Increased Axial Length Corresponds to Decreased Retinal Light Dose: A Parsimonious Explanation for Decreasing AMD Risk in Myopia

Abstract

Purpose: A recent systematic review indicated that higher sunlight exposure increased risk of AMD. The Beaver Dam study and the Pathologies Oculaires Liées à L'âge study both noted that wearing hats and/or sunglasses significantly decrease some AMD lesions, suggesting that reduced retinal light dose (RLD) may be related to reduced AMD risk. Given that myopes also have reduced AMD risk, we hypothesize its link to decreased RLD.

Methods: Using a one-surface schematic eye and ray-tracing, spectacle power, vertex distance, corneal power, anterior chamber depth, and axial length to calculate relative light flux through the pupil and resultant image size on the retina in a randomly selected group of 71 eyes from the Reykjavik Eye Study. Pupil size is unaffected by refractive error; thus, RLD can be calculated. We verified this using a more complete set of ocular biometric variables and ray-tracing included in an optical design software (Opticsoft II).

Results: RLD is inversely proportional to axial length. Comparing the two methods for calculating RLD using a Bland-Altman plot demonstrated equivalence. The ray-tracing method indicated that the retina of a hyperope with a 21-mm axial length would always be receiving 1.8× more photons per square millimeter than the retina of a myope with a 27-mm axial length.

Conclusions: RLD is inversely proportional to axial length, as is AMD risk. The RLD for our 21-mm axial length wearing a pair of inexpensive commercial sunglasses would be equivalent to the RLD for a 27-mm myope. This may explain the decreased AMD risk in highly myopic individuals.

High levels of sunlight exposure have been suggested to be a possible cause of AMD, and indeed, a recent systematic review and meta-analysis on this question¹ indicated that individuals with higher levels of sunlight exposure are at significantly increased risk of AMD. Despite this analysis, the link between AMD and sun exposure remains controversial, as two of the best-known studies included in the review, The Beaver Dam study (BDES)² and the Pathologies Oculaires Liées à L'âge (POLA) study,³ concluded that there were few significant relationships between environmental exposure to light and the 10-year incidence and progression of AMD. Both studies did acknowledge that subjects wearing hats and/or sunglasses, which would decrease retinal light dose (RLD), significantly decreased some of the early retinal lesions seen in AMD. Additionally, when controlling for light exposure, the BDES² noted that men exposed to the highest levels of light were protected from some AMD lesions by myopia. 

Maltzman et al.⁴ first noted an association between refractive error and AMD with myopic individuals being at lower risk compared with their hyperopic counterparts. A large population-based study has also shown this to be a dose-dependent phenomenon: the greater the hyperopia or the shorter the axial length, the greater the risk for early AMD.⁵ This increase of early AMD risk was associated with increases in both hyperopic refraction and decreases in axial length, by up to 8% and 29%, respectively. Myopes and hyperopes are different insofar as their refractive components are concerned (spectacle correction, corneal power, anterior chamber depth, lens power, and axial length).⁶ Given the decreased AMD risk afforded by the wearing of hats and sunglasses (presumably by decreasing RLD), this begs the question as to whether the decreased AMD risk seen in myopia could also be caused by a decrease in RLD as a result of the optical component differences between eyes of different refractive errors. Of particular significance in considering the issue of RLD is that pupil size is independent of refractive error,⁷ and therefore it will not be a confounder when calculating light dose. If RLD is indeed playing a role in the genesis and evolution of AMD, then we expect that a lower RLD will exist in myopes as compared with hyperopes. 

Methods

Two methods were used to calculate retinal light dose in our random sample of 71 eyes from the Reykjavik Eye Study, which provided refractive and biometric data on a population-based sample of residents in Reykjavik, Iceland.⁶ The sample size was in part determined through power analysis, whereby correlation coefficients of r = 0.33 or higher would be statistically significant with n = 70 at p = 0.05 and a power of 0.80, which is a value deemed sufficient by the team members for this analysis. In addition, data processing fees for the involved engineer limited the sample to n = 71, given the institutional funds available to the principal investigator. The first involves ray-tracing with a simple one-surface schematic eye using the most important refractive variables, and the second uses the ray-tracing feature inside optical design software (Optisoft II; Optisoft Ltd, Huntington, York, UK),⁸ which used all of the eye's refractive variables. 

