The study included 180 eyes (88 right ones and 92 left ones) from 98 subjects. All of them were Spaniards and most of the younger ones (less than 24 years old) were students from Murcia's School of Optometry. Age ranged from 20 to 58, with a corresponding mean and standard deviation of 35.6±12.3 and 35.0±12.4 years for eyes and subjects, respectively. Subjects read and signed an informed consent form that had been previously approved by the Ethics Committee of the University of Murcia. They also approved the experiment itself, which followed the tenets of the Declaration of Helsinki.

Subjects who normally wear contact lenses removed their lenses at least 24 (hydrophilic contact lenses) or 48 hours (gas-permeable contact lenses) before the experiment. A preliminary optometric study was performed, which included autorefraction by means of a Canon Tomey TL-1000®, retinoscopy, and subjective refraction by with trial lenses. The results of this preliminary test, together with a questionnaire, provided a basis for exclusion from the experiment. The exclusion criteria were: amblyopia; any kind of accommodative anomaly; continuous pharmacological treatment that could modify the accommodative mechanism; any kind of ocular surgery (including refractive surgery); and any kind of ocular disease that could affect vision or the accommodative mechanism (such us glaucoma, conjunctivitis or cataracts). These criteria were applied after the subject's medical history. Accommodation anomalies in young subjects were detected when the accommodative amplitude measured using negative lenses (see next subsection) was always smaller than the minimum one suggested by the Hofstetter formula22 and the accommodative flexibility was smaller than 11 cycles per minute, under monocular conditions.

From the initial 99 subjects not excluded by these criteria, only one could not accommodate to the visual target used in our experiments, so the data from that subject were removed from the study.

The subjective measurement of the amplitude of accommodation was performed using a custom-made Badal Optometer which was integrated in the refraction unit. As accommodation stimulus, it used a Bailey-Lovie chart (luminance = 100 cd/m2) located 6m (20 feet) away from the subject. The Badal optometer consisted of two well-aligned achromatic doublets (focal length = 100mm) as described elsewhere.21 One of the lenses was located at a distance of 100mm from the eye's entrance pupil. This distance was fixed for each subject using a focusing camera that displayed a clear image of the subject's iris when the correct distance was achieved. The second doublet was free to be moved axially during the experiment by the observer, by means of a joystick. The precision of the positioning of the lens was 1mm, corresponding to 0.1 D of target vergence. The subject's head was fixed using a chin rest. The subject's astigmatism, if any, was corrected by means of a trial lens placed 1.2cm in front of the eye. The possible refraction changes due to the astigmatism and the distance between the target and the moving achromatic lens was taken into account in the calculations.

The subject's task was to find the two extreme positions of the Badal optometer's mobile lens where the line containing 0.8 VA (20/25) letters was still clear without any perceptible blur. We refer to this dioptric range between near-point and far-point as the subjective AA. For comparison purposes, we also used a standard clinical method for determining the subjective AA, in which minus lenses are used to induce changes in vergence of a target located at a fixed viewing distance. As in the Badal method, astigmatism was corrected using trial lenses.

All the subjective measurements were performed by the same researcher, who was a trained optometrist. Distances between negative lenses and the eye were also taken into account in the calculations. The measurements were performed 3 times for the minus-lens method and 5 times for the Badal method (5 for the far point and another 5 for the near point). Mean and standard deviation of the measurements was similar for both methods, so we adopted the Badal optometer technique as our preferred method because it was more precise due to its superior resolution (0.1 D, compared to 0.25 D for the minus-lens method). Moreover, the Badal-optometer experiment was quicker and simpler than the measurements done with the minus-lenses method.

When a subject views a target intended to stimulate accommodation (seeFigure 1), we anticipate that the eye's refractive state will change in order to clearly focus the target onto the foveal cone apertures. To the extent that this does not happen, the eye exhibits an error of accommodation. Our goal was to measure the maximum change in the refractive state that could be achieved by manipulating target vergence. To quantify this outcome, we used the wavefront aberrometer to determine the eye's refractive state, defined here as the target vergence required to maximize retinal image quality. This definition is based on the fundamental principle that to optimize retinal image quality, the target must be optically conjugate to the cone photoreceptor apertures. Thus, the vergence of the plane that is optically conjugate to the retina determines the eye's refractive state. When accommodation is fully relaxed, the eye's refractive state is also its refractive error, both of which are equal to the vergence of the far point (i.e. the retinal conjugate of the relaxed eye). These various distances and their corresponding vergences are summarized with reference to figure 1 as follows:

• 1.

