Elastic Properties of Human Lens Zonules as a Function of Age in Presbyopes

Abstract

Purpose.: To investigate the elastic properties of human lens zonules as a function of age in presbyopes.

Methods.: We studied 16 presbyopic human donor eyes (ages 47–97). Anterior eye sections with crystalline lens, zonules, ciliary body, and sclera were stretched radially. The stretching device consisted of a chamber filled with balanced salt solution and eight radial hooks to hold the anterior eye section. Radial stretching was created with a stepper motor connected to a digital outside micrometer for linear displacement and digital balance for force measurement. Three eye globes were used to test our methodology. For 13 eye globes, the spring constant, elastic modulus of the zonular system, and Young's modulus of the zonules were calculated.

Results.: We found linear dependence for force-elongation and force-strain relationships at all ages. In young presbyopic eyes (ages 47–60), the Young's modulus of the zonules was 340 mN/mm², whereas in older eyes (ages 83–97) it was significantly lower at 270 mN/mm². However, the correlation coefficient between Young's modulus and age (47–97 years) was not significant with P = 0.063.

Conclusions.: The zonular system in presbyopic eyes was linear elastic, and the Young's modulus of the zonules decreased 20% from presbyopic age to late presbyopic age. However, there was no significant correlation between Young's modulus and age in presbyopes.

Introduction

The human accommodative system consists of four structures: crystalline lens, zonules of Zinn, ciliary muscle, and choroid. To understand how this system functions, it is necessary to know the mechanical properties of each part. 

Various mechanical characteristics must be estimated. The primary characteristic is force-elongation, with its slope representing the spring constant (mN/mm). The force-strain characteristic, which is the force as a function of relative elongation, has a slope that describes an elastic modulus of the tissue as a whole (mN). We call this the elastic modulus of the zonular system. The force per cross-sectional area as a function of relative elongation is the stress-strain characteristic, whose slope is the elastic modulus of the material, the Young's modulus (mN/mm²). 

Two earlier studies estimated the Young's modulus of zonular fibers. Fisher estimated indirectly the force applied to the lens and zonules by comparing lens thickness changes during stretching of anterior eye sections to the thickness changes during lens spinning experiments. ^1,2 He determined Young's modulus for the zonules to be 350 mN/mm², which was constant for eyes from donors between ages 15 and 45. ³ Van Alphen and Graebel used a two-arm stretching device with one force sensor, and determined Young's modulus for the zonules to be 1000 mN/mm² for 20-year-old donor eyes and 1500 mN/mm² for 60-year-old donor eyes. ⁴  

The objective of our study was to measure the mechanical properties of the zonules using our own stretching device and to determine possible changes with age in presbyopes. Such changes with age would impact several strategies to restore accommodation in presbyopes. 

Methods

We studied 16 presbyopic human donor eyes (ages 47–97). The donor eyes were provided by the Banco de Ojos para Tratamientos de la Ceguera (Barcelona, Spain) and had a mean postmortem time of 2.5 days (range 1–6 days). The donor eyes were kept constantly at 8°C. Typical cause of death was multiorgan or cardiorespiratory failure. We excluded donors with cancer, HIV positivity, and hepatitis, as well as those with anterior eye segment abnormalities found during tissue preparation or with previous ocular surgery. Research using this material was in conformance with the Declaration of Helsinki, and complied with Spanish regulations on the use of human postmortem and donor organs. 

The stretching device (Fig. 1A) consisted of a chamber filled with balanced salt solution (at 25°C) and eight radial hooks to hold the anterior eye section (Fig. 1B). The hooks were connected to a force sensor below the chamber by means of 6-0 Prolene, blue monofilament suture (Figs. 1C, 1D). We used the force sensor from a commercial digital precision balance (Precisa BJ 210C; Precisa Gravimetrics AG, Dietikon, Switzerland) that allows force readings at 0.1 mN precision. It was calibrated with eight microextension springs (spring constant 10.0 mN/mm each). The sensor was mounted on top of a digital outside micrometer. The outside micrometer could move the force sensor up and down by means of a high-torque stepper motor. The up and down movement of the force sensor with the connected eight sutures produced the stretching and relaxing of the anterior eye section in the experimental chamber while measuring the applied force (Fig. 1C). One step movement by the motor produced a radial displacement of approximately 1.25 μm in the experimental chamber. Stepper motor, force sensor, and digital micrometer were controlled by Labview software (Labview 6; National Instruments, Austin, Texas). 

Figure 1. 

