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Shear modulus measurements on isolated human lens nuclei
Experimental Eye Research 103 (2012) 78e81
Contents lists available at SciVerse ScienceDirect
Experimental Eye Research journal homepage: www.elsevier.com/locate/yexer
Short communication
Shear modulus measurements on isolated human lens nuclei C.-K. Chai, H.J. Burd*, G.S. Wilde Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 March 2012 Accepted in revised form 10 August 2012 Available online 21 August 2012
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. Ó 2012 Published by Elsevier Ltd.
Keywords: lens nucleus shear modulus
The accommodation process in humans is driven by the shape changes that develop in the lens in response to contraction of the ciliary muscle. As a consequence of natural ageing processes, however, it is generally understood that the stiffness of the lens increases with age. These age-related stiffness changes are widely assumed to be a key contributor to presbyopia. Previous researchers have collected data on lens stiffness using a variety of experimental techniques such as spinning lens testing (Fisher, 1971; Burd et al., 2011; Wilde et al., 2012), dynamic mechanical analysis (Weeber et al., 2005) indentation testing (Weeber et al., 2007; Heys et al., 2004, 2007) and acoustic methods (Hollman et al., 2007). These data indicate a similar broad tendency for the stiffness of the lens to increase with increasing age. However, they have generally been obtained at different geometric locations in the lens and with different lens storage, experimental and data analysis protocols. These issues, together with the fact that the number of experimental data sets on lens stiffness currently available in the literature is relatively small, mean that it is not yet possible to achieve a detailed description of the mechanical changes that occur within the lens as a consequence of ageing. A program of research has been recently completed at Oxford University on the use of a spinning lens test procedure, based on the approach initially devised by Fisher (1971), to collect new data on the stiffness characteristics (specifically, the shear modulus) of the human lens substance. The spinning lens rig and associated experimental protocols for conducting these measurements on the
* Corresponding author. E-mail address: [email protected] (H.J. Burd). 0014-4835/$ e see front matter Ó 2012 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.exer.2012.08.003
human lens is specified in Burd et al. (2011) and a program of stiffness tests on eye bank lenses of a range of ages is described in Wilde et al. (2012). The current paper describes a series of tests that are closely related to the experiments in Wilde et al. (2012). The tests described in Wilde et al. (2012) are referred to below as the ‘main tests’; the tests described in the current paper are referred to as the ‘subsidiary’ tests. An important feature of the spinning lens approach (in the form described by Burd et al., 2011) is that it is necessary to infer data on lens shear modulus on the basis of observations on the surface deformations induced in the lens when it is spun about its polar axis. Since the lens is widely regarded as being non-homogeneous, this requires a data processing protocol in which the spatial variation of shear modulus within the lens is specified in an appropriate parameterized form, with the values of the relevant parameters being determined using finite element inverse analysis. This test protocol has certain advantages over alternative procedures (e.g. the risk of mechanical damage to the lens associated with the sample preparation methods required for indentation testing is avoided). Nevertheless, it is noted that the spinning lens test is an indirect method in the sense that it relies on the use of a mathematical optimization procedure to infer spatial variations of shear modulus from the observed geometric changes that occur during the test. The results of the main test program were interpreted using different assumed forms of spatial variation of shear modulus within the lens. In one of these forms e Model D e the lens is represented as a two-compartment model comprising a mechanically homogeneous nucleus embedded in a mechanically homogeneous cortex. The values of nucleus shear modulus determined
C.-K. Chai et al. / Experimental Eye Research 103 (2012) 78e81
for Model D have particular relevance to the nucleus stiffness data presented in the current paper. A curve fitted to these data (Wilde et al., 2012) determined for lenses in the main test program, age range 12e58 years, is:
log10
GN c
¼
A A* b1 A A * * A>A* b2 A A
(1)
