Influence of Ametropia and Its Correction on Measurement of Accommodation

Abstract

Purpose: Amplitude of accommodation (AA) is reportedly greater for myopic eyes than for hyperopic eyes. We investigated potential explanations for this difference.

Methods: Analytical analysis and computer ray tracing were performed on two schematic eye models of axial ametropia. Using paraxial and nonparaxial approaches, AA was specified for the naked and the corrected eye using the anterior corneal surface as the reference plane.

Results: Assuming that axial myopia is due entirely to an increase in vitreous chamber depth, AA increases with the amount of myopia for two reasons that have not always been taken into account. First is the choice of reference location for specifying refractive error and AA in diopters. When specified relative to the cornea, AA increases with the degree of myopia more than when specified relative to the eye's first Gaussian principal plane. The second factor is movement of the eye's second Gaussian principal plane toward the retina during accommodation, which has a larger dioptric effect in shorter eyes.

Conclusions: Using the corneal plane (placed at the corneal vertex) as the reference plane for specifying accommodation, AA depends slightly on the axial length of the eye's vitreous chamber. This dependency can be reduced significantly by using a reference plane located 4 mm posterior to the corneal plane. A simple formula is provided to help clinicians and researchers obtain a value of AA that closely reflects power changes of the crystalline lens, independent of axial ametropia and its correction with lenses.

The human eye has a maximum capacity of accommodation, known as accommodative amplitude (AA), which depends mainly on age.¹ Ametropia is a potential factor influencing AA but studies have produced conflicting results. Some studies have reported greater AA in myopes than in emmetropes and hyperopes,^2–6 while others found the opposite^7,8 or no differences.⁹ Table 1 summarizes the results found in those studies. Although most studies show that AA is larger in myopic eyes, results are not fully comparable because of differences in methodologies between studies. Different choices of reference plane for specifying vergence of the far and near points are of particular importance for our investigation of potential optical explanations for variability in results reported in the literature. 

Table 1

 

Summary of Studies Comparing the Amplitudes of Accommodation of Myopes (M), Emmetropes (E), and Hyperopes (H)

 

Accommodative amplitude is usually measured clinically when the patient is wearing habitual correcting lenses. The optical effect of the correcting lens when stimulus vergence varies has to be taken into account to give the appropriate measure of the accommodation response. We will refer to this well-known effect of correcting lenses as the “external effect,” which has been studied previously^10,11 and is normally taken into account.^12,13 

In general, empirical studies of AA are hampered by the lack of a universally accepted definition of accommodation or AA. Perhaps the simplest definition of AA is the maximum achievable increase from the relaxed state of the lens equivalent power, where equivalent power is the refracting power associated with the principal planes of a Gaussian paraxial model of the crystalline lens. This definition would be useful, for example, for investigating the effect of ametropia on the capacity of the ciliary muscle to modify the lens. Alternatively, AA can be defined by the maximum achievable increase of equivalent power of the eye's complete optical system, which would be useful for investigating the relative contributions of refractive and axial components of refractive error. Yet another definition is a change in the eye's refractive state, as specified by the difference in vergence of the near point and far point relative to the anterior principal plane. This definition has the advantage of specifying the range of target vergences that can be brought into clear focus by accommodation. These three definitions can yield significant differences in AA. For example, in the Bennett and Rabbetts schematic eye model,¹⁰ assuming that the distance between principal planes of the cornea and the lens remains constant during accommodation, a 10-diopter (D) change in refractive state is produced by a 12.95-D increase in equivalent power of the crystalline lens, which changes the equivalent power of the whole eye by 11.12 D. Although all three values are equally valid, they have different interpretations. A potential for confusion arises if a researcher or clinician uses one definition for measurement but a different definition for application (or interpretation) of that measurement, and consequently comes to an erroneous conclusion. 

All three of the definitions of AA summarized above are difficult to put into practice, however, because they stipulate that equivalent power and the far and near points be referenced to the eye's principal planes, which are theoretical constructs within the framework of Gaussian optics that cannot be easily located in a laboratory or clinical setting. For this reason, it is common practice to approximate AA as the difference in the vergences of the far point (R) and the near point (P) in object space, both relative to a physical reference plane such as the corneal plane (placed at the corneal vertex) or spectacle plane. Thus, in practice, AA is defined as R − P diopters. However, the use of such a practical definition can be difficult to connect with the more theoretical definitions of AA because of movement of the principal planes relative to the fixed anatomic location of the cornea. To take such movement into account requires knowledge of other parameters of the eye that are usually unknown (e.g., corneal power, anterior chamber depth, axial length). Thus, measurements of AA in two eyes with crystalline lenses that change the same way during accommodation may be different if, for instance, the axial lengths of the two eyes are different. 

