Physiology-like crystalline lens modelling for children

Qing Li; Fengzhou Fang; Fengzhou Fang

doi:10.1364/OE.402372

1. Introduction

Understanding the patterns in the change of human eye with respect to age is fundamental in developing effective methods to control myopia, which is typically developed and progressed during the age range of 6 to 15 years old [1]. As the most complex component of the human eye, crystalline lens undergoes a significant change in that period, which is different from the change in adulthood. Specifically, the shape, dimension and refractive index distribution of the crystalline lens change in a coordinated way accompanied by the enlargement of the vitreous chamber so that the eye maintains emmetropic.

Contributing to around one third of the total optical power of the eye, crystalline lens has a unique optical structure. It is composed of a series of shell-like layers of transparent crystalline fiber cells [2]. Based on the time of formation, the crystalline lens contains a core formed through prenatal period and the outer layers formed after birth. The epithelial cells located at the equatorial area of the lens differentiate into fiber cells. Those fiber cells elongate beneath the external surface of the lens, extend both anteriorly and posteriorly towards the lens axis, and stop growing when meeting each other at the lens axis [2]. In this way, the lens is constantly growing layer by layer throughout the whole lifespan, while the refractive index of each layer tends to increase as it is embedded inside the lens. Meanwhile, the fiber cells are also under modification during childhood. It has been found that the nucleus of the lens undergoes compaction during childhood [3], and thus a decrease in axial lens thickness during before age 11 has been found by many studies [4–6].

Various models were proposed in the past to represent different aspects of the crystalline lens with gradient refractive index distribution (GRIN). Based on the perspective of the mathematical description, those models can be classified into three categories.

(1) External-surface-following lens models
In this category, the external lens surface and the internal iso-indicial surfaces are represented by the same type of equation, and the refractive index is uniform across the external surface. The relationship between GRIN and the corresponding iso-indicial surfaces is generally described by the exponential function:
$(1)$$n = {n_0} + ({n_s} - {n_0}) \cdot {\gamma ^p}$$$ where ${n_0}$ and ${n_s}$ are the refractive indices at the lens center and surface, respectively; $\gamma $ is the normalized distance from lens center, generally defined along the lens axis. The equations of all the lens surfaces are thus defined using $\gamma $ as a magnifying variable, where each value of $\gamma $ corresponds to a particular iso-indicial surface. The external lens surface is a special case when $\gamma = 1$.
Typical lens models belonging to this category include those proposed by Navarro, et al. [7], Mehdi, et al. [8] and Conor, et al. [9]. The lens model proposed by Navarro, et al. [7] is composed of a series of concentric conicoid surfaces, in which the geometry of the iso-indicial surfaces is proportional to each other by the factor of ${\gamma ^2}$. That ‘invariant surface geometry’ is also presented in the GIGL model developed by Mehd, et al. [8], where an extra third-order term is induced to achieve smooth connection at the lens equatorial zone. The AVOCADO model presented by Conor, et al. [9] is an updated version of the GIGL model, which is parametrized to decouple the exponent factor p along the axial and equatorial index profiles.
(2) Polynomial-defined lens models
The refractive index distribution of the lens models in this category is described by a polynomial with respect to the axial (Z) and equatorial (Y) coordinates of the lens, in a general form [10]:
$(2)$$n(Y,Z) = {N_0}(Z) + {N_1}(Z){Y^2} + {N_2}(Z){Y^4} \ldots ,$$$ where $\begin{array}{l} {N_0}(Z) = {N_{0,0}} + {N_{0,1}}Z + {N_{0,2}}{Z^2} \ldots \\ {N_1}(Z) = {N_{1,0}} + {N_{1,1}}Z + {N_{1,2}}{Z^2} \ldots \\ {N_2}(Z) = {N_{2,0}} + {N_{2,1}}Z + {N_{2,2}}{Z^2} \ldots \end{array}$
Proposed first by Gullstrand [11], the polynomial of fourth-order is used to describe the GRIN features of the lens. In this lens model, the external lens surface is independent from the inner structure. Similarly, a second order polynomial was applied by Liou, et al. [12] to describe the lens internal gradient index structure. However, both models were presented as a single lens model without age-dependent adaptability. The coefficients of the polynomials are not correlated with the other parameters of the lens.
Based on the same polynomial equation, Goncharov, et al. [13] presented a fourth order polynomial in which the coefficients depend on the external surface parameters of the lens. It should be noticed that the model is constructed to make the external iso-indicial surface coincident with the external surface of the lens. Thus, the coefficients of the polynomial were derived as functions of the geometric parameters of the lens.
For models within this category, the capability of representing age-dependent lenticular properties is constrained by the complexities implicit in the polynomial description. Moreover, the coefficients of the function have no direct physical meanings. As a result, extra effort is needed to build the relationship between GRIN and the external surface geometry.
(3) Surface-geometry-centered models

There are also models [14,15] proposed to describe the geometry of the lenticular surfaces with higher accuracy. To fit the external surface geometry as measured on the lens, various mathematical equations were proposed and evaluated. However, the best equation for description should depend on the application. For the purpose of optical analysis, the improvement of accuracy produced by complicated equations may not be significant, while often extra effort has to be made to apply those equations in ray tracing or optical computation.

It should be noticed that most of the previous lens models assume a uniform refractive index across the external surface of the lens. However, as revealed by measurement data [16–18], there exists a significant difference in the refractive indices at the equatorial lens surface with those at the anterior and posterior surface vertices. This discrepancy not only affects the distribution of the refractive index at the lens surface, but also impact the inner gradient index distribution. As a result, even though the optical performance of the lens is mostly determined by its central region, the inner GRIN structure can still have significant optical contribution. Moreover, from the physiological point of view, there are multiple layers of fiber cells that are still elongating near the lens surface. If the fact that the inner or older fiber cells tend to have higher refractive index works for the whole fiber cells, there should be a differentiation of the refractive indices in those elongating fibers beneath the lens surface.

In the application of lens models to represent the age-dependent and/or accommodation-dependent properties of the human lens, much attention is paid on establishing the relationship between structural and optical properties of the lens. As data on some lenticular parameters are difficult to obtain, those parameters are often used as adjustable variables to produce the targeted optical performance. As a result, more independent parameters of the lens model provide more freedom to predict the optical properties as measured from the lens. Therefore, the development of the lens models is moving toward a more realistic description of the lens structure. Specifically, if the optical properties in peripheral fields are to be analyzed, the requirement in the degree of freedom for definable parameters or anatomical similarity is even higher. In that situation, the model constructed only for predicting axial optical properties can have significant limitations.

In this paper, we propose a physiology-like crystalline lens (PCL) model to describe the age-dependent properties of the crystalline lens of children. To our knowledge, it is the first lens model that has the ability to involve almost all the properties measurable on the in vivo human lens. The optical analysis of the lens model has revealed many insights on the relationship between those parameters with the optical performance of the lens. It is believed that the proposed PCL model will serve as a powerful tool for investigating the growth of human eye and help understand the mechanisms of myopic development and progression.

2. Theoretic approaches

2.1 Mathematical description of the PCL model

The physiology-like lens model is established with the aim of simulating the realistic distribution of the lenticular fiber cells, based on the following assumptions.

• The refractive index within each layer of fiber cells is homogenous;
• The refractive indices of the outer fiber cells are lower than those of the inner fiber cells, namely refractive index increases towards the center of the lens;
• The refractive indices of the elongating fiber cells can be differentiated by layers and the values are lower for those at more external locations.

Figure 1 shows the structure of the PCL model, which is constructed by cutting out a lens shell from the GRIN structure.

Fig. 1. Schematic of the axial section of the PCL lens model. The solid lines represent the external surface of the lens. The iso-indicial surfaces are represented by the dashed lines. $D = 2A$ is the equatorial diameter of the lens; ${T_a}$ and ${T_p}$ are the anterior and posterior axial thickness from surface vertex to lens center, respectively; ${z_{0a}}$ and ${z_{0p}}$ are the anterior and posterior axial position of the boundaries of central conic and peripheral third-order polynomial external surfaces; ${B_a}$ and ${B_p}$ are the anterior and posterior axial semi-diameters for the outermost virtual iso-indicial surface.