where GN is nucleus shear modulus, A is age in years and the parameters A*, c, b1, b2 are 27.5 years, 68.8 Pa, 0.00562 years1 and 0.0767 years1 respectively. The data processing procedures adopted in the main test program required assumptions to be made on the dimensions of the nucleus. These assumptions (see Burd et al., 2011) were based principally on information gleaned from the literature. However, to provide additional data on nucleus geometry, attempts were made to isolate the nuclei in some of the lenses tested in the main program once testing on each lens had been completed; dimensional measurements could then be made on the nuclear core that had been isolated in this way. Furthermore, it proved to be possible to employ the spinning lens rig used in the main test program to obtain an independent measurement of shear modulus for these isolated nuclei. The current paper describes these subsidiary tests on the isolated lens nuclei. It is emphasised that the principal purpose of these subsidiary tests was to provide a serendipitous check on the data obtained in the main test program rather than a new, independent, data set. The experimental procedure adopted in the subsidiary tests is as follows. Immediately after completion of some of the tests in the main test program, the de-capsulated lens was placed in water (rather than the isotonic solution used to transport and store the lenses for the main tests) for a suitable length of time (average 1 h 46 min). During this period, the lens was occasionally agitated, gently, with an ophthalmic spear; this caused a gradual swelling of the cortical layers of the lens fibres, which then tended to slough off to reveal a relatively stable inner core. This inner core generally showed a greater resistance to the swelling and sloughing off process than the surrounding material. Any loose material that remained attached to the stable inner core after this process was completed was carefully removed prior to testing. This process is similar to the water dissection procedure that has been previously described by Augusteyn (2010). It is noted that the lens has a complex internal structure. Although the term ‘nucleus’ is widely used in the literature to describe an inner portion of the lens, previous authors have used different nomenclatures and definitions to determine the geometrical boundary of the nucleus (e.g. Augusteyn, 2010). As a consequence, the term ‘nucleus’ does not currently have an unambiguous anatomical meaning. Augusteyn (2010) uses the term ‘nuclear core’ to describe the stable core of material that remains after the cortex has been removed by in vitro water dissection; we therefore adopt the nomenclature ‘nuclear core’ to describe the stable inner core of material obtained by our own dissection process. All instances in this paper where the term ‘nucleus’ or ‘nuclei’ is used to describe this inner core of the lens substance should be interpreted as referring to the nuclear core as described by Augusteyn (2010). Once the nuclear core had been extracted it was returned to the spinning lens rig and tests were conducted using protocols that correspond closely to those adopted in the main test program. Each nucleus was placed carefully, with the anterior pole uppermost, on the lens support ring. Ophthalmic spears were used to manipulate the nucleus so that the polar axis coincided with the axis of rotation. The accuracy with which the nucleus was located on the
79
support ring was checked by eye. Once the nucleus had been correctly mounted on the rotor, the lens box was replaced and gauze fixed to the sides of the box was moistened. The nucleus was then subjected to a test sequence involving reference tests conducted at 70 rpm together with tests at higher speeds for the purpose of inducing measurable deformations in the nucleus. Without exception, 70 rpm reference tests were conducted before, and after, each higher speed test. These reference tests were for the purpose of establishing a ‘reference outline’ for the nucleus. (The reference outline is an axisymmetric representation of the nucleus cross-section when the effects of rotational body forces are negligible, see Burd et al., 2011.) A typical test sequence (that was applied to 10 of the nuclei in the current study) is shown in Table 1. Alternative sequences (which were based on a similar general pattern to the sequence shown in Table 1) were employed for the other 12 nuclei that were tested; these alternative sequences were intended to investigate, in a non-systematic way, the influence of rotational speed on the mechanical response of the nucleus. In some cases, the sequence included tests at 1000 rpm. For the two 63-year nuclei, the test sequence that was employed included testing at 3465 rpm. Importantly, however, all of the test sequences employed in the study included a 2000 rpm test; this particular speed is therefore adopted as the standard condition for the determination of the nucleus shear modulus data presented in the current paper. In some cases the nuclei fell off the support ring when the rotational speed was increased beyond 2000 rpm. In