Theoretical and ray-tracing calculations on an accommodative eye model based on published anatomical data, together with wave-front experimental results on 15 eyes, are computed to study the change of spherical aberration during accommodation and its influence on the accommodation response. The three methodologies show that primary spherical aberration should decrease during accommodation, while secondary spherical aberration should increase. The hyperbolic shape of the lens' surfaces is the main factor responsible for the change of those aberrations during accommodation. Assuming that the eye accommodated to optimize image quality by minimizing the RMS of the wave front, it is shown that primary spherical aberration decreases the accommodation response, while secondary spherical aberration slightly increases it. The total effect of the spherical aberration is a reduction of around 1/7 D per diopter of stimulus approximation, although that value depends on the pupil size and its reduction during accommodation. The apparent accommodation error (lead and lag), typically present in the accommodation/response curve, could then be explained as a consequence of the strategy used by the visual system, and the apparatus of measurement, to select the best image plane that can be affected by the change of the spherical aberration during accommodation.

*in vivo*(Jones, Atchison, Meder, & Pope, 2005; Koretz, Bertasso, Neider, True-Gabelt, Kaufman, 1987; Neider, Crawford, Kaufman, & Bito, 1990; Strenk et al., 1999) and

*in vitro*(Glasser & Campbell, 1998; Manns et al., 2007) measurements using different techniques that show, with great detail, the changes of the lens during accommodation. Wave-front technology has been mainly used during the last 15 years in the eye to get objective

*in vivo*measurements of how the power of the eye changes when the light passes through different parts of the pupil (Howland & Howland, 1977; Liang, Grimm, Goelz, & Bille, 1994; Sminov, 1961). This technology has also been applied to the accommodated eye, showing a curious effect: the power of the eye, which is usually larger in the periphery of the pupil than at its center, varies during accommodation in such a way that in the accommodated eye it is larger in the center than in the periphery of the pupil (Atchison, Collins, Wildsoet, Christensen, & Waterworth, 1995; Buehren & Collins, 2006; Cheng et al., 2004; He, Burns, & Marcos, 2000; Ivanoff, 1947; Kooman, Tousey, & Scolnik, 1949; López-Gil et al., 2008; Ninomiya et al., 2002; Plainis, Ginis, & Pallikaris, 2005; Radhakrishnan & Charman, 2007; Tscherning, 1900). That power distribution is related to the radial symmetrical high-order terms of the ocular wave front, usually known as spherical aberration (SA).

*r*is the radial coordinate of the entrance pupil, and

*r*

_{0}is the actual entrance pupil radius.

*A*

_{d}corresponds to the coefficient to the defocus, while

*A*

_{s}and

*B*

_{s}correspond to primary (SA4) and secondary (SA6) spherical aberrations, respectively.

*ρ*=

*A*

_{d}) and Zernike defocus (

*B*

_{s}) and sixth-order Zernike SA (

*A*

_{s}(or

*a*

_{4}

^{0}) during accommodation; in fact, this effect was already known by researchers in the first half of the last century. As a matter of fact, Young (1801), and later Tscherning (1900), using what can be assumed to be the first subjective aberroscope, showed that the eye had positive primary spherical aberration (

*A*

_{s}), whose magnitude decreases during accommodation. Similar results using other techniques were obtained by Ames and Proctor (1921), Ivanoff (1947), Jenkins (1963), Kooman et al. (1949), and Sminov (1961). In the last two decades, several other studies, most of which made use of aberrometric techniques, showed similar results both after

*in vivo*measurements (Atchison et al., 1995; Buehren & Collins, 2006; Cheng et al., 2004; He et al., 2000; López-Gil et al., 2008; Ninomiya et al., 2002; Plainis et al., 2005; Radhakrishnan & Charman, 2007) as well as after

*in vitro*ones (carried out by stretching the lens; Glasser & Campbell, 1998; Manns al., 2007). In some cases, the authors have found a decrease in

*A*

_{s}or

*in vivo*measurements, little has been said about how much it really benefits the accommodation response. Moreover, as far as we know, except for three studies—two in humans (Lopez-Gil, Lara, & Fernandez-Sanchez, 2006; Ninomiya et al., 2002) and another one in pigs (Roorda & Glasser, 2004)—there are no other studies that investigated the possible increase or decrease in the secondary SA, and its influence on the accommodation response has never been investigated. In the following, we will give an answer to these points by means of theoretical calculations, computer simulations, and experimental measurements.

