1. Introduction

Ageing manifests in different parts of the eye at varying rates. These include the anterior and posterior asphericities of the cornea [1], the anterior chamber depth [2], and most pertinently, the crystalline lens [3, 4] which changes with age as it grows. Early eye models, such as Helmholtz [5], Gullstrand [6], Emsley [7] and Le Grand [8], used a limited number of spherical surfaces to model the real eye and their purpose is to analyze first-order aberrations. Subsequent models, such as the wide-angle systems of Lotmar [9], Drasdo and Fowler [10], Kooijman [11], Navarro et al. [12], Liou and Brennan [13], Norrby [14] and Goncharov and Dainty [15], introduced aspherical surfaces to provide improved fits to real data as well as for other applications that predict retinal image size [10] or analyze the light level on the retina [11]. Some of these models incorporate a GRIN lens [12, 13, 15], are dependent on accommodation level [12, 14] or on age [14].

The lens thickness, and anterior and posterior lens curvatures increase with age [3]. With no alterations in the refractive index such geometric changes with age would lead to myopia. This has been deemed the lens paradox [16]. A theory was proposed to explain that the refractive index may alter with age to counter any myopic change [17]. This was subsequently supported in a number of models [18–21].

One of the lens models that incorporates shells of varying refractive index was that of Pomerantzeff et al. [21]. In this model the surface of the lens and the nucleus have different curvatures and the surface shape of the lens differs from the biological lens with anterior and posterior surfaces meeting at a point at the equator. Subsequently, Sheil [22] suggested that changing the internal curvatures of GRIN shells can be more effective than altering the GRIN profile when counteracting curvature changes. The links between internal indicial contours and internal surface curvatures are not directly related according to the AVOCADO model [23]. The relationship between curvatures and refractive index contours of the lens requires further analysis.

The present work investigates the paraxial properties of the eye, applying a general approach to different lens models. This method is based on the ABCD system matrix [24, 25] as this allows an analysis of the effects of the internal structures on the entire system. It is shown here that changing the inner curvatures of the indicial contours of the lens can counteract shape changes with age and offset an increase in myopia.

2. Method

2.1 Eye model

An age-related eye model was constructed using parameters and external lens curvatures from the study by Atchison et al. [19]. These are given in Table. 1. The ratio between anterior lens thickness and total lens thickness is taken as 0.5399 [4].

Table 1. Data for the eye model.

View Table

The refractive index n of the lens was created using the GRIN power law equation [4]:

2.2 System matrix

To analyze the effects of the GRIN and the internal curvatures of the lens on the paraxial properties, the system matrix approach was used [24, 25]. The ABCD system matrix M of the eye model was constructed by multiplying refractive and transference matrices in reverse order to obtain

Matrix L represents the lens and its GRIN structure, and is constructed from a succession of refractive and transference matrices, multiplied in reverse from the posterior to the anterior of the lens. According to Eq. (1) the lens is divided into a series of N shells which makes it possible to identify an index n for every i-th, and $\left(i+1\right)$-th shell, a thickness t of the i-th shell, and therefore the position of the shell within the eye model, as well as the radius of curvature r of the i-th shell.

In Fig. 1 the dioptric power of the eye model starts to approach stability when the number of shells exceeds 100 with a difference of 0.0037 D for 100 shells. To eliminate any variation, $\text{N}=200$ was chosen for the eye model.

 

Fig. 1 Power of the eye model against the number of shells inside the lens (age 20 years).

Download Full Size | PDF

A range of different models for the internal curvature were used, with power laws representing the index of refraction. For the i-th shell the radius is:

 

Fig. 2 Magnitudes of the radii of curvature of the internal shells of the lens against normalized distance d. Four different power laws (1, 3, 5, and 7) are presented. Parameters are based on an eye of age 20 years.

Download Full Size | PDF

The internal nucleus of the lens is modelled to be a single entity i.e. homogenous refractive index and equiconvex, and therefore excludes power law 1. This adheres to previous studies that show a nucleus of uniform refractive index magnitude with no asymmetries between anterior and posterior contours [4, 26]. These empirical findings contrast with previous models such as the AVOCADO [23] or modified Navarro model [27] where the nucleus has different curvatures for its anterior and posterior surfaces. The power law is treated as unchanging with age.

To clarify the method used for the construction of matrix L, Fig. 3 shows the construction of the i-th shell. Each shell has the same index of refraction, however, whilst the lens nucleus is equiconvex, the radii for anterior and posterior profiles increase with progression towards the lens surface. This increase is greater for the anterior than for the posterior radii.

 

Fig. 3 The index of refraction $n_i$, position d inside the lens, and magnitude of the radii of curvature $r_i$ for anterior and posterior part of the i-th shell for an eye of age 20 years and using power law 5.