all these cases, however, a complete data set had already been obtained for the 2000 rpm test. No further tests were conducted on nuclei that had fallen off the support ring in this way. Processing of the images collected in the tests followed closely the procedures described in Burd et al. (2011). Initially, a custom process, based on MATLABÒ, is used to determine the edges of the nucleus from each of the 16 images collected from the two reference tests that straddle the test at 2000 rpm (for example, in Table 1 the relevant reference tests are NR1 and NR2). These nucleus outlines are then used to define a cubic spline representation of the reference outline. An axisymmetric finite element mesh is then generated on this reference outline. These meshes consisted of fifteennoded triangles with, in the region of, 12,000 nodes and 1500 elements. Next, the edges of the nucleus from each of the 8 images collected from the test at 2000 rpm (test reference NT1 in the sequence shown in Table 1) is used to determine an axisymmetric spline representation of the target outline. The shear modulus for the nucleus is determined using finite element inverse analysis. In this approach the nucleus is represented as a homogeneous nearly-incompressible neo-Hookean material with a shear modulus G. A forward finite element analysis of the test is conducted, based on an assumed value of G; this provides a computed outline for the nucleus. An optimisation process is then used to determine the value of nucleus shear modulus that provides a best fit between the computed outline and the target outline, specifically by minimising the area enclosed between the two outlines. These procedures are based on the computational methods described in Burd et al. (2011). In conducting this optimization procedure, some uncertainty exists on the most appropriate support conditions to be assumed at Table 1 Typical test sequence. The resulting image sets are referred to by the label given in the top row, with R standing for ‘reference’ and T standing for ‘test’. Test reference
NR1
NT1
NR2
NT2
NR3
NT3
NR4
Rotational speed (rpm)
70
2000
70
2450
70
3000
70
80
C.-K. Chai et al. / Experimental Eye Research 103 (2012) 78e81
the interface between the nucleus and the lens support ring (Burd et al., 2011). Two separate conditions have been assumed in the current study. In one case e termed constraint F e the nodes in the nucleus mesh in contact with the support ring are assumed to be fully fixed. In the other case e termed constraint S e the nodes in contact with the support ring are allowed to slide, with no friction or cohesion, over the support ring. These conditions have been applied separately to determine two, independent, values of shear modulus for each nucleus. It seems likely that the conditions implied by constraints F and S correspond to the limits of plausible behaviour at the nucleus/ring interface. A total of 22 lens nuclei has been investigated in this way. These lens nuclei had ages ranging from 34 to 63 years. Most of these nuclei were drawn from the group (termed Set G in Wilde et al., 2012) that comprised lenses that were regarded as being of sufficiently high quality to merit inclusion in main test program. In some cases, lenses that had been excluded from Set G on the basis of the criteria adopted in the main tests for swelling, surface fluid or superficial mechanical damage were nevertheless included in the current study. In these cases, any disturbance to the tissue was assumed to have been confined to the outer layers of the lens; the nucleus would therefore have been unaffected. Of the 22 lens nuclei included in the current study, 12 were extracted from lenses that are included in Set G. Data on the equatorial diameter and polar thickness of the nuclei, determined from the reference outlines, are plotted in Fig. 1. These data do not show any significant dependence on age. The dimensions are consistent with the observations reported by Augusteyn (2010) that water dissection may be used to obtain a “relatively stable core measuring 6.5e7 3 mm”. It is noted, in passing, that the data also correspond, closely, to the dimensions of the apparent diffusion barrier in the lens reported by Sweeney and Truscott (1998). The nucleus dimensions plotted in Fig. 1 correspond closely to the geometric parameters assumed for the nucleus in Model D (Burd et al., 2011). Since the dimensions of lens nucleus determined from these subsidiary tests were used to inform the choice of nucleus dimension adopted in the main test program, this reasonably close correspondence is not unexpected. Importantly, this similarity between the geometric model for the nucleus used in the main tests and the dimensions of the nuclear cores that are investigated in the current subsidiary tests allows a fair comparison
Fig. 2. A 41-year nuclear core rotating at 70 rpm (left) and spinning at 3000 rpm (right).