*In vitro*measurements of young subjects' lenses also confirmed that the lens' front surface undergoes a bigger curvature change than the posterior surface after stretching (Glasser & Campbell, 1998; Manns et al., 2007).

*R*

_{1L}and

*R*

_{2L}and conic constants

*k*

_{1L}and

*k*

_{2L}, respectively. 1 shows the mathematical calculations used to find the dependency of the eye's wave front with

*k*

_{1L}and

*k*

_{2L}(Equation A8), as well as the change of

*A*

_{s}and

*B*

_{s}during accommodation (Equations A13 and A14).

*A*

_{s}should change in the negative direction (decrease) during accommodation since both surfaces of the lens contribute to it. In particular, for the front surface

*R*

_{1L}decreases (Dubbelman, van der Heidje, & Weeber, 2005),

*β*

_{1L}(the magnification between the plane of the lens' front surface and the eye's entrance pupil plane) is positive and remains nearly constant, and

*k*

_{1L}is negative (hyperbolic shape; Dubbelman et al., 2005). For the back surface, a similar situation occurs, because although the change of index of refraction is negative, the radius of curvature (

*R*

_{2L}) is also negative and so is

*k*

_{2L}. (In fact,

*k*

_{1L}and

*k*

_{2L}seem to become even more negative (Dubbelman et al., 2005), while

*β*

_{1L}and

*β*

_{2L}decrease only slightly during accommodation (see Computer simulations section).) On the other hand, Equation A11 shows that

*B*

_{s}should change in the positive direction (increase) after accommodation, since its value depends on

*A*

_{d}, for a pupil radius

*r*

_{0}corresponds to −2

*A*

_{d}/

*A*

_{s}, the best image plane in terms of minimum variance of the wave-front aberration (RMS) will be observed in the presence of a defocus

*A*

_{s}(Mahajan, 1991). 2 shows an extension of this effect in the presence of primary (

*A*

_{s}) and secondary (

*B*

_{s}) spherical aberrations. In that case, the defocus necessary to minimize RMS is:

*A*

_{s}− (9/10)

*B*

_{s}. Therefore, from Equations A9 and A10, the refractive effect of primary and secondary SAs on the refractive state would correspond to

*W*

_{c}(

*r*) in Equation A8 represents the wave-front variation induced by the cornea. Assuming that there is no any change in the cornea during accommodation, the contribution of the cornea to the change of the refractive state during accommodation would come from the change of the defocus originated by the change of cornea's primary and secondary SAs undergone when the pupil decreases its size. Then, after Equations 5–7, and assuming that neither the indices of refraction (Hermans, Dubbelman, Van der Heidje, & Heethaar, 2008) nor

*β*

_{1L}and

*β*

_{2L}vary during accommodation (see Computer simulations section), the accommodation (refractive change from a relaxed state to a given accommodation state) can be expressed as

*R*

_{1L}and 1/

*R*

_{2L}; in agreement with experimental measurements; Koretz, Cook, & Kaufman, 2002) and is the main factor responsible for the refractive change of the eye during accommodation. However, since both surfaces of the lens have a negative asphericity, the last two brackets also add some amount of accommodation (third and fourth brackets). The bracket corresponding to the effect of the primary spherical aberration (second bracket) produces a decrease in the accommodation, while the one corresponding to the secondary spherical aberration (third bracket) produces an increase in the accommodation. In total, the effect of spherical aberration results in a reduction of the accommodation since the second bracket has a larger weight than the third one.

_{P}, pass from a more hypermetropic to a more myopic position with respect O

_{RMS}after accommodation (Figure 1).

*in vivo*measurements with a wavelength of 0.587

*μ*m and an accommodation stimulus range between 0 and 7 D. We have assumed two types of pupil size: a 4-mm fixed pupil and a linear decrease in the pupil diameter during accommodation of 0.1 mm/D, starting from a diameter of 4.5 mm for the unaccommodated state.