To assess the magnitudes of discrepancies identified above, we undertook a theoretical analysis of AA for two schematic eye models of accommodation. Our results provide insight into the optical reasons for reports of increased AA measured in myopic eyes, and their potential clinical consequences. 

Methods

In order to compare AA of a naked eye, AA_N, using the corneal plane as reference plane for eyes with different axial refractive errors, we performed computer ray tracing simulations as well as analytical calculations under the assumption of paraxial optics. Ametropia was introduced into these models by changing the axial location of the retina relative to the second surface of the lens without changing the eye's optics. 

To find the axial length corresponding to a given degree of ametropia we computed, in the unaccommodated model eye, the position of the plane conjugate to an object situated at vergences ranging from −10 to +10 D in steps of 1 D measured from the corneal plane. 

Computation

We used commercial ray tracing software (Zemax OpticStudio; Radiant Zemax, Kirkland, WA, USA) to calculate the paraxial and nonparaxial refractive state of the eye using a minimum root mean square (RMS) criterion (i.e., a Zernike refraction¹⁴). Paraxial analysis was performed for two different eye models, and nonparaxial analysis for one of them. This strategy enabled an assessment of how much the results depend on the specific eye model used and the Gaussian approximations in paraxial optics. We chose the eye models of Le Grand¹⁵ and Navarro et al.¹⁶ because they are parameterized for both the accommodated and the relaxed states. The value of AA_N in the Le Grand eye model is approximately 7 D, so we selected that value as the accommodation value for the Navarro eye model as well. Axial ametropia was introduced into the models by changing the axial position of the retina. All simulations were carried out using ray tracing with a 5-mm entrance pupil diameter and a wavelength of 550 nm. 

Calculations

When calculating AA_N for the Gaussian approximation, we took into account the axial movement of principal planes of the eye's optical system that occurs when the lens accommodates. Formulas used for all calculations can be found in Appendix A. The analytical equations were verified by application to the two eye models, the results of which agreed with paraxial ray tracing results from Radiant Zemax. 

Results

The effects of axial refractive errors on AA are shown in Figure 1 for both eye models using paraxial analysis and nonparaxial analysis for the Navarro eye model. In all three cases, AA is well fit by a straight line (R² > 0.999) with a negative slope indicating larger AA_N in myopes than in emmetropes, and these in turn larger than in hyperopes. The unitless slope of the fitted lines ranged from −0.053 to −0.059. 

Figure 1

 

Accommodative amplitude computed from eye models versus axial refractive error. Negative values of refractive error indicate myopia, and positive values indicate hyperopia. Open symbols are from paraxial analysis and filled symbols are from the nonparaxial analysis.

Difference in AA_N determined by paraxial and nonparaxial analyses of the Navarro eye models is 0.60 D. Estimated AA_N values for the nonparaxial eye model are lower because of the presence of positive spherical aberration in the relaxed eye that decreases during accommodation, thereby resulting in a decrease in the accommodation response.^14,17 For smaller pupil diameters, spherical aberration has less effect on the Zernike refraction, resulting in a shift in AA_N values toward the paraxial result. 

From analytical calculations under the Gaussian optics approximation (Appendix A), we derived the following equation to calculate the AA_N from the corneal plane, C′, as a function of axial length:  

 

where: A_D and A_A are the vergences of the far and near points measured from the first principal plane (H) of the eye; is the displacement of the first principal plane from relaxed to the accommodated state; and are the distances from the first principal plane to the corneal plane of unaccommodated and accommodated eyes, respectively; φ_A and φ_D represent the powers of the eye when accommodating and in the relaxed state, respectively; and c is defined by Equation A9. Equation 1 has a dependency on the axial length expressed as the distance between the secondary principal plane and the image plane in the relaxed eye, .  

