August 2015
Volume 56, Issue 9
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Lens  |   August 2015
Measurement of Ex Vivo Porcine Lens Shape During Simulated Accommodation, Before and After fs-Laser Treatment
Author Affiliations & Notes
  • Jan Hahn
    Laser Zentrum Hannover e.V. Hanover, Germany
  • Michael Fromm
    Laser Zentrum Hannover e.V. Hanover, Germany
    ROWIAK GmbH, Hanover, Germany
  • Fedaa AL Halabi
    Laser Zentrum Hannover e.V. Hanover, Germany
  • Silke Besdo
    Institute of Continuum Mechanics, Leibniz Universität Hannover, Hanover, Germany
  • Holger Lubatschowski
    ROWIAK GmbH, Hanover, Germany
  • Tammo Ripken
    Laser Zentrum Hannover e.V. Hanover, Germany
  • Alexander Krüger
    Laser Zentrum Hannover e.V. Hanover, Germany
  • Correspondence: Jan Hahn, Laser Zentrum Hannover e.V., Hollerithallee 8, 30419 Hanover, Germany; j.hahn@lzh.de
  • Footnotes
     Current affiliation: *Institute for Multiphase Processes, Leibniz Universität Hannover, Hanover, Germany.
Investigative Ophthalmology & Visual Science August 2015, Vol.56, 5332-5343. doi:10.1167/iovs.14-16185
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      Jan Hahn, Michael Fromm, Fedaa AL Halabi, Silke Besdo, Holger Lubatschowski, Tammo Ripken, Alexander Krüger; Measurement of Ex Vivo Porcine Lens Shape During Simulated Accommodation, Before and After fs-Laser Treatment. Invest. Ophthalmol. Vis. Sci. 2015;56(9):5332-5343. doi: 10.1167/iovs.14-16185.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

Purpose: According to Helmholtz, accommodation is based on the flexibility of the crystalline lens, which decreases with age, causing presbyopia. With femtosecond (fs)-lentotomy treatment, it is possible to restore the flexibility of presbyopic lenses. The efficiency of the treatment can be systematically evaluated using the finite element method based on experimental data. The purpose of this study was to quantify the shape change of ex vivo lenses in different accommodation states according to the fs-lentotomy treatment.

Methods: Five lenses with ciliary body excised from ex vivo porcine eyes (age: approximately 6 months, exact age unknown) were stretched in an accommodation device before and after laser treatment. Depending on the accommodation state, the lens shape, reconstructed from lens thickness, diameter, and anterior and posterior curvature, was measured using optical coherence tomography (OCT). The complete lens shape was parameterized and each measured parameter was compared to the results of a control group (n = 5, age: approximately 6 months, exact age unknown) without treatment.

Results: The amplitudes of the parameters thickness (+140%), diameter (+54%), and anterior radius of curvature (+57%) significantly increased after treatment (P < 0.05), and showed no significant change for the control group. By contrast, the amplitude of the posterior radius of curvature showed no change after treatment (P > 0.05).

Conclusions: Measurement of the lens shape in different accommodation states was successful and showed significant changes after the treatment. The resulting data will be utilized as input for a finite element model to systematically evaluate the effect of fs-lentotomy treatment in future work.

Today there are increasing efforts to find treatment methods for presbyopia, the decrease in accommodation that happens to everyone from the age of 40 onward. According to von Helmholtz, accommodation is a deformation of the crystalline lens caused by the contraction of the ciliary muscle and the restoring forces of the lens capsule.1 The lens grows throughout its lifetime2,3 and the lens fibers become more and more compact,4 while the ciliary muscle stays active.5 The commonly accepted major cause for the development of presbyopia is the hardening of the crystalline lens with age, which leads to a loss of accommodation. Investigations on this hardening of the lens with age deviate strongly with the measurement method.610 
Possible methods to counter the effect of presbyopia so far are monovision LASIK,11 PresbyLASIK,12 insertion of intracorneal inlays13 or intraocular lenses,1416 and lens refilling.17,18 Based on the idea of Krueger et al.,19 we developed a femtosecond (fs)-laser treatment method, called fs-lentotomy,20,21 to increase lens flexibility leading to a partial regain of accommodation ability. In fs-lentotomy, ultrashort laser pulses are used to make three-dimensional (3-D) microcuts in the lens tissue without inducing cataract formation or retina damage.22,23 The fs-lentotomy treatment causes a clearly measurable static and dynamic change in thickness in ex vivo human lenses.24 Ackermann et al. have also shown an increase of lens flexibility caused by fs-laser treatment on ex vivo porcine lenses (Ackermann R, et al. IOVS 2009;50:ARVO E-Abstract 6134). 
A simple approach to measure the change of flexibility of the lenses due to the laser treatment is the Fisher spinning test,6 where the crystalline lens is located on a platform and rotated around its polar axis. The resulting centrifugal forces flatten the lens, and the degree of flattening can be used to determine the lens flexibility. The curvatures are measured using a camera during rotation. Due to the mass gradient inside the lens and the dependency of these forces from the distance to the rotation axis, the resulting flattened lens shape does not correspond to the in vivo situation, where the contact points of the stretching forces are located on the outside of the lens. To simulate accommodation in accordance with the classical Helmholtz theory, it is necessary to deform the lens by placing forces on the lens capsule via zonules.25 For this purpose, a lens stretching device can be used, as reported by several authors for human lenses,2630 monkey lenses,26,3032 and porcine lenses.33 In this manner, it is possible to maintain the lens with ciliary body in medium during the simulation of accommodation, to keep it from drying, and to minimize its weight and also enhance accuracy of measuring lens curvature when using optical coherence tomography (OCT).3438 
The number of possible patterns for fs-lentotomy is far too high to be tested for effectivity in patients. In the past, different variations of the so-called steering wheel cutting pattern (described in detail in the section “Femtosecond-Laser Cutting”) were tested using Fisher spinning tests.21 In order to evaluate cutting patterns before experimental testing, a computer simulation of the continuum mechanics is appropriate, which describes the lens accommodation and the tissue cuts of fs-lentotomy using finite element (FE) methods.39 The geometrical changes observed and quantified in this study will serve as input parameters (native unstretched lens) and ground truth for the evaluation of the calculated geometries (native stretched, treated unstretched, and treated stretched) of such a simulation. 
This study reports on the quantification of the change in lens shape in the different states of accommodation before and after fs-lentotomy treatment of ex vivo porcine lenses. For this purpose, a lens stretching device that simulates accommodation was combined with an OCT system, which allows visualization of the ex vivo porcine lenses within the stretching device. The measured geometrical parameters, the differences between accommodation states, and the results of the treatment effect were analyzed. To make sure the measured changes of geometry were based on the laser treatment, a control group underwent the same treatment protocol except for the laser treatment, including applanation of the lens. 
Materials and Methods
Porcine Lenses
Porcine lenses (n = 5 treatment group and n = 5 control group, approximately 6 months, exact age unknown) used in this study were obtained from a local slaughterhouse (Schlachthof Hannover GmbH, Hanover, Germany). All lenses were stored in Roswell Park Memorial Institute medium (RPMI) (RPMI 1640 without phenol red; Biochrom GmbH, Berlin, Germany) at room temperature immediately after the dissection, as established by Augusteyn et al.40 Postmortem time was approximately 5 hours. For preparation, the posterior half of the globe, the cornea, and the iris were removed with a scalpel. Special care was taken not to destroy the zonules or the lens capsule during preparation. The anterior half of the eyeball including zonules, ciliary body, and sclera was mounted in a custom-made lens stretching device (see “Lens Stretcher” section) and fixated with brackets. The lenses were immersed in RPMI solution during the whole process. 
Lens Stretcher
The lens stretching device (see Fig. 1) consists of a removable inlay for mounting the segmented anterior half of the eyeball and a device equipped with eight stepper motors for conducting the stretching and unstretching process over the inlay onto the lens. The maximum stretching state of the sclera is 2.5 mm in radius, in which the inlay can be locked for better handling. 
Figure 1
 