Radius of the front surface of the cornea | 7.87 |

Asphericity of the front surface of the cornea | 0.85 |

Corneal refractive index | 1.376 |

Corneal thickness | 0.574 |

Radius of the back surface of the cornea | 6.4 |

Asphericity of the back surface of the cornea | 0.82 |

Refractive index of the aqueous humor (n _{a}) | 1.336 |

Anterior chamber depth | 2.996 − 0.036D |

Entrance pupil radius | 2 or 2.25 − 0.05 * D |

Radius of the front surface of the lens (R _{L1}) | 1 / (0.0894 + 0.0067D) |

Asphericity of the front surface of the lens (k _{L1}) | −4.5 − 0.5D |

Magnification between EP and L1 (β _{L1}) | 1.1269 |

Refractive index of the lens (n _{L}) | 1.4293 |

Lens thickness | 3.638 + 0.043D |

Radius of the back surface of the lens (R _{L2}) | −1 / (0.1712 + 0.0037D) |

Asphericity of the back surface of the lens (k _{L2}) | −1.43 |

Magnification between EP and L2 (β _{L2}) | 1.2945 |

Refractive index of the vitreous humor (n _{v}) | 1.336 |

Axial length | 24 |

*β*

_{1L}and

*β*

_{2L}, corresponding, respectively, to the pupil magnification of the front and back surfaces of the lens, depend both on the accommodation state. However, the ray-tracing simulations have demonstrated that their value decreases very little during accommodation. For instance, when the entrance pupil was set to 4 mm (fixed value), we found:

*β*

_{1L}= 1.1269 − 0.0006

*D*(

*R*

^{2}= 0.9998), and

*β*

_{2L}= 1.2945 − 0.0014

*D*(

*R*

^{2}= 0.9987), for the front and back surfaces of the lens, respectively. Thus, even when it comes to sixth-order spherical aberration, which includes a term in

*β*

^{−6}(see Equation 8), the value of that terms does not change more than 1.6% across the abovementioned 7-D accommodation range. That is the reason why in Table 1,

*β*

_{1L}and

*β*

_{2L}are considered to be constants.

*In vivo*ocular wave-front change during accommodation

^{2}.

*A*

_{s}]

_{T}= 0.429

*μ*m and [

*B*

_{s}]

_{T}= 0.081

*μ*m for a 4.5-mm pupil) to the values shown in Table 1 permits us to obtain the theoretical change of

*A*

_{s}and

*B*

_{s}during accommodation. (Changes in Zernike instead of Seidel during accommodation can simply be calculated taking into account Equations 3 and 4.)

*A*

_{s}and

*B*

_{s}were obtained, including in Equations 3 and 4 the Zernike values calculated by the Zemax-EE software for a wavelength of 0.587

*μ*m (Equations 3 and 4).

*R*), accommodation response to a certain vergence of the object,

*X*(corresponding to a stimulus accommodation of

*R*−

*X*), is defined by the change of refractive state of the eye. For instance, for a myope with a refraction

*R*= −2 D, which changes the refractive state from −2 D (relaxed eye) to −4 D when seeing an object placed at 20 cm (−5 D of vergence), it will be said that for a stimulus accommodation of 3 D (= −2 D − (−5 D)), the eye is presenting an accommodation response of 2 D (= −2 D − (−4 D)). Figure 3 shows the resulting theoretical accommodation response, obtained based on the data of Table 1 and Equation 8. The contribution of each surface of the lens is shown separately, and so are the different effects on the accommodation response of defocus and fourth- and sixth-order spherical aberrations.

*In vivo*ocular wave-front changes during accommodation

*A*

_{s},

*B*

_{s}, and pupil diameter during accommodation for the 15 subjects included in the study. (Although the computer simulations and

*in vivo*measurements have been obtained discretely by using a stimulus that approaches to the eye by steps of 0.5 D, a continuous line has been used to connect the results in the figures, for clarity.)