Equation 1 shows that the AA_N is the sum of three contributions. The first term corresponds to the change in power of the eye due to the lens when accommodating, and takes into account the distance between the corneal and principal planes. The second term in the equation is due principally to the movement of the secondary principal plane during accommodation. This term is always negative and, in absolute value, is smaller for myopes, so its effect is to reduce the AA_N more in hyperopes. The last term in the equation is related to the movement of the first principal plane with accommodation, and its change during accommodation for different axial lengths is negligible compared to the other two terms (Fig. 2). All terms in Equation 1 are affected by the choice of reference plane C′ for specifying vergences. 

Figure 2

 

Contribution of each term in Equation 1 to the change of AA for different axial refractive errors using Le Grand eye model. First term is represented by blue squares, second term by green triangles, and third term by red circles. Regression coefficient R² = 1 for all three data sets.

Discussion

The motivation for our theoretical analysis was to determine the magnitude of changes in AA associated with degree of ametropia that can be attributed to different conventions for specifying AA. Results from paraxial and nonparaxial models are in agreement that there is a slight dependence of AA on axial length (Fig. 1) when the corneal plane is used as reference origin for specifying vergences. There are two reasons for this result (see Appendix A). The first reason is related to the decision (made for practical reasons mentioned in the introduction) to move the reference origin for vergences from the anterior principal plane of the eye (H, inside the eye) to the corneal plane, C′, for the purpose of defining AA. The second reason is related to the displacement of the posterior principal plane of the eye, H′, toward the retina when the eye accommodates (see Appendix A for supporting data). For the Le Grand eye model, the first reason accounts for approximately 52% of the change in the AA with axial refractive error shown in Figure 1, while the second reason accounts for most (45%) of the remaining 48%. While the two reasons have independent origins, they are related to each other since the distance changes during accommodation due to the displacement of H toward the retina.  

Both explanations for why measured AA varies with refractive error involve intraocular changes, so we refer to them as “internal effects.” These internal effects appear to have been overlooked previously, perhaps because attention has concentrated on the well-known external effect of moving the reference origin for vergence from the corneal plane to the spectacle plane. Although Le Grand was aware that AA varied with the axial refractive error, and found an equation that relates the AA of an emmetrope with that of an ametrope, he did not explain the origin of the variation.¹⁵ The values obtained by Le Grand are lower than those obtained in this work because he used the anterior principal point, H, as the origin of vergences. In 1864, using Helmholtz's diagrammatic eye, Donders⁸ calculated the near and far points of three eyes with different axial lengths. Contrary to our findings, Donders found that hyperopes accommodate more than myopes. Based on our calculations, Donders' results can be explained by the fact that he used the nodal point of the unaccommodated eye, N, as origin of vergence, which is closer to the retina than H. Thus an opposite change in AA with axial refractive error compared to the one shown in Figure 1 can be expected. These results reveal the importance of choosing the origin of measurement appropriately. 

A novel choice of reference origin for vergence is that plane behind the corneal plane where the first term of Equation 1 changes its slope with axial refractive error (slope in the top line in Figure 2 becomes positive), canceling the variation generated by the second term (lower line in Fig. 2). That null point may be a preferred choice of origin for specifying AA, although it depends on several factors such as the displacement of H and H′ toward the retina during accommodation, on the choice of eye model, and on the degree of accommodation. Paraxial analysis of the Le Grand model indicates that this null position is 3.68 mm behind the corneal plane. Paraxial analysis of the Navarro eye model indicates the plane to be 4.49 mm behind the corneal plane for 10 D of accommodation and 4.29 mm behind for just 1 D of accommodation. Thus, a reference plane located 4 mm behind the corneal plane for specifying accommodation will strongly reduce the dependency of estimated values of AA on the axial refractive error of the eye. A practical advantage of this proposed reference plane is its close proximity to the eye's entrance pupil, which is normally approximately 3 mm posterior to the corneal plane. Thus when a Badal optometer is used to measure accommodation, and the focal plane of the Badal lens is positioned to coincide with the eye's entrance pupil, then measurements of AA should be largely unaffected by the internal effects described above. That expectation is consistent with the findings of Fisher et al.,⁹ who reported that AA was independent of refractive error when specified relative to the eye's entrance pupil. However, this advantage of a Badal optometer would be lost for other configurations that do not use the entrance pupil of the eye for specifying AA. 