Top view of anterior half of porcine eyeball covered with RPMI medium and mounted on the inlay (light gray) of the motorized lens stretching device (left). Construction of the removable inlay with a schematic half of the eyeball in the middle from different views (right).
Figure 1
 
Top view of anterior half of porcine eyeball covered with RPMI medium and mounted on the inlay (light gray) of the motorized lens stretching device (left). Construction of the removable inlay with a schematic half of the eyeball in the middle from different views (right).
Before starting the measurements, the lens was radially stretched and unstretched once to make sure it was fixed correctly. 
OCT Imaging
To measure the geometry of the crystalline lens, spectral-domain OCT (see Fig. 2), which has been described in detail by Donner et al.,41 was used. In brief, the light of a broadband superluminescent diode (SLD) (Superlum Diodes, Ltd., Carrigtwohill, Ireland) is transmitted through a single mode fiber into a Michelson interferometer with an 80:20 nonpolarizing beam splitter (NPBS). Here the light is split into a reference and a sample arm, which contains an xy galvanometer scanner (XY) (6210H; Cambridge Technology, Inc., Bedford, MA, USA), a dichroic mirror (M2), and a Hastings-triplet-lens (HT2), which is used as scanning lens. The interferometer and scanner are mounted as compound units on a translation stage for shifting (ΔzSamp) the OCT focus in z-direction (depth) into the sample. The dispersion of the scanning lens is compensated by a second Hastings-triplet-lens (HT1) in the reference arm. The increase in the optical path length caused by the medium used was compensated by lengthening the reference arm (ΔzRef) with a motorized mirror (M1). The reference signal is analyzed using an OCT spectrometer with a line camera (runner; Basler AG, Ahrensburg, Germany). 
Figure 2
 
Optical coherence tomography setup with the lens mounted in the stretcher inlay covered with RPMI solution. SLD, superluminescent diode; C, polarization-sensitive circulator; CL, collimating lens; NPBS, nonpolarizing beam splitter; HT1/HT2, Hastings-triplet-lens; M1/M2, silver mirror; XY, galvanometer scanner system; L, lens; DG, diffraction grating; CMOS, line image sensor.
Figure 2
 