to be made between nucleus shear modulus data obtained from both studies. Typical images recorded during a test sequence are illustrated in Fig. 2. The left image, obtained at a speed of 70 rpm, shows half of a typical image used to establish the reference outline. The right image shows the nucleus when spinning at 3000 rpm; the deformations associated with the rotational motion are clearly visible. Data on shear modulus for the set of 22 lenses included in the current study, determined at a rotational speed of 2000 rpm, are shown in Fig. 3. A pair of shear modulus data for each nucleus e shown connected by a vertical line e are plotted in this figure. One value was obtained using constraint S in the finite element analysis and the other using constraint F. Shear modulus values obtained using constraint S were invariably larger than the data obtained using constraint F. Also shown in Fig. 3 is the age-stiffness model for Model D determined from the main test program. Individual data on Model D nucleus shear modulus, GN, obtained from the main tests (Fig. 8b
Nucleus shear modulus Model D GN
10000 Shear modulus (Pa)
10
Dimension (mm)
8 6 4 2
1000
Nucleus diameter Polar thickness
0 30
40
50 Age (years)
60
100 30
40
50
60
70
Age (years) Fig. 1. Data on equatorial diameter and polar thickness of the nuclear core determined from the reference outlines. The dashed lines show the values of nucleus diameter and polar thickness (6.9 mm and 2.83 mm respectively, taken from Gullapalli et al., 1995) adopted in the main test program for the determination of shear modulus parameters for Model D.
Fig. 3. Pairs of nucleus shear modulus data (shown as filled circles) for the twenty-two lens nuclei included in the current study. Also shown, from the main test program (Wilde et al., 2012) is the ageestiffness relationship for Model D (dashed line) and individual nucleus shear modulus data GN (shown as crosses).
C.-K. Chai et al. / Experimental Eye Research 103 (2012) 78e81
of Wilde et al., 2012) are also included in the plot. The current data correlate reasonably well with the Model D relationship and the main test data, although the quality of this correlation does appear to deteriorate with increasing age. It is noted that one of the data pairs in Fig. 3 (for a nucleus of age 48 years) lies well below the general trend implied by the rest of the data. This particular nucleus was not included in Set G and so comparisons cannot be made with the results of the main test program. One possible explanation for the anomalous behaviour of this lens is that the age of the donor may have been incorrectly transcribed in the documentation received from the eye bank. However, no evidence of any error of this sort could be found in the documentation made available to the authors and so the matter remains unresolved. In most cases, tests at 1000 rpm did not deform the nucleus sufficiently to permit a useful analysis, but data from tests conducted at high speeds (i.e. 2450 rpm, 3000 rpm and 3465 rpm) generally accord well with those collected at 2000 rpm. These higher speeds may in fact be more reliable when testing older, stiffer nuclei as the larger deformations induced are likely to lead to greater accuracy in the analysis. The nucleus stiffness data presented in this paper were obtained as an extension to the main set of spinning lens tests on the lens substance (Wilde et al., 2012). In this main test program, considerable care was taken to minimise the possibility of lens swelling prior to the test and to exclude lenses for which mechanical or swelling damage was suspected. In contrast, the data on lens nucleus presented in the current paper were obtained by mechanical tests on a lens that had been deliberately