1) than negative (shaded red in Fig. 1), indicating a positive overall character for c2 0.
Fig. 1 Defocus aberration coefficient, c2 0, as a function of visible wavelengths. Positive c2 0 values are shaded blue and negative c2 0 values are shaded red.
3.2. Modulation transfer functions (MTFs)
Figure 2A shows 2-dimensional polychromatic modulation transfer functions (MTFs) calculated for distance vision of an eye without higher-order aberrations. MTFs are shown without spherical aberration, with positive overall ocular residual spherical aberration ( + 0.05 µm and + 0.10 µm), and with negative overall ocular residual spherical aberration (−0.05 µm and −0.10 µm). Qualitatively, leaving + 0.05 µm of residual spherical aberration yielded the best modulation transfer function, and leaving −0.10 µm of residual spherical aberration yielded the worst modulation transfer function.
Fig. 2 Modulation transfer functions and spherical aberration. A. Theoretical 2-dimensional polychromatic modulation transfer functions (MTFs) for a source at distance (optical infinity), with varying values of overall ocular spherical aberration. B. Theoretical volumes of the MTFs, calculated with limits of 50 lp/mm and 100 lp/mm.
Figure 2B shows quantitative values for the volumes of the 3-dimensional MTFs represented by the 2-dimensional MTFs in Fig. 2A. For the blue curve of MTF volumes at the 50 lp/mm limit (representing 20/40 Snellen distance vision), a slight maximum existed at the x-value of + 0.05 µm spherical aberration. The maximum y-value was 5% larger than the MTF volume with 0 µm of spherical aberration and 1% larger than the MTF volume with + 0.10 µm of spherical aberration. For the red curve of MTF volumes with the 100 lp/mm limit, representing 20/20 Snellen distance vision, a more distinct maximum MTF volume occurred at the x-value of + 0.05 µm of spherical aberration. This maximum y-value was 8% higher than the MTF volume with 0 µm of spherical aberration, 14% higher than the MTF volume with + 0.10 µm of spherical aberration, and 126% higher than the minimum calculated MTF volume at the x-value of −0.10 µm spherical aberration.
3.3. Effect of chromatic aberration on individual eyes
The data for case #1 were obtained from the eye of a man who was 72 years old, with a corneal wavefront pattern as shown in Fig. 3A . Using an idealistic mathematical model with an average focusing wavelength of 550 nm, the resultant MTF volumes with the 100 lp/mm limit (red curve, Fig. 3B) indicated that the optimal overall ocular spherical aberration for this eye was ~0 to + 0.05 µm. Using the realistic mathematical model with an average focusing wavelength of 575 nm with the data for this eye, the resultant MTF volumes with the 100 lp/mm limit (red curve, Fig. 3C) exhibited a maximum at the x-value of + 0.10 µm of spherical aberration.
Fig. 3 Case #1 – Neutral overall spherical aberration indicated by idealistic model. A. Corneal wavefront pattern in Zernike view. B. Volumes of the resultant modulation transfer function (MTF), plotted as a function of residual overall ocular spherical aberration, using an idealistic model with an average focusing wavelength of 550 nm. C. Volumes of MTFs versus overall ocular spherical aberration, using a realistic model with an average focusing wavelength of 575 nm.
The data for case #2 were obtained from a man who was 44 years old, with a corneal wavefront pattern as shown in Fig. 4A . Using an idealistic mathematical model with an average focusing wavelength of 550 nm, the resultant MTF volumes with the 100 lp/mm limit (red curve, Fig. 4B) indicated that the optimal overall ocular spherical aberration for this eye was + 0.20 µm. Using the realistic mathematical model with an average focusing wavelength of 575 nm with the data for this eye, the resultant MTF volumes with the 100 lp/mm limit (red curve, Fig. 4C) exhibited a maximum at the x value of + 0.25 µm of spherical aberration.
Fig. 4 Case #2 – Positive overall spherical aberration indicated by idealistic model. A. Corneal wavefront pattern in Zernike view. B. Volumes of the resultant modulation transfer function (MTF), plotted as a function of residual overall ocular spherical aberration, using an idealistic model with an average focusing wavelength of 550 nm. C. Volumes of MTFs versus overall ocular spherical aberration, using a realistic model with an average focusing wavelength of 575 nm.
