Experimental Protocols for Ex Vivo Lens Stretching Tests to Investigate the Biomechanics of the Human Accommodation Apparatus

Abstract

Purpose.: To explore alternative experimental protocols to investigate the biomechanical behavior of the crystalline lens and zonules using ex vivo stretching.

Methods.: Radial stretching tests were conducted on the anterior segment (consisting of lens, zonules, ciliary body, and sclera) of four pairs of presbyopic human donor eyes. A simple mechanical model is used to describe the behavior of the anterior segment when tested in this way. Each pair of samples was initially stretched with the ciliary body intact. One sample was retested after cutting the ciliary body radially, and the other sample was retested after removing the lens.

Results.: The external forces needed to stretch the sample with the ciliary body intact were significantly greater than for the tests in which the ciliary body had been cut. The forces measured with the ciliary body intact and lens in situ were comparable to the sum of the forces measured in the tests in which the ciliary body had been cut (lens in situ) and the forces measured in the tests on the intact ciliary body with the lens removed.

Conclusions.: When stretching tests are conducted on the anterior segment, significant circumferential tensions develop in the ciliary body. This means that the forces applied to the lens and zonules cannot be related directly to the forces applied by the external loading system. If radial cuts are introduced in the ciliary body prior to testing, however, then this difficulty does not arise.

Ex vivo studies on the mechanical performance of the human lens during the accommodation process are typically conducted using experiments in which radial forces or displacements are imposed onto anterior eye sections of human donor globes. Experimental studies of this sort are typically undertaken to investigate the function of the normal accommodation apparatus and to observe the way in which the performance of the accommodation mechanism changes with age. Alternatively, experiments of this sort provide possibilities for the assessment of potential treatments for presbyopia, for instance, refilling the lens capsule with a soft gel^1,2 or using femtosecond laser treatments to increase the compliance of the crystalline lens.³ 

Fisher⁴ was probably the first to design a device for mounting anterior eye segments and imposing on them a prescribed radial stretch. Fisher's design imposed radial stretching via eight arms. Similar devices have been developed by Pierscionek,⁵ Glasser and Campbell,⁶ Koopmans et al.,⁷ and Schachar.⁸ These devices were designed to make measurements on the optical and kinematic characteristics of the anterior segment samples when subjected to an externally applied radial stretch; they did not include the measurement of the forces that were applied to the sample during the stretching process. Fisher,^4,9 however, estimated the radial force applied to the lens indirectly, by comparing observed lens thickness changes during stretching with the thickness changes observed in a spinning lens experiment. 

Enhanced forms of lens stretching apparatus have been developed to include procedures to measure the external force that is applied to the sample during the test. A rudimentary two-arm stretching device, incorporating a single force sensor, is described by van Alphen and Graebel.¹⁰ A more advanced system, consisting of an eight-arm stretcher (named EVAS) incorporating a single external force measurement, is described by Manns et al.¹¹ The EVAS system was used by Augusteyn et al.¹² to conduct an extensive series of tests on human donated eyes. Ehrmann et al.¹³ describe an enhanced form of the EVAS system, named EVAS II. Reilly et al.¹⁴ designed a similar stretcher incorporating eight independent arms. 

Lens stretching tests are typically conducted with radial cuts made to the sclera between the loading points.^11–13,15 However, when test protocols are used in which the sample being tested includes the intact ciliary body, the externally applied radial force is distributed between the lens–zonule system and the ciliary body in an indeterminate way. This is a consequence of the fact that the forces within the sample are statically indeterminate; that is, the distribution of the applied force between a circumferential force in the ciliary body and a radial force in the lens–zonule system cannot be determined using the equations of static equilibrium alone.¹⁶ 

As a consequence of this lack of statical determinacy, the forces actually applied to the lens during stretching cannot be related directly to the measured external force applied to the sample (for an intact ciliary body). Manns et al.¹¹ and Ehrmann et al.¹³ acknowledge this aspect of the standard form of the test. Although in the studies described by Manns et al.¹¹ and Ehrmann et al.¹³ the majority of the samples were tested with the ciliary body intact, a few subsidiary tests were conducted in which the lens was removed from the sample and the stretching experiment repeated. Manns et al.¹¹ concluded, on the basis of tests on cynomolgus eyes, that “The force needed to stretch the ciliary body alone was found to range between 22% and 29% of the force required to stretch the ciliary body with the lens attached” (p. 3267). Ehrmann et al.¹³ conducted a similar set of subsidiary experiments on human and cynomolgus eyes and concluded, “The results indicate that around 30 per cent of the stretching force can be attributed to the circumferential ciliary body extension” (p. 310). It is presumed (although the authors do not state this explicitly) that these force comparisons are on the basis of the same value of the externally applied radial displacement. 