Optical coherence tomography setup with the lens mounted in the stretcher inlay covered with RPMI solution. SLD, superluminescent diode; C, polarization-sensitive circulator; CL, collimating lens; NPBS, nonpolarizing beam splitter; HT1/HT2, Hastings-triplet-lens; M1/M2, silver mirror; XY, galvanometer scanner system; L, lens; DG, diffraction grating; CMOS, line image sensor.
The OCT system is able to acquire 3-D stacks (800 × 800 × 800 voxels, total volume 6.43 × 6.43 × 2.57 mm3 in air), composed of 800 B-mode (brightness mode) images along the y-direction. To measure anterior and posterior curvature of the lens directly without distortions due to the spherical lens contour or the internal refractive index gradient, the inlay of the stretching device can be turned around to measure both (anterior and posterior) surfaces. In addition to anterior and posterior curvature determination in the fixed unstretched and stretched state, lens thickness and equatorial diameter were also measured using OCT by shifting the image region, which means shifting the OCT focus in z-direction and adjusting the reference arm length. Next, the inlay of the stretching device was placed under the laser system in the stretched state. The OCT measurement protocol was performed before and after fs-lentotomy for the treatment group. The control group underwent the same protocol including OCT measurement, applanation of the lens without laser treatment, and again OCT measurement. 
Femtosecond-Laser Cutting
For the fs-lentotomy treatment, a 100-kHz laser system (FCPA μJewel D-400; IMRA America, Inc., Ann Arbor, MI, USA) was used (λ = 1041 nm; pulse duration: 306 fs; further details have been described by Schumacher et al.22). The 3-D cutting pattern used in this study was a modified steering wheel pattern described by Ripken et al.21 with 12 triangular planes without top or bottom. The cuts are made at a depth of 0.8 mm. It was realized using a scanner (Intelliscan 20; Scanlab AG, Puchheim, Germany) with a pair of galvanometer scanners (Dynaxis L; Scanlab AG) in the x- and y-plane and a linear translation stage in the z-direction (depth). Every lens was applanated under a glass plate during the laser process, which took approximately 30 seconds. The pulse energy applied to the lenses was around 1.4 μJ. 
Photography of Isolated Lens
At the end of each experiment, after stretching the treated lens, zonules and the ciliary body were removed carefully from the lens (treatment and control group). The lens thickness (apex to apex) tIso and the thickness of the anterior half zIso of every isolated lens were determined using photographs (see Fig. 3). A glass bead (SiLiBeads Type P; Sigmund Lindner GmbH, Warmensteinach, Germany) with the diameter tRef = 7.00 ± 0.01 mm was used to scale the lens dimensions in a glass cuvette filled with RPMI. The ratio Display FormulaImage not available was assumed to be a constant for the lens in the isolated, unstretched and stretched state, and used to determine the level of equator for modeling the lens shape.  
Figure 3
 
Photograph of an isolated lens (control group: lens 4) placed in a glass cuvette together with a precision glass bead as dimension reference at the end of the experiment. The photograph was used to determine the ratio σ = zIso/tIso, which is assumed to be constant in the isolated, unstretched and stretched state for the specific lens.
Figure 3
 
Photograph of an isolated lens (control group: lens 4) placed in a glass cuvette together with a precision glass bead as dimension reference at the end of the experiment. The photograph was used to determine the ratio σ = zIso/tIso, which is assumed to be constant in the isolated, unstretched and stretched state for the specific lens.
Extraction and Parameterization of Lens Shape
First, the lens thickness (apex to apex) values of the four states, measured using OCT, were corrected by the mean paraxial refractive index of n = 1.5197 given for young porcine lenses.33 With the corrected lens thickness t and the ratio σ taken from photographs of the isolated lens, the distance zapex from the anterior lens apex to the position of the equator plane (assumed at z = 0) could be estimated to be zapex = . A substack (Δy = 80.375 μm) of 10 B-scans around the lens apex was extracted from the OCT volume data. This substack was projected using the median in order to find the position of the lens apex, represented by the height maximum of the resulting curved line. After rescaling the z-axis by a factor found through calibration measurements with OCT on the glass bead with 7.00 ± 0.01 mm in diameter in the same medium (RPMI), the central curvatures were extracted from the projected layer. The peripheral portion was in part shadowed by the lens stretching device, and also partly insufficiently contrasted in OCT because of the drop of backscattering at higher angles of incidence of the probe beam on the lens surface (see grayscale inverted B-mode image stack median projections in Fig. 4). 
Figure 4
 
Median projection of 10 inverted B-mode images acquired with OCT of anterior and posterior lens with circle fits at the apex to determine central curvature c. Deviation of the circle fit is described by point P on lens shape through asphericity k (Equation 2). With measured relation of apex to thickness of the isolated lens measured in photographs, the z-position of the apex is determined, and the imaged lens shape in the central zone can be described (Equation 1). Inversing of Equation 1 to Equation 3 can determine ceq and keq to fit the lens shape beyond point P in the marginal zone.
Figure 4
 