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The lens is assumed to be rotationally symmetric with respect to the axis Z which is pointing towards the retina, while the radial axis Y points towards the lens equator. The lens structure is divided into anterior and posterior parts by the equatorial plane, where the iso-indicial surfaces of both sections smoothly intersect each other. For the simplicity of the mathematical presentation, the origin of the coordinate system is set at the lens center, namely the center point of the equatorial section of the lens.

2.1.1 Dimension and shape of the external surface

The ideal mathematical form to describe the external lens surface has been found to be fairly complex. However, for the purpose of optical analysis, conic surfaces generally suffice to represent the central zone near the optical axis. As a result, data of axial radius of curvature (R) and asphericity (Q) for the external surfaces at the axial zone are widely measured and published in literature. To apply those data in the lens model while maintaining a smooth connection at the equatorial edge, the external lens surface of the PCL model is composed of two types of surfaces—the conic surface for the central zone and the polynomial surface of third order for the peripheral zone. It should be mentioned that the third-order polynomial surface is derived by smooth connection with conic surface and a specific equatorial diameter and thus may not be perfect in describing the external surface at the equatorial zone. However, since it is blocked by the iris, its surface shape has insignificant impacts on the optical performance of the lens. The boundary of the two types of surfaces are defined by angle ${\theta _a}$ and ${\theta _p}$, which are the angles subtended from the lenticular central point by the anterior and posterior axial conic surfaces, respectively.

Given axial thickness (T), ratio of anterior and total axial thicknesses (Ta/T), vertex radius of curvature (Ra, Rp), and asphericity (Qa, Qp), the lens external surface at the central zone can be defined by:

(3.a)$$z = \left\{ \begin{array}{l} \frac{{({x^2} + {y^2})/{R_a}}}{{1 + \sqrt {1 - (1 + {Q_a})({x^2} + {y^2})/R_a^2} }} - {T_a},( - {T_a} \le z \le 0)\\ \frac{{ - ({x^2} + {y^2})/{R_p}}}{{1 + \sqrt {1 - (1 + {Q_p})({x^2} + {y^2})/R_p^2} }} + {T_p},(0 \le z \le {T_p}) \end{array} \right.,$$

or equivalently,

(3.b)$${\omega ^2} = {x^2} + {y^2} = {[{f_{axial}}(z)]^2} = \left\{ \begin{array}{l} 2(z + {T_a}){R_a} - (1 + {Q_a}){(z + {T_a})^2},( - {T_a} \le z \le 0)\\ 2( - z + {T_p}){R_p} - (1 + {Q_p}){( - z + {T_p})^2},(0 \le z \le {T_p}) \end{array} \right.,$$

where ${T_a}$ and ${T_p}$ are the anterior and posterior axial distances from the anterior and posterior vertices to the equatorial plane of the lens, respectively.

Given equatorial lens diameter (D = 2A) and the boundary angles ${\theta _a}$ and ${\theta _p}$, the external surface at the peripheral zone is represented by Eq. (4):

(4)$${\omega ^2} = {x^2} + {y^2} = {[{f_{peripheral}}(z)]^2} = \left\{ \begin{array}{l} {C_{1a}}{z^3} + {C_{2a}}{z^2} + {A^2},( - {T_a} \le z \le 0)\\ - {C_{1p}}{z^3} + {C_{2p}}{z^2} + {A^2},(0 \le z \le {T_p}) \end{array} \right.,$$

where ${C_{1a}},\,{C_{2a}},\,{C_{1p}}$ and ${C_{2p}}$ are derived by the assumed boundary conditions, namely solving the equations:

(5)$$\left\{ \begin{array}{l} {f_{peripheral}}({z_0}) = {f_{axial}}({z_0})\\ {f_{peripheral}}(0) = {A^2}\\ {{f^{\prime}}_{peripheral}}({z_0}) = {{f^{\prime}}_{axial}}({z_0})\\ {{f^{\prime}}_{peripheral}}(0) = 0 \end{array} \right.,$$

where ${z_0}$ is the axial coordinate of the intersection of the axial conic and peripheral third-order polynomial (as shown in Fig. 1). Notice that for the posterior part of the lens, ${z_0} ={-} {z_{0p}}$.

In summary, the external surface of the lens is defined by nine independent parameters—Ra, Rp, Qa, Qp, T, Ta/T, D, ${\theta _a},\,\,{\theta _p}$, which are all measurable by the current techniques such as MRI [16].

2.1.2 Structure of the refractive index distribution

Unlike the case for the external surface where no limits are put on the values of asphericity, the inner iso-indicial surfaces are defined by ellipsoids. It is assumed that the anterior and posterior parts of the iso-indicial surfaces intersect each other smoothly at the equatorial plane. In this way, each part is represented by a specific half-ellipsoid:

(6)$$\frac{{{z^2}}}{{{b^2}}} + \frac{{{x^2} + {y^2}}}{{{a^2}}} = 1,$$

where b and a are the semi-diameters of the ellipsoid along the axial and equatorial axes, respectively. The axial radius of curvature and asphericity for those surfaces are derived by

(7)$$r = \frac{{{a^2}}}{b},\textrm{ }Q = \frac{{{a^2}}}{{{b^2}}} - 1.$$

As a result, the asphericity for all the iso-indicial surfaces is larger than -1.

The refractive index of the lens model is described as a function of the three-dimensional coordinate of the position inside the lens, namely $n(x,y,z)$. The refractive index of each iso-indicial layer inside the lens can be described along the Z axis from $z ={-} {B_a}$ to $z = {B_p}$ (Fig. 1) by the function

(8)$$n(0,0,z) = {n_0} + ({n_{s\_e}} - {n_0}) \times {\gamma ^p},$$

where ${n_0} = n(0,0,0)$, ${n_{s\_e}} = n({\pm} A,0,0) = n(0, \pm A,0)$ are the refractive index at the lens center and the edge of the lens equator, respectively. Here we assume the lowest refractive index of the lens locates at the equatorial edge, as found by MRI measurement [16–18]. The total normalized distance $\gamma $ for all the iso-indicial surfaces inside the lens is defined as

(9)$$\gamma = \frac{{{b_a}}}{{{B_a}}} = \frac{{{b_p}}}{{{B_p}}},\textrm{ }0 \le {b_a} \le {B_a},\textrm{ }0 \le {b_p} \le {B_p},$$

where ${B_a} \ge {T_a}$, ${B_p} \ge {T_p}$ are the axial semi-diameters for the outermost iso-indicial surface for the anterior and posterior parts, respectively (Fig. 1). In this way, the surfaces corresponding to the same refractive index are represented by the same $\gamma $. Specifically, the external iso-indicial surface approaches $\gamma = 1$, which represes the external layer at the equatorial edge of the lens.