 

Top view of stretching device for anterior eye sections (A), experimental chamber with eight hooks (B), schematic of main components (C), and photograph of main components (D).

The human donor eye globes were prepared as follows: We used the inked border of a test tube to mark a concentric circle on the sclera around the cornea, approximately 2 mm from the limbus. We then marked eight radial lines on the center of the cornea with an inked keratectomy marker (Fig. 2A). Subsequently, we used 7-0 Prolene suture to attach eight glass beads to the sclera around the marked circumference aligned with the eight radial marks. The cornea then was trephined with a 10 mm trephine (Fig. 2B), and complete removal was performed with scissors. The iris was removed entirely with scissors, and an anterior eye section was separated from the eye globe by cutting the sclera 360° with scissors just below the beads (Fig. 2C). This anterior eye section was placed inside the experimental chamber, which was filled with balanced salt solution, and the glass beads were connected to the metal hooks of the stretching device (Fig. 2D). Once the anterior eye section was in position, the sclera and ciliary body were divided into eight separate segments by making eight radial incisions between the beads. 

Figure 2. 

 

Main surgical steps of the preparation of the anterior eye section, with the glass beads sutured to the sclera (A), trephination of the cornea (B), separation of the specimen from the rest of the eye globe using Westcott scissors (C), and anterior eye section fitted into the stretching device submerged in balanced salt solution (D).

Before performing the measurement, the resting state of the anterior eye section had to be determined by stretching and relaxing the hooks until no force variation was detected by the force sensor. Each measurement was obtained from three cycles consisting of 10 stretching pulses followed by 10 relaxing pulses. Each pulse included a move of 200 stepper motor steps. This gave a total overall stretch distance per cycle of 2500 μm. The time between each stretching pulse and the force measurement was 10 seconds; the time between the force measurement and the next stretching pulse was 5 seconds. Three stretching cycles were necessary because the first cycle was needed to “condition” the eye section and release some internal friction. The third cycle was useful to check the repeatability of the measurement. The data from the second cycle were used for assessment. 

We placed a webcam with Carl Zeiss Tesar optics (Carl Zeiss, Jena, Germany) and a native 2-megapixel sensor above the experimental chamber to monitor the coronal diameter, and radial changes of the crystalline lens and the ciliary body during stretching (Fig. 3). Radial extension of the anterior eye section stretches the crystalline lens, zonules, and ciliary body. Since the ciliary body and the sclera were divided into 8 separate segments, each of the segments could be represented as three springs in series (Fig. 3, upper part). For such a system in series, the force acting on all components is the same, making it possible to estimate the mechanical properties of each component independently. For the zonules, this results in the following equation:  

 

where F is the applied force, k_Z the spring constant of the zonular apparatus, R_L the radius of the crystalline lens, and R_CB the ciliary body radius; Δ stands for the difference in radius before and after stretching. The radii R_L and R_CB were measured in our coronal diameter images taken by the webcam and the mean of the eight measurements from each hook were used in our calculations. Since the total length of the zonules (R_Z) cannot be seen in our coronal photographs, we estimated it as the sum of the space between the lens equator and the ciliary body, the circumlental space (R_CB – R_L), and the distance from the insertion of the zonules into the lens equator, the zonular insertion distance (R_ZID). The zonular insertion distance R_ZID was calculated according to the literature.^5,6 

 

 

 

If the ciliary body is not cut radially, this would result in an additional circumferential tension (lower part, Fig. 3). In this case we would expect an increased force during the stretching of the anterior eye segment compared to the situation when the ciliary body is cut radially (upper part, Fig. 3). To test this hypothesis, we performed an experiment with 3 donor eyes (75, 81, and 81 years; postmortem time 2–3 days). The specimen was prepared as described above, but the ciliary body was kept intact and a first measurement set with three stretching cycles was done. Then, we cut the ciliary body radially inside the experimental chamber and kept all other settings unchanged, and performed a second measurement set. 

Figure 3. 

 

Measured distances, radius of the ciliary body (R_CB ) and radius of the crystalline lens (R_L ) next to a schematic of crystalline lens, zonules, and ciliary body as springs in series in our main stretching experiments (upper part). The effect of keeping the ciliary body intact is drawn in the lower part of the figure.