Median projection of 10 inverted B-mode images acquired with OCT of anterior and posterior lens with circle fits at the apex to determine central curvature c. Deviation of the circle fit is described by point P on lens shape through asphericity k (Equation 2). With measured relation of apex to thickness of the isolated lens measured in photographs, the z-position of the apex is determined, and the imaged lens shape in the central zone can be described (Equation 1). Inversing of Equation 1 to Equation 3 can determine ceq and keq to fit the lens shape beyond point P in the marginal zone.
Assuming a rotational symmetry, the detected curvature is described as a function of x, the distance from the apex to apex axis, which is defined as z-axis with zero at the lens equator (see Fig. 4). Therefore, the curvature can be described by the following equation42:    
The curvature c was determined using a manual circle fit at the lens apex with the radius of curvature c−1; the asphericity k refers to a point P located on the lens shape, and zapex is the z-position of the apex. The asphericity can be calculated using xp and zp of point P with the formula    
Since only the central zone could be imaged with the OCT system, the marginal zone had to be modeled with two conics (anterior and posterior), with their apex at the equator of the lens (see Fig. 4). 
The peripheral curvature ceq and asphericity keq (beyond Pant/post) can then be calculated by inserting the following inverse formula42 in Equation 1 by use of continuity conditions at point P and parameter xeq, which is half of the lens diameter:    
The derivation of the function describing the central region is equal to the derivation describing the marginal area in point P. The distance of P to the optical axis was set to xP,ant = 2.5 mm for the anterior half, and to xP,post = 2.0 mm for the posterior half. These maximum x values for point P refer to the maximum reflecting angle of 45° for the OCT beam. For the anterior image, a much smaller value was chosen in order not to place point P outside the high-contrast imaging range. 
Based on the assumed constant ratio of the lens thickness to the apex-to-apex distance, the volume of the lens at every accommodation state was calculated via the cross-sectional area bordered by the determined lens shape. This was necessary to make sure there was no change in lens volume due to the applanation process or the laser treatment itself. 
Simulation of Back Focal Distance and Calculation of Optical Power
The back focal distance (BFD), given by the distance from the posterior surface to the focal point, was determined using ray tracing software (ZEMAX-EE 2011; Radiant ZEMAX LLC, Kirkland, WA, USA). First, the central lens part was modeled on the measured parameters thickness t, anterior curvature cant, and posterior curvature cpost, with the corresponding asphericities kant and kpost for each state. Then the quick focus function was used to find the focal point with the minimum spot diameter for apertures from near 0 to 5 mm in diameter. 
Since the gradient refractive index (GRIN)43 for the lenses was unknown, due to their individual inner structure, an equivalent refractive index of neq = 1.4955 for the lens44 was used. The outer medium is supposed to be aqueous and vitreous humor45 each with nh = 1.336. 
The optical power D of the lenses regarding the effective focal length EFL, which corresponds to the paraxial case with the aperture of 0, is calculated using the following formulas:     
Statistics
The geometric parameters were tested for significant results, and a two-tailed t-test (paired) was conducted to evaluate these 11 hypotheses. 
Hypothesis 1 states that the lens thickness decreases significantly when stretched. Hypotheses 2 through 4 say that lens diameter (2) and anterior (3) and posterior radius (4) of curvature increase in the stretching process. Because of the stated changes of the lens shape, the calculated optical power also decreases (hypothesis 5). Hypotheses 1 through 5 are null hypotheses. 
The following hypotheses concerning the influence of fs-lentotomy treatment on the porcine lenses in simulated accommodation were made. The difference of the measured values between the stretched and the unstretched state is called amplitude. Hypothesis 6 says that the laser treatment leads to higher amplitude in thickness when stretched. Furthermore, the laser treatment increases the amplitude of diameter during simulated accommodation (hypothesis 7) and leads to higher amplitude in anterior (hypothesis 8) and posterior radius of curvature (hypothesis 9) when stretched. With the supposed changes in the lens geometry (hypotheses 6–9), the amplitude of the calculated optical power also has to be increased by the laser treatment in the unstretched as well as in the stretched state (hypothesis 10). Finally, the protocol without laser treatment in the control group leads to no significant changes of the above parameters when remeasured (hypothesis 11). 
Mean values are given with standard deviation; P < 0.05 means statistical significance; P < 0.001 means extreme statistical significance; and P > 0.05 means no statistical significance throughout this article. 
Results
The laser procedure could be conducted successfully on all lenses. There was no damage or detachment of lens capsule visible on the OCT images after the procedure. 
Unstretched-Stretched (Native)
In Table 1, the mean values of the geometrical parameters thickness, diameter, and anterior and posterior curvature and the optical power, the latter calculated following Equations 4 and 5 with use of the simulated BFD, are shown with standard deviation for the porcine lenses (n = 5) before laser treatment for the unstretched and stretched state. The curvature is given in form of the radius of curvature (reciprocal value). Furthermore, the mean values of the change of all parameters, due to the stretching process, are quoted with the standard deviations. 
Table 1
 
Mean of Measured Parameters Including Calculated Optical Power for the Lenses in Native Unstretched and Stretched State (n = 5), With SD Denoting Standard Deviation
Table 1
 
Mean of Measured Parameters Including Calculated Optical Power for the Lenses in Native Unstretched and Stretched State (n = 5), With SD Denoting Standard Deviation
The mean thickness value changed significantly (hypothesis 1 confirmed) in the negative direction. The means of the measured geometrical parameters diameter and anterior and posterior radius of curvature increased significantly for all lenses (hypotheses 2–4 confirmed). Furthermore, the calculated optical power decreased significantly (hypothesis 5 confirmed). 
Amplitudes (Unstretched-Stretched) Before and After Laser Treatment
In Figure 5, the mean amplitude values of all geometrical parameters (amplitudes from unstretched to stretched state) are shown with standard deviation for porcine lenses 1 to 5 before and after the laser treatment. The curvature is given in form of the radius of curvature (reciprocal value). All values for the posterior radius of curvature and the calculated optical power of lens 3 had to be excluded because the OCT data for the stretched, treated state were missing. Therefore the mean values for the posterior radius of curvature and the calculated optical power were calculated with n = 4 lenses (lens 3 excepted). Consequently the t-tests were also conducted with n = 4 lenses. 
Figure 5
 
Results of study on ex vivo porcine lenses (n = 5) with mean amplitudes of thickness, diameter, anterior radius of curvature, posterior radius of curvature, and calculated optical power before (native) and after (treated) fs-laser treatment. **Statistical significance with P < 0.001. *Statistical significance with P < 0.05. †Lens 3 was omitted from statistics (n = 4).
Figure 5
 
Results of study on ex vivo porcine lenses (n = 5) with mean amplitudes of thickness, diameter, anterior radius of curvature, posterior radius of curvature, and calculated optical power before (native) and after (treated) fs-laser treatment. **Statistical significance with P < 0.001. *Statistical significance with P < 0.05. †Lens 3 was omitted from statistics (n = 4).
The amplitudes of the measured geometrical parameters thickness, diameter, and anterior radius of curvature increased significantly for all treated lenses (hypotheses 6–8 confirmed). The amplitude of the posterior radius of curvature and the amplitude of the calculated optical power did not change significantly (hypotheses 9 and 10 disconfirmed). The control group showed no significant change in amplitudes for all measured or calculated parameters after the conducted protocol without laser treatment (hypothesis 11 confirmed, values not shown). 
The absolute mean amplitude of central thickness changed from 0.167 ± 0.041 to 0.401 ± 0.033 mm after the fs-lentotomy treatment, an increase of 140% (P < 0.001, Table 2). The mean amplitude of the lens diameter increased by 54% from 0.396 ± 0.096 to 0.610 ± 0.093 mm (P < 0.05), and the mean amplitude of the anterior radius of curvature increased by 57% from 0.778 ± 0.372 to 1.220 ± 0.458 mm (P < 0.05). The mean amplitude of the posterior radius of curvature and the absolute mean amplitude of the calculated optical power did not change significantly (P > 0.05) after the laser treatment. 
Table 2
 
