Finite element modelling of radial lentotomy cuts to improve the accommodation performance of the human lens.

PURPOSE: The use of a femtosecond laser to form planes of cavitation bubbles within the ocular lens has been proposed as a potential treatment for presbyopia. The intended purpose of these planes of cavitation bubbles (referred to in this paper as 'cutting planes') is to increase the compliance of the lens, with a consequential increase in the amplitude of accommodation. The current paper describes a computational modelling study, based on three-dimensional finite element analysis, to investigate the relationship between the geometric arrangement of the cutting planes and the resulting improvement in lens accommodation performance. The study is limited to radial cutting planes. METHODS: The effectiveness of a variety of cutting plane geometries was investigated by means of modelling studies conducted on a 45-year human lens. RESULTS: The results obtained from the analyses depend on the particular modelling procedures that are employed. When the lens substance is modelled as an incompressible material, radial cutting planes are found to be ineffective. However, when a poroelastic model is employed for the lens substance, radial cuts are shown to cause an increase in the computed accommodation performance of the lens. In this case, radial cuts made in the peripheral regions of the lens have a relatively small influence on the accommodation performance of the lens; the lentotomy process is seen to be more effective when cuts are made near to the polar axis. CONCLUSIONS: When the lens substance is modelled as a poroelastic material, the computational results suggest that useful improvements in lens accommodation performance can be achieved, provided that the radial cuts are extended to the polar axis. Radial cuts are ineffective when the lens substance is modelled as an incompressible material. Significant challenges remain in developing a safe and effective surgical procedure based on this lentotomy technique.

Shear modulus measurements on isolated human lens nuclei.

The use of a spinning lens test to determine ex vivo the shear modulus of 22 isolated human lens nuclei with ages ranging from 34 to 63 years is described. In this test procedure, the lens nucleus is spun about its polar axis. Images of the nucleus viewed from directions perpendicular to the polar axis are collected; these are used to quantify the deformations induced in the nucleus by the rotational motion. Data on these deformations are used to infer, by applying finite element inverse analysis, values for the shear modulus of the nucleus. The data on shear modulus obtained from this test program indicate that the nucleus stiffens very rapidly with age. These data are shown to compare well with the results of a related study (Wilde et al., 2012) in which the shear modulus of the nucleus is determined by similar spinning lens tests conducted on the entire lens substance.

Shear modulus data for the human lens determined from a spinning lens test.

The paper describes a program of mechanical testing on donated human eye bank lenses. The principal purpose of the tests was to obtain experimental data on the shear modulus of the lens for use in future computational models of the accommodation process. Testing was conducted using a procedure in which deformations are induced in the lens by spinning it about its polar axis. Shear modulus data were inferred from these observed deformations by means of a finite element inverse analysis procedure in which the spatial variation of the shear modulus within the lens is represented by an appropriate function (see Burd et al., 2011 for a detailed specification of the design of the spinning lens test rig, experimental protocols and associated data analysis procedures that were employed in the tests). Inferred data on lens shear modulus are presented for a set of twenty-nine lenses in the age range 12 years to 58 years. The lenses were tested between 47 h and 110 h from the time of death (average post-mortem time 74 h). Care was taken to exclude any lenses that had been affected by excessive post-mortem swelling, or any lenses that had suffered mechanical damage during storage, transit or the testing process. The experimental data on shear modulus indicate that, for young lenses, the cortex is stiffer than the nucleus. The shear modulus of the nucleus and cortex both increase with increasing age. The shear modulus of the nucleus increases more rapidly than the cortex with the consequence that from an age of about 45 years onwards the nucleus is stiffer than the cortex. The principal shear modulus data presented in the paper were obtained by testing at a rotational speed of 1,000 rpm. Supplementary tests were conducted at rotational speeds of 700 rpm and 1,400 rpm. The results from these supplementary tests are in good agreement with the data obtained from the principal 1,000 rpm tests. Studies on the possible effects of lens drying during the test suggested that this factor is unlikely to have led to significant errors in the experimental determination of the shear modulus. The shear modulus data presented in the paper are used to develop 'age-stiffness' models to represent the shear modulus of the lens as a function of age. These models are in a form that may be readily incorporated in a finite element model of the accommodation process. A comparison is attempted between the shear modulus data presented in the current paper and equivalent data published by previous authors. This comparison highlights various limitations and inconsistencies in the data sets.

An improved spinning lens test to determine the stiffness of the human lens.

It is widely accepted that age-related changes in lens stiffness are significant for the development of presbyopia. However, precise details on the relative importance of age-related changes in the stiffness of the lens, in comparison with other potential mechanisms for the development of presbyopia, have not yet been established. One contributing factor to this uncertainty is the paucity and variability of experimental data on lens stiffness. The available published data generally indicate that stiffness varies spatially within the lens and that stiffness parameters tend to increase with age. However, considerable differences exist between these published data sets, both qualitatively and quantitatively. The current paper describes new and improved methods, based on the spinning lens approach pioneered by Fisher, R.F. (1971) 'The elastic constants of the human lens', Journal of Physiology, 212, 147-180, to make measurements on the stiffness of the human lens. These new procedures have been developed in an attempt to eliminate, or at least substantially reduce, various systematic errors in Fisher's original experiment. An improved test rig has been constructed and a new modelling procedure for determining lens stiffness parameters from observations made during the test has been devised. The experiment involves mounting a human lens on a vertical rotor so that the lens spins on its optical axis (typically at 1000 rpm). An automatic imaging system is used to capture the outline of the lens, while it is rotating, at pre-determined angular orientations. These images are used to quantify the deformations developed in the lens as a consequence of the centripetal forces induced by the rotation. Lens stiffness is inferred using axisymmetric finite element inverse analysis in which a nearly-incompressible neo-Hookean constitutive model is used to represent the mechanics of the lens. A numerical optimisation procedure is used to determine the stiffness parameters that provide a best fit between the finite element model and the experimental data. Sample results are presented for a human lens of age 33 years.

Can reliable values of Young's modulus be deduced from Fisher's (1971) spinning lens measurements?

The current textbook view of the causes of presbyopia rests very largely on a series of experiments reported by R.F. Fisher some three decades ago, and in particular on the values of lens Young's modulus inferred from the deformation caused by spinning excised lenses about their optical axis (Fisher 1971) We studied the extent to which inferred values of Young's modulus are influenced by assumptions inherent in the mathematical procedures used by Fisher to interpret the test and we investigated several alternative interpretation methods. The results suggest that modelling assumptions inherent in Fisher's original method may have led to systematic errors in the determination of the Young's modulus of the cortex and nucleus. Fisher's conclusion that the cortex is stiffer than the nucleus, particularly in middle age, may be an artefact associated with these systematic errors. Moreover, none of the models we explored are able to account for Fisher's claim that the removal of the capsule has only a modest effect on the deformations induced in the spinning lens.