In addition to this lack of statical determinacy, it is important to recognize that the mechanical processes that occur during ex vivo lens stretching tests (for an intact ciliary body) are not entirely analogous to those that actually act during in vivo accommodation. In the lens stretching test, the ciliary body behaves as a passive element, increasing in circumference in response to an external force. Conversely, in the living eye, accommodation is driven by active contraction of the ciliary body without the application of an external force. These considerations suggest that the mechanical behavior of the postmortem ciliary body in the lens stretching test is unlikely to relate in any meaningful way to its in vivo characteristics. In contrast, however, the lens and zonules are passive elements both in in vivo accommodation and in ex vivo stretching. The behavior of the lens and zonules in a lens stretching test is therefore, in principle, likely to be representative of their performance in vivo. 

The current paper is concerned with the development of lens stretching protocols that take account of the lack of statical determinacy within the sample (for an intact ciliary body) and the nonrepresentative mechanical behavior of the postmortem ciliary body. These protocols recognize the need to remove the confounding influence of the ciliary body and to adopt procedures that allow meaningful measurements to be made on the mechanical behavior of the lens-zonule. A one-dimensional mechanical model of the lens stretching test is proposed; this model provides a theoretical basis for the experimental protocols. Three separate testing modes are explored: mode A, lens, zonules, and intact ciliary body; mode B, lens, zonules with radial cuts in the ciliary body (to remove circumferential forces developed in the ciliary body); and mode C, ciliary body alone. The study is concerned only with the behavior of the sample when the stretching forces are applied at a sufficiently slow rate for dynamic effects to be negligible. 

Data on four pairs of donated eyes are presented. These data are not intended to represent a complete study of the biomechanics of the lens-zonule. Instead, they provide an example set of data on the anterior segment when tested in each of the three modes; these data are used to explore the extent to which the confounding influence of the postmortem ciliary muscle can be excluded from the test. The measured data are also used to assess the extent to which the proposed one-dimensional mechanical model provides a realistic representation of the lens stretching test. 

Methods

Mechanical Model to Represent the Lens Stretching Test

The proposed model for the mechanical performance of the intact lens–zonule–ciliary body system when tested in a lens stretcher consists of four lumped mechanical components as shown in Figure 1A. The force labeled F_EXT is the total radial force applied to the sample by the external loading system, and Δ_EXT is the corresponding radial displacement of the locations on the ciliary body, or sclera, where the external loading is applied. 

Figure 1

 

Top: Human anterior eye sections inside the experimental chamber showing the three test modes; mode A with ciliary body intact (A), mode B with ciliary body cut radially (B), and mode C with ciliary body intact and the lens removed (C). The white elements represent the stiffness of the labeled parts of the eye. R_CBE is the distance between the lens polar axis and the ciliary body inner edge. This distance was measured for the eight scleral segments (asterisk) and then averaged. The dashed red line symbolizes the radial cut of the ciliary body. Bottom: One-dimensional mechanical model to represent the mechanics of ex vivo lens stretching tests for each mode. The parameter k_CBC represents the circumferential stiffness of the ciliary body, and the parameter k_CBR represents the radial stiffness of the ciliary body. The parameters k_Z and k_L represent the radial stiffness of the zonules and lens, respectively. Δ_CBE is the change of R_CBE during stretching. Δ_L is the radial displacement of the lens, and Δ_EXT is the radial displacement of the point of attachment of the external force, F_EXT.

The model shown in Figure 1A is based on the assumption that the loading and the geometry of the lens–zonule–ciliary body system are axially symmetric around the polar axis of the lens. This assumption is, in fact, implicit in the data presented by Augusteyn et al.¹² and Manns et al.¹¹ On this basis, the radial displacement of the lens equator, indicated as Δ_L, is assumed to be the same at all points around the equatorial perimeter. Similarly, the radial displacement of the inner edge of the ciliary body, indicated as Δ_CBE, is assumed not to vary with circumferential position. 