Measured Parameters Including Calculated Optical Power for Unstretched and Stretched in Native and Treated State (n = 5), With SD Denoting Standard Deviation and P Denoting Probability
Table 2
 
Measured Parameters Including Calculated Optical Power for Unstretched and Stretched in Native and Treated State (n = 5), With SD Denoting Standard Deviation and P Denoting Probability
Unstretched (Native-Treated)
The lens thickness of the treated lenses increased by 0.105 ± 0.058 mm on average, from 5.432 ± 0.515 to 5.537 ± 0.555 mm in the state of simulated near accommodation (P < 0.05, Table 3). The lens diameter did not change significantly in the state of near vision (P > 0.05). The anterior radius of curvature decreased significantly by 0.560 ± 0.290 mm from 6.988 ± 0.542 to 6.428 ± 0.448 mm (P < 0.05). The posterior radius of curvature and the calculated optical power did not change significantly (P > 0.05) in the unstretched state. 
Table 3
 
Measured Parameters Including Calculated Optical Power for Native-Treated in Unstretched and Stretched State (n = 5), With SD Denoting Standard Deviation and P Denoting Probability
Table 3
 
Measured Parameters Including Calculated Optical Power for Native-Treated in Unstretched and Stretched State (n = 5), With SD Denoting Standard Deviation and P Denoting Probability
Stretched (Native-Treated)
Furthermore, Table 3 shows that in the disaccommodated state, the lens thickness decreased by 0.129 ± 0.033 mm from 5.265 ± 0.515 to 5.136 ± 0.515 mm (P < 0.05) after treatment. The increasing diameter of 0.102 ± 0.044 mm from 9.254 ± 0.563 to 9.356 ± 0.605 mm shows statistical significance (P < 0.05) after the laser treatment in the disaccommodated state. The change of anterior and posterior radius of curvature and calculated optical power was not statistically significant in the state of distance vision (P > 0.05). 
Altogether, the lenses, on average, became thicker and more spherical (anterior radius of curvature) in the unstretched state, and thinner and broader in the stretched state due to the laser treatment. 
Back Focal Distance (Spherical Aberration)
Figure 6 shows the simulated BFD for the four states of lens 1, referenced to their equatorial planes with an aperture of 3 mm in diameter. 
Figure 6
 
Simulated ray tracing (ZEMAX) of applying different apertures with the found parameters anterior and posterior curvatures, describing the central zone (near 0–5 mm in diameter). Here the states (a–d) of lens 1 are shown with the exemplary aperture of 3 mm in diameter, and the equatorial plane as common reference. Refractive index of the lens is neq = 1.4955 and surrounding medium is nh = 1.336. Back focal distance was determined by the quick focus function.
Figure 6
 
Simulated ray tracing (ZEMAX) of applying different apertures with the found parameters anterior and posterior curvatures, describing the central zone (near 0–5 mm in diameter). Here the states (a–d) of lens 1 are shown with the exemplary aperture of 3 mm in diameter, and the equatorial plane as common reference. Refractive index of the lens is neq = 1.4955 and surrounding medium is nh = 1.336. Back focal distance was determined by the quick focus function.
The simulated BFD with different apertures (near 0–5 mm in diameter) of lenses 1, 2, 4, and 5 in all four states are shown in Figure 7. The BFD of all lenses decreased with increasing aperture in the unstretched and stretched native state. This indicates a monotone shift of focus to shorter BFD for higher apertures, which points to a positive spherical aberration for the unstretched and stretched native state. Therefore the spherical aberration remained positive in both stretching states after the laser treatment. 
Figure 7
 
Simulated back focal distances (BFD) for different apertures in diameter of the parametrized lens contour describing the central zone. For each lens, the BFD is plotted in the stretched (diamond), unstretched (circle), native (gray line), and treated (black dashed line) state.
Figure 7
 
Simulated back focal distances (BFD) for different apertures in diameter of the parametrized lens contour describing the central zone. For each lens, the BFD is plotted in the stretched (diamond), unstretched (circle), native (gray line), and treated (black dashed line) state.
In Figure 8, the differences (Δ) between the native and treated state for lenses 1, 2, 4, and 5 of the simulated BFD are shown for the unstretched and stretched state. 
Figure 8
 
Differences between the native and treated state of the simulated back focal distances (BFD) over the aperture. For each of lenses 1, 2, 4, and 5, the Deltas of the BFD are shown for the unstretched and stretched state.
Figure 8
 
Differences between the native and treated state of the simulated back focal distances (BFD) over the aperture. For each of lenses 1, 2, 4, and 5, the Deltas of the BFD are shown for the unstretched and stretched state.
Although all lenses showed a positive spherical aberration, there was a quantitative change of positive spherical aberration due to laser treatment. In some cases (lens 1 unstretched, lens 5 unstretched, and lens 4 stretched), the positive spherical aberration increased with the treatment, while lens 1 (stretched) and lens 4 (unstretched) showed a decrease in positive spherical aberration. Lenses 2 and 5 (stretched) stayed unchanged in positive spherical aberration. The change of lens 2 (unstretched) showed no clear direction. 
Complete Lens Shape and Volume
With the determined and fitted parameters, the complete lens shape can be described following Equations 1 and 3. The contours of all four states are presented by lens 1 in Figure 9
Figure 9
 