Download Full Size | PDF

Thus far, we have parameters for the eye as a function of age based on empirical data (Table 1) and relationships for the internal structure of the lens. Equation (1) gives us the refractive index and Eq. (6) gives us the radii of curvature for each shell. In the next section, we explore the effect of four power laws and five refractive indices for the lens nucleus $n_{\text{max}}$ to obtain the combination that best compensates for the lens paradox with 200 shells.

Using MATLAB (MathWorks, Inc.) the GRIN profiles for the model eye for each of the four powers and five indices of refraction are derived and for each combination, we obtain the system matrix M and seek the $r_{\text{N}}$ in each case that results in emmetropia, i.e. when $A=0$ [24, 25], for the age range 20 to 70 years. The results are compared to the empirical GRIN profiles of the lenses [4, 26] and an optimal combination of $r_{\text{N}}$, $n_{\text{max}}$ and power law suggested.

3. Results

Using a homogenous index of refraction for the lens and the age-related changes in the parameters given in Table 1, the change in power of the eye from age 20 to 70 years is around 1.63 D with a shift towards myopia. This is illustrated by the red line in Fig. 5. However, it is known from clinical experience that the eye does not become myopic with age, giving rise to the lens paradox.

According to the eye model and matrix optics described in the previous section and parameters given in Table 1, Fig. 4 shows the models for a 20 year old, 45 year old, and 70 year old lens, using power law 5 and $n_{\text{max}}\left(Age\right)$ given by Eq. (2) for the index of the nucleus. Since we are not interested in wide-angle analysis, asphericity coefficients are not taken into account and all the surfaces in Fig. 4 are spherical. For the model presented as an example in Fig. 4 the system matrices are:

 

Fig. 4 Lens model of (a) 20 year old, (b) 45 year old, and (c) 70 year old, with lens nucleus of index of refraction $n_{\text{max}}\left(Age\right)$ (Eq. (2) and power law 5. All measurements are in mm.

Download Full Size | PDF

 

Fig. 5 Power of the eye model as a function of age according to parameters given in Table 1 and $n_{\text{max}}=1.43$. The red dashed line gives the power of the eye model with power law 1 and the black solid line to power law 5 and using $n_{\text{max}}\left(Age\right)$.

Download Full Size | PDF

The differences between models shown in Fig. 5 arise because of the contribution of different power laws and of differences in maximum refractive index. The red dashed line in Fig. 5, using power law 1 and a maximum refractive index $n_{\text{max}}=1.43$, does not satisfy the required power for maintaining emmetropia with age. However, using power law 5 and $n_{\text{max}}\left(Age\right)$ (solid black line in Fig. 5) we are able to compensate for the lens paradox and the eye is emmetropic for ages 20 to 70 years. The system matrices M for the eye for ages 20, 45 and 70 years, given by Eqs. (7) to (9), all show that $A=0$ i.e. they are all emmetropic.

In order to compensate for the changing power of the eye with age, as seen by the black line in Fig. 5, appropriate internal curvatures of the nucleus $r_{\text{N}}$ were sought. With the application of different power laws, the nuclear radius of curvature was found to increase with age (Fig. 6). In young lenses in particular, the curvature of the nucleus is steeper compared to both anterior and posterior surfaces of the lens, indicating that the lens resembles an avocado [23]. With age this alters and in older lenses the nucleus is flatter than the posterior surface.

 

Fig. 6 Radius of curvature of the nucleus $r_{\text{N}}$as a function of age for different power laws. Radii of curvature of the anterior (rla - red dotted) and magnitude of the posterior (rlp - black dotted) lens surfaces are also given. The linear fits for $r_{\text{N}}$ are given. The $R^2$ values are 0.9949, 0.9962 and 0.9968 for power laws 3, 5 and 7, respectively. $p<0.001$ for all $r_{\text{N}}$ fits.

Download Full Size | PDF

Figure 6 shows the fits for the different power laws and indicates that as the power law increases, the slope of the line becomes flatter.

In order to analyze the effects of different nuclear indices of refraction, power law 5 was used to describe the internal GRIN structure of the lens. The relationships are given in Fig. 7.

 

Fig. 7 Radius of curvature of the nucleus as a function of age for five different nuclear maximum index values using power law 5. The linear fits for $r_{\text{N}}$ are given. Radii of curvature of the anterior (rla - red dotted) and magnitude of the posterior (rlp - black dotted) lens surfaces are also given. The $R^2$ values are 0.9951, 0,9957, 0.9962, 0.9965 and 0.9964 for $n_{\text{max}}$ values of 1,41, 1.42, 1.43, $n_{\text{max}}\left(Age\right)$ and 1.44, respectively. $p<0.001$ for all $r_{\text{N}}$ fits

Download Full Size | PDF

From Fig. 7 we see that the internal nuclear radius of curvature increases with age and with $n_{\text{max}}$. When comparing Figs. 6 and 7, it is apparent that the slope has a stronger dependence on the power law than on $n_{\text{max}}$.

