A study for accommodating the human crystalline lens by finite element simulation

doi:10.1016/j.compmedimag.2006.09.008

Computerized Medical Imaging and Graphics

Volume 30, Issues 6–7, September–October 2006, Pages 371-376

https://doi.org/10.1016/j.compmedimag.2006.09.008 Get rights and content

Abstract

This paper constructs two finite element models of human crystalline lens and zonules based on published clinical data. Displacement and pressure were applied to study the mechanism of vision accommodation. The simulation results show that, in Model A, under the pull of the zonules, the thickness of the lens decreased linearly, and the lens diameter increased linearly. The optical power of the lens increased as the zonules displacement increased. Furthermore, the pressure had a remarkable influence on the shape of the lens and the optical power. The lens also became thinner and flatter as the pressure increased. The optical power increased when the pressure increased. In Model B, the lens became thicker and optical power increased as the equatorial zonules stretched. It is basically consistent with Schachar's hypothesis. The outcome of this paper proved that the analytical model presented in this paper can be used in the theoretical study of the accommodation mechanism of the lens.

Introduction

The deformation of the human crystalline lens has been considered as the physiological basis of vision accommodation, which is believed to be due to the contracting and relaxing of the ciliary muscles and the zonules. However, the mechanism of lens accommodation is still not thoroughly clear [1], [2]. The most direct and accurate method to study the lens accommodation is to measure the accommodating lens in vivo, but the lens is normally partially obscured by the iris and direct measurements of changes in ciliary body and lens during the accommodation process are difficult. Recently, theoretical analysis methods based on the mathematical model of the lens have been used to study the mechanism of lens accommodation. Sophisticated mechanical analysis becomes available by using the computer-aided design and finite element analysis. Schachar et al. [3] used a mathematical method to study the accommodating lens in order to prove Schachar's hypothesis of accommodation. Burd et al. [4] constructed a finite element model of the lens and the zonules to study the mechanism of accommodation. Shung [5] examined the deformation effect of the lens when a few periodical radial point pulls were applied at the lens equator using his finite element model. Theoretical analysis provides great possibilities that are not available in experimental studies, which makes it a useful supplement to experimental studies of the accommodation mechanism.

The purpose of this paper is to construct analytical models of the lens and the zonules to study the deformation of accommodating the lens. Two different models were developed to simulate the accommodation process according to different hypothesis of mechanism. In Section 2, these two finite element models of the human crystalline lens were constructed on the basis of experimental data. Section 3 uses the presented models to simulate the deformation of the lens and the zonules during accommodating and analyze the simulation results in details. Section 4 gives out the conclusions of modeling and simulation.

Section snippets

Geometry and material properties of the lens

In this study and all previous ones the lens is assumed to be axisymmetrical. Under this assumption, only the profile data are needed to construct a lens. The measurement and mathematical description of the lens profile are very important to model the lens, but this is beyond the scope of this paper. In this paper clinical data measured or calculated, respectively, by Fincham [6], Brown [7], Strenk et al. [8], Burd et al. [4] and Fisher and Pettet [9] are used to describe the lens geometric

Model A: Deformation of the lens under the pull of the zonules

To study the relationship between the deformation of the lens and the displacement of the ciliary body, the following parameters were calculated: lens thickness (t), lens radius (R), the shift of lens equator plane (Shift), curvature radii of the anterior and posterior surfaces (r_a and r_p), the optical power (OP) and the force applied by ciliary body to cause the deformation (CF). Calculations were conducted by applying a displacement (D) to the ciliary body point (point C in Fig. 2) that would

Conclusions

In this paper, two different finite element model, Model A and Model B of the human crystalline lens have been presented to simulate the accommodation process. By using Model A, we find that the lens's thickness and its optical power decreased as the ciliary body moved away from the lens. This conclusion can support the Helmholtz's hypothesis partially. Furthermore, the influence of the eye pressure on the lens by modeling and simulation has been studied for the first time. Our results show

Acknowledgement

The publication has been supported by the National Natural Science Foundation of China (No. 60371012).