Method 1

The eye's refractive components, their respective refractive indices, and the spacing of these components will determine the number of light rays (photon equivalents) passing through a given pupil (henceforth called flux). This flux will be most influenced by spectacle power and vertex distance, the corneal power, and the anterior chamber depth. Similarly, the distribution of this flux on the retina (henceforth called image magnification) will be influenced by these variables in addition to the crystalline lens and the axial length. From this information, the RLD can be calculated. 

Flux to the Pupil

We created a simplified one-surface schematic eye to calculate light flux through the pupil (Fig. 1), where h is paraxial ray height, u is angle at the spectacle lens, h′ is the paraxial ray height at the cornea, h″ is the half-pupil size (a constant that does not change with changes in refractive error), u′ is the angle at the spectacle lens following refraction, u″ is the angle after refraction by the cornea, R is the spectacle lens power, ac is the anterior chamber depth, K is the corneal power, and N is the refractive index of the cornea and anterior chamber (1.33). Then it follows that  

\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\unicode[Times]{x1D6C2}}\)\(\def\bupbeta{\unicode[Times]{x1D6C3}}\)\(\def\bupgamma{\unicode[Times]{x1D6C4}}\)\(\def\bupdelta{\unicode[Times]{x1D6C5}}\)\(\def\bupepsilon{\unicode[Times]{x1D6C6}}\)\(\def\bupvarepsilon{\unicode[Times]{x1D6DC}}\)\(\def\bupzeta{\unicode[Times]{x1D6C7}}\)\(\def\bupeta{\unicode[Times]{x1D6C8}}\)\(\def\buptheta{\unicode[Times]{x1D6C9}}\)\(\def\bupiota{\unicode[Times]{x1D6CA}}\)\(\def\bupkappa{\unicode[Times]{x1D6CB}}\)\(\def\buplambda{\unicode[Times]{x1D6CC}}\)\(\def\bupmu{\unicode[Times]{x1D6CD}}\)\(\def\bupnu{\unicode[Times]{x1D6CE}}\)\(\def\bupxi{\unicode[Times]{x1D6CF}}\)\(\def\bupomicron{\unicode[Times]{x1D6D0}}\)\(\def\buppi{\unicode[Times]{x1D6D1}}\)\(\def\buprho{\unicode[Times]{x1D6D2}}\)\(\def\bupsigma{\unicode[Times]{x1D6D4}}\)\(\def\buptau{\unicode[Times]{x1D6D5}}\)\(\def\bupupsilon{\unicode[Times]{x1D6D6}}\)\(\def\bupphi{\unicode[Times]{x1D6D7}}\)\(\def\bupchi{\unicode[Times]{x1D6D8}}\)\(\def\buppsy{\unicode[Times]{x1D6D9}}\)\(\def\bupomega{\unicode[Times]{x1D6DA}}\)\(\def\bupvartheta{\unicode[Times]{x1D6DD}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bUpsilon{\bf{\Upsilon}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\(\def\iGamma{\unicode[Times]{x1D6E4}}\)\(\def\iDelta{\unicode[Times]{x1D6E5}}\)\(\def\iTheta{\unicode[Times]{x1D6E9}}\)\(\def\iLambda{\unicode[Times]{x1D6EC}}\)\(\def\iXi{\unicode[Times]{x1D6EF}}\)\(\def\iPi{\unicode[Times]{x1D6F1}}\)\(\def\iSigma{\unicode[Times]{x1D6F4}}\)\(\def\iUpsilon{\unicode[Times]{x1D6F6}}\)\(\def\iPhi{\unicode[Times]{x1D6F7}}\)\(\def\iPsi{\unicode[Times]{x1D6F9}}\)\(\def\iOmega{\unicode[Times]{x1D6FA}}\)\(\def\biGamma{\unicode[Times]{x1D71E}}\)\(\def\biDelta{\unicode[Times]{x1D71F}}\)\(\def\biTheta{\unicode[Times]{x1D723}}\)\(\def\biLambda{\unicode[Times]{x1D726}}\)\(\def\biXi{\unicode[Times]{x1D729}}\)\(\def\biPi{\unicode[Times]{x1D72B}}\)\(\def\biSigma{\unicode[Times]{x1D72E}}\)\(\def\biUpsilon{\unicode[Times]{x1D730}}\)\(\def\biPhi{\unicode[Times]{x1D731}}\)\(\def\biPsi{\unicode[Times]{x1D733}}\)\(\def\biOmega{\unicode[Times]{x1D734}}\)\begin{equation}u^{\prime} = u + h*R = h*R\end{equation}