The refractive index at the axial vertex point of the lens surface is thus

(10.a)$${n_{s\_a}} = n(0,0, - {T_a}) = {n_0} + ({n_{s\_e}} - {n_0}) \times {({t_a})^p},$$

(10.b)$${n_{s\_p}} = n(0,0,{T_p}) = {n_0} + ({n_{s\_e}} - {n_0}) \times {({t_p})^p},$$

for anterior and posterior parts, respectively. For the ease of parametrization, two parameters ${t_a}$ and ${t_p}$ are introduced to replace ${B_a}$ and ${B_p}$ so that

(11)$${t_a} = \frac{{{T_a}}}{{{B_a}}},\textrm{ }{t_p} = \frac{{{T_p}}}{{{B_p}}},\textrm{ }0 < {t_a},\textrm{ }{t_p} \le 1.$$

In this way, the axial half diameter of the iso-indicial surfaces can be derived as:

(12)$${b_a} = {T_a} \times \frac{\gamma }{{{t_a}}},\textrm{ }{b_p} = {T_p} \times \frac{\gamma }{{{t_p}}}.$$

The axial radii of curvature of the anterior and posterior iso-indicial surfaces are defined by

(13.a)$${r_a}(\gamma ) = {s_a} \times {R_a} \times {(\frac{{{b_a}}}{{{T_a}}})^q},\textrm{ }0 < {s_a} \le 1,$$

(13.b)$${r_p}(\gamma ) = {s_p} \times {R_p} \times {(\frac{{{b_p}}}{{{T_p}}})^q},\textrm{ }0 < {s_p} \le 1.)$$

Here ${s_a}$ and ${s_p}$ represent the ratio of the axial radius of curvature of the inner iso-indicial surface at the lens vertex to that of the external surface, namely:

(14)$${s_a} = \frac{{{r_a}({b_a} = {T_a})}}{{{R_a}}},\textrm{ }{\textrm{s}_p} = \frac{{{r_p}({b_p} = {T_p})}}{{{R_p}}}.$$

Here ${s_a}$ and ${s_p}$ are constrained within 1 to maintain a realistic lenticular structure.

The physical meaning of the axial curvature exponent q in Eq. (13) is displayed in Fig. 2 for the anterior part of the lens, while the case for the posterior part is equivalent. The value of q determines the trend of the gradient in the decline of ${r_a}$ towards the center of the lens. As shown in Fig. 2(B), the shapes of the inner surfaces deviate more from the external surface for larger q values. Moreover, ${r_a}$ and ${r_p}$ for the iso-indicial surfaces that intersect the lens axis at points inside the lens are not larger than Ra and Rp, respectively, as long as q ≥ 0.

Fig. 2. The impacts of q on the lenticular gradient index structure for the case where ${k_a}$ are set as 1. (A) Trendline of radius of curvature of the anterior iso-indicial surface with respect to the normalized axial distance $\gamma $. The range of $\gamma $ for the iso-indicial surfaces whose axial vertex points are inside the lens is from ${t_a}$ to 0. (B) The impacts of q on the gradient index structure. Larger q produces a more convex shape.

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To make sure the anterior and posterior iso-indicial surfaces for the same refractive index intersect each other smoothly at the equatorial plane, we assume that

(15)$${a_a}(\gamma ) = {a_p}(\gamma ).$$

For the external surface of the lens, we have

(16)$${a_a}(1) = {a_p}(1) = A.$$

Combining Eqs. (11)–(16) we have

(17)$${s_a} = \frac{{t_a^{q + 1}{A^2}}}{{{R_a}{T_a}}},\textrm{ }{\textrm{s}_p} = \frac{{t_p^{q + 1}{A^2}}}{{{R_p}{T_p}}}.$$

From Eqs. (8)–(11), we get

(18)$${t_a} = {(\frac{{{n_{s\_a}} - {n_0}}}{{{n_{s\_e}} - {n_0}}})^{\frac{1}{p}}},\textrm{ }{t_p} = {(\frac{{{n_{s\_p}} - {n_0}}}{{{n_{s\_e}} - {n_0}}})^{\frac{1}{p}}}.$$

As a result, the refractive index distribution of the lens is determined by the nine external lens surface parameters along with six additional parameters—${n_0},\,{n_{s\_a}},\,{n_{s\_p}},\,{n_{s\_e}},\,p,\,q$. It should be mentioned that under the realistic modelling condition of ${n_0}$ being larger than the refractive indices at lens surface, all the 15 determinative parameters of the PCL lens model are independent to each other, and can be measured in vivo with the current measurement methods such as MRI [16].

Combining all the equations above, the equation describing the GRIN of the PCL lens model is derived as follows:

(19)$$\left\{ \begin{array}{l} \frac{{{z^2}}}{{{{({T_a} \times \sqrt[p]{{\frac{{{n_{s\_e}} - {n_0}}}{{{n_{s\_a}} - {n_0}}}}} \times \gamma )}^2}}} + \frac{{{x^2} + {y^2}}}{{{A^2}{\gamma^{q + 1}}}} = 1,\textrm{ }z \le 0\\ \frac{{{z^2}}}{{{{({T_p} \times \sqrt[p]{{\frac{{{n_{s\_e}} - {n_0}}}{{{n_{s\_p}} - {n_0}}}}} \times \gamma )}^2}}} + \frac{{{x^2} + {y^2}}}{{{A^2}{\gamma^{q + 1}}}} = 1,\textrm{ }z \ge 0 \end{array} \right.,\textrm{ where }\gamma = \sqrt[p]{{\frac{{n - {n_0}}}{{{n_{s\_e}} - {n_0}}}}}.$$

where (x,y,z) represent the coordinate of any point inside the lens.

From this equation, each specific refractive index is directly related to the geometry of the corresponding iso-indicial surface. The geometry of any iso-indicial surface can be derived from Eq. (19) by given a specific refractive index. The intersection points of the peripheral iso-indicial surfaces with the external lenticular surface are derived by combining Eq. (19) with Eq. (3) or Eq. (4).

2.1.3 Axial and equatorial index profiles

To derive the refractive index profile along the axial and equatorial direction, the anterior axial, posterior axial and equatorial normalized radii are represented respectively as:

(20.a)$${\gamma _a} = \frac{{{b_a}}}{{{T_a}}},\textrm{ }0 \le {\gamma _a} \le 1,$$

(20.b)$${\gamma _p} = \frac{{{b_p}}}{{{T_p}}},\textrm{ }0 \le {\gamma _p} \le 1,$$

(20.c)$${\gamma _e} = \frac{{{a_a}}}{A},\textrm{ }0 \le {\gamma _e} \le 1..c)$$

Combining the previous Eqs. (7), (12), (13), (15), (17), the relationship between those realistic normalized radii and the global normalized radius $\gamma$ are:

(21)$${\gamma _a} = \frac{\gamma }{{{t_a}}},\textrm{ }{\gamma _p} = \frac{\gamma }{{{t_p}}},\textrm{ }{\gamma _e} = {\gamma ^{\frac{{q + 1}}{2}}}.$$

Therefore, the index profile along those three directions can be derived as

(22.a)$$n(0,0,z) = {n_0} + ({n_{s\_a}} - {n_0})\gamma _a^p,\textrm{ } - {T_a} \le z \le 0,$$

(22.b)$$n(0,0,z) = {n_0} + ({n_{s\_p}} - {n_0})\gamma _p^p,\textrm{ }0 \le z \le {T_p},.b)$$

(22.c)$$n(x,y,0) = {n_0} + ({n_{s\_e}} - {n_0})\gamma _e^{\frac{{2p}}{{q + 1}}},\textrm{ }z = 0..c)$$

As can be seen, the exponents of the index profiles along the axial and equatorial directions are p and 2p/(1+q), respectively. This means that the exponent parameter q of the axial curvature is directly linked to the equatorial index profile. Reversely, q can be determined given the exponent parameter (${p_{equatorial}}$) for the equatorial index profile. As a result, even though ${p_{equatorial}}$ is not among the 15 parameters to define the PCL model, it can replace the parameter q by $q = \frac{{2p}}{{{p_{equatorial}}}} - 1$.

2.2 Lens parameters from biometric data

The application of the PCL model for children is achieved based on a systematic analysis of the available experimental data. For the purpose of representing the growth pattern with age, the criteria for data selection is as follows:

(1) Data measured on in vivo lenses are prioritized over that obtained by in vitro studies.
(2) The age range is targeted for 6 to 15 years old, hence data within this range is prioritized. For those parameters that are not available on this age group, boundary values close to age 6 or 15 years are considered to extrapolate the values of the parameters.
(3) Data on the accommodated lenses are not considered since there are significant changes in lens shape, dimension and gradient index distribution with accommodation.

The final selected values and age-dependent functions of the total 15 lens parameters for the age-dependent PCL model are summarized in Table 1.

Table 1. Lens parameters for the age-dependent PCL lens model for children aged 6 to 15.