subjected to an uncertain process in which swelling was induced in the cortex, so that the cortex fibres could be removed to reveal the nuclear core. The quality of the nuclei tested in the current test program is therefore regarded as being less certain than the quality of lenses tested in the main test program. It is perhaps surprising, therefore, that reasonable agreement is achieved between the current tests and the data obtained from the main test program. It is noted that both sets of tests indicate that the nucleus stiffness increases very substantially with age (for example, the data in Fig. 3 suggest that the shear modulus increases
81
by about an order of magnitude between the ages of 40 years and 60 years). The exponential nature of the relationship between age and shear modulus means that any small differences in the rate of ageing between individuals are likely to be amplified, leading to relatively large differences in nucleus shear modulus when those individuals share the same age. It seems plausible to the authors that effects of this sort will tend to swamp any minor adjustments to the stiffness of the nucleus (e.g. as a consequence of water absorption during the process of isolating the nucleus) that occur due to shortcomings in the preparation of the nucleus prior to the test. The project was supported financially by the Wellcome Trust. The assistance of Dr Val. Smith and colleagues at the Bristol Eye Bank is gratefully acknowledged. References Augusteyn, R.C., 2010. On the growth and internal structure of the human lens. Exp. Eye Res. 90 (6), 643e654. Burd, H.J., Wilde, G.S., Judge, S.J., 2011. An improved spinning lens test to determine the stiffness of the human lens. Exp. Eye Res. 92 (1), 28e39. Fisher, R.F., 1971. The elastic constants of the human lens. J. Physiol. 212 (1), 147e 180. Gullapalli, V.K., Murthy, P.R., Murthy, K.R., 1995. Colour of the nucleus as a marker of nuclear hardness, diameter and central thickness. Indian J. Ophthalmol. 43, 181e184. Heys, K.R., Cram, S.L., Truscott, R.J.W., 2004. Massive increase in the stiffness of the human lens nucleus with age: the basis for presbyopia? Mol. Vis. 10, 956e963. Heys, K.R., Friedrich, M.G., Truscott, R.J.W., 2007. Presbyopia and heat: changes associated with aging of the human lens suggest a functional role for the small heat shock protein, a-crystallin, in maintaining lens flexibility. Aging Cell 6 (6), 807e815. Hollman, K.W., O’Donnell, M., Erpelding, T.N., 2007. Mapping elasticity in human lenses using bubble-based acoustic radiation force. Exp. Eye Res. 85 (6), 890e 893. Sweeney, M.H.J., Truscott, R.J.W., 1998. An impediment to glutathione diffusion in older normal human lenses: a possible precondition for nuclear cataract. Exp. Eye Res. 67 (5), 587e595. Weeber, H.A., Eckert, G., Soergel, F., Meyer, C.H., Pechhold, W., van der Heijde, R.G.L., 2005. Dynamic mechanical properties of human lenses. Exp. Eye Res. 80 (3), 425e434. Weeber, H.A., Eckert, G., Pechhold, W., van der Heijde, R.G.L., 2007. Stiffness gradient in the crystalline lens. Graefe’s Arch. Clin. Exp. Ophthalmol. 245 (9), 1357e1366. Wilde, G.S., Burd, H.J., Judge, S.J., 2012. Shear modulus data for the human lens determined from a spinning lens test. Exp. Eye Res. 97, 36e48.
Contents lists available at SciVerse ScienceDirect
Experimental Eye Research journal homepage: www.elsevier.com/locate/yexer
Short communication
Shear modulus measurements on isolated human lens nuclei C.-K. Chai, H.J. Burd*, G.S. Wilde Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 March 2012 Accepted in revised form 10 August 2012 Available online 21 August 2012
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. Ó 2012 Published by Elsevier Ltd.