Porcine lens 1 before (left side) and after (right side) laser treatment in unstretched (black line) and stretched (gray line) states. Dashed lines mark the fitted part of the lens shape.
Figure 9
 
Porcine lens 1 before (left side) and after (right side) laser treatment in unstretched (black line) and stretched (gray line) states. Dashed lines mark the fitted part of the lens shape.
The mean calculated lens volume, based on the extracted lens shape, before and after the laser treatment changed by 0.203 ± 2.389 mm3 (from 198.224 ± 48.479 to 198.021 ± 50.559 mm3) in the unstretched state (0.3%) and by 3.753 ± 3.688 mm3 (from 222.141 ± 54.278 to 218.387 ± 57.753 mm3) in the stretched state (2.0%), which was not statistically significant. This means that no measurable inflation of the lenses takes place due to the fs-laser pulses, but a real increase of deformability of the tissue. Furthermore, potential compression of the lenses during the applanation process can be eliminated. 
Discussion
Porcine Lenses as a Model for Accommodation
In Germany, human eyeballs are available from foreign eye banks, but only after cornea extraction. Virological testing and transport of the globes takes 2 to 4 days, so ciliary body and zonules do not stay intact. Thus, these lenses are inadequate for use in a lens stretching device. Even though pigs are not able to accommodate, porcine lenses are widely used as a model for accommodation or even presbyopia studies.33,46 The protein content and distribution in porcine lenses are comparable to those in human lenses,47,48 and porcine lenses are easily obtainable. Domestic pigs show a relatively small body weight until the age of 50 days, after which body weight increases.49 This could imply that the hardening of the lens proceeds in a similar way from the age of 50 days on, which would mean that lenses from older pigs (late puberty reached with 180 days)50 should be a good model for presbyopia studies.33 Kammel et al.33 showed that the change of optical power with stretching is approximately five times higher for young porcine lenses (150–180 days) than for sows (270–720 days), but without knowing the individual ages. Unfortunately, there was (and is) no possibility to obtain individual ages of the pigs from our local slaughterhouse. We only could assume that lenses of different sizes were from animals of different ages, but so far there has been no possibility to do a comparative study with lenses from pigs of a known older age. 
Validity of the Unstretched Lens Dimensions and Optical Power
Our measured values (n = 5) for native, unstretched porcine lens thickness (5.432 ± 0.515 mm), diameter (8.858 ± 0.608 mm), and anterior (6.988 ± 0.542 mm) and posterior radius of curvature (3.516 ± 0.151 mm) are below those measured by Kammel et al.33 (7.7 ± 0.2 mm thickness, 10.0 ± 0.3 mm diameter) and Wong et al.51 (lens thickness: 7.4 ± 0.1 mm, anterior radius of curvature: 7.08 ± 0.35 mm, posterior radius of curvature: 5.08 ± 0.14 mm) at zero stretching. The smaller dimensions in our study might be attributed to a younger average age of the pigs. The high standard deviation could indicate a higher variance in age, due to some older animals among the young pigs. 
Our calculated values for optical power of the native, unstretched porcine lens (64.564 ± 2.475 D) are above those measured by other groups. Again the smaller dimensions of our lenses and globes could be the reason. Wong et al.51 measured 49.9 ± 1.5 diopters (D) for the refractive power (6-mm aperture) of a porcine eye lens51 in solution, and Vilupuru et al.44 measured 25.89-mm effective focal length, which corresponds to a refractive power of 51.6 D (surrounding refractive index of n = 1.336) in solution. Birkenfeld et al.43 used a laser ray tracing system to measure the back focal length of porcine lenses (2-mm ring of light), corresponding to a higher optical power of 61.9 ± 2.4 D in solution, calculated by inverting the back focal length multiplied by the refractive index of the medium.43 
In this study, the optical power was calculated from the individual simulated BFD, based on the measured lens shape and an equivalent refractive index, without taking into account the existing GRIN of the crystalline lens. The equivalent refractive index neq = 1.4955 taken from literature might be too high, and the mean thickness of our lenses is relatively small, so that the ZEMAX simulation overestimates the refractive power of the lens. 
Accommodation Amplitude of Untreated Eyes
Our calculated absolute mean amplitude value of 13.979 ± 0.562 D for the unstretched to stretched state (64.564 ± 2.475 D − 50.585 ± 2.501 D) does not correspond well to the mean value of Kammel et al.33 for young lenses (3.5 D), but is in the range of Wong et al.51 (11.899 D). 
Kammel et al.33 also worked with porcine lenses in a lens stretching device, and measured a mean optical power for young porcine lenses of 39 ± 0.8 D (unstretched) and 35.5 ± 0.9 D (2 mm stretched).33 The same calculations for the optical power with an equivalent refractive index of Display FormulaImage not available = 1.4686 from Wong et al.,51 found by comparing measurements using scanning laser ray tracing and simulations using mathematical ray tracing on a porcine lens refilled with silicone oil, led to 54.700 ± 1.988 and 42.801 ± 2.134 D for the unstretched and stretched state, respectively.  
Treatment Effect in the Stretching Device
The increase of the measured amplitudes for central thickness (+140%), diameter (+54%), and anterior radius of curvature (+57%) after the fs-lentotomy treatment shows a clear influence of the laser treatment on the behavior of the porcine lenses in simulated accommodation (P < 0.05). The laser cuts cause a higher deformation ability of the tissue, which leads to the lens forming a more spherical shape in the state of simulated near vision and a more flat shape in the state of simulated distance vision. Only the amplitude of the smaller posterior radius of curvature decreases by 10% after laser treatment, which is not statistically significant (P > 0.05). The accommodation amplitude of the calculated optical power increases nonsignificantly by 1% on average (P > 0.05). 
This calculated value is small in comparison to the increase of deformation ability, due to the decrease of the amplitude of the posterior radius of curvature. The unexpected flattening of the posterior radius of curvature in the unstretched state after laser treatment compensates the decreasing anterior radius of curvature. Since the calculated lens volume does not change during treatment, we have no satisfying explanation for this behavior at the moment. 
The BFD simulation shows a positive spherical aberration in all cases. In some cases, the spherical aberration increases (lenses 1 and 5 in the unstretched, lens 4 in the stretched state) due to the laser treatment, meaning a decrease of the optical power in the unstretched state for higher aperture values (lenses 1 and 5 in the unstretched state). We found no relation to a specific geometric parameter of the lens shape. The effect of laser treatment on the spherical aberration of the lens and the possibility of a beneficial influence will be investigated in more detail in our future work using a laser ray tracing setup. 
Comparison to Results With Fisher Spinning Tests
These are the first results showing a clear influence of fs-lentotomy treatment on the complete lens shape during simulated accommodation in a lens stretcher. 
In our past work, we have shown a static molding of human lenses (thickness increase: 97 ± 14 μm) immediately after laser treatment, with an increase of deformation ability (+16%) and accommodation amplitude for human lenses (+59% change in calculated normalized optical power)24 due to laser treatment measured with the Fisher spinning test. For porcine lenses, we have shown an increase of deformation ability of 26.7% based on the amplitude of the normalized central lens thickness by inducing the steering wheel pattern with 12 planes plus a conical top and bottom.21 All these experiments were conducted on isolated lenses without zonules and ciliary body. Compared to our former results on isolated porcine lenses obtained with the Fisher spinning test, our current study would show an even higher increase in deformation ability as calculated by Ripken et al.21 Strictly speaking, such a calculation of the deformation ability cannot be performed in the same way, because in the Fisher spinning test, the entire curve of stretch versus angular velocity (and the resulting centrifugal force on the volume) was used to evaluate the deformation ability of the isolated lens, while in the case of the current study, there was a constant amplitude radial stretch of the lens together with the entire ciliary body (and the resulting tensile stress on the lens), mimicking an isometric amplitude of ciliary muscle contraction. In terms of material properties of the lens, the Fisher spinning test might be suitable, but in terms of the real anatomical situation in the eye, the stretcher experiments are more meaningful. A more general discussion of the validity of the Fisher spinning test is given by Burd et al.52 
In conclusion, so far there is no treatment available for the cause of presbyopia, and the mechanism of accommodation is not completely understood. Femtosecond-lentotomy is a promising tool to treat presbyopia, but it is still necessary to systematically evaluate the effect of treatment on the accommodating lens. In this work, we have successfully measured the porcine lens shape during simulated accommodation in a lens stretcher before and after laser treatment with OCT. The lens shape changes significantly when stretched. Furthermore, changes of the parameters central thickness, anterior radius of curvature, and diameter show a clear influence from laser treatment, whereas the optical power is hardly affected. The marginal zone of the lens was modeled with two conics (anterior and posterior) with their apex at the equator of the lens. The measured lens shape in the four states (unstretched and stretched, before and after treatment) is suitable to serve as input data and ground truth for a realistic FE model of the accommodating crystalline lens with fs-lentotomy cuts. 
These methods have to be transferred to presbyopic ex vivo lenses, which are stiffer than young porcine lenses, also taking into account the GRIN. With such a model, we would be able to systematically investigate the effect of the cut position and geometry in future work. The output of this simulation in turn will be experimentally validated in the stretching device using laser ray tracing and 3-D OCT measurements for GRIN reconstruction. 
The overall effectiveness of treatment can ultimately be proven only in clinical studies. Nevertheless, lens stretching experiments with measurement of the change of optical power will be necessary to further develop and improve the cutting shapes and parameters, and to elucidate the impact of surgery on the gradient refractive index contribution to the accommodation amplitude. 
Acknowledgments
Presented at the Russian-Bavarian/Russian-German Conference, Hanover, Germany, October 2013, and at the BMT 2014 - 48th DGBMT Annual Conference, Hanover, Germany, October 2014. 
Supported by German Research Foundation (DFG; Bonn, Germany) Grant LU 498/9-1. Holger Lubatschowski is CEO of ROWIAK GmbH. 
Disclosure: J. Hahn, None; M. Fromm, None; ROWIAK GmbH (E); F. AL Halabi, None; S. Besdo, None; H. Lubatschowski, ROWIAK GmbH (I, E), P; T. Ripken, P; A. Krüger, None 
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Figure 1
 