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2.2.1 Apex radius of curvature and axial lens thickness

The apex radii of curvature (Ra, Rp) and axial thickness (T) are determined from the regression lines published by Mutti, et al. [4]. This might be the only study that measured the lenticular apical radius of curvature on a large number of children (Sampling size: 994) covering a wide range of age from 6 to 15 yr, which also provided the regression trendlines of the measured parameters with age. To maintain a reasonable anatomic reality, parameters not measured in the study were determined based on the pertinent data from other studies, except for q which is optimized to match the age-dependent change in lens power of the PCL model to that provided by Mutti, et al. (the regression line of the calculated lens power) [4].

2.2.2 Surface asphericity

The in vivo studies reporting the asphericity of the anterior and posterior external surfaces (Qa, Qp) are scarce and there is a large variation both between studies and between individual subjects. Using Scheimpflug imaging, 102 subjects aged 16 to 65 yr were measured [17]. It was found that the conic constants can vary across a wide range (${p_a}:\textrm{ } - 23\sim 7;\,{p_p}:\textrm{ } - 14\sim 4$) and their correlation with age were not significant. Transferred from conic constant to asphericity (p = 1+Q), the mean and standard deviation (SD) for Qa and Qp are $- 5\, \pm \,4.7$ and $- 4\, \pm \,3.6$, respectively. On the other hand, Ishii, et al. found a much narrower values on 25 children aged from 1 month to 6 years old as $4.60 \times {10^{ - 4}} \pm 9.21 \times {10^{ - 5}}$ (Qa) and $2.49 \times {10^{ - 4}} \pm 1.63 \times {10^{ - 4}}$(Qp), respectively [19]. And the correlation with age was found insignificant as well. Here we chose the data measured by Ishii, et al. for two reasons: (1) their data were acquired through magnetic resonance imaging (MRI) which is free from optical distortion; and (2) the less negative asphericity is related to a more rounded shape which may be closer to the shape of the lens of children. However, the impacts of the measured variation of Q on the optical performance of the lens will be analyzed between age 6 and 15 years old.

As the asphericity measured by Ishii, et al. [19] was the calculated by fitting a ellipse for the anterior and posterior lens surface within the central 120 zone, the subtended angle parameters ${\theta _a}$ and ${\theta _p}$ are set as 120 as well.

2.2.3 Equatorial diameter

An increase with age in the equatorial diameter (D) of the lens has been found among in vivo studies [19–21]. However, none of them reported D over the age range of 6 to 15 yr. As a result, the measured values close to age 6 and 15 are used as approximate boundaries. In this study, it is assumed that D increases with age within an approximate interval of 8∼9 mm during the age range of 6∼15, which is consistent with the data from the in vivo measurement (Table 2). Therefore, we use the regression equation provided by Ishii, et al which predict an increment in D with age from 8.20 mm (6 yr) to 8.76 mm (15 yr) [19].

Table 2. Equatorial diameter (D) obtained by in vivo measurement.

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2.2.4 Parameters of the refractive index distribution

In the PCL model, the refractive index distribution is directly defined by six parameters— refractive index at lens center (${n_0}$), anterior surface vertex (${n_{s\_a}}$), posterior surface vertex (${n_{s\_p}}$) and equatorial edge (${n_{s\_e}}$), exponent of axial index profile (p) and exponent of axial change of curvature (q). All the parameters are defined based on available data, except q being adjusted to fit the trendline of the lens power with age of 6 to 15 years old [4].

Several interesting facts were found by previous studies regarding the lenticular refractive index:

• A decline of ${n_0}$ with age was observed by most in vitro studies [17,24–27], while in vivo studies [22,28] found the correlation with age was insignificant.
• The dependence of refractive index at the lens external surface on age was not found in most studies [26–29], except an in vitro study [25] which found an increase with age. Specifically, Pierscionek, et al. [18] found an increase with age for ${n_{s\_e}}$ and ${n_{s\_p}}$, and insignificant age-related change in ${n_{s\_a}}$ by piercing through 14 donor lenses with reflectometric fiber optic sensor.
• The refractive index at the equatorial edge was reported generally less than that at the axial surface vertices [16–18], while there are a few subjects showing an opposite pattern in one another study [27].
• The exponent parameter p along the lens axis was observed to increase with age by some studies [24,26,28], while other studies [17,25] found no significant correlation.
• The exponent parameter p along the equatorial direction was found to increase with age by most studies [17,18,24–29].

The refractive index distribution across the lenticular external surface were reported only by Pierscionek, et al. [30], who found no topographical variations of refractive index over both anterior and posterior surfaces. However, their measurement was conducted with capsulated lenses using a reflectometric optic fiber, which was found to be too sensitive that even thinning or abrasion on the surface can affect the results. Therefore, their findings are likely for the index distribution for the lens capsule, rather than that for the inner lens fiber cells.

It should be mentioned that all of the previous studies measuring the lenticular refractive index are conducted on a small sample size, and the population of children are not sufficient enough to derive a particular pattern of change during this age range. In this study, the average data published by a recent in vivo study [16] on the young adults are utilized as a basis for analysis. The impact on optical performance by other values of those parameters as found in other studies will be analyzed in the next section.

2.2.5 Evaluation of the optical contribution of the lenticular parameters

There are significant variations in the measured parameters between both individual subjects and different measurement methods. Therefore, the same optical performance is likely to be produced by lenses with different combinations of the values of its parameters. It is worthwhile to study how much the variations found in various parameters can affect the lenticular optical properties. Based on the data from literature, the range of variations to be evaluated for certain lens parameters are listed in Table 3. The optical impacts of each parameter are analyzed by assuming the rest of the lens parameters are fixed and as set in Table 1. Under that assumption, some extreme values found in literature may produce a model with unrealistic structure. Therefore, the range of some parameters for analysis in this paper is constrained to a narrower interval than the that found in literature. On the other hand, for the parameters lack of sufficient measurement data, the range for analysis is adjusted to a wider interval.

Table 3. Range of variation in the lens parameters for analysis.

View Table | View all tables in this article

2.3 Optical Analysis

Optical analysis for the lens model is performed based on three-dimensional ray tracing [33] through a custom-built program coded in MATLAB. The lens model is assigned with a finite number of iso-indicial lens layers. In this paper, the lens model is set as 200 iso-indicial layers which is approximately a continuous gradient index structure.

Corresponding to the ‘calculated lens power’ [4], the optical power of the lens model is defined by paraxial Gaussian optical power—obtained by finitely tracing a paraxial ray parallel to the lens axis as shown in Fig. 3. The principal point P’ and focus point F’ are found by the intersection points of the ray emerged from lens with the input parallel ray and optical axis. After obtaining principal and focal points of the lens, the paraxial lens power is derived by

(23)$${P_{lens}} = \frac{{{n_{vitreous}}}}{{|{P^{\prime}F^{\prime}} |}} ={-} \frac{{{n_{aqueous}}}}{{|{PF} |}},$$

where the refractive indices of aqueous and vitreous humor are set as 4/3.

Fig. 3. Paraxial optical properties of the lens derived by tracing a ray close to the optical axis, with the height set as 0.01 mm.

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Considering the fact that the wavefront aberration of the human eye is generally analyzed for rays coming out of the eye, the wavefront aberration of the lens model is analyzed in the same direction for rays emit from the posterior focal point of the lens. The wavefront aberration is calculated for the wavefront emerged at the anterior lens vertex, which is approximately the location of iris and also the exit pupil of the lens. Aberrations are reported as Zernike polynomials following the ISO standard 24157:2008. Three pupil diameters (3, 4 and 5 mm) have been analyzed. Specifically, the spherical aberration is represented by the Zernike coefficient $C_4^0$.