The force in each of the four mechanical components in Figure 1 is related to the extension, which is defined as the difference in the displacement developed across the element. For the zonules, for example, the extension, Δ_Z, is Δ_Z = Δ_CBE − Δ_L. In the simplest form of the model, each element is assumed to be linear elastic. In this case the force in each element is directly proportional to the extension; the constant of proportionality is termed the “stiffness.” If each element is linear, then the response of the whole system must also be linear. Although some of the reported experimental data in Manns et al.¹¹ and Ehrmann et al.¹³ indicate a nonlinear response, the assumption of linearity is typically adopted in the analysis and discussion of lens stretching data.¹² 

Equations relating the forces that act in the model may be determined using the principle of equilibrium for a static system.¹⁷ A key feature of the model in Figure 1A is that the external force is related to the internal forces by  

 

where F_C represents the circumferential force in the ciliary body and F_L is the force in the zonules and lens. This equilibrium equation applies regardless of whether the individual elements in the system behave in a linear or nonlinear manner. For simplicity, the discussion given below is on the basis that each element in the system is linear and can therefore be assigned an elastic stiffness as indicated in Figure 1. The model can be extended, however, to incorporate nonlinear elements.  

The parameters k_Z and k_L in Figure 1 represent the radial stiffness of the zonules and lens, respectively. It is noted that the zonules connect with the ciliary body in a complex manner; they do not simply connect with the ciliary body on its inner boundary. The value of k_z should therefore be regarded as representing an overall equivalent stiffness of the zonules, rather than being related, directly, to the combined stiffness of the individual zonular fibers. 

The stiffness k_CBC in Figure 1 represents the circumferential stiffness of the ciliary body. It should be carefully noted that the value of k_CBC that would be appropriate to represent the ciliary body in the lens stretching test is unlikely to relate in any meaningful way to the in vivo mechanical behavior of the ciliary body. The parameter k_CBC relates to the passive circumferential stiffness of the ciliary body when it is subject to external stretching forces. In the in vivo system, however, the ciliary body functions as an active component; there is no obvious link between the in vivo mechanics of the ciliary body and the passive stiffness of the postmortem tissue. 

The parameter k_CBR in Figure 1 represents the radial stiffness of the ciliary body and (when present) the sclera. The value of k_CBR is determined partly by the intrinsic stiffness of the relevant tissue and partly by the detailed design of the connection between the external loading system and the sample. These considerations, taken together with the likelihood that any radial deformations induced in the ciliary body–sclera will not be representative of the deformations that occur in vivo, suggest that the value of k_CBR that is appropriate in a model of the lens stretching test will have no particular relevance for the in vivo system. 

Testing Protocols

A commonly adopted test protocol as, for example, adopted by Manns et al.,¹¹ Ehrmann et al.,¹³ and Augusteyn et al.,¹² involves testing the sample with the ciliary body intact (but with radial cuts in the sclera). This form of test (which in this paper is referred to as mode A) is represented by the mechanical model shown in Figure 1A. 

Mode A is a convenient approach from a practical perspective. However, the internal indeterminacy that is inherent in this configuration means that the external force, F_EXT (which is measured), cannot be related directly to the force, F_L, actually applied to the lens during the test. An alternative configuration, termed mode B, is therefore proposed in which radial cuts are introduced in the ciliary body between each part of the sclera where the external loading is applied. When the ciliary body is cut radially, the circumferential force in the ciliary body automatically falls to zero and the element representing the circumferential stiffness of the ciliary body, k_CBC, is removed in the mechanical model (Fig. 1B). The radial stiffness element, k_CBR, remains, however, as this represents the radial stiffness of the remaining connection between the loading points and the zonules. 

In a third configuration, termed mode C, the lens is removed and the ciliary body is left intact. As compared with mode A, both stiffness elements of the ciliary body remain in the mechanical model, but the element representing the crystalline lens is removed (Fig. 1C). This third configuration corresponds to the subsidiary tests described by Manns et al.¹¹ and Ehrmann et al.¹³ 

Modes B and C have the considerable advantage that the mechanical elements are arranged in series. This means, for example, that in mode B the external force is transmitted directly to the lens and zonules, and in mode C the external force is transmitted directly to the ciliary body. 