As an example of the relevance of the internal effects described above, let us assume that an eye with 10-D axial myopia corrected with a spectacle lens accommodates 7 D. After correcting for the external effect of the correcting spectacle lens, the myopic eye would still have an AA 0.6 D larger than an emmetropic eye with the same optical system (see Fig. 1). Since we have assumed that the optical system is the same for both eyes in this example, the difference in AA does not reflect a difference in accommodation ability of the crystalline lens. Instead, the difference is a manifestation of the internal effect described above. A method for correcting both external and internal effects is presented in Appendix B, along with reanalysis of data from studies summarized in Table 1. 

Our calculations may explain, in part, the experimental results obtained by other researchers summarized in Table 1. Unfortunately, most of the latter studies (with the exception of Fisher et al.⁹) do not include sufficient methodological details to allow correction of reported results for the internal effects described above. Nevertheless, we estimate the order of magnitude that might be expected for such corrections (assuming typical values for missing parameters) to be the same order of magnitude as the differences reported between myopic and emmetropic eyes (see Table B1 at the end of Appendix B). Future studies of accommodation should take into account such methodological details that appear to have been overlooked previously in order to accurately estimate AAs. 

To properly compare accommodation performances among groups with different axial refractive errors, a common origin of vergences should be used. Although the spectacle plane is a useful plane of reference (most of the rulers in clinical phoropters use that plane to obtain the near point), the external and internal effects may give a very different value of accommodation in the hypothetical case of two eyes with the same lens that changes exactly in the same way during accommodation. To solve that problem, we propose one practical equation (Equation 2 below; also see Appendix B, Equation B11) to obtain AA_N from the value of the vergence to the near point (assuming that the vergence of the far point is 0), which is referred to the plane of correction (ophthalmic or contact lens):  

 

where d is the distance between the correction plane and the corneal plane in meters, P_L is the power of the correction lens in diopters, and P_C is the vergence in diopters of the corrected eye's near point as measured from the correction plane (with positive sign for real objects). As an example of application of Equation 2, let us assume a 5-D myope (P_L = −5 D), corrected with spectacle lenses placed at 14 mm of the corneal plane (d = 0.014 m). If the near point of this subject is located 12 cm from the spectacle plane (P_C = 1/(0.12 m) = 8.33 D), the AA will be 6.17 D.  

In conclusion, our study shows that optical effects related to the axial location of Gaussian principal planes internal to the eye produce a small (on the order of tenths of a diopter) difference in the AA among subjects with different axial refractive errors with an identical change in power of the lens during accommodation. We provide a tool to transfer results to standardized measures from the spectacle plane that includes a reference plane 4 mm inside the eye as origin of vergences to handle the internal effect. That choice of reference, rather than the corneal plane, makes measurement of accommodation more independent of the axial length of the eye. 

Acknowledgments

The authors thank Elvira Pérez Jiménez for her initial calculations. 

Presented in part at the VPOptics meeting in Wrowclaw, Poland, August 2014, and at the 15th International Myopia Conference in Wenzhou, P.R. China, September 2015. 

Supported by Fundación Séneca de la Región de Murcia (15312/PI/10) and European Grant ERC-2012-StG 309416-SACCO. 

Disclosure: P. Bernal-Molina, None; F. Vargas-Martín, None; L.N. Thibos, None; N. López-Gil, None 

Appendix A. Analytical Calculation of the Internal Effect

In this appendix we will obtain an analytical expression of the amplitude of accommodation of the uncorrected eye as a function of its axial length under Gauss approximation. 

Figure A1 shows the distances in the relaxed (half-upper side) and accommodated eye (half-lower side) of the far (FP) and near point (NP) to the corneal plane C′ and principal planes H_D and H_A. Relative distances from the principal planes to the cornea and the retina are also shown. Eye refraction is irrelevant in this demonstration, and a myopic eye is chosen for clarity in the diagram. Figure A1 also shows schematically the displacement of the principal planes toward the retina in the accommodated eye. These displacements are real and result from the increase in power of the eye during accommodation due to the increase in power of the lens. Values of those displacements depend on the choice of eye model and on the amount of accommodation, but not on the axial length of the eye, L. 