3. Results

3.1 Age-dependent PCL lens model

Based on the parameters set in Table 1, the axial sections of the model for age 6, 8, 10, 12 and 14 are shown in Fig. 4. The dot pairs on the lens surface signify the boundaries of the conic surface zone. As can be seen, by setting the angles subtended from lens center as 120 deg, the conic surfaces cover around 4 to 5 mm in diameter. The peripheral zone is smoothed naturally by the third-order polynomials. The two red curves in each lens represent the inner complete iso-indicial surfaces that are outermost with respect to the external surfaces at anterior and posterior vertex, respectively. The layers external to the red curves represent those elongating fiber cells that have not yet reached the lens sutures. As shown in Fig. 4, there are several layers of lens fiber cells that have reached the suture and stopped elongating at one side of the lens but are still elongating on the other side of the lens.

Fig. 4. Axial sections of the age-dependent PCL models for age 6, 8, 10, 12 and 14 yr, with the number of iso-indicial layers set as 200 (A) and 10 (B).

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The impact of the number of iso-indicial lens layers ($Len{s_N}$) on lens power is shown in Fig. 5. As can be seen, the lens power increases for larger number of lens layers, and the difference is most significant for lower number of layers around 10 to 100. As the layers larger than 170, the change in lens power with $Len{s_N}$ is almost insignificant. The impact of $Len{s_N}$ on spherical aberration (SA) is displayed in Fig. 6 for an exit pupil diameter of 5 mm. SA decreases rapidly as $Len{s_N}$ increases from 10 to 30 and then there is an insignificant but opposite trend for more lens layers. Moreover, the fluctuation of SA within the age range is the most significant for lower $Len{s_N}$. As the lens approaches a gradient index distribution, the change in SA with age is smoother.

Fig. 5. Change of lens power with respect to age for lens models with different number of iso-indicial lens layers ($Len{s_N}$). The red line represents the trendline for model with $Len{s_N} = 200$ which coincides with the regression line of ‘calculated lens power’ fit by Mutti, et al. [4].

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Fig. 6. Change of lens spherical aberration with respect to age for lenses with different number of iso-indicial layers ($\textrm{Len}{\textrm{s}_\textrm{N}}$); (B) is a localized enlargement of (A). Exit pupil diameter is set as 5 mm for aberration analysis.

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The characteristics of the gradient index distribution derived in this section is shown in Fig. 7. By setting the axial exponent as the constant of 3.312, the model predicts a decrease in q from age 6 to around 9 yr and an increase afterwards. As determined by Eq. (2)0, exponent parameter along the equatorial axis shows an opposite trendline waving within the interval of 2.5 to 2.75. That range is significantly lower than what has been found in the study by Adnan, et al. [16], which were measured as 4.3358 ± 0.2568 on young adults aged 18∼29 yr. The results in this study suggest that the exponent parameter p for the equatorial index profile may be less than p of the axial profile among children, which is consistent with the fact that the nucleus of the younger population tend to have a lower equatorial diameter as revealed by the axial sections obtained from in vitro studies [17,26,29]. As revealed by this model, that effect is closely related to the axial radii of curvature for the inner iso-indicial surfaces. As shown in Fig. 7(B), based on the settings in Table 1, the inner iso-indicial surfaces of the younger lens tend to have lower radius of curvature. Moreover, it is interesting to notice that the change of the axial radius with axial distance is nonlinear for the whole age range of 6 to 15, as shown in Fig. 7(B).

Fig. 7. Characteristics of the lenticular gradient index distribution with respect to age. (A) Age-dependent change of the exponents for axial index profile (p), axial radii of curvature (q) and equatorial index profile (2p/(q+1)). (B) Change in axial radius of curvature with respect to the axial distance for the iso-indicial surfaces of the PCL lens model with age 6, 8, 10, 12 and 14 yr, where dashed lines represent the peripheral iso-indicial surfaces that intersect with the external lens surface.

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The change in spherical aberration (SA) with age for different pupil sizes is shown in Fig. 8. The trendlines show an approximate U-shaped curve which peaks at around 9 years old. The magnitude in the change of SA is significantly larger and more negative for larger exit pupil diameter. Based on the assumption of parameters set in Table 1, our model predicts negative SA for pupil diameter of 5 mm, which reaches nearly zero at around 9 years old. It should be noted that the shape of SA approximately matches the opposite of the curve for parameter q (Fig. 7(A)) suggesting that q may have a major role on the SA of the lens.

Fig. 8. Change in spherical aberration $C_4^0$ with age.

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3.2 Optical impacts of the variations in lens parameters

The optical contribution of each lens parameter is analyzed individually based on the range in Table 3, while the rest of the parameters are set in Table 1. In this section, all the results on spherical aberrations are presented for the exit pupil diameter of 5 mm. To have a wider view of the patterns of the trendlines, the optical change with respect to each parameter is evaluated based on five lens models with age set as 6, 8, 10, 12 and 14, respectively.

The optical impact of the anterior (Ra) and posterior (Rp) radius of curvature is shown in Fig. 9. It has been found that Rp generally has a larger impact on lens power compared with Ra. An increase of 1 mm in Rp will result in a decrease of around 0.9 D/mm, while the decrease caused by the same change in Ra is only around 0.2 D/mm. A positive correlation with spherical aberration (SA) is found for Ra, with an increase of around 0.02 µm as Ra increases from 11 to 15 mm. However, there is no significant pattern in the trendlines of SA with respect to Rp, and the shape of the trendlines are different between the lenses with different ages.

Fig. 9. Change of lens power and spherical aberration with respect to Ra and Rp.

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Consistent with the optical theory, the asphericity of both anterior (Qa) and posterior lens surfaces (Qp) does not influence the lens power. On the other hand, a positive correlation with SA is found for both Qa and Qp. As shown in Fig. 10, the relative increase in SA with respect to Qp is significantly larger than that for Qa. Across the designated range of variation, Qp can cause an increment in SA of approximate 0.5 $\mathrm{\mu}\textrm{m}$ which is more than 5 times larger than the increase induced by Qa.

Fig. 10. Change of lenticular spherical aberration with respect to Qa (A) and Qp (B).

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The optical influence of the axial thickness (T), ratio of anterior axial thickness to total axial thickness (Ta/T) and equatorial diameter (D) is shown in Fig. 11. As can be seen, both optical power and SA increase with larger T. The variation of T between 3 to 4.1 mm among children can produce a maximum change of around 4 D in lens power and 0.1 µm in SA. On the contrary, the variation of Ta/T does not have a significant effect on lens power, which is found to change within 0.25 D for the range of 0.4 to 0.52. A decline in lens power with increased D has been found across lenses of different ages. As shown in Fig. 11(C), as D increases from 8 to 8.8 mm, lens power decreases nearly linearly by around 2.5 D.

Fig. 11. Change of optical power and spherical aberration with respect to T, Ta/T and D.

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As shown in Fig. 12, (a) positive correlation with lens power is found for the refractive index at the anterior lens surface vertex (${n_{s\_a}}$), posterior surface vertex (${n_{s\_p}}$) and lens center (${n_0}$), while an inverse trend is observed for the refractive index at the equatorial edge (${n_{s\_e}}$). The results show that ${n_{s\_e}}$ and ${n_0}$ have significantly larger impact on lens power than that of ${n_{s\_a}}$ and ${n_{s\_p}}$. Moreover, the trendlines of all the four index parameters are nearly parallel across lenses with different ages, indicating that the relative change in lens power is generally independent from the settings of the other lens parameters.

Fig. 12. Change of lens power with respect to ${n_{s\_a}}$(A), ${n_{s\_p}}$(B), ${n_{s\_e}}$(C) and ${n_0}$(D).

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Fig. 13. Change in lens spherical aberration with respect to ${n_{s\_a}}$(A), ${n_{s\_p}}$(B), ${n_{s\_e}}$(C) and ${n_0}$(D).

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On the other hand, the effects of ${n_{s\_a}},\,{n_{s\_p}}$ and ${n_0}$ on SA turn out to be almost insignificant (Fig. 13), while ${n_{s\_e}}$ has a strong impact on SA. For the lens aged 6 yr, SA increases by 0.1 µm as ${n_{s\_e}}$ increases from 1.34 to 1.36.