Stretching experiments were conducted to investigate the performance of anterior segment samples when tested in each of these configurations. The results of these tests were used to investigate the extent to which the mechanical model specified in Figure 1 provides a useful representation of the behavior of the sample. 

Stretching Tests

Four pairs of presbyopic human donor eyes (42, 51, 74, and 85 years of age) were provided by the Banco de Ojos para Tratamiento de la Ceguera (Barcelona, Spain). The 42-year-old donor had a transparent lens; the 51-year-old lens had some very minor cortical opacities; the 74- and the 85-year-old lenses had some nuclear coloration and opacity, as is to be expected for their age. The samples had a mean postmortem time of 63 hours (range, 32–90 hours). The donor eyes were kept at 8°C. Research was conducted under the tenets of the Declaration of Helsinki and according to Spanish regulations for the use of human tissues from organ donors. 

The stretching device¹⁵ consists of a chamber filled with balanced salt solution (BSS) in which the anterior segment is loaded by eight metal loading hooks. The hooks are connected to a force sensor located below the chamber using a 6-0 Prolene monofilament suture (Ethicon, LLC, San Lorenzo, Puerto Rico, USA). The force sensor (Precisa BJ 210C; Precisa Gravimetrics AG, Dietikon, Switzerland) is mounted on top of a digital outside micrometer (Digimatic micrometer Series 293; Mitutoyo Corporation, Kawasaki, Japan) that translates vertically by means of a high-torque stepper motor. Each step by the motor produces a radial displacement of the loading hooks of 1.25 μm in the chamber, resulting in the stretching or relaxing of the anterior segment. A high-resolution webcam (QuickCam Pro for Notebooks; Logitech, Lausanne, Switzerland) is located above the chamber to record the movement of the anterior segment. The loading system was controlled using LabView.¹⁵ 

The force sensor was calibrated with eight micro springs (average individual spring constant of 10 mN/mm) placed in the experimental chamber and connected in the center to each other and on the outside to the eight loading hooks. The measurement precision of the sensor was 0.1 mN. 

The tissue was prepared as follows. Initially, eight glass beads were sutured to the sclera at 3.5 mm off the limbus with a 7-0 Prolene monofilament suture. The cornea was trephined, the iris was removed, and the sclera was cut circumferentially below the beads to obtain the anterior eye segment. The anterior segment was placed into the chamber and the sclera was cut, radially, to divide it into eight independent scleral segments (one segment per bead). Each independent segment was connected to a loading hook (Fig. 1). A more detailed description of the stretching device and the tissue preparation has been published previously.¹⁵ 

Before making the measurements, the resting state of the anterior segment was determined by stretching the sample and then relaxing it until the force detected by the force sensor was approximately zero. The experiment then consisted of three cycles of measurements, each cycle consisting of 10 stretching increments and 10 relaxing increments. Each increment corresponds to 200 steps of the motor. The first cycle was to condition the tissue; the second cycle was when the measurements were taken; and the third cycle was used to confirm the reproducibility. 

The radius of the inner edge of the ciliary body was measured manually on the recorded photographs taken for each stretching increment printed on A4 paper. A central ciliary process in each segment was chosen, and care was taken to always follow the same ciliary process to measure the distance R_CBE to the center of the lens (Fig. 1A). These distance measurements from the eight segments were averaged. The measurements were calibrated using an image of a surgical ruler placed inside the experimental chamber. The estimated measurement precision is 50 μm. Values of the displacement of the inner edge of the ciliary body, Δ_CBE, were determined with reference to the measured values of R_CBE in the resting state. 

To investigate the performance of the anterior segments, the following procedures were conducted on each pair of eyes: modes A and either B or C. Initially, each pair of samples was stretched with the ciliary body intact and the lens in place (mode A). After each mode A test had been completed, in one sample (I), radial cuts were introduced in the ciliary body and the sample was retested (mode B). In the contralateral eye (II), the sample was retested after the lens had been removed, but keeping the ciliary body intact (mode C). On the basis of the proposed mechanical model, it would be expected that for the same value of inner ciliary body displacement, Δ_CBE, the external force measured in mode A should be equal to the sum of the forces measured in modes B and C. 