Top view of anterior half of porcine eyeball covered with RPMI medium and mounted on the inlay (light gray) of the motorized lens stretching device (left). Construction of the removable inlay with a schematic half of the eyeball in the middle from different views (right).
Figure 1
 
Top view of anterior half of porcine eyeball covered with RPMI medium and mounted on the inlay (light gray) of the motorized lens stretching device (left). Construction of the removable inlay with a schematic half of the eyeball in the middle from different views (right).
Figure 2
 
Optical coherence tomography setup with the lens mounted in the stretcher inlay covered with RPMI solution. SLD, superluminescent diode; C, polarization-sensitive circulator; CL, collimating lens; NPBS, nonpolarizing beam splitter; HT1/HT2, Hastings-triplet-lens; M1/M2, silver mirror; XY, galvanometer scanner system; L, lens; DG, diffraction grating; CMOS, line image sensor.
Figure 2
 
Optical coherence tomography setup with the lens mounted in the stretcher inlay covered with RPMI solution. SLD, superluminescent diode; C, polarization-sensitive circulator; CL, collimating lens; NPBS, nonpolarizing beam splitter; HT1/HT2, Hastings-triplet-lens; M1/M2, silver mirror; XY, galvanometer scanner system; L, lens; DG, diffraction grating; CMOS, line image sensor.
Figure 3
 