 

\begin{equation}h^{\prime} = h - z*u^{\prime} \end{equation}

 

\begin{equation}u^{\prime\prime} = u^{\prime} + h^{\prime} *K/N\end{equation}

 

\begin{equation}h'^{\prime} = h^{\prime} - ac*u^{\prime\prime} \end{equation}

 

\begin{equation}h^{\prime\prime} = h - z*\left( {h*R} \right) - ac*\left[ {h*R + \left( {h - z*h*R} \right)*K/N} \right]\end{equation}

 

\begin{equation}h^{\prime\prime} = h*\left\{ {1 - z*R - ac*\left[ {R + K/N*\left( {1 - z*R} \right)} \right]} \right\}\end{equation}

 

\begin{equation}h = h^{\prime\prime} /\left\{ {1 - z*R - ac*\left[ {R + K/N*\left( {1 - z*R} \right)} \right]} \right\}\end{equation}

 

Figure 1

 

Schematic for calculation of light rays through the pupil.

In this linear model, the flux through the pupil (h′′) will be proportional to h, and the flux through a round pupil will be proportional to h². 

Image Magnification

The image magnification (hence the density of photons on the retina) will be proportional to the product of the spectacle magnification and the magnification induced by the eye itself (ocular magnification). Spectacle magnification is calculated from simple formulas that include spectacle power and vertex distance (14 mm).⁹ The spectacle lens magnification (M) can be derived from the formulas U + R = V and M = U/V, where U is object vergence at the lens, V is image vergence at the lens, R is lens power, and M is magnification. By the sign convention, all vergences are negative in these examples. We have chosen to conduct our calculations using spectacle lenses. The use of contact lenses yields the same RLD results (not shown here), because the flux changed caused by the removal of the spectacle lens would be cancelled by the retinal image magnification change. 

Previous work has shown that the ocular magnification calculation does not require corneal power, ac depth, or lens parameters, but only the axial length and is accepted as being proportional to the eye's axial length (L) in millimeters minus 1.82 (a correction factor).^10,11 The ocular magnification in the above references notes that ocular magnification for a retinal object is inversely proportional to L − 1.82. This principle has been used accurately to help calculate the size of a retinal object from retinal photographs. In our case, the image is in the retinal plane; hence, magnification induced by the eye will now be directly proportional to L − 1.82 mm. 

The total magnification of such a system will be proportional to the product of the spectacle magnification and the ocular magnification. 

RLD

From the above information of flux through the pupil and its distribution on the retina, the light rays (photon equivalents/mm) can be calculated. We assume that all subjects will be exposed to the same intensity and duration of ambient light. Transposing these calculations to a two-dimensional model with a round pupil onto the retinal plane, RLD will be proportional to the linear flux²/linear magnification². Please note that we calculate not an absolute but rather the RLD of one eye in relation to another. 

Method 2

To verify the results of RLD calculations from our simple one-surface schematic eye, we used the ray-tracing feature inside of an optical design software (Opticsoft II), the same refractive variables as in our first method, in addition to corneal thickness, posterior corneal surface power, and crystalline lens power. The corneal thickness was fixed at 550 μm for all eyes, and the posterior corneal surface radius was set at 0.85× that of the anterior surface.¹² The software was used to derive the anterior and posterior lens powers while respecting their normal ratio (the anterior surface having 35% of the lens refractive power and the posterior surface 65%).¹³ We cross-checked our lens power results with the calculated lens power from the Reykjavik eye study (which used lens power calculating ray-tracing software PhacoOptics) and found that they correlated highly (r = 0.98). Standard refractive indices of ocular tissues were used for these calculations¹⁴ (air N = 1.00, spectacle lens N = 1.59, cornea N = 1.38, crystalline lens N = 1.39, and vitreous N = 1.34). 

