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To determine the relationship between accommodative amplitude and central lens thickness/equatorial lens diameter (CLT/ELD) ratio in vertebrates.
Midsagittal sections of lenses from fixed, post mortem eyes from 125 different vertebrate species were photographed. Their CLT/ELD ratios were correlated with independently published measurements of their accommodative amplitudes. Using the non‐linear finite element method (FEM), the efficiency of zonular traction (the absolute change in central radius of curvature per unit force [|ΔCR|/F]) for model lenses with CLT/ELD ratios from 0.45 to 0.9 was determined.
Vertebrates with CLT/ELD ratios 0.6 have high accommodative amplitudes. Zonular traction was found to be most efficient for those model lenses having CLT/ELD ratios 0.6.
Vertebrates with lenses that have CLT/ELD ratios 0.6 – i.e. “long oval” shapes – have the greatest accommodative amplitudes; e.g. primates, diving birds and diurnal birds of prey. Vertebrates that have oval or spherical shaped lenses, like owls and most mammals, have low accommodative amplitudes. Zonular traction was found to be most efficient when applied to model lenses with CLT/ELD ratios 0.6. The implications of these findings on the mechanism of accommodation are discussed.
The mechanism of vertebrate accommodation has fascinated scientists for hundreds of years. Multiple evolutionary mechanisms have developed to enable different species to accommodate: amphibians and snakes move their lenses forward; cyclostomes and teleosts move theirs backwards and in mammals the curvature of the lens changes.1 This latter means, particularly in man, is the most efficient accommodative mechanism of all.2,3,4,5,6,7
Diving birds and primates, most notably humans, exemplify the efficiency and speed of accommodation induced by deformation of the lens. Diving birds and primates, such as the Rhesus monkey, accommodate up to a remarkable 40 to 60 diopters.8 On the other hand, most other vertebrates with anatomically similar eyes and normal functioning ciliary muscles have minimal accommodative amplitude.2,3,4,5,6,7,9,10,11
It has been consistently observed that in those vertebrates that rely on a change in equatorial diameter for accommodation, the presence of a spherical lens usually indicates low accommodative amplitude,2,3,4,5,6,7 while rcently, it has been proposed that quantification of the ellipticity of vertebrate lenses may be used to differentiate vertebrates with high amplitudes of accommodation.12 The present study examines this hypothesis.
Using the non‐linear finite element method (FEM), the efficiency of zonular traction was tested by determining how the absolute change in central radius of curvature per unit force (|ΔCR|/F) varies with the ratio of central lens thickness to equatorial lens diameter (CLT/ELD). In addition, the CLT/ELD ratios of post mortem vertebrate eyes were compared to their accommodative amplitudes as published in the literature.
A non‐linear FEM, axisymmetrical generic two‐dimensional model lens was constructed using ABAQUS (Version 6.5, ABAQUS, Inc., Pawtucket, RI, 2005). Standard and hybrid quadrilateral elements were used for the capsule and lens, respectively. The number of lenticular elements was varied, until 1345 lenticular and 76 capsular elements were found to consistently result in a converged solution (fig 1A1A).). In general, a three‐dimensional model is more realistic; however, since the lens is axisymmetric, and proper constitutive relationships and a hybrid formulation were used, a two‐dimensional model is sufficient.13
For simplicity, the baseline profile of the model lens was a symmetrical ellipse. The baseline CLT/ELD ratio was varied from 0.45 to 0.9 by holding the equatorial diameter constant at 3.0 mm and changing the CLT. The lens capsule was assigned an elastic modulus of 1.5 MPa, a uniform thickness of 20 μm, and a Poisson's ratio of 0.47.13 The lens tissue was considered incompressible, with a Poisson's ratio equal to 0.5 and an elastic modulus of 150 Pa.13 Contact elements with a stiffness of 0.1 N/mm and a frictional coefficient of 0.005 were placed between the capsule and stroma.13 The zonules were modelled with an elastic modulus of 1.5 MPa.13 Incremental displacements of the zonules were applied and the forces were obtained from the analyses. The central radius of curvatures were obtained by determining the radii of the spheres that best fitted the central lenticular surfaces of each model lens within a 1.1 mm aperture before and after the application of zonular traction (fig 1B1B).13 The |ΔCR|/F, required to obtain a 0.06 mm (2%) increase in the ELD was quantified.
Paraffin blocks of the midsagittal sections of 125 vertebrate eyes from vertebrates that were at least 1 month of age were obtained from the Zoological Society of San Diego and photographed on a copy stand using a digital camera with a macro lens. A millimeter ruler was included in all photographs for calibration. The photographs were downloaded onto a computer and triplicate measurements, in pixels (Pixel Stick, version 1.1, www.pixelatedsoftware.com), of central thickness and equatorial diameter of each lens were made (fig 1C1C).). The person making the measurements (RAS) did not know the species at the time of the measurements, but did know the working hypothesis. The means and standard deviations of the CLT and ELD of the lenses, and their ratios, were calculated. The vertebrates were assigned high or low accommodative amplitudes according to literature reports based on their accommodation (high if >5 diopters) or classification and visual needs based on eating requirements.2,3,4,5,6,7,8,9,10,11 Accommodation induced by a change in corneal curvature, forward or backward movement of the lens, or which involved a system of force mediated by a structure other than the ciliary muscle were not included. Statistical analysis was performed using two binomial distributions: one for the vertebrates with high accommodative amplitudes and one for the vertebrates with low accommodative amplitudes.