The data for case #3 were obtained from a woman who was 69 years old, with a corneal wavefront pattern as shown in Fig. 5A . Using an idealistic mathematical model with an average focusing wavelength of 550 nm, the resultant MTF volumes with the 100 lp/mm limit (red curve, Fig. 5B) indicated that the optimal overall ocular spherical aberration for this eye was −0.20 to −0.15 µm. Using the realistic mathematical model with an average focusing wavelength of 575 nm with the data for this eye, the resultant MTF volumes with the 100 lp/mm limit (red curve, Fig. 5C) exhibited a maximum at the x-value of −0.15 µm of spherical aberration.
Fig. 5 Case #3 – Negative overall spherical aberration indicated by idealistic model. A. Corneal wavefront pattern in Zernike view. B. Volumes of the resultant modulation transfer function (MTF), plotted as a function of residual overall ocular spherical aberration, using an idealistic model with an average focusing wavelength of 550 nm. C. Volumes of MTFs versus overall ocular spherical aberration, using a realistic model with an average focusing wavelength of 575 nm.
For these 3 cases, the new realistic mathematical model indicated that the optimal overall ocular spherical aberration should be an average of 0.05 μm more positive than the optimal overall ocular spherical aberration indicated by the idealistic model.
4. Discussion
In an idealistic model of a human eye without longitudinal chromatic aberration, the average focusing wavelength would be 550 nm, near the maximum photopic retinal response at 555 nm [20]. The new realistic mathematical model presented here, which included the contribution of longitudinal chromatic aberration, found that the defocus aberration coefficient at that wavelength would be + 0.16 µm, corresponding to 0.125 D of myopia. The realistic mathematical model indicated that the average focusing wavelength was not 550 nm, but instead was 575 nm. The selection of focusing wavelength (575 nm) in this study is consistent with the values used in other publications: 575 nm by Le Grand [21], 570 nm by Marcos et al [15], 589 nm by Thibos et al [22], and 575 nm when extracted from an equation used by Dai [18]. Published hypotheses about the optimal focusing wavelength include color fringe reduction [21], highest luminance efficiency [21,23], sparing of accommodation [21,24], and receptor spacing and color detection [25].
At the average focusing wavelength of 575 nm, the defocus aberration coefficient was zero, corresponding to emmetropia. Considered over the visible spectrum, the polychromatic defocus aberration function had a generally positive character. Accordingly, the polychromatic MTF volume was best in a theoretical human eye with no higher-order aberrations when Z4 0 (spherical aberration) was + 0.05 µm. The slightly positive character of the optimal overall ocular spherical aberration was ascribed to the contribution of the generally positive c2 0 value.
The effect of longitudinal chromatic aberration is factored into the defocus term for each wavelength via the r2 relationship, as indicated by Eq. (3). In this study, the shift of optimal spherical aberration was directly related to ocular longitudinal chromatic aberration because the contributions from other high-order aberrations on the defocus were identical. This study paradigm provides a way to study the causal relationship between optimal spherical aberration and longitudinal chromatic aberration.
It is also imperative to note that Zernike defocus might be different from clinical defocus. This is often referred as the difference between objective and subjective measurements. Other Zernike terms, such as Z2 2 astigmatism, Z2 −2 astigmatism, and Z4 0 spherical aberration, also contribute to the traditional r2 defocus term, which was related to Seidel defocus. The topic of determining optimal clinical defocus has been thoroughly discussed in other publications (e.g. Thibos et al [26]). Nonetheless, the choice of best focus does not affect the conclusions of this study.