Results

Experimental data on the relationship between the external force, F_EXT, and the radial displacement of the inner edge of the ciliary body, Δ_CBE, are shown in Figure 2 as circles (sample I) and triangles (sample II). The data from the mode A and C tests appear to be approximately linear for small values of displacement for all four eyes; at larger values of Δ_CBE, the external force increases with ciliary body displacement in a nonlinear way. The data from the mode B tests, however, appear to be approximately linear for the full range of ciliary body displacements developed in the test. These data suggest that the lens and zonules behave, essentially, as linear elements, whereas the ciliary body behaves in a nonlinear manner with the stiffness of the tissue tending to increase with increasing ciliary body displacement. This observed nonlinearity of the force–displacement response from the mode A tests is consistent with previous research.^11,13 

Figure 2

 

Measured external force, F_EXT, versus displacement of the inner edge of the ciliary body, Δ_CBE, for the four pairs of human donor eyes tested in this study. Measured data from one eye (sample I, circle) and the contralateral eye (sample II, triangle) with best-fit curves. The horizontal error bars indicate the standard deviation. Postmortem times in hours are indicated.

Each set of experimental data has been fitted with a function that has a linear part and an exponential part:  

 

, where the parameters a, b, c, and δ have been determined using a least-squares process assuming that the displacements (and not the forces) are subject to experimental error. The parameter δ is interpreted as an offset associated with measurement errors in the initial value of R_CBE. The resulting best-fit curves (R² values between 0.907 and 0.994) are shown in Figure 2.  

The results of the mode A tests conducted on each sample, plotted in Figure 2, generally indicate that, for inner ciliary body displacements of up to approximately 600 μm, the paired samples from each donor exhibit responses that appear similar. For higher values of displacement, however, the responses of the 42-year and 85-year paired samples are significantly different. It is noticeable that, within each pair, the sample with the lower postmortem time generally has a stiffer response (i.e., the external force is larger for any particular value of ciliary body displacement). 

Discussion

In the in vivo eye, the inner ciliary body is unlikely to displace radially by more than 500 μm^18,19 during the accommodation process. The discussion given below is, therefore, confined to the behavior of the lens–zonule–ciliary body system for values of inner ciliary body displacement of 500 μm and below. 

The mode A tests were always the first tests to be conducted in the same way on each sample. However, comparison of the results for the mode A tests conducted on each pair of samples of the same age shows a variation in external force of up to 21% (based on the best-fit curves) for Δ_CBE = 500 μm (Table, A_I and A_II). This variation is consistent with an expectation that the use of postmortem tissue will introduce some variation between samples, especially when muscle tissue is involved. 

Table

 

External Force for a Radial Displacement of the Inner Edge of the Ciliary Body of Δ_CBE = 500 μm

 

To quantify the variations in the external force induced in each of the three test modes, estimates of the external force required to induce an inner ciliary body displacement of 500 μm were made (Table) based on the best-fit curves. The external force needed to stretch the ciliary body alone (mode C_II) was found to be between 44% and 62% of the external force required to stretch the corresponding intact system (mode A_II). The external force needed to stretch the lens-zonules (mode B_I) was found to be between 39% and 68% of the force required to stretch the corresponding intact system (mode A_I). 

The external force needed to stretch the ciliary muscle alone (mode C_II), averaged over all of the samples that were tested, was found to be 51% of the average force required to stretch the corresponding intact system (mode A_II) for Δ_CBE = 500 μm (Table). This result differs from previous studies^11,13 that indicate that the ciliary muscle contributes between 22% and 30% of the external force for human and cynomolgus eyes. It should be noted, however, that the Manns et al.¹¹ and Ehrmann et al.¹³ comparisons appear to have been made on the basis of the same imposed value of external displacement, Δ_EXT, whereas in the current paper, the applied forces are compared on the basis of the same value of ciliary body displacement, Δ_CBE. It is suggested, on the basis of the model in Figure 1, that Δ_CBE is a more robust parameter than Δ_EXT for comparisons of this sort. Also, as a consequence of the nonlinearity of the system, the precise contribution of the ciliary body will depend on the magnitude of the displacement actually being applied in the experiment. It is also noted that variations would be expected on the basis that the thickness of the ciliary body changes with age²⁰ and varies between individuals.^21,22 