Keywords: lens nucleus shear modulus
The accommodation process in humans is driven by the shape changes that develop in the lens in response to contraction of the ciliary muscle. As a consequence of natural ageing processes, however, it is generally understood that the stiffness of the lens increases with age. These age-related stiffness changes are widely assumed to be a key contributor to presbyopia. Previous researchers have collected data on lens stiffness using a variety of experimental techniques such as spinning lens testing (Fisher, 1971; Burd et al., 2011; Wilde et al., 2012), dynamic mechanical analysis (Weeber et al., 2005) indentation testing (Weeber et al., 2007; Heys et al., 2004, 2007) and acoustic methods (Hollman et al., 2007). These data indicate a similar broad tendency for the stiffness of the lens to increase with increasing age. However, they have generally been obtained at different geometric locations in the lens and with different lens storage, experimental and data analysis protocols. These issues, together with the fact that the number of experimental data sets on lens stiffness currently available in the literature is relatively small, mean that it is not yet possible to achieve a detailed description of the mechanical changes that occur within the lens as a consequence of ageing. A program of research has been recently completed at Oxford University on the use of a spinning lens test procedure, based on the approach initially devised by Fisher (1971), to collect new data on the stiffness characteristics (specifically, the shear modulus) of the human lens substance. The spinning lens rig and associated experimental protocols for conducting these measurements on the
* Corresponding author. E-mail address: [email protected] (H.J. Burd). 0014-4835/$ e see front matter Ó 2012 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.exer.2012.08.003
human lens is specified in Burd et al. (2011) and a program of stiffness tests on eye bank lenses of a range of ages is described in Wilde et al. (2012). The current paper describes a series of tests that are closely related to the experiments in Wilde et al. (2012). The tests described in Wilde et al. (2012) are referred to below as the ‘main tests’; the tests described in the current paper are referred to as the ‘subsidiary’ tests. An important feature of the spinning lens approach (in the form described by Burd et al., 2011) is that it is necessary to infer data on lens shear modulus on the basis of observations on the surface deformations induced in the lens when it is spun about its polar axis. Since the lens is widely regarded as being non-homogeneous, this requires a data processing protocol in which the spatial variation of shear modulus within the lens is specified in an appropriate parameterized form, with the values of the relevant parameters being determined using finite element inverse analysis. This test protocol has certain advantages over alternative procedures (e.g. the risk of mechanical damage to the lens associated with the sample preparation methods required for indentation testing is avoided). Nevertheless, it is noted that the spinning lens test is an indirect method in the sense that it relies on the use of a mathematical optimization procedure to infer spatial variations of shear modulus from the observed geometric changes that occur during the test. The results of the main test program were interpreted using different assumed forms of spatial variation of shear modulus within the lens. In one of these forms e Model D e the lens is represented as a two-compartment model comprising a mechanically homogeneous nucleus embedded in a mechanically homogeneous cortex. The values of nucleus shear modulus determined
C.-K. Chai et al. / Experimental Eye Research 103 (2012) 78e81
for Model D have particular relevance to the nucleus stiffness data presented in the current paper. A curve fitted to these data (Wilde et al., 2012) determined for lenses in the main test program, age range 12e58 years, is:
log10
GN c
¼
A A* b1 A A * * A>A* b2 A A
(1)
where GN is nucleus shear modulus, A is age in years and the parameters A*, c, b1, b2 are 27.5 years, 68.8 Pa, 0.00562 years1 and 0.0767 years1 respectively. The data processing procedures adopted in the main test program required assumptions to be made on the dimensions of the nucleus. These assumptions (see Burd et al., 2011) were based principally on information gleaned from the literature. However, to provide additional data on nucleus geometry, attempts were made to isolate the nuclei in some of the lenses tested in the main program once testing on each lens had been completed; dimensional measurements could then be made on the nuclear core that had been isolated in this way. Furthermore, it proved to be possible to employ the spinning lens rig used in the main test program to obtain an independent measurement of shear modulus for these isolated nuclei. The current paper describes these subsidiary tests on the isolated lens nuclei. It is emphasised that the principal purpose of these subsidiary tests was to provide a serendipitous check on the data obtained in the main test program rather than a new, independent, data set. The experimental