  Target vergence: T = 1/target distance = 1/t

• 2.

  Refractive state of accommodated eye: A = 1/a

• 3.

  Refractive error of relaxed eye: R = 1/r

Figure 1.

Definition of terms. Target vergence T is equal to the inverse of the viewing distance “t”. Distance “r” is equal to the axial position of the plane conjugate to the foveal cone apertures when accommodation is relaxed. The axial location of this retinal conjugate plane is called the “far point”. The refractive error of the eye equals the vergence (1/r) of the far point. The refractive state of the accommodated eye is defined as the vergence (1/a) of the retinal conjugate plane. Wavefront refraction is based on the assumption that the vergence of the retinal conjugate plane is the same as the target vergence required to maximize retinal image quality.

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All distances are referred to the corneal plane and, according to the standard ophthalmic sign convention, they are negatives if the corresponding points are located in front of the eye, as in figure 1. Values for T, A, and R are equal to the power K of an ideal lens placed in the pupil plane that forms a virtual image at infinity of the target plane, the retinal conjugates when eye accommodates, or far point, respectively. Accommodation stimulus, AS, is defined as the change in target vergence: AS = R-T. Accommodation response, AR, is defined as the change in refractive state of the accommodated eye compared to the relaxed eye: AR = R-A. Thus, to find the accommodation response we have to find the power of two ideal lenses, which correspond to the refraction, R, and to the refractive state of the accommodated eye, A. This is achieved by means of a through-focus calculation described next.

Since the power of the eye varies across the pupil area when higher-order aberrations are present, different conjugate foveal planes would be found depending on the pupil zone used and, therefore, the value of K would change across the pupil. The average K value that produces a “good image” for an ideal point object placed at the fovea and a natural pupil (assumed to have a circular shape), for a particular accommodation state, will depend on the particular definition of “good image” that we use. In this sense, we need to choose a metric that quantifies image quality and then we have to look for the target vergence that maximizes that metric. A variety of optical-quality metrics are available for this purpose, many of which were evaluated by Thibos et al.22 for predicting the refractive state of the relaxed eye. Some metrics quantify the quality of the eye's wavefront aberration function, whereas others quantify the quality of the point spread function or of the optical transfer function associated with the retinal image. For a detailed mathematical description of these metrics see Appendix A in reference Thibos et al.22

We have applied all 32 independent metrics described by Thibos et al.22 to determine the refractive state of the eye at various levels of accommodative response, including the relaxed eye. Appendix A shows a table with the names and some information of all the metrics used. The procedure followed for each subject consisted of two steps. In the first step, 3 wavefront aberration functions were obtained for the un-accommodated eye using the IRX3 aberrometer (Imagine Eyes, France). To ensure that the eye is unaccommodated, the aberrometer uses a fogging method in which the target distance is increased in 1 D steps while the eye's refractive state is continuously monitored. When the eye fails to change its refractive state in the hyperopic direction in response to the change in target vergence, then accommodation is assumed to be fully relaxed. From three consecutive measurements taken over 10 s in this relaxed state, the more hyperopic one was selected as the reference to be used when computing the accommodative response. In this case, the refraction is obtained from the second-order Zernike coefficient C20 obtained from a least-squares fit of a sphero-cylindrical surface to the wavefront. The wavefront corresponding to the unaccommodated eye is named W0(T0).

In the second step, three repeated sets of wavefronts were recorded for the accommodated eye by using a stimulus vergence that changes automatically in steps of 0.5 D. After each movement of the target, there was a delay of 1.5 s before a wavefront measurement was made, in order to give the subject enough time to accommodate. From the refraction obtained in the first step, we added 0.5 D to the nearest hyperopic vergence multiple of 0.5 D (for instance, if the Zernike refractive error was 2.17 D, we obtained 2.5D + 0.5 D = 3 D; or if was -2.34 D we used -2 D + 0.5 D=-1.5 D). This value became the vergence of the first position of the stimulus in a series of accommodation measurements. Before collecting the data, we ran at least three trials, so that the subject could understand the task well and with the aim of getting an idea about the range of accommodation. This stimulus accommodation range was calculated for each subject according to Hofstetter formula20 and was increased by at least 2.5 D to guarantee that the stimulus would cover the subject's full range of accommodation. For subjects younger than 35 years old, the range used was 10 D. In the few cases where the accommodation response measured by the aberrometer did not show a maximum value (indicating that the target failed to get closer than the near point), the stimulus accommodation range was increased by 2 D, and the set of accommodation measurements was repeated.