The effect of the c2 0 (polychromatic aberration defocus coefficient) on the Z4 0 (spherical aberration term) can be considered as a wavefront-flattening effect. A perfect optical system has a flat wavefront aberration map [27]. Longitudinal chromatic aberration adds a positive (myopic) shift to the defocus term, yielding a wavefront curve as shown in Fig. 5A. A common metric of wavefront flatness is the difference in phase between the highest peak and the deepest valley on the wavefront surface [27]. The curve in Fig. 6A has a large difference between the central valley and the higher edges, with steep changes from center to edge. This aberrated wavefront can be ameliorated by adding a positive spherical aberration, such as the wavefront curve shown in Fig. 6B. The superposition of the waves in Fig. 5A and Fig. 5B yields a much flatter central wavefront, as shown in Fig. 6C. This interrelation between the defocus term and the spherical aberration term agrees with other computational models, which have suggested that ophthalmic wave aberrations are interdependent in ways that improve the MTF of the eye and thus improve the resultant image quality at the retina [26,28,29].
Fig. 6 Example wavefront curves. A. A wavefront curve with a myopic (positive) defocus aberration coefficient. B. A wavefront curve with a positive spherical aberration. C. A combined wavefront, with positive defocus aberration and positive spherical aberration.
When the general principles of this study were applied to 3 case studies representing the positive, neutral, and negative range of spherical aberration, the overall ocular spherical aberration calculated with the contribution of longitudinal chromatic aberration was ~0.05 μm more positive than the optimal overall ocular spherical aberration indicated by an idealistic model without the contribution of longitudinal chromatic aberration. These 3 specific results represent the overall results of previous and current studies. In a prior report of simulated customizations of IOL asphericity for 94 patients, most eyes seemed to need −0.05 μm to −0.10 μm of overall ocular aberration, depending on which metric of optical quality was used [11]. The current study indicates that those results were biased in the negative direction by omitting the contribution of longitudinal chromatic aberration.
The current study also agrees with clinical results indicating that positive overall ocular aberration was associated with good vision [6–8] or supernormal vision [9] in healthy phakic subjects, and was associated improved contrast sensitivity for pseudophakic eyes versus pseudophakic eyes with neutral spherical aberration [4,5,13,14,30]. The current work supports the concept that selecting an IOL that leaves a slightly positive overall ocular spherical aberration may yield better vision for the patient.
Achieving this slightly positive value of overall ocular spherical aberration will depend on the right combination of the eye and the IOL. A Tecnis IOL (−0.27 µm spherical aberration) could be useful for a patient with a large amount of corneal spherical aberration (eg, >0.32 µm). An AcrySof IOL (−0.20 µm spherical aberration) would be useful for the average cornea. A SofPort AO IOL (0 µm spherical aberration) could be most useful for an eye with a very small amount of corneal spherical aberration (eg, <10 µm), such as an eye that has been previously treated with hyperopic laser in situ keratomileusis (LASIK) [31].
For cataract patients, visual benefits with optimal selection of spherical aberration will likely be relevant only in dim lighting. Our model assumed a 6-mm pupil, which would normally occur only under mesopic or scotopic conditions. For example, in one study of aspheric versus nonaspheric IOLs after implantation in cataract patients with an average age of 70 years, the eyes with aspheric IOLs had pupils that were 3.6 ± 0.5 mm under photopic conditions, 4.1 ± 0.5 mm under mesopic conditions, and 4.7 ± 0.6 mm under scotopic conditions [32]. A different study, concerning a cataract patient population with an average age of 66 years, reported that the eyes with aspheric IOLs had pupils that were 4.6 ± 0.8 mm under mesopic conditions and 5.4 ± 1.0 mm under scotopic conditions [33]. Accordingly, in both studies, contrast sensitivity was significantly better with aspheric IOLs than with nonaspheric IOLs under mesopic conditions, but not under photopic conditions [32,33].
In conclusion, this study presented a new mathematical model that demonstrated the importance of the polychromatic defocus aberration coefficient in identifying the optimal overall ocular spherical aberration as + 0.05 µm. Comprehensive mathematical models to understand the optical system of the eye, and clinical versus mathematical comparisons, should be ongoing and important goals in ophthalmology.
Acknowledgments
The authors thank Dr. Li Wang and Dr. Douglas Koch (both from Baylor University, Ophthalmology Department, Houston, TX) for providing corneal wavefront data and for helpful discussions on this topic. This work was funded by Alcon Research, Ltd., which also provided writing assistance for this manuscript.
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