From Instituto Oftalmológico Amigó, Tenerife, Spain (Amigo, Bonaque); Instituto Universitario de Investigación en Envejecimiento, University of Murcia, Spain (López-Gil); and Indiana University School of Optometry, Bloomington, Indiana (Thibos).
The authors have no financial interest in the materials presented herein.
Study concept and design (A.A., S.B., N.L.G.); data collection (A.A., S.B.); analysis and interpretation of data (A.A., S.B., N.L.G., L.T.); drafting of the manuscript (A.A., S.B., N.L.G., L.T.); critical revision of the manuscript (A.A., S.B., N.L.G., L.T.); administrative, technical, or material support (A.A., S.B.); supervision (A.A., S.B.)
Correspondence: Alfredo Amigo, MD, PhD, C/El Humo 1-1°A, 38003 Tenerife, Spain. Tel: 34 62 966 8820; E-mail: email@example.com
Altering physiological corneal asphericity has been proposed as a compensating mechanism for presbyopia.1–4 The rationale for such treatment is that the degree of asphericity determines the degree of optical spherical aberration that, in turn, determines the presbyopic eye’s depth-offield (ie, the range of target distances for which retinal image quality is acceptable). However, spherical aberration also shifts the center of the depth-of-field by an amount that depends on the magnitude and sign of spherical aberration.5 If the center of the depth-of-field is specified by the target distance that maximizes retinal image quality, the corresponding target vergence becomes a functional definition of the presbyopic eye’s refractive state. In this study, we use a quantitative model of the eye’s optical system to predict how the eye’s refractive state will depend on the degree of corneal asphericity and pupil diameter.
The anterior surface of an aspheric cornea is frequently described mathematically as a conic of revolution, which includes an asphericity factor, Q.6 It may be demonstrated that the following approximate relationship exists between spherical aberration in terms of Zernike fourth order (spherical aberration) and asphericity of the anterior corneal surface7–10:
is the corneal refraction index, r is the pupil radius, and Rc
are the apical radius and asphericity of the anterior surface, respectively. Thus, it can be shown that for a distance object and Q=1/nc2
, the surface is free from spherical aberration, whereas for Q<1/nc2
, spherical aberration is <0, and when Q>−1/nc2
, spherical aberration is >0. The normal shape of the anterior corneal surface is similar to a prolate ellipse with an average Q-factor between −0.2 and −0.267 and nc
=1.337, thus the average first surface of the normal cornea has positive spherical aberration.11
The eye’s other refracting surfaces generate negative spherical aberration, which tends to compensate for the positive spherical aberration generated by the front surface of the cornea, although the total balance is still positive in the population average when accommodation is relaxed.1
The present study evaluates the theoretical refractive shift created by a change in asphericity, and thus the efficacy and limitations of this type of treatment to compensate for presbyopia.
Materials and Methods
Standard Aberration Nomenclature
In this report, we use standard nomenclature for reporting optical aberrations of eyes that quantifies the strength of spherical aberration by the Zernike coefficient and the corresponding Zernike polynomial12 as . The presence of positive spherical aberration tends to make an eye with a large pupil appear relatively myopic compared to the paraxial refractive state.13 Conversely, negative spherical aberration tends to make an eye appear hyperopic compared to the paraxial refractive state.13 These effects depend, strongly on the value of the aberration coefficient , which in turn depends on pupil diameter even if the dioptric properties of an eye are fixed.14,15 Therefore, it is expected that in the presence of spherical aberration a change in pupil diameter will produce a shift in the eye’s refractive state, as mentioned previously in the context of accommodation.8 Previous theoretical analysis of this problem indicated that the loss of accommodative response of the crystalline lens in a presbyopic eye is partially mitigated by pupil miosis only in eyes with negative spherical aberration, which accounts for the near vision improvement in the hyperprolate presbyopic eye.1–4
Ray Tracing Simulations
To determine the spherical aberration and refractive state of the whole eye, Navarro’s model eye16 was implemented with ray tracing software (ZEMAX-EE; ZEMAX Development Corp, San Diego, California). This model is known to represent well the aberrations present in the human eye.16,17 The ZEMAX-EE model provided control of pupil size and asphericity of all ocular surfaces when computing the system wavefront error expressed as Zernike coeffcients.12 Simulations were carried out for a wavelength of 587 μm following the next three steps:
The Navarro eye model (which included a front corneal surface with Q=−0.26) was implemented. For some calculations, the axial length was adjusted to keep the eye nominally emmetropic (in the Zernike sense) by finding the retinal position that minimizes the root-mean-square of the eye when the object is at infinity.
The front surface corneal Qc value was modified as well as the pupil radius (r). No adjustments to the corneal apical radius of curvature were made in this study, but would be appropriate when modeling specific treatment modalities.
The Zernike coefficients provided by ZEMAX for each new Qc and r were used to find the refractive state of the eye using different ocular quality metrics.
The methodology used was “wavefront refraction”13,18 based on a definition of refractive state as the power of the correcting lens required to optimize the eye’s optical quality. A variety of metrics for wavefront quality and retinal image quality are available for this computation.