The model in Figure 1 implies that, for the same value of inner ciliary body displacement, the sum of the measured external force from mode B and mode C tests should be equal to the external force measured in a mode A test. An exercise was conducted to investigate the extent to which the current experimental data were consistent with this feature of the model. Since the mode B tests were conducted on sample I and the mode C tests on sample II, it is thought appropriate to compare the sum of the external forces measured in mode B_I and C_II with the external forces (termed mode A_mean) determined from the average of the mode A tests conducted on samples I and II. Data (computed using the best-fit curves) on A_mean and the sum of the mode B_I and C_II data are shown in Figure 3. The force difference (at 300-μm displacement) between the two curves is on average 7.5% (range, 4.3%–10%). For each age, these two sets of data are therefore seen to be comparable. This observation supports the assumptions inherent in the model in Figure 1. 

Figure 3

 

External force, F_EXT, versus displacement of the inner edge of the ciliary body, Δ_CBE, determined from the best-fit curves. Data are shown for the mean of mode A_I and A_II and for the sum of mode B_I and C_II.

The current data confirm that, when tested in mode A, the ciliary body provides a significant contribution to the total external force. Since the mechanical performance of the ciliary body when tested ex vivo in radial stretching does not relate directly to its natural physiological function, the magnitude of the applied external force in a mode A test does not have any precise physiological significance. For detailed studies of the biomechanics of the lens–zonule system (for which the lens force, F_L, actually applied to the lens-zonule needs to be determined), two alternative test protocols appear to be available. The test could be conducted, directly, in mode B (in which case F_L = F_EXT). Alternatively, an indirect approach could be adopted in which mode A and mode C tests are performed on the same sample; the measured external forces are then subtracted in an appropriate way (i.e., at the same values of Δ_CBE) to determine the forces, F_L, actually being applied to the lens-zonule. The mechanical behavior of the lens-zonule determined using both protocols (based on the best-fit curves) for the current data is plotted in Figure 4. The solid line shows the mode B_I data (corresponding to the direct approach); the dashed line shows the expected behavior of the lens–zonule system (for sample II) determined, indirectly, by subtracting the mode C_II data from the mode A_II data. The force difference (at 300-μm displacement) between the two curves in Figure 4 is on average 11.9% (range, 1.2%–27%). Therefore, the two sets of data are comparable (especially for the two younger pairs of eyes). Some variation is expected since the data relate to postmortem tissue and involves comparing the results of three different measurements (mode B_I, A_II, and C_II). 

Figure 4

 

Lens force, F_L, versus displacement of the inner boundary of the ciliary body, Δ_CBE, determined from the best-fit curves. Data are shown for mode B_I and values estimated from the difference between mode A_II and mode C_II.

When the direct approach is adopted (by conducting the test in mode B), the possibility exists that the zonules may be damaged when the ciliary body is cut. In addition, the potential benefits of the ring action of the ciliary body in increasing the uniformity of the forces applied to the lens are lost. However, when the indirect approach is used (by conducting mode A and mode C tests and then subtracting the results), any experimental errors are amplified by the process of subtracting two sets of data, since two possible sets of errors are being combined. The indirect approach also requires the use of appropriate data processing procedures (e.g., involving the development of best-fit curves) to facilitate the subtraction of force data at corresponding values of inner ciliary body displacement. It is suggested that to investigate the biomechanics of the lens–zonule system, the use of a direct measurement is preferable to an indirect measurement. 

When stretching tests are conducted on the anterior segment, significant circumferential tensions develop in the ciliary body. This means that the forces applied to the lens and zonules cannot be related directly to the forces applied by the external loading system. If radial cuts are introduced in the ciliary body prior to testing, however, then this difficulty does not arise. Precise knowledge of the forces applied to the lens during accommodation will be important for the development of computational models of the accommodative system and eventually for the mechanical design of accommodating intraocular lenses or lens refilling. 

Acknowledgments

Disclosure: L. Pinilla Cortés, None; H.J. Burd, None; G.A. Montenegro, None; J. Christopher D'Antin, None; M. Mikielewicz, None; R.I. Barraquer, None; R. Michael, None