Photograph of an isolated lens (control group: lens 4) placed in a glass cuvette together with a precision glass bead as dimension reference at the end of the experiment. The photograph was used to determine the ratio σ = zIso/tIso, which is assumed to be constant in the isolated, unstretched and stretched state for the specific lens.
Figure 3
 
Photograph of an isolated lens (control group: lens 4) placed in a glass cuvette together with a precision glass bead as dimension reference at the end of the experiment. The photograph was used to determine the ratio σ = zIso/tIso, which is assumed to be constant in the isolated, unstretched and stretched state for the specific lens.
Figure 4
 
Median projection of 10 inverted B-mode images acquired with OCT of anterior and posterior lens with circle fits at the apex to determine central curvature c. Deviation of the circle fit is described by point P on lens shape through asphericity k (Equation 2). With measured relation of apex to thickness of the isolated lens measured in photographs, the z-position of the apex is determined, and the imaged lens shape in the central zone can be described (Equation 1). Inversing of Equation 1 to Equation 3 can determine ceq and keq to fit the lens shape beyond point P in the marginal zone.
Figure 4
 
Median projection of 10 inverted B-mode images acquired with OCT of anterior and posterior lens with circle fits at the apex to determine central curvature c. Deviation of the circle fit is described by point P on lens shape through asphericity k (Equation 2). With measured relation of apex to thickness of the isolated lens measured in photographs, the z-position of the apex is determined, and the imaged lens shape in the central zone can be described (Equation 1). Inversing of Equation 1 to Equation 3 can determine ceq and keq to fit the lens shape beyond point P in the marginal zone.
Figure 5
 
Results of study on ex vivo porcine lenses (n = 5) with mean amplitudes of thickness, diameter, anterior radius of curvature, posterior radius of curvature, and calculated optical power before (native) and after (treated) fs-laser treatment. **Statistical significance with P < 0.001. *Statistical significance with P < 0.05. †Lens 3 was omitted from statistics (n = 4).
Figure 5
 
Results of study on ex vivo porcine lenses (n = 5) with mean amplitudes of thickness, diameter, anterior radius of curvature, posterior radius of curvature, and calculated optical power before (native) and after (treated) fs-laser treatment. **Statistical significance with P < 0.001. *Statistical significance with P < 0.05. †Lens 3 was omitted from statistics (n = 4).
Figure 6
 
Simulated ray tracing (ZEMAX) of applying different apertures with the found parameters anterior and posterior curvatures, describing the central zone (near 0–5 mm in diameter). Here the states (a–d) of lens 1 are shown with the exemplary aperture of 3 mm in diameter, and the equatorial plane as common reference. Refractive index of the lens is neq = 1.4955 and surrounding medium is nh = 1.336. Back focal distance was determined by the quick focus function.
Figure 6
 
Simulated ray tracing (ZEMAX) of applying different apertures with the found parameters anterior and posterior curvatures, describing the central zone (near 0–5 mm in diameter). Here the states (a–d) of lens 1 are shown with the exemplary aperture of 3 mm in diameter, and the equatorial plane as common reference. Refractive index of the lens is neq = 1.4955 and surrounding medium is nh = 1.336. Back focal distance was determined by the quick focus function.
Figure 7
 
Simulated back focal distances (BFD) for different apertures in diameter of the parametrized lens contour describing the central zone. For each lens, the BFD is plotted in the stretched (diamond), unstretched (circle), native (gray line), and treated (black dashed line) state.
Figure 7
 
Simulated back focal distances (BFD) for different apertures in diameter of the parametrized lens contour describing the central zone. For each lens, the BFD is plotted in the stretched (diamond), unstretched (circle), native (gray line), and treated (black dashed line) state.
Figure 8
 
Differences between the native and treated state of the simulated back focal distances (BFD) over the aperture. For each of lenses 1, 2, 4, and 5, the Deltas of the BFD are shown for the unstretched and stretched state.
Figure 8
 
Differences between the native and treated state of the simulated back focal distances (BFD) over the aperture. For each of lenses 1, 2, 4, and 5, the Deltas of the BFD are shown for the unstretched and stretched state.
Figure 9
 
Porcine lens 1 before (left side) and after (right side) laser treatment in unstretched (black line) and stretched (gray line) states. Dashed lines mark the fitted part of the lens shape.
Figure 9
 
Porcine lens 1 before (left side) and after (right side) laser treatment in unstretched (black line) and stretched (gray line) states. Dashed lines mark the fitted part of the lens shape.
Table 1
 
Mean of Measured Parameters Including Calculated Optical Power for the Lenses in Native Unstretched and Stretched State (n = 5), With SD Denoting Standard Deviation
Table 1
 
Mean of Measured Parameters Including Calculated Optical Power for the Lenses in Native Unstretched and Stretched State (n = 5), With SD Denoting Standard Deviation
Table 2
 
Measured Parameters Including Calculated Optical Power for Unstretched and Stretched in Native and Treated State (n = 5), With SD Denoting Standard Deviation and P Denoting Probability
Table 2
 
Measured Parameters Including Calculated Optical Power for Unstretched and Stretched in Native and Treated State (n = 5), With SD Denoting Standard Deviation and P Denoting Probability
Table 3
 
Measured Parameters Including Calculated Optical Power for Native-Treated in Unstretched and Stretched State (n = 5), With SD Denoting Standard Deviation and P Denoting Probability
Table 3
 
Measured Parameters Including Calculated Optical Power for Native-Treated in Unstretched and Stretched State (n = 5), With SD Denoting Standard Deviation and P Denoting Probability
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