*β*

_{1L}and

*β*

_{2L}, but in reality, they are not constant (see Computer simulations section). Third, the modification of the wave front after its propagation from the second surface of the lens to the first one has been ignored, since it is expected to play a very small role.

*μ*m (86%) is due to the lens' front surface and 0.01

*μ*m (14%) to its back surface.

*A*

_{s}equal to 0.94

*μ*m, that is, Δ

*A*

_{s}= −0.99

*μ*m (which corresponds to Δ

*a*

_{4}

^{0}= −0.056

*μ*m; Figure 2A). That value is a little bit higher than the one yielded by the simulations: Δ

*A*

_{s}= −0.87

*μ*m (Δ

*a*

_{4}

^{0}= −0.052

*μ*m) for the eye model based on the data of Table 1 (Figure 2A), or Δ

*a*

_{4}

^{0}= −0.05

*μ*m when using Navarro's eye model (Navarro, Santamaría, & Bescós, 1985; López‐Gil et al., 2008). Similar values have also been found experimentally by independent and previous studies performed by various authors: Cheng et al. (2004), Δ

*μ*m; Plainis et al. (2005), about Δ

*μ*m; or López-Gil et al. (2008), Δ

*μ*m. Atchison et al. (1995) found a change of fourth-order spherical aberration in terms of defocus of −0.34 D, which corresponds to a Δ

*μ*m for a 4-mm pupil. Radhakrishnan and Charman (2007) also found a value of Δ

*μ*m, although he used a natural pupil diameter.

*a*

_{6}

^{0}during accommodation for a fixed pupil size correspond to Ninomiya et al. (2002). Their data, although it cannot be directly compared with our calculations (Figure 2B) because the pupil diameter values are different, show a similar trend. Figure 4B clearly shows that, on average, there is an increase in

*r*′ <

*r*), we can expect a smaller change of

*A*

_{s}and

*B*

_{s}during accommodation, relative to the situation when the pupil size remains constant (

*r*′ =

*r*). This fact is reflected in Figure 2A, where the slope of Δ

*a*

_{4}

^{0}is bigger for the natural pupil condition than for the 4-mm-diameter pupil until about 5 D of stimulus accommodation, for which the natural pupil size is about 4 mm. Then, Equations A21 and A22 indicate that the change of

*a*

_{4}

^{0}and

*a*

_{6}

^{0}during accommodation will be bigger for larger pupils, which is in good agreement with the experimental data shown by López-Gil et al. (2008) and Radhakrishnan and Charman (2007). As an example, Figure 7 shows that when the pupil does not decrease during accommodation (stimulus accommodation between 0 and 2.5 D) or even when it increases (around 4 D of stimulus accommodation) the value of SA4 decreases rapidly.

*A*

_{s}and an increase in

*B*

_{s}. As indicated by other researches (Cheng et al., 2004; López-Gil et al., 2008; Plainis et al., 2005), most of the subjects have a positive

*A*

_{s}(

*A*

_{s}(so

*A*

_{s}has a negative value across the whole accommodation range, as shown by the brown line in Figure 4A, corresponding to subject #6. In these cases where

*A*

_{s}< 0 in the relaxed eye, the minimum SA4 was usually more negative than in the cases where the relaxed eye presented a positive SA4 value.

*a*

_{4}

^{0}for the relaxed eye, and this value barely decreases during accommodation (see brown line at the bottom of Figure 5A). However SA4 has a clear negative impact on his accommodation, as shown in Figure 6B. However, the same subject shows a large increase in

*a*

_{6}

^{0}(see brown line at the bottom of Figure 5B) resulting in a benefit on the AA of about 0.5 D (Figure 6B).

*X*-axis) corresponding to the paraxial far point (maybe preferred subjectively), the accommodation response would present a lead of accommodation if the relaxed eye has a positive SA4 (as is the case in most of the subjects analyzed, see Figure 4A). Then, Figure 5 shows not only a lag but also a lead of accommodation. Figure 1 also shows schematically the presence of a lead of accommodation, which corresponds to the absolute difference in diopters between the vergences of O

_{P}and O

_{RMS}in the relaxed eye, while the lag of accommodation correspond to the same difference but in the accommodated eye (with an opposite sign than the lead). However, the presence of a lead or lag of accommodation would depend on the sign of SA4 in the relaxed eye. Thus, for SA4 = 0, no lead could be expected, while for SA4 < 0 a lag of accommodation could be expected in the whole range of accommodation, being much larger in the accommodated eye than in the relaxed eye as predicted by Equation 8.