Figure A1

 

Schematic myopic eye relaxed (top) and accommodated (bottom). C′ is the corneal plane. r and p are the distances from the corneal plane to the far (FP) and near (NP) points, respectively. a_D and a_A are the distances from the first principal plane (H_D when relaxed and H_A when accommodated) to the far and near points, respectively. and are the distances from the second principal plane ( when relaxed and when accommodated) to the retina when the eye is relaxed and accommodating, respectively. L is the axial length of the eye.

Assuming a first order (Gaussian) approximation, we have for the relaxed eye:  

 

and for the accommodated eye:  

 

where n and n′ are the indices of refraction outside the eye and the vitreous humor, respectively, and and ( < ) are the image focal lengths of the relaxed and accommodated eye, respectively. and are positive values, while a and a′ values are negative for distances in front of the eye and positive for virtual objects. Equations A1 and A2 can be also rewritten in terms of vergences by defining , , , and , as:  

 

and  

 

The distance in Equations A1 and A2 can be related to the axial length as:  

 

If n = 1, we have from Figure A1:  

 

and  

 

By means of Equations A3, A4, A6, and A7, the AA can be expressed as:  

 

where  

 

 

Appendix B. Formula for Correcting the External and Internal Effects

This appendix develops a formula to compensate AA measurements for the external effect of correcting lenses and for internal effect related to the axial displacement of Gaussian principal planes during accommodation. 

Figure B1 shows a myopic eye without correction (top) and with an optical system (OS) with principal planes H and H′ for correcting the eye's myopia (bottom). Distance from H′ to the corneal plane, C′, is d and has a positive value. 

Figure B1

 

Schematic myopic eye without correction (top) and with correction (bottom). C′ is the corneal plane. Subscript N indicates naked eye and C indicates corrected eye. r and p are the distances from the corneal plane to the far (FP) and near (NP) points, respectively. H and H′ are the principal planes of the correction system. C is the position of the object plane whose image is C′.

Assuming that C is the position of the object whose image is C′ (C and C′ are conjugate through the OS), we can apply generalized Gaussian correspondence laws taking C and C′ as reference planes to compute the positions of the images of the far and near points of the corrected eye. Assuming that the lateral magnification of the OS for C and C′ is β′_c, and f ′ the focal distance of the OS, we have for the far point:  

 

and for the near point:  

 

where n′ is referred to the image space of the OS. Subtracting Equation B1 from Equation B2 yields:  

 

Now if the accommodative amplitude of the naked eye, referred to C′, is defined as the difference between near-point vergence and far-point vergence,  

 

and that of the corrected eye, referred to C, is similarly defined as  

 

then we finally have  

 

 

represents the amplitude of accommodation with respect to plane C but we want to obtain the relation with origin on C′, . In a practical case, usually a thin ophthalmic lens is used in air at a distance d in front of the eye for its correction. In that case, taking into account that d′ is the distance between the spectacle lens and C, and that R represents the refractive error of the naked eye that is perfectly corrected with a lens with power P_L, we can use the following relations:  

 

to obtain  

 

So, finally we obtain  

 

 

If the distance between C and C′ in Figure B1 is very small compared with r or p, then ≈ and Equation B8 can be approximated with  

 

 

However, when measuring AA clinically, the origin of measurements is typically the spectacle plane where the correction for ametropia is located. Calculating the propagation of AA of the corrected eye from C to the correction plane, an expression relating AA (as defined in B4) of the naked eye with that measured clinically, can be found:  

 

 

 

where P_C is the vergence of the near point with respect to the correction plane in absolute value.  

Now, in order to reduce the internal effect, we have to minimize the dependence of AA with axial refractive error. As stated in the Discussion, we recommend a reference plane placed 4 mm behind the corneal plane. That is, in Equation B10 the parameter d should now correspond to the distance from the correction to the corneal plane plus 4 mm:  

 

 

If, for instance, the correction is made by contact lenses, d = 0 m in Equation B11, P_L is the power of the contact lens and P_C must be the vergence of the near point to the corneal plane (which is the correction plane in this case) in absolute value. 

Application of Equation B11 to previous results (Table 1) is shown in Table B1. 

Table B1

 

Summary of the Information Provided in Each Experimental Study Mentioned in the Introduction Section and a Prediction (Last Column) After Assuming Some Values Not Provided by the Authors (in Bold), Together With the Assumption That Subjects Have Similar Optics and Different Refractions Are Caused by Different Axial Lengths