The spring constant and the elastic modulus of the zonular system were calculated from the force-elongation and force-strain data (Figs. 4, 5). To calculate the Young's modulus of the zonules, we needed the number and diameter of the zonular fibers. These data were obtained from Weeber and van der Heijde, ⁷ who based their calculations on the observations by Kaczurowski ⁸ : 100 anterior fibers with 50 μm diameter and 50 equatorial and 135 posterior fibers, both with 40 μm diameters. This resulted in a total cross-sectional area for all zonular fibers of 0.427 mm². 

Figure 4. 

 

Force as a function of radial zonular elongation. The slope represents the spring constant of the zonular system.

Figure 5. 

 

Force as a function of radial zonular strain. The slope represents the elastic modulus of the zonular system.

Student's t-test was applied to compare the following groups: (1) the two conditions of the ciliary body in three lenses of the pilot experiment (Table 1), (2) the five youngest eyes (ages 47–60) with the five oldest eyes (ages 83–97) from the main experiment (Table 2); we refer to these groups as “young presbyopic eyes” and “older eyes” in this text, and (3) we compared eyes with short postmortem times (1–2.5 days, n = 9) to eyes with longer postmortem times (3.5–6 days, n = 4) from the main experiment (Table 3). The Pearson correlation coefficient was calculated for the relationship between Young's modulus and age. The significance level for Student's t-test and Pearson correlation was set to 0.05. 

Table 1. 

 

Zonular Elastic Properties for Different Specimen Preparation (Tested in Three Extra Donor Eyes)

	Ciliary Body	Mean	t-Test; P Value
Spring constant	Intact	119	0.007
(mN/mm)	Cut	47.6
Elongation at 30 mN	Intact	257	0.009
(μm)	Cut	651
System modulus	Intact	249	0.010
(mN)	Cut	104
Force at 10% strain	Intact	24.9	0.010
(mN)	Cut	10.4
Young's modulus	Intact	582	0.010
(mN/mm2)	Cut	243

Table 2. 

 

Zonular Elastic Properties of the Five Youngest Eyes (Ages 47–60) and Five Oldest Eyes (Ages 83–97)

	Age	Mean	t-Test; P Value
Spring constant	Young	80.3	0.003
(mN/mm)	Old	53.2
Elongation at 30 mN	Young	379	0.003
(μm)	Old	574
System modulus	Young	146	0.013
(mN)	Old	114
Force at 10% strain	Young	14.6	0.013
(mN)	Old	11.4
Young's modulus	Young	342	0.013
(mN/mm2)	Old	268

Table 3. 

 

Zonular Elastic Properties for Different Postmortem Times

	Postmortem Time	Mean	t-Test; P Value
Spring constant	Short	70.7	0.805
(mN/mm)	Long	68.1
Elongation at 30 mN	Short	445	0.688
(μm)	Long	475
System modulus	Short	132	0.797
(mN)	Long	136
Force at 10% strain	Short	13.3	0.797
(mN)	Long	13.6
Young's modulus	Short	308	0.797
(mN/mm2)	Long	317

 Short postmortem times 1 to 2.5 days (n = 9) and long postmortem times 3.5 to 6 days (n = 4).

Results

Our test with respect to the radial cutting of the ciliary body revealed that the mean force at 10% strain of the zonules was 25 mN with the ciliary body kept intact, whereas with ciliary body cut radially the force was 10 mN (Table 1). Also all other parameters calculated were significantly different for both conditions. For the main experiment we always cut the ciliary body radially. 

Before each measurement, the resting state of the anterior eye section was set by stretching and relaxing the system until no force variation was detected by the force sensor. This resulted in a mean circumlental space of 1046 μm (SD 270 μm), which was constant in both age groups. 

We found a linear dependence for the force-elongation relationship of the zonular system at all ages (Fig. 4). Young presbyopic eyes had a higher spring constant (slope in Fig. 4) than older eyes (Table 2). Mean zonular elongation with 30 mN force was 380 μm in young presbyopic eyes and 575 μm in older eyes. 

Converting the data into relative elongation or strain changed the linear relationship only slightly, bringing the lines for different ages somewhat closer together (Fig. 5). The slope of this force-strain relationship gives the elastic modulus of the zonular system, which remained significantly different in both age groups (Table 2). The mean force at 10% strain was 14.6 mN in younger presbyopic eyes and 11.4 mN in older eyes. 

The Young's modulus for the zonules shows weak dependence on age (Fig. 6). In young presbyopic eyes (ages 47–60), the Young's modulus of the zonules was 340 mN/mm², whereas in older eyes (ages 83–97), it was significantly lower at 270 mN/mm² (Table 2). However, the Pearson correlation coefficient was not significant (r = −0.529 and P = 0.63). The Pearson correlation test was selected because the Shapiro-Wilk test for small samples confirmed that the data were distributed normally (P = 0.743). 