procedure adopted in the subsidiary tests is as follows. Immediately after completion of some of the tests in the main test program, the de-capsulated lens was placed in water (rather than the isotonic solution used to transport and store the lenses for the main tests) for a suitable length of time (average 1 h 46 min). During this period, the lens was occasionally agitated, gently, with an ophthalmic spear; this caused a gradual swelling of the cortical layers of the lens fibres, which then tended to slough off to reveal a relatively stable inner core. This inner core generally showed a greater resistance to the swelling and sloughing off process than the surrounding material. Any loose material that remained attached to the stable inner core after this process was completed was carefully removed prior to testing. This process is similar to the water dissection procedure that has been previously described by Augusteyn (2010). It is noted that the lens has a complex internal structure. Although the term ‘nucleus’ is widely used in the literature to describe an inner portion of the lens, previous authors have used different nomenclatures and definitions to determine the geometrical boundary of the nucleus (e.g. Augusteyn, 2010). As a consequence, the term ‘nucleus’ does not currently have an unambiguous anatomical meaning. Augusteyn (2010) uses the term ‘nuclear core’ to describe the stable core of material that remains after the cortex has been removed by in vitro water dissection; we therefore adopt the nomenclature ‘nuclear core’ to describe the stable inner core of material obtained by our own dissection process. All instances in this paper where the term ‘nucleus’ or ‘nuclei’ is used to describe this inner core of the lens substance should be interpreted as referring to the nuclear core as described by Augusteyn (2010). Once the nuclear core had been extracted it was returned to the spinning lens rig and tests were conducted using protocols that correspond closely to those adopted in the main test program. Each nucleus was placed carefully, with the anterior pole uppermost, on the lens support ring. Ophthalmic spears were used to manipulate the nucleus so that the polar axis coincided with the axis of rotation. The accuracy with which the nucleus was located on the
79
support ring was checked by eye. Once the nucleus had been correctly mounted on the rotor, the lens box was replaced and gauze fixed to the sides of the box was moistened. The nucleus was then subjected to a test sequence involving reference tests conducted at 70 rpm together with tests at higher speeds for the purpose of inducing measurable deformations in the nucleus. Without exception, 70 rpm reference tests were conducted before, and after, each higher speed test. These reference tests were for the purpose of establishing a ‘reference outline’ for the nucleus. (The reference outline is an axisymmetric representation of the nucleus cross-section when the effects of rotational body forces are negligible, see Burd et al., 2011.) A typical test sequence (that was applied to 10 of the nuclei in the current study) is shown in Table 1. Alternative sequences (which were based on a similar general pattern to the sequence shown in Table 1) were employed for the other 12 nuclei that were tested; these alternative sequences were intended to investigate, in a non-systematic way, the influence of rotational speed on the mechanical response of the nucleus. In some cases, the sequence included tests at 1000 rpm. For the two 63-year nuclei, the test sequence that was employed included testing at 3465 rpm. Importantly, however, all of the test sequences employed in the study included a 2000 rpm test; this particular speed is therefore adopted as the standard condition for the determination of the nucleus shear modulus data presented in the current paper. In some cases the nuclei fell off the support ring when the rotational speed was increased beyond 2000 rpm. In all these cases, however, a complete data set had already been obtained for the 2000 rpm test. No further tests were conducted on nuclei that had fallen off the support ring in this way. Processing of the images collected in the tests followed closely the procedures described in Burd et al. (2011). Initially, a custom process, based on MATLABÒ, is used to determine the edges of the nucleus from each of the 16 images collected from the two reference tests that straddle the test at 2000 rpm (for example, in Table 1 the relevant reference tests are NR1 and NR2). These nucleus outlines are then used to define a cubic spline representation of the reference outline. An axisymmetric finite element mesh is then generated on this reference outline. These meshes consisted of fifteennoded triangles with, in the region of, 12,000 nodes and 1500 elements. Next, the edges of the nucleus from each of the 8 images collected from the test at 2000 rpm (test reference NT1 in the sequence shown in Table 1) is used to determine an axisymmetric spline representation of the target outline. The shear modulus for the nucleus is determined using finite element inverse analysis. In this approach the nucleus is represented as a homogeneous nearly-incompressible neo-Hookean material with