According to the intervals as set in Table 3, the exponent parameters of axial index profile (p), axial curvature profile (q), and equatorial index profile (2p/(q+1)) have the largest impact on the optical performance of the lens compared with other lenticular parameters. As shown in Fig. 14, lens power is significantly larger for lower values of p and 2p/(q+1), with a decline of around 15 D for p from 2 to 3, and a decline of 8 D for 2p/(q+1) increasing from 2.2 to 3.2. The trendlines are almost parallel across different lenses, showing that the contribution of other lenticular parameters is not significant enough to make a difference. Moreover, q has a significant influence on the lens power, and it turns more powerful for larger q values. An increment of more than 8 D in lens power can be induced by increasing q from 1 to 2. Meanwhile, the magnitude of the change in SA by changing those exponent parameters are also much larger than that induced by most other lenticular parameters, as displayed in Figs. 14(D), 14(E), and 14(F). It can be seen that the increase in either of the exponent parameters for the axial and equatorial index profiles can cause a positive change in SA. Moreover, the correlations with SA and lens power are opposite for all the three exponent parameters.

Fig. 14. The change of lens power and spherical aberration with respect to p, q, and 2p/(q+1).

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3.3 Optical impacts of differentiating refractive index distribution in the peripheral zone of elongating fiber layers

To evaluate the optical contribution of the zone composed of the elongating fiber layers, the PCL model is compared with a simplified version, i.e., by assuming that the refractive index of the outermost shell is homogenous and equal to the refractive index at the anterior vertex (${n_{s\_a}}$). The outermost shell in the simplified model is defined by the outermost complete iso-indicial surface of the PCL lens model, namely the inner curve of the two red curves shown in Fig. 4(A). The comparison of the axial section of the PCL model and its simplified version is illustrated in Fig. 15. Both models share the same setting of lenticular parameters. The external surface geometry and dimension of both models are set the same, and the GRIN structure of the inner layers of the simplified model is identical to that of the PCL model. As can be seen in Fig. 15(B), there is a significant deviation in geometry of the external surface with the outermost inner iso-indicial surface, which is mainly caused by the different asphericity values of both surfaces.

Fig. 15. Axial section of the PCL model (A) and the simplified PCL model (B).

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Using the parameters set in Table 1, the difference in lens power and spherical aberration between the PCL lens model and its simplified version is shown in Fig. 16. As can be seen, the lens power of the simplified model is generally around 4 D larger than that of the PCL model, while the difference in SA is around 0.25 µm, with the simplified model showing more negative values of SA.

Fig. 16. The change of lens power (A) and spherical aberration (B) with respect to age for the PCL model and the simplified PCL model.

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Specifically, Fig. 17 illustrates the comparison in total wavefront aberration and higher order aberration for the case of age 11 yr. It is apparent that the external zone in the PCL model has a significant role in lowering the wavefront aberration of the lens.

Fig. 17. The total and higher order wavefront aberrations for the PCL model(A) and the simplified PCL model(B) of age 11.

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4. Discussion

4.1 Age-dependent properties of the crystalline lens for children

Due to insufficient experimental data, the age-dependent lens model constructed in section 2.2 assumes that the refractive indices at the four key locations and exponent of the axial profile do not change with age. As revealed by this study, ${n_{s\_e}},\,{n_0}$, and p have a significant impact on the optical performance of the lens. The reported variation of those parameters would result in an exaggerating change in the lens power. Moreover, many in vitro and in vivo studies showed significant change in those parameters with age for the adult population, especially ${n_0}$ and the exponent parameter for equatorial index profile [17,18,24–29]. Therefore, it is highly possible that those parameters also go through a significant change during the age range of 6 to 15 years old. That assumption can be further verified when more data are available in future.

In this study, the curvature exponent parameter q is found to vary between 1.4 to 1.6 during the age range of 6 to 15 yr. It reveals that the apical radius of curvature of the iso-indicial surfaces (r) inside the lens follows an exponential function rather than linear function with respect to the axial distance (z). In the PCL model, r is defined by:

(24.a)$${r_a} = {R_a} \cdot {s_a} \cdot {( - \frac{z}{{{T_a}}})^q},\textrm{ } - {T_a} \le z \le 0,\textrm{ }x = y = 0,$$

(24.b)$${r_p} = {R_p} \cdot {s_p} \cdot {( - \frac{z}{{{T_p}}})^q},\textrm{ 0} \le z \le {T_p},\textrm{ }x = y = 0..b)$$

As can be seen, the dependence of r with respect to the axial distance is only linear when q equals to 1. In a theoretical study by Navarro, et al. [34], the gradient of curvature (G) was found to have a significant impact on lens power by assuming a linear correlation of r with z. The results show that this linear correlation is not sufficient to describe the power of the lens of children, which are generally higher compared with that of the adults. Furthermore, the exponent parameter along the equatorial index profile is predicted to be 2p/(1+q) by the PCL model. That explains the role of the equatorial index distribution on the lens power by its connection with axial curvatures, which was proved theoretically by Smith [10]. A similar connection was also present in the AVOCADO lens model [9] albeit only theoretically, since the equatorial index profile derived in the AVOCADO model is located in the plane intersecting lens center, which is different from the realistic equatorial plane that passes through the equatorial edge of the lens.

The number of the iso-indicial lens shells has been found to significantly affect the optical performance. By assuming the number of layers as 200, the index distribution within the lens model can be considered as gradient. The result has shown that the gradient distribution can rise lens power by 2 D compared with the same model of 10 layers. Moreover, the gradient lens is also found to have lower spherical aberration compared with that of the stepped lens model. The difference could reach more than 0.5 µm compared with the case of the 10-layer structure. As a result, the advantage of the gradient index distribution over the stepped shell structure is evident. However, some studies [35–37] showed that the crystalline lens could be composed of several zones of discontinuity. Using X-ray Talbot interferometry, Mehdi, et al. [36] proposed that the zones of discontinuity observed inside the lens might be caused by the light scattered from the subtle fluctuations in some iso-indicial surfaces. However, the localized fluctuations over some iso-indicial surfaces is too small and too random to be described as a stepped index profile. Therefore, the discontinuity of lens zones found by those studies may have more impact on light-scattering than optical power and aberration.

The spherical aberration (SA) is predicted to waving within the interval of around $- 0.1\sim 0\textrm{ }\mathrm{\mu}\textrm{m}$ for pupil diameter of 5 mm by the age-dependent PCL model. That range of SA is consistent with the SA inferred from in vivo measurements [38,39] which was derived as the subtraction of corneal SA from the total ocular SA. Specifically, in a study on a large sample of 675 subjects, Philip, et al. [39] found that the lenticular SA of adolescents aged 16.9 ± 0.7 yr could range from -0.15 to 0.01 µm for a pupil diameter of 5 mm. Furthermore, the analysis in this study has found that the pupil diameter has a significant impact on the sign of SA. For pupil diameter within 4 mm, SA is positive across the whole age range. As pupil size increases, SA becomes more negative. However, the change of SA with respect to age predicted by the age-dependent model is influenced by the assumptions on the change of lens parameters with age. It should be noted that the current data over the change of SA with age is still too scarce to determine any pattern regarding the lenses of children.

4.2 Relative contributions of the lens parameters to the optical performance

The parameters of the age-dependent PCL model represent an average trend among the population. The variations between individual subjects with the same age were reported by previous studies. Moreover, there are also significant variations in the results published by different studies or obtained from different measurement methods. Therefore, it is useful to understand how those variations in the structural parameters could affect the optical performance of the lens. In this paper, we use the age-dependent lens model as a starting point, and then change only the value of a certain lens parameter to observe the corresponding change in the optical properties across different ages.