To compare the two methods, we applied the analysis suggested by Altman and Bland,¹⁵ whereby the difference in the values obtained between the two methods are plotted against the average of the values obtained by the two methods. Any systematic differences between methods 1 and 2 are reflected in the mean difference score. The width of the limits of agreement (LOA = mean value + [1.96 × the SD of difference scores]) defines the range over which 95% of the difference scores are distributed. In the context of Bland-Altman plots, accuracy refers to the placement of the LOA in relation to zero. Ideally, the LOAs are centered on or near and contain zero, indicating equivalence of the two techniques. The width of the LOA is used as a measure of measurement precision, whereby narrower ranges indicate higher precision. Their interpretation is often based on clinical relevance and does not rely on P values. 

Results

A regression plot using the RLD calculated from the one surface schematic eye model demonstrate a linear relationship between RLD and axial length (L), with RLDα − 8.5 × L + 293, and the software model yielded a regression line with RLDα − 7.55 × L + 270 (Fig. 2). 

Figure 2

 

Axial length as a function of retinal light dose for both method 1 (ray-tracing with a simple one-surface schematic eye) and method 2 (ray-tracing feature inside of an optical design software). The overlap of the data indicates the comparability of the two methods.

The residual plots done for method 1 (the simple ray-tracing model) demonstrate that the regression line estimation of RLD is robust with the estimated RLD closely approximating the calculated RLD (Fig. 3). Based on this model, a 21-mm eye would receive 1.3× more photons per square millimeter at the retina compared with a 24-mm eye and 1.8× more than a 27-mm eye. For the software model, these multiples would be 1.3× and 1.7×, respectively. 

Figure 3

 

Bland-Altman plot of mean RLD as a function of the difference in RLD between the two presented calculation methods (method 1 – method 2). The solid lines indicate the limits of agreement (±2 SDs) from the mean difference, containing 98% of the data points.

The RLD estimates from the two methods were compared using a Bland-Altman plot,¹⁵ where values from method 2 were subtracted from values derived in method 1. Negative difference scores indicate that the estimated values of RLD were larger in the simplified method 1. With a mean value for the difference scores of −0.56 (SD = 2.42), their distribution is centered close to zero, indicating an approximate equivalence of the two calculation methods (Fig. 4). 

Figure 4

 

RLD as a function of its residuals in method 1. The regression line estimation of RLD is robust with the estimated RLD closely approximating the calculated RLD.

The ray-tracing model eye (Fig. 1) permits us to visualize and understand the optical processes of photon flux through the pupil and image magnification. As one increases the spectacle power (more positive lens R), it is evident that the height of a luminous bar h will increase, resulting in more photons entering the pupil h″ of fixed height. Hence, the photon flux through a pupil will be greater for hyperopes than for myopes. Corneal power and ac depth will modify this to a lesser degree as they have less “optical leverage.” If we simplify the flux calculation, using h′ as the pupil and use only the first two equations, ignoring the corneal and ac effect results in an RLD for a 21-mm axial length eye that will be 1.5× that of a 24-mm one and 2× a 27-mm one. This demonstrates the much stronger leverage of the spectacle (R) over the vertex distance of 14 mm compared with the cornea and ac. 

Insofar as magnification is concerned, the retinal image will be projected onto a smaller retinal arc for a short axial length eye compared with a long one, and hence, photon density will be greater in the former. This effect of increasing photon density will be somewhat decreased with hyperopic spectacle refraction magnifying the retinal image and decreasing the photon density, whereas myopic spectacles will have the opposite effect. In short, these concepts are confirmed with the results with short axial length eyes having more flux through the pupil and this flux being distributed over a smaller area resulting in an increased number of photons per square millimeter in this group of eyes. 

Discussion

The goal of the present study was to calculate the retinal light dose in a normal population randomly chosen from the Reykjavik Eye Study to test the hypothesis that long-axial length myopes receive a decreased light dose to their maculas compared with short-axial length hyperopes. Given that in vivo measurements of RLD in subjects are not possible, we used two methods to calculate RLD: a method using a one-surface schematic eye and the most important ocular refractive variables, and ray-tracing software and a more complete set of these variables. RLD is inversely proportional to axial length. Every millimeter decrease in axial length significantly increases RLD. The ray-tracing method indicated that the retina of a 21-mm axial length eye would always be receiving 1.8× more photons per square millimeter than a 27-mm axial length one. 