a shear modulus G. A forward finite element analysis of the test is conducted, based on an assumed value of G; this provides a computed outline for the nucleus. An optimisation process is then used to determine the value of nucleus shear modulus that provides a best fit between the computed outline and the target outline, specifically by minimising the area enclosed between the two outlines. These procedures are based on the computational methods described in Burd et al. (2011). In conducting this optimization procedure, some uncertainty exists on the most appropriate support conditions to be assumed at Table 1 Typical test sequence. The resulting image sets are referred to by the label given in the top row, with R standing for ‘reference’ and T standing for ‘test’. Test reference
NR1
NT1
NR2
NT2
NR3
NT3
NR4
Rotational speed (rpm)
70
2000
70
2450
70
3000
70
80
C.-K. Chai et al. / Experimental Eye Research 103 (2012) 78e81
the interface between the nucleus and the lens support ring (Burd et al., 2011). Two separate conditions have been assumed in the current study. In one case e termed constraint F e the nodes in the nucleus mesh in contact with the support ring are assumed to be fully fixed. In the other case e termed constraint S e the nodes in contact with the support ring are allowed to slide, with no friction or cohesion, over the support ring. These conditions have been applied separately to determine two, independent, values of shear modulus for each nucleus. It seems likely that the conditions implied by constraints F and S correspond to the limits of plausible behaviour at the nucleus/ring interface. A total of 22 lens nuclei has been investigated in this way. These lens nuclei had ages ranging from 34 to 63 years. Most of these nuclei were drawn from the group (termed Set G in Wilde et al., 2012) that comprised lenses that were regarded as being of sufficiently high quality to merit inclusion in main test program. In some cases, lenses that had been excluded from Set G on the basis of the criteria adopted in the main tests for swelling, surface fluid or superficial mechanical damage were nevertheless included in the current study. In these cases, any disturbance to the tissue was assumed to have been confined to the outer layers of the lens; the nucleus would therefore have been unaffected. Of the 22 lens nuclei included in the current study, 12 were extracted from lenses that are included in Set G. Data on the equatorial diameter and polar thickness of the nuclei, determined from the reference outlines, are plotted in Fig. 1. These data do not show any significant dependence on age. The dimensions are consistent with the observations reported by Augusteyn (2010) that water dissection may be used to obtain a “relatively stable core measuring 6.5e7 3 mm”. It is noted, in passing, that the data also correspond, closely, to the dimensions of the apparent diffusion barrier in the lens reported by Sweeney and Truscott (1998). The nucleus dimensions plotted in Fig. 1 correspond closely to the geometric parameters assumed for the nucleus in Model D (Burd et al., 2011). Since the dimensions of lens nucleus determined from these subsidiary tests were used to inform the choice of nucleus dimension adopted in the main test program, this reasonably close correspondence is not unexpected. Importantly, this similarity between the geometric model for the nucleus used in the main tests and the dimensions of the nuclear cores that are investigated in the current subsidiary tests allows a fair comparison
Fig. 2. A 41-year nuclear core rotating at 70 rpm (left) and spinning at 3000 rpm (right).
to be made between nucleus shear modulus data obtained from both studies. Typical images recorded during a test sequence are illustrated in Fig. 2. The left image, obtained at a speed of 70 rpm, shows half of a typical image used to establish the reference outline. The right image shows the nucleus when spinning at 3000 rpm; the deformations associated with the rotational motion are clearly visible. Data on shear modulus for the set of 22 lenses included in the current study, determined at a rotational speed of 2000 rpm, are shown in Fig. 3. A pair of shear modulus data for each nucleus e shown connected by a vertical line e are plotted in this figure. One value was obtained using constraint S in the finite element analysis and the other using constraint F. Shear modulus values obtained using constraint S were invariably larger than the data obtained using constraint F. Also shown in Fig. 3 is the age-stiffness model for Model D determined from the main test program. Individual data on Model D nucleus shear modulus, GN, obtained from the main tests (Fig. 8b
Nucleus shear modulus Model D GN
10000 Shear modulus (Pa)
10
Dimension (mm)
8 6 4 2
1000
Nucleus diameter Polar thickness
0 30
40
50 Age (years)
60
100 30
40
50
60
70
Age (years) Fig. 1. Data on equatorial diameter and polar thickness of the nuclear core determined from the reference outlines. The dashed lines show the values of nucleus diameter and polar thickness (6.9 mm and 2.83 mm respectively, taken from Gullapalli et al., 1995) adopted in the main test program for the determination of shear modulus parameters for Model D.
Fig. 