From the previous analysis, the changes in lens power with respect to radius of curvature at the anterior surface vertex (Ra) and posterior surface vertex (Rp), axial thickness (T) and equatorial diameter (D) are all found to be approximately linear and similar across lenses of different ages. As shown in Fig. 18, (T) has the largest influence on lens power, followed by D, Rp and Ra, respectively. The result indicates that D is an important factor in the contribution of lens power, which was generally ignored by previous studies. In the PCL lens model, D defines the equatorial boundary of the iso-indicial surfaces. As defined in Eqs. (11) and (15), the decline in D would result in a faster decrease in the radius of curvature of the inner iso-indicial surfaces, which leads to an increase in lens power.

Fig. 18. Relative change in lens power induced by 1 mm increase of Ra, Rp, T and D.

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Figure 19 shows the relative change in lens power induced by increasing 0.01 of refractive index at the anterior surface vertex (${n_{s\_a}}$), posterior surface vertex (${n_{s\_p}}$), equatorial edge (${n_{s\_e}}$), and lens center (${n_0}$). The results show that ${n_0}$ and ${n_{s\_e}}$ have a significant contribution to the optical power while the impacts of ${n_{s\_a}}$ and ${n_{s\_p}}$ are relatively insignificant. As a result, the variations of ${n_0}$ and ${n_{s\_e}}$ measured by previous studies can cause a huge difference in the lens modelling, and this is also the case for the exponent parameters for the axial (p) and equatorial (2p/(1+q)) index profiles as well. These findings suggest that more measurement data of those parameters are required to build a realistic age-dependent lens model.

Fig. 19. Relative change in lens power induced by 0.01 increase of ${n_{s\_a}},\,{n_{s\_p}},\,{n_{s\_e}}$, ${n_{s\_p}}$, and ${n_0}$.

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As for the spherical aberration, Qp, p and q (equivalently 2p/(q+1)) are found to be the most powerful contributing factors according to the data variations, which can enlarge the magnitude of SA by 6-8 times. The rest of the lens parameters could also induce a change in SA, but in a much smaller level within 0.1 µm. The finding indicates that Qp is an appropriate choice to adjust SA when an accurate measurement of Qp is not possible.

4.3 Significance of the peripheral zone of the elongating fiber layers

In this study, the optical significance of the peripheral zone composed of elongating fiber cells is evaluated by substituting it into a single homogeneous layer with refractive index set as ${n_{s\_a}}$. In this way, the refractive index distribution and the surface geometry for the inner zone is exactly the same for both PCL model and its simplified version. The analysis in section 3.3 shows that the differences between the two cases are large for both lens power and SA. The differences in the optical properties are mainly caused by two aspects: (1) the gradient refractive index distribution over the external zone composed of elongating fiber cells, and (2) the deviation in the shape of the external lens surface and the inner iso-indicial surfaces. As shown by Eq. (24), the axial radius of curvature of the external surface and the inner iso-indicial surfaces change continuously along the optical axis in the PCL model. Therefore, the deviation in the external surface geometry is caused by the discontinuity in asphericity of external surface with inner iso-indicial surfaces. Specifically, by incorporating the elongating fiber zone, the PCL model reveals the fact that the geometry of the inner iso-indicial contours may not follow the external lens surface in a continuous way. Therefore, the differences in refractive index over equatorial edge and surface vertex as found in measurement studies should not be ignored in lens modelling.

There are still some assumptions in the current PCL model that can be improved in future studies. In this paper, the junctions of anterior and posterior surfaces are assumed to locate at the same equatorial plane. It has been found among adults that this junction tends to move backward with aging [17], so that the intersection points are instead situated at a curved surface. Furthermore, it has been found that the exponent parameter p of anterior and posterior axial index profiles is different [16], while they are the same in the current model construction. Therefore, there is still some space of improvement of the current model, which can be achieved by inducing more physical parameters and using more complex surface geometry. As more measurement data are available, more accurate lens models will be built in future.

Furthermore, the optical performance of the lens is analyzed only for the central field. The relationship between peripheral optical properties and the PCL structure will be further investigated. As the peripheral zone of the PCL model is definable by the measurement data, it is believed that the PCL model has significant potential and advantage in producing more realistic peripheral optical performance.

5. Conclusion

In the present study, we have proposed a lens model dedicated to representing the age-dependent properties of crystalline lens for children. The proposed lens model has unique advantages as follows:

(1) It is capable of modeling the different surface refractive indices at anterior vertex, posterior vertex and equatorial edge, and achieving the physiological structure of the elongating fiber cells.
(2) It involves most of the parameters (15 in total) that can be measured with the current techniques, and all the parameters have direct physical meanings so that age-dependent functions can be defined easily.
(3) The widely-used conic surface and its simplicity are retained while maintaining a relatively realistic external surface geometry with the use of an additional third order polynomial surface at the peripheral zone.

The model also retains some important features, such as i) dependence of the inner gradient index distribution on the external lens surface; ii) smooth connection of iso-indicial surfaces at the lens equatorial plane; and iii) independence of the exponent parameter p for the axial and equatorial index profiles. All of those advantages make the model very suitable to represent the age-dependent changes in the crystalline lens of children.

It should be noted that all of the parameters of the PCL model have direct physical meanings and can be measured in vivo. For the purpose of describing the age-dependent properties of the lens of children, an age-dependent model will be built easily without much effort of optimization if given more biometric data. Therefore, it is expected that more parameters should be measured in future studies to have a more realistic model. On the other hand, if the purpose is for predicting certain optical properties only, optimization process can be done on specific parameters, while other parameters can be assumed within the range of variation as found from literature. Having a large number of definable parameters help maintain a realistic lens structure.

Furthermore, optical analysis of the PCL model has revealed some insightful conclusions:

(1) The lens model with gradient refractive index distribution has significantly higher optical power and lower spherical aberration compared with the shell model having stepped distribution of refractive index.
(2) The constructed age-dependent PCL model shows realistic magnitude of the spherical aberration as measured, proving the capability of the model for modelling the lens of children.
(3) The exponent parameter of the axial radius of curvature for the iso-indicial surfaces follow a nonlinear correlation with respect to the axial distance; a linear correlation is not sufficient to model the high optical power of children.
(4) The equatorial diameter, refractive index at the equatorial edge, exponent parameters for the axial and equatorial index profiles and the peripheral zone of elongating fiber cells have significant optical contribution.

With more realistic structure and more complete definable parameters, this newly developed lens model can be used as a powerful tool for modelling the peripheral optical properties of the lens under accommodation, which will be achieved in future work.

Funding

Ministry of Education of the People's Republic of China (No. B07014); Science Foundation Ireland (15, B3208, RP); State Administration of Foreign Experts Affairs ("111" Project).

Acknowledgments

The authors would like to thank the financial support from the Science Foundation Ireland (SFI) (No. 15/RP/B3208) and the “111” Project by the State Administration of Foreign Experts Affairs and the Ministry of Education of China (No. B07014).

Disclosures

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Parameter	Value	Source
Ra	$11.45 + 0.151 \times (A g e - 10) - 0.021 \times (A g e - 10)^{2}$	Mutti, et al. (1998) [4]
Rp	$6.236 + 0.063 \times (A g e - 10) - 0.004 \times (A g e - 10)^{2}$	Mutti, et al. (1998) [4]
T	$\begin{array}{l} 3.428 - 0.0111 \times (A g e - 10) + 0.0055 \times (A g e - 10)^{2} \\ - {5.0}^{- 4} \times (A g e - 10)^{3} \end{array}$	Mutti, et al. (1998) [4]
Qa	$4.6 \times 10^{- 4}$	Ishii, et al. (2013) [19]
Qp	$2.49 \times 10^{- 4}$	Ishii, et al. (2013) [19]
D	$0.613 \times \ln (A g e) + 7.104$	Ishii, et al. (2013) [19]
Ta/T	0.45	Adjusted from Ishii, et al. (2013) [19]
$n_{0}$ ^a	1.4012	Adnan Khan, et al. (2018) [16]
$n_{s_a}$ ^a	1.3663	Adnan Khan, et al. (2018) [16]
$n_{s_p}$ ^a	1.3801	Adnan Khan, et al. (2018) [16]
$n_{s_e}$ ^a	1.3560	Adnan Khan, et al. (2018) [16]
p^a	3.3120	Adnan Khan, et al. (2018) [16]
q	$\begin{array}{l} - 2.188 \times 10^{- 5} \times A g e^{4} - 0.0002985 \times A g e^{3} \\ + 0 .02786 \times A g e^{2} - 0.3632 \times A g e \\ + 2.836 \end{array}$	Optimized to provide the calculated lens power given by Mutti, et al. (1998) [4]
$θ_{a}$	120 deg	Set to cover the optical zone.
$θ_{p}$	120 deg	Set to cover the optical zone.