*r*

^{2}) that the polynomial includes to balance the term in

*r*

^{4}(Equation 2), thus minimizing the RMS. However, that only happens for relatively large amounts of induced

*μ*m), the visual system does not have its refraction modified, as has been pointed out recently by Cheng, Bradley, Ravikumar, and Thibos (2010). When Seidel's primary spherical aberration is added (Seidel SA, which is similar to an “unbalanced Zernike spherical aberration”), the opposite effect is expected. That is, low values of Seidel SA will modify the refraction, while large values will hardly modify it. In particular, positive small amounts of Seidel SA induce a negative defocus (hyperopic eye) while a negative small amount of Seidel SA induces a positive defocus (myopic eye).

*μ*m (0.06

*μ*m for SA6), which is larger than 0.1

*μ*m, indicating that probably the refraction change produced by the effect of SA4 (Figures 3 and 8) could be overestimated. That is probably the main discrepancy between the experimental accommodation response (Figure 5) and the calculated one (Figure 8). However, even in the case that only half of that SA4 was really used to change the refraction state of the eye, it also causes the decrease in AA by about 0.7 D (for a mean pupil size of 5 mm), which is a large value (10% of the 1:1 response). Moreover, we should keep in mind that the change of SA4 during accommodation is continuous, so its effect could be different from the case where SA4 is added instantaneously, as in the studies previously (Benard et al., submitted for publication; Rocha et al., 2009) undertaken.

*R*

_{1L}and

*k*

_{1L}are the radius of curvature and the conic constant of the surface, respectively (Figure A1).

*x*

_{0}=

*y*

_{0}and

*z*

_{0}=

*R*

_{1L}, we can rewrite Equation A1 in polar coordinates as

*r*

_{1i}represents the radial coordinate at the iris plane.

*z,*we have

*r*

_{1i}, Equation A3 can then be expressed as (see Roorda & Glasser, 2004 for a similar result for a calculation up to fourth order)

*n*

_{L}and

*n*

_{a}represent the index of refraction of the aqueous humor and the lens, respectively, already taking into account the sign of the wave front when it is getting out of the eye (Rocha et al., 2009).

*β*

_{1L}=

*r*/

*r*

_{1i}and

*β*

_{2L}=

*r*/

*r*

_{2i}for the front and back surfaces, respectively, the total radial dependence of the wave front at the eye's entrance pupil would be, after Equation A5:

*W*

_{C}(

*r*) represents the radial contribution of the cornea, up to the sixth order (

*n*= 2, 4, or 6), and where we have assumed that

*n*

_{a}=

*n*

_{v}=

*n*. The term related to the first bracket corresponds to defocus:

*A*

_{s}) and secondary (

*B*

_{s}) spherical aberrations generated by the lens, respectively:

*a*

_{4}

^{0}would also be affected by the term

*W*

_{C}(see Equation A8). In that case,

*A*

_{s}]

_{T}and [

*A*

_{s}]

_{C}are the values of the fourth-order spherical aberration of the whole eye and that corresponding to the cornea, respectively. Thus, the change of the total fourth-order spherical aberration during accommodation is given by

*r*′ =

*r*; for instance, by using an artificial pupil in front of the eye), the last equations can be rewritten as

*β*

_{1L}and

*β*

_{2L}do not change during accommodation, this leads to

*A*

_{s}by

*B*

_{s}by

*A*

_{d}necessary to balance the effect of the presence of primary and secondary spherical aberrations, in terms of minimum RMS, will be obtained for an

*A*

_{d}value that minimizes the function 〈

*W*(

*ρ*)

^{2}〉 − 〈

*W*(

*ρ*)〉

^{2}with

*A*

_{d}that minimized Equation B2 is obtained using

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