The FEM predicted that when zonular traction was applied to model lenses with CLT/ELT ratios 0.6, the |ΔCR|/F was 0.13 mm/mN, and when the CLT/ELD ratio was 0.7, the |ΔCR|/F was 0.04 mm/mN (fig 2A2A).
Post mortem vertebrate lenses with CLT/ELD ratios 0.6 had high amplitudes of accommodation and lenses with ratios >0.6 had low accommodative amplitudes ((figsfigs 2B and 33;; ;tablestables 1 and 22).
Given the lack of overlap in CLT/ELD ratios between those with high accommodative amplitudes (n=22) and those with low accommodative amplitudes (n=103), it can be said with 95% confidence (calculated from two exact binomial 97.5% confidence intervals, one for the high accommodators and one for the low accommodators) that any point of separation between the CLT/ELD ratios of 0.58 and 0.61 (e.g. a CLT/ELD ratio =0.60) is simultaneously above the 84.6th percentile of the target population of the high accommodative amplitude group and below the 3.5th percentile of the target population of the low accommodative amplitude group.
Many vertebrates with minimal lenticular accommodation possess ciliary musculature with associated neural‐ciliary muscle function, zonules, lens capsules and lenticular material properties that do not significantly differ from vertebrates with large amplitudes of accommodation.2,3,4,5,6,7,9,10,11 Remaining possible explanations for the large differences in accommodative amplitudes between vertebrates with high and low amplitudes are geometric factors: the size and shape of their lenses and the relative size of the eye. In the rodent eye, for example, the lens occupies most of the interior of the eyeball, leaving little room for accommodation. Clearly such eyes have low or no accommodation. However, even when the lens does not occupy the bulk of the eyeball there will be variations in accommodative amplitude that are based on the geometry of the lens itself. This study has considered the effect of lens shape on accommodation.
The FEM analysis demonstrated that for a fixed increase in ELD, model lenses with the CLT/ELD ratios 0.60 responded at least three times more efficiently to zonular traction than lenses with CLT/ELD ratios 0.7 (fig 2A2A).). These results may not apply to all vertebrates because the FEM model lens was generic, symmetrical and assumed that the nucleus and cortex of the accommodating lens have the same material properties. However, midsagittal sections of vertebrate eyes revealed that vertebrates with CLT/ELD ratios 0.6 have high amplitudes of accommodation, while those eyes with CLT/ELD ratios >0.60 have low accommodative amplitudes. Only 125 vertebrate eyes were studied. Although there was a strong statistical association between shape and accommodative amplitude, some experimental error may have been arisen from fixation artifacts and deviation from perfect midsagittal sectioning. In addition, the ages of lenses varied and hence every lens can be treated representative of the species at that given age.
The qualitative accommodative amplitudes of vertebrates can be predicted from the CLT/ELD ratios of their lenses. Interestingly, it has been observed that the suture patterns of some vertebrate lenses relate to accommodative amplitude.14 This could be expected since the lens suture pattern directly relates to dimensions of the lens and therefore to the CLT/ELD ratio. However, it is unlikely that either the pattern of lens sutures or the innate structure of the sutures has a direct mechanical influence on accommodative amplitude. The lens sutures have a preferential orientation. Therefore, if the lens sutures had a direct effect on the mechanical response of the lens to zonular traction, the viscoelastic shear properties of the lens should be affected by lens orientation. No such relation has been found. Furthermore, it has been demonstrated that the shear modulus of the lens tissue is low and that the lens tissue is more viscous than elastic.15 Consequently, it is improbable that the relatively small area occupied by the lens sutures induces sufficient anisotropy in the lens to significantly affect its overall mechanical response to zonular traction.
The finding that vertebrate lenses with CLT/ELD ratios 0.6 – i.e. a “long oval” shape12 – have high amplitudes of accommodation may have implications for understanding the mechanism of accommodation. According to Helmholtz's theory of accommodation, the post mortem lens is expected to be in its most accommodated state given that the tension imposed by the ciliary muscle via the zonule is no longer functional. If this is the case, then post mortem lenses with the highest accommodative amplitudes could be expected to be more spherical than lenses from eyes with less or no accommodation. The opposite has been found. Post mortem lenses with a “long oval” shape have the largest accommodative amplitudes.
We wish to thank the Zoological Society of San Diego for supplying the vertebrate specimens and Rebecca Papendick, DVM and her staff for their outstanding assistance.
Competing interests: RAS has a financial interest in the surgical reversal of presbyopia