Author (year)	Measurement method	Age range (Sample size: N)	Results
Xue, et al. (2019) [20]	Ultrasonography	21∼60 yr (N = 79)	D increases with age at the interval between 8.75mm and 9.75mm; D is within 8.75∼9 mm for the subjects aged near to 21.
Ishii, et al. (2013) [19]	MRI	1 month∼6 yr (N = 25)	$0.613 \times \ln (A g e) + 7.104$ ; D is around 8.2 mm at age 6.
Atchison, et al. (2008) [21]	MRI	18∼69 yr (N = 106)	For the young group (aged 19∼28 yr), the mean ± SD^a of D were $9.03 \pm 0.34$ mm and $9.51 \pm 0.26$ mm for the transverse axial section and vertical sagittal section, respectively.
Jones, et al. (2007) [22]	MRI	18∼59 yr (N = 44)	Mean ± SD^a for the whole sample were $9.33 \pm 0.33$ mm; D ranges from 8.6 to 9.9 mm for subjects aged 18.
Strenk, et al. (1999) [23]	MRI	22∼83 yr (N = 25)	D ranges from 8.5 mm to 10 mm for the whole sample; D ranges from 8.5 mm to 9.3 mm for subjects aged close to 22 yr.

Parameter	Range for analysis	Approximate^a interval found from literature (Total age range)	Source
Ra	11∼15 mm	6.4∼15.8 mm (6∼15 yr)	Mutti, et al. (1998) [4]
Rp	6∼8.5 mm	4.5∼8.3 mm (6∼15 yr)	Mutti, et al. (1998) [4]
T	3∼4.1 mm	2.9∼4.1 mm (6∼15 yr)	Mutti, et al. (1998) [4]
Qa	-22∼5	-22.5∼5.5 (16∼65 yr)	Dubbleman, et al. (2001) [31]
Qp	-15∼3	-14∼3 (16∼65 yr)	Dubbleman, et al. (2001) [31]
D	8∼8.8 mm (0.1)	6.9∼9.1 mm (4∼6 yr)	Ishii, et al. (2013) [19]
D	8∼8.8 mm (0.1)	8.6∼9.9 mm (18∼20 yr)	Jones, et al. (2007) [22]
Ta/T	0.4∼0.52	0.36∼0.5 (0∼56 yr)	^bEduardo, et al. (2020) [32];
Ta/T	0.4∼0.52	0.46∼0.48 (4∼6 yr)	Ishii, et al. (2013) [19]
$n_{0}$	1.4∼1.41	1.403∼1.422 (18∼20 yr)	Jones, et al. (2007) [22]
$n_{0}$	1.4∼1.41	1.4008∼1.4016 (18∼29 yr)	Adnan Khan, et al. (2018) [16]
$n_{s_a}$	1.36∼1.38	1.3637∼1.3689 (18∼29 yr)	Adnan Khan, et al. (2018) [16]
$n_{s_p}$	1.376∼1.382	1.3784∼1.3818 (18∼29 yr)	Adnan Khan, et al. (2018) [16]
$n_{s_e}$	1.34∼1.62	1.3541∼1.3579 (18∼29 yr)	Adnan Khan, et al. (2018) [16]
p	2∼4.2	2.9152∼3.7088 (18∼29 yr)	Adnan Khan, et al. (2018) [16]

Parameter	Value	Source
Ra	$11.45 + 0.151 \times (A g e - 10) - 0.021 \times (A g e - 10)^{2}$	Mutti, et al. (1998) [4]
Rp	$6.236 + 0.063 \times (A g e - 10) - 0.004 \times (A g e - 10)^{2}$	Mutti, et al. (1998) [4]
T	$\begin{array}{l} 3.428 - 0.0111 \times (A g e - 10) + 0.0055 \times (A g e - 10)^{2} \\ - {5.0}^{- 4} \times (A g e - 10)^{3} \end{array}$	Mutti, et al. (1998) [4]
Qa	$4.6 \times 10^{- 4}$	Ishii, et al. (2013) [19]
Qp	$2.49 \times 10^{- 4}$	Ishii, et al. (2013) [19]
D	$0.613 \times \ln (A g e) + 7.104$	Ishii, et al. (2013) [19]
Ta/T	0.45	Adjusted from Ishii, et al. (2013) [19]
$n_{0}$ ^a	1.4012	Adnan Khan, et al. (2018) [16]
$n_{s_a}$ ^a	1.3663	Adnan Khan, et al. (2018) [16]
$n_{s_p}$ ^a	1.3801	Adnan Khan, et al. (2018) [16]
$n_{s_e}$ ^a	1.3560	Adnan Khan, et al. (2018) [16]
p^a	3.3120	Adnan Khan, et al. (2018) [16]
q	$\begin{array}{l} - 2.188 \times 10^{- 5} \times A g e^{4} - 0.0002985 \times A g e^{3} \\ + 0 .02786 \times A g e^{2} - 0.3632 \times A g e \\ + 2.836 \end{array}$	Optimized to provide the calculated lens power given by Mutti, et al. (1998) [4]
$θ_{a}$	120 deg	Set to cover the optical zone.
$θ_{p}$	120 deg	Set to cover the optical zone.

Author (year)	Measurement method	Age range (Sample size: N)	Results
Xue, et al. (2019) [20]	Ultrasonography	21∼60 yr (N = 79)	D increases with age at the interval between 8.75mm and 9.75mm; D is within 8.75∼9 mm for the subjects aged near to 21.
Ishii, et al. (2013) [19]	MRI	1 month∼6 yr (N = 25)	$0.613 \times \ln (A g e) + 7.104$ ; D is around 8.2 mm at age 6.
Atchison, et al. (2008) [21]	MRI	18∼69 yr (N = 106)	For the young group (aged 19∼28 yr), the mean ± SD^a of D were $9.03 \pm 0.34$ mm and $9.51 \pm 0.26$ mm for the transverse axial section and vertical sagittal section, respectively.
Jones, et al. (2007) [22]	MRI	18∼59 yr (N = 44)	Mean ± SD^a for the whole sample were $9.33 \pm 0.33$ mm; D ranges from 8.6 to 9.9 mm for subjects aged 18.
Strenk, et al. (1999) [23]	MRI	22∼83 yr (N = 25)	D ranges from 8.5 mm to 10 mm for the whole sample; D ranges from 8.5 mm to 9.3 mm for subjects aged close to 22 yr.

Physiology-like crystalline lens modelling for children

Abstract

1. Introduction

2. Theoretic approaches

2.1 Mathematical description of the PCL model

2.1.1 Dimension and shape of the external surface

2.1.2 Structure of the refractive index distribution

2.1.3 Axial and equatorial index profiles

2.2 Lens parameters from biometric data

2.2.1 Apex radius of curvature and axial lens thickness

2.2.2 Surface asphericity

2.2.3 Equatorial diameter

2.2.4 Parameters of the refractive index distribution

2.2.5 Evaluation of the optical contribution of the lenticular parameters

2.3 Optical Analysis

3. Results

3.1 Age-dependent PCL lens model

3.2 Optical impacts of the variations in lens parameters

3.3 Optical impacts of differentiating refractive index distribution in the peripheral zone of elongating fiber layers

4. Discussion

4.1 Age-dependent properties of the crystalline lens for children

4.2 Relative contributions of the lens parameters to the optical performance

4.3 Significance of the peripheral zone of the elongating fiber layers

5. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (19)

Tables (3)

Equations (33)

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