Liu Zhuo received his MD from National University of Defense Technology (N.U.D.T.), P.R. China in 2002, and achieved PhD in Medical Electronic Engineering form N.U.D.T. in 2006. He now works at computerized image analyzing, data processing of human physical ability test.

References (15)

H.J. Burd et al.
Numerical modeling of the accommodating lens
Vision Res
(2002)
N. Brown
The change in shape and internal form of the lens of the eye on accommodation
Exp Eye Res
(1973)
S. Krag et al.
Mechanical properties of the human lens capsule
Prog Retinal Eye Res
(2003)
H.V. Helmholtz
R.A. Schachar
Is Helmholtz's theory of accommodation correct?
Ann Ophthalmol
(1999)
R.A. Schachar et al.
Mathematic proof of Schachar's hypothesis of accommodation
Ann Ophthalmol
(1993)
V.W. Shung
An analysis of a crystalline lens subjected to equatorial periodic pulls
(August 2002)

There are more references available in the full text version of this article.

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A set of 3D Finite Element models of the anterior eye together with a custom developed pre-stress modelling approach was proposed to simulate vision for distant objects (the unaccommodated state) to vision for near objects (accommodation). One of the five zonular groups was cut off in sequence creating five models with different zonular arrangements, the contribution of each zonular group was analysed by comparing results of each specific zonular-cut model with those from the all-zonules model in terms of lens shape and zonular tensions.
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These opposing effects need to be considered in AIOL designs, highlighting yet another complexity associated with these devices. To our knowledge, this is the first model of the post-surgical lens capsule with implanted AIOL, but other studies have modeled the native human lens (Burd et al., 2002; Cabeza-Gil et al., 2021; Hermans et al., 2008; Liu et al., 2006; Ljubimova et al., 2008; Wang et al., 2016). These studies have demonstrated important relationships between zonular traction and changes in lens shape/optical power.
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During capsulotomy, the force applied to the anterior capsule is a crucial parameter controlling capsule tears, that affects the clinical performance. This study aims to investigate the tear force in capsulotomy and analyze the effects of different tearing conditions on the tear force.
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The mechanical behaviour of zonular fibres greatly affects the accommodation process in mammalian eyes. This paper introduces a detailed measurement procedure for the purpose of obtaining the force–displacement diagram necessary to evaluate the mechanical properties of porcine zonular fibres in situ. It is a complex technique, keeping the integrity of the zonular bundles between the crystalline lens and the ciliary muscle cells. We present a brief description of the measurement procedure both in theory and in practice, along with the force–displacement diagrams acquired from a porcine sample group. The strengths of this newly developed method are the unequivocal force transmission between the sample and the transducer, and the intact connection between the ciliary body and the crystalline lens via zonular fibres. With the aid of these measurements, we define an estimated material model for the zonular apparatus both analytically and using the finite element method. The two different evaluation methods show close agreement in the calculated Young׳s modulus for the zonular fibres. The range of the calculated elastic modulus is 200–250 kPa. This new measuring method is adaptable to human specimens. Despite its complexity, the entire procedure and the evaluation part are reproducible. The constitutive model aims to shed light on the mechanics of the accommodation process.

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Prof. Boliang Wang has been working at medical electronic engineering for over twenty-years and possesses four patents in medic mechanism. He is taking the lead of virtual Chinese human eye and organ's group project in Xiamen University, P.R. China.

Xiuying Xu received her MD in computer technology from Xiamen University, P.R. China in 2006. She is good at computerized geometry modeling and graphics programming.

Cheng Wang received his PhD from National University of Defense Technology in 2003. He is interested in automated target recognition and image processing.

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A study for accommodating the human crystalline lens by finite element simulation

Abstract

Introduction

Section snippets

Geometry and material properties of the lens

Model A: Deformation of the lens under the pull of the zonules

Conclusions

Acknowledgement

Vision Res

Exp Eye Res

Prog Retinal Eye Res

Is Helmholtz's theory of accommodation correct?

Ann Ophthalmol

Mathematic proof of Schachar's hypothesis of accommodation

Ann Ophthalmol

An analysis of a crystalline lens subjected to equatorial periodic pulls