At the end of each measurement of the set of accommodation wavefronts, the aberrometer displayed the accommodation response computed using the same procedure that had been previously employed to obtain the initial refraction. Among the three repeated set of measurements, we selected again the one with the largest AA. In general, the differences between all of them were usually smaller than 1 D, although it depends on the particular subject's ability to accommodate (see López-Gil et al.17, for details on the repeatability of the measurement). The set of wavefronts corresponding to the accommodated eye are named Wi(Ti), with i=1 to i≤21.

The Zernike expansion depended on the size of the subject's pupil. For most of subjects it was made up to the 8th order, with the minimum being 5th order (corresponding to small pupils under accommodation of some subjects). To mimic as much as possible natural viewing conditions, no mydriatic drugs were used and wavefronts were obtained under natural pupil size, so the study has taken into account natural accommodative miosis. For most of the subjects, when the target was closer than their near point, a pupil diameter increase was observed due to a relaxation in their accommodation. Prior to the accommodation measurements, the subjects were initially instructed to keep the target as clear as possible all the time as well as to blink frequently (the aberrometer software automatically eliminates Shack-Hartmann images affected by blinks). The visual target was a figure of a balloon at the end of a road. The stimulus had a luminance of 50 cd/m2, was polychromatic and contained multiple spatial frequencies.

For each optical quality metric, Mj, we computed the refractive error of the relaxed eye (R0j=Mj[W0(T0)]), and the refractive state of the accommodated eye for each value of stimulus vergence, Ti, (Rij=Mj[Wi(Ti)]). The computation finds the lens power that, when added to the eye's wavefront aberration, maximizes the value of the metric. Then, taking into account that the response of the unaccommodated eye is zero, the accommodation response for a certain normalized stimulus accommodation, Ti-R0j, corresponds to Rij-R0j. AA was then estimated for this same metric as the maximum range of accommodation response, that is, the maximum value of the eye's refractive state minus the minimum value of the refractive state.

All subjective measurements were carried out by the same researcher, who was different from the researcher involved in the objective measurements. Each researcher was aware of the results obtained by his colleague. For each subject, the experiment took about 2 hours including a break in between of about 20’. The whole experiment took about two years and a half.

During accommodation there is a well-known decrease in fourth-order spherical aberration (z40) in young subjects, typically resulting in a change in sign, going from positive to negative. It has been pointed out by Collins et al.32 and by López-Gil et al.33 that this change could help the eye to accommodate. In particular, López-Gil et al.33 compared the use of RMSw (which is an image quality metric typically used by ray tracing software for optimization and which only depends on the defocus term, z20) with PARAX (which also takes into account the spherical aberration terms, z40 and z60) for the calculation of the accommodative response in a group of 15 young subjects. Equations 1 and 2 show how the AA is calculated for the RMSw and the PARAX metrics, respectively:

The present results shown in figure 6 add further evidence to the fact that, if it can be assumed that Eq. 2 represents a good metrics to predict the refraction, as have been showed before,22 spherical aberration does extend the eye's AA. For our study population the PARAX metric yielded a larger AA (0.42 D more, on average) than the RMSw metric did. As indicated by Eq. 1,2 the value of this difference will depend on the amount of spherical aberration, which, in turn, depends on the maximum AA. Thus, we would expect the difference in AA (between the value determined by PARAX and the value obtained by means of RMSw) to be greater for younger subjects. This prediction is confirmed in figure 9. The data have been fitted using a logarithmic function which approaches a zero value for the older subjects:

Figure 9.

Difference between the AA obtained using PARAX and that obtained using RMSw as metric, plotted as a function of age. Fitting curve: y = -0.7596 ln(x) + 3.0878; r2 = 0.224.

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Although the regression coefficient of the fit was not high (R2= 0.22), it can be seen that for subjects under 25 years old, an average increase of 0.75 D in the AA was obtained when the effect of spherical aberration was included in the calculation. Figure 9 shows that for most of our subjects the use of PARAX metric gave a larger AA than RMSw, this difference being in some cases as high as 2.3 D. This can be explained by taking a look at Eq. 1, 2 and by taking into account the fact that during accommodation, Z40 decreases and eventually changes sign, while Z60 usually increases.33

It is worth stressing that these two metrics, RMSw and PARAX, are the only metrics used in this article that gave, in a very direct way (Eq. 1 and 2), the accommodation state of the eye from the Zernike coefficients and the associated pupil radius. Both are metrics applied at the pupil plane directly to wavefront error functions and, therefore, are indirect measures of the retinal image quality.