Interestingly, in our example, the RLD for a hyperopic 21-mm eye wearing a pair of inexpensive commercial sunglasses (which filter approximately 40% of visible light¹⁶) would be equivalent to the RLD received by a 27-mm myopic eye without sunglasses. Note that in the BDES, the protective effect of sunglasses and the wearing of hats was based on self-reporting their use more than 50% of the time, whereas in our examples, an eye of long axial length would be receiving the lower RLD 100% of the time owing to the constant presence of this anatomic feature. The BDES and POLA studies both failed to show a significant sun exposure–AMD association, although a systematic review and meta-analysis of the literature exploring sunlight exposure–AMD risk has done that. In fact, the POLA study found just the opposite with subjects, whereby participants who reported higher exposure to sunlight also demonstrate fewer retinal pigmentary abnormalities and early signs of AMD. Darzins et al.¹⁷ showed that sensitivity to glare and poor tanning ability were in fact markers for increased AMD risk, and hence, sun sensitivity could be regarded as a confounder in AMD–sun exposure studies. When the eye is exposed to bright light, pupils constrict, and a number of behavioral modifications occur including lid closing, gaze aversion, and the wearing of hats and sunglasses.¹⁸ Calculating cumulative retinal light exposure from ambient exposure is fraught for the above reasons as some of above-mentioned confounders would be virtually impossible to measure. This uncertainty in ascertaining cumulative sunlight exposure and consequent retinal light dose with the different epidemiologic studies may explain their different conclusions. However, the calculation of the decrease in eye light exposure with the reported wearing of hats and sunglasses would be much simpler: the filtering capacity of the glass and the time spent wearing them outdoors being the determinants. 

Nevertheless, our findings do not prove cause and effect for excess retinal light exposure and AMD association in this population (short axial length hyperopes), and they may represent a simple epiphenomenon. Alternate theories to explain increased AMD risk in short axial length hyperopes summarized by Pan et al.¹⁹ include increased sclera rigidity, higher intraocular VEGF concentration, and lower incidence of posterior vitreous detachment (PVD) in these eyes. Although the higher incidence of spectacle wearing in the myopic population was also advanced as a reason for myopes to be protected from AMD, presumably by decreasing light exposure, spectacles do not filter²⁰ the short wave-length visible light that is felt to cause the most retinal damage.²¹ Additionally, the epidemiology shows a consistent increase in AMD risk with increasing hypermetropia. We would expect an interruption in this trend if indeed glasses were playing a role in altering retinal light exposure in populations more likely to wear them (myopes and higher hyperopes).⁵ 

Apart from placing a measuring device in the retinal plane to record light dose, we cannot be sure how much light is reaching the retina. However, we demonstrate in our model that an eye's RLD will be inversely proportional to its axial length. This same inversely proportional relationship has been noted for the epidemiology of AMD, with shorter axial length increasing AMD risk. Animal models²² and the theoretical AMD disease mechanisms^23,24 all appear supportive of an AMD–light exposure link. Rather than invoke alternate hypotheses to explain the AMD–axial length relationship, the logical and parsimonious explanation for short axial length hyperopes being more at risk for AMD is excess light dose to the macula. Reflecting on the concept of the reciprocal nature of retinal phototoxicity,¹⁷ the shorter the axial length the more frequently the retina would be exposed to toxic levels of light. If indeed this is the mechanism for increased risk of AMD in hyperopes, then it can be seen that relatively small decreases in macular light exposure could result in important risk reduction for AMD. We believe not only does our study provide an explanation for the protective effect of myopia in AMD but also an additional rationale for eye protection from light toxicity. It seems only logical that the use of sunglasses and hats should be strongly advocated, thereby decreasing light impinging on the macula and minimizing eye exposure to more damaging shorter wavelength²¹ light, both indoors and outdoors. 

Acknowledgments

The authors thank Thomas Olsen for generously providing access to the Reykjavik Eye Study raw data and Pierre Lachapelle and Len Levin for helpful editorial review of the manuscript. MGQ received Canada and Quebec R&D tax credits. 

Disclosure: M.G. Quigley, None; I. Powell, None; W. Wittich, None