Figure 7 shows the $r_{\text{N}}$ as a function of age ranging from 20 to 70 years for a selection of five nuclear refractive indices using power law 5 and compares these to the anterior and posterior lens surface radii (Table 1). With the lowest values of $n_{\text{max}}$ equal to 1.41 and 1.42, at every age $r_{\text{N}}$ is steeper than both the anterior and posterior surfaces. When the value of $n_{\text{max}}$ is equal to 1.43, the equator exhibits a steeper curvature compared to both anterior and posterior surfaces for younger people until the age of 45. After that, the curvature of the nucleus becomes flatter than the posterior surface. With $n_{\text{max}}\left(Age\right)$ according to Eq. (2) the equator is flatter than the posterior surface from age 27 but steeper than the anterior surface. With the highest refractive index value of 1.44, the nucleus is flatter than the posterior surface and becomes flatter than the anterior surface at the age of 53.

4. Discussion and conclusions

In order to investigate how the GRIN may offset changes in curvature with age and provide further insight into solving the lens paradox, an eye model based on experimental biometric data [19] was created. The ABCD system matrix [24, 25] formalism was used to obtain the $r_{\text{N}}$ that results in emmetropia and, hence, addresses the lens paradox.

According to data for refractive index from in-vitro lenses [4, 26] and modelling the nucleus as a single entity rather than constructed from two halves, the curvatures of the nucleus were treated as equiconvex. This distinguishes the presented model from other models such as the AVOCADO [23] or the modified Navarro model [27].

This work shows the results of modelling the internal curvatures of the lens shells using four different power laws (i.e. 1, 3, 5, and 7) and, for every shell, a specific radius of curvature and index of refraction could be calculated for the normalized distance in the lens, constructing the system matrix M for the model eye. From Fig. 2 and Fig. 5 we saw that power law 1 follows a straight line profile for the change in radius of curvature and does not compensate for the lens paradox. For the GRIN profile chosen and for a fixed value of $n_{\text{max}}$ with age, only a power law higher than 3 can counteract the increases in curvature with age in order to prevent myopia.

Changing the power law affects the nuclear central curvature with age. From Fig. 6 we see that the higher the power law, the flatter is the slope of the change of the nucleus curvature with age. Hence, with high power laws, the changes in nuclear curvature with age are minimal. Conversely, with low power laws the so-called avocado shape of the nucleus in younger lenses becomes evident.

The power law and the internal index of refraction of the nucleus $n_{\text{max}}$ affect the shape of the lens. Choosing $n_{\text{max}}$ equal to 1.41, 1.42, 1.43, 1.44 or $n_{\text{max}}\left(Age\right)$, an internal index that varies with age according to Pierscionek et al [4], we see from Fig. 7 that the nuclear internal curvature $r_{\text{N}}$ becomes flatter with age. In particular, for low values of $n_{\text{max}}$ that is 1.41 and 1.42, the nuclear curvature is always steeper than both external surfaces of the lens. With the mid-range values, that is 1.43 and $n_{\text{max}}\left(Age\right)$, the internal surface of the nucleus becomes flatter than the posterior surface of the lens and closely resembles empirical findings [4] for power laws 5 and 7. For 1.44 the internal surface of the lens nucleus is flatter than the posterior lens surface and after the age of 53, is also flatter than the anterior surface of the lens. The $n_{\text{max}}\left(Age\right)$ combined with power law 5 provides the closest resemblance to the empirical findings of Pierscionek et al [4].

With age the anterior chamber depth decreases, the lens thickness and axial length increase, and the anterior radius of curvature of the lens becomes steeper [19]. According to the results of this work, the nucleus becomes flatter with age. The present study shows how the combination of a suitable nuclear refractive index with changes in internal curvatures of the lens may explain the lens paradox, and add to developing a better understanding of how the lens may alter with age. Alteration of the GRIN profile has been shown to offset a potential myopic shift resulting from curvature increase [28]. It should be noted that the lens is a biological structure and local alterations in refractive index, necessitating alterations in protein/water relationships [29], are likely to affect cell shape and hence curvature. Changes in refractive index with age can result from alterations in state of water that manifest in more protein bound water becoming free water [29,30] and can also occur in response to metabolic or glycation-like changes such as with hyperglycaemia as found in diabetic patients who have been shown to experience a hyperopic shift in response to elevated blood glucose [31]. Hyperopic shifts caused by alteration in refractive index were also observed as transient occurrences after initiating therapy for hyperglycaemia [32, 33] and in longer duration Type 2 diabetes [34]. It is further pertinent that lens surface shape varies in individuals and this can mask age-related changes. These individual variations and the biology that may underlie them requires further investigation in order to broaden the knowledge of optics of the eye and to advance the potential for personalized eye care.