Figure 6. 

 

Young's modulus of the zonules calculated as a function of age.

There was no significant difference in the elastic properties of the eyes with respect to postmortem time (1–2.5 days vs. 3.5–6 days, Table 3). 

Discussion

Our objective was to measure the elastic properties of the human lens zonules within a physiologic range. In ex vivo experiments, Fisher reported a change of 220 μm in the human lens radius during accommodation in 20-year-olds, which declines linearly to 0 μm in 50-year-olds. ³ He found a change of 790 μm in the ciliary body radius during accommodation, which was independent of age. More recent studies by Stachs et al. ⁹ and Strenk et al. ¹⁰ using ultrasound revealed that the anterior contour point of the ciliary muscle moves approximately 360 μm during pharmacologically-induced accommodation in young subjects (34 years) and 180 μm in older subjects (71 years). The circumlental space in young eyes (mean 26 years) has been estimated to be approximately 2000 μm using magnetic resonance imaging. ¹⁰ A more recent study estimated it to be approximately 1000 μm in young eyes (mean 22 years) and approximately 650 μm in older eyes (mean 63 years). ¹¹ Therefore, we should have covered the physiologic range with our stretching experiments, with a mean circumlental space of 1046 μm and experimental zonular elongation between 0 and approximately 700 μm (Fig. 4). 

The difference between our measurement temperature of 25°C and normal body temperature of 36°C may have influenced our results. Because of the somewhat colder environment, our tissue probably was somewhat stiffer or rigid as it may be in the living human. With respect to postmortem time, our retrospective comparison of the elastic properties estimated showed no difference between eyes with short postmortem times (1–2.5 days) and long postmortem times (3.5–6 days, Table 3). 

Another concern was the integrity of the ciliary body. Keeping the ciliary body radially intact results in a more homogeneous stretch of the anterior eye section and none of the zonular fibers is lost by the cutting procedure. On the other hand, the intact ciliary body must increase its diameter during stretching, which induces a circumferential tension (Fig. 3). We have shown this with an extra experiment (Table 1). The main problem with the described circumferential tension is that it cannot be predicted with a trivial spring model. When we cut the ciliary body, we had the lens, zonules, and ciliary body/sclera as springs in series. The intact ciliary body would be an extra spring component circumferentially between our 8 scleral segments (Fig. 3). Therefore, we decided to cut the ciliary body radially accepting the risk to cut some single zonular fibers and having a less homogeneous stretching shape of our anterior eye sections. 

It is interesting that we found a clear linear characteristic for the force-elongation relationship. This linear elastic behavior means that elongation is proportional to the loading force, and the elongation returns to zero after unloading. Such behavior would be expected in a structure that has to mediate a force between the crystalline lens and ciliary body with the choroid. Because of this linear relationship it makes sense to give the spring constant for our experiments (Tables 1–3). The value of elongation that is induced by a force of 30 mN is given for a better intuitive understanding. 

We also calculated the elastic modulus of the zonular system, which is interesting for mechanical modeling of the accommodation system. It excludes the errors introduced into the calculation of Young's modulus. This error originates from the uncertain number, diameter, and length of the zonules. One part of the length of the zonules, the zonular insertion distance, could only be estimated with data from the literature as explained in the methods. The length of the zonules within the ciliary body remains unknown as the length of the zonule portion entering the capsule. Therefore, we consider our absolute values for Young's modulus of the zonules only a rough estimate. Still, it allows valid relative comparison of different age groups. Our results are very similar to those of Fisher, who found the Young's modulus of the zonules of 350 mN/mm² to be constant between ages 15 and 45. ³ Our results are lower than the value reported by van Alphen and Graebel, who found 1500 mN/mm² for 60-year-old eyes. ⁴ This difference might be explained by the considerable deformation in a two-arm stretching device, with quite uneven stretching of the anterior eye section in different quadrants. 

We found that the Young's modulus of the zonules decreases from presbyopic age to late presbyopic age by 20%. However, there was no significant correlation between Young's modulus and age in presbyopes. A follow-up study with younger human donor eyes (ages 20–40) is under preparation to establish the relation of the Young's modulus with age, including the period with accommodation. 

Footnotes

 Disclosure: R. Michael, None; M. Mikielewicz, None; C. Gordillo, None; G.A. Montenegro, None; L. Pinilla Cortés, None; R.I. Barraquer, None