Performing wavefront refraction for most metrics of wavefront or retinal image quality requires numerical optimization of the correcting lens power.13,18 A simple optimization algorithm was used in which a series of defocus values were added to the Zernike coefficient of the eye. The defocus value that maximized any particular metric was taken as the refractive state according to that metric. Using a variety of image quality metrics listed in (available as supplemental material in the PDF version of this article), how refractive state changes in response to changes in corneal asphericity was determined. Calculations were carried out using a MATLAB application allowing the processing of Zernike polynomials up to the 7th order obtained from the ZEMAX modeling. The independent variables were Q factor (ranging from 0 to −1.50 in 0.25 steps) and pupil diameter (ranging from 2 to 6 mm in 1-mm steps).
To report the effect of pupil diameter on refractive state, results were averaged for the following metrics known from previous experiments5,13,19–22 to account well for changes in refractive state due to accommodation: PFSt, EW, SRX, LIB, STD, NS, VSX, SFcMTF, AreaMTF, SROTF, VOTF, VSOTF, SRMTF (see for definitions).
Effect of Corneal Asphericity on Ocular Spherical Aberration
Figure 1 shows the effect of corneal Q factor on ocular spherical aberration. For 2- and 3-mm pupil diameters, ocular spherical aberration is close to zero, regardless of corneal asphericity because of the strong dependency of on pupil radius (Eq.). For larger pupil diameters, spherical aberration varies almost linearly with corneal asphericity, as predicted by Eq.(). Spherical aberration is zero for Q=−0.6, regardless of the pupil diameter.
Figure 1. Spherical aberration values resulting from the modification of corneal Q for different pupil diameters in the Navarro eye model. P = pupil size (in mm)
Effect of Corneal Asphericity on Refractive Error
Figure 2 shows the effect of manipulating corneal asphericity on refractive error of the whole eye with pupil diameter of 6 mm. When Q=−0.6, the spherical aberration of the whole eye vanishes and thus the computed refractive error is exactly the same for all metrics. When Q>−0.6, the eye’s refractive error changes in the hyperopic direction according to most metrics. Conversely, when Q<−0.6, the eye’s refractive error changes in the myopic direction for most metrics.
Figure 2. Refraction for each of the 32 metrics as a function of corneal asphericity for a 6-mm pupil size. The metrics are defined in (available online in the PDF version of this article).
Figure 3 shows the average change of refractive state produced by changes in corneal Q factor for different pupil sizes. For all pupil sizes, the ocular refractive error changes in the hyperopic direction as Q becomes more negative. However, the rate at which this change takes place depends on pupil diameter.
Figure 3. Relationship between the refraction of an emmetropic eye (Q=−0.26) and corneal asphericity for different pupil diameters (P [in mm]). In this case, the eye was emmetropized by changing the axial length when Q=−0.26 for each pupil size simulated.
Figure 4 shows how refractive state changes with Q values and pupil ranges. Accommodation is calculated by pupil miosis when the pupil decreases from 6 mm to 2, 3, 4, or 5 mm. For a fixed Q value, decreasing pupil size may generate a change of refractive state that may aid accommodation. In this figure, a negative value on the y-axis represents an increase in effective ocular power generated by the presence of spherical aberration induced by corneal asphericity.
Figure 4. Pseudoaccommodation values obtained from pupil dynamics from an emmetropic eye starting with A) a 6-mm pupil that decreases to 5, 4, 3, or 2 mm; B) a 5-mm pupil that decreases to 4, 3, or 2 mm; C) a 4-mm pupil that decreases to 3 or 2 mm; and D) a 3-mm pupil that decreases to 2 mm. All of these values were assessed for different corneal Q factor values.
Treatment of presbyopia by increasing corneal prolateness (ie, making the corneal Q factor more negative) is based on obtaining a myopic refractive change in the eye during convergence miosis. This myopic refractive change is a consequence of surgically induced negative spherical aberration for the whole eye together with a convergence miosis still present in the presbyopic eye. The phenomenon needs a significant change in pupil diameter to produce a useful change in refraction.
The changes in refractive state produced by pupil miosis described in this study refer to an axial shift in the target location that maximizes retinal image quality. Such changes are often called “pseudoaccommodation” because the mechanism does not involve changes in the crystalline lens. However, the term pseudoaccommodation has also been used to describe increased size of the depth-of-field when the pupil constricts. A larger depth-of-field makes it possible to move a target closer to the eye without a significant loss of visual quality, thus presenting a false indication of accommodation. To avoid confusion, we recommend against using the term pseudoaccommodation to describe the real changes in optimum target vergence caused by changes in pupil diameter. Instead, the term “accommodation (or disaccommodation) by pupil miosis” better captures the notion of a functional (and potentially useful) change in refractive state caused by a mechanism distinct from changes in the crystalline lens. It is interesting to note that Eq.() indicates that not only depends on the corneal Q-factor, but also on the radius of the surface. For instance, after a 3.00-D myopic LASIK treatment, a slight decrease of 0.04 μm of as Rc increases would be expected. However, our goal was not to model a specific mode of refractive therapy including all relevant parameters, but to show the potential change of refractive state of the eye when just corneal asphericity is changed.
Maximum Limit of Preferred Q
Due to the difference in ocular physiological parameters between patients, it is not possible to determine a unique Q value that provides a visual advantage for all cases, because what influences the final result is the spherical aberration present in the eye. For that reason, the maximum limit of Q must be calculated individually for each surgery.
On the other hand, an excessive increase in spherical aberration has been shown to induce halo vision and/or blurring, especially under scotopic conditions where pupils are large.11,23 In general, it seems that small values of spherical aberration commonly found in eyes decrease acuity by a relatively small amount under optimum-focus conditions.24,25 Nevertheless, the present results show that, for 6-mm pupils, exceeding a more negative spherical aberration than −0.4 μm (corresponding to a Q value more negative than −1.25) does not increase the amount of accommodation by pupil miosis (see Fig 4A). For smaller pupils (see Figs 4B, 4C, and 4D), Q values even more negative than −1.50 may be induced and, theoretically, the eye’s refractive state would still change in the myopic direction with pupil miosis. However, it is advisable to be more conservative with regards to increasing the negative Q value in those cases where residual accommodation still exists, as it is a well-known fact that spherical aberration becomes more negative as the eye accommodates,8 and in those cases, a physiological change in aberration due to the patient’s age might be expected.26
Refractive Effect of an Increase in Spherical Aberration in Far Vision
Increasing corneal asphericity induces a proportional hypermetropization that must be compensated for when planning the surgical treatment (see Fig 3). The amount of sphere to be compensated will vary according to the algorithm of the laser equipment used, and therefore the present results may not be directly extrapolated to any specific laser platform. It may also be observed that beyond −0.4 μm spherical aberration, hypermetropization does not increase.
Refractive Effect of an Increase in Spherical Aberration in Near Vision
Figure 4 shows that the wider the physiological pupil range, the more the refractive state can change when increasing the negative value of Q. Because the average eye has a positive spherical aberration,14 reducing spherical aberration implies an improvement in near vision because the negative effect of spherical aberration >0 in presbyopic patients obliges them to accommodate more than the change in stimulus vergence indicates. A myopic refractive change is not obtained until net negative spherical aberration is induced. The sum of both mechanisms may provide a maximum addition of 1.00 D, for a wide pupil range and a preexisting average spherical aberration, which can be added to the degree of monovision tolerated by the patient. Minimum pupil diameter is important because, for a given pupil range, smaller pupils are more effective at inducing a focus shift during miosis.
It is interesting to note that for small Q-factor and a large pupil with a small change in its diameter, the eye could become slightly hyperopic (see Fig 4A for Q=−1.5 and change in pupil size from 6 to 5 mm). In that case, the large amount of induced negative spherical aberration creates a similar image quality for different refractive states, producing a slightly hyperopic shift in the refraction.
Influence of Preexisting Refractive State in the Final Result
Theoretically, a patient with high preexisting spherical aberration >0 will benefit the most from an increase in corneal prolateness because eliminating the hyperopia that is produced by convergence miosis diminishes the symptoms of presbyopia and the retinal image quality increases. A patient with a preexisting spherical aberration near zero may be equipped with a maximum accommodation by pupil miosis. A presbyopic patient with a high preexisting spherical aberration <0 would have the worst refractive outcome because the amount of induced accommodation by pupil miosis would be more restricted.
Results obtained do not depend on the degree of the patient’s refraction because that factor only adds or subtracts a constant value on the defocus, and the refractive state of the eye generated by the amount of defocus does not change with pupil diameter.
With regards to monocular/binocular treatment, although a monocular approach has been proposed,4 it is likely that bilateral symmetric treatment would be more appropriate given the advantages that a symmetrical Q rise provides to binocular summation.27 It has been shown that a monocular change in the Q value of 0.1 could generate a significant change in the spatial distribution of the retinal image, which could modify binocular vision such as stereopsis or binocular summation.28 In this sense, a relative large modification on the Q value after surgery in one eye should be followed by a similar modification in the contralateral eye to result in minimal deterioration in the binocular vision.
From the results of the present study, the theoretical effect of hyperprolateness on depth-of-focus may not be inferred, although the refractive change due to the pupil dynamics is now well described to be applied in the case of corneal laser surgery or other corneal techniques such as contact lenses or corneal refractive therapy. The change in Q-factor can also be performed on the surfaces of an intraocular lens (IOL) to generate a total negative spherical aberration in the eye. In this case, the results shown herein could be potentially applied, not only at the corneal plane, but also in IOL designs.
Simulations performed in a model eye show that pupil miosis in the presence of negative spherical aberration causes a myopic change in the refractive state of the eye according to numerous metrics of image quality. Our results could be applied to treating presbyopia by means of corneal hyperprolateness. In that case, the goal of optimal asphericity is determined by preoperative corneal and total aberrations as well as a wide enough pupil range, so that total postoperative spherical aberration does not exceed −0.4 μm, as visual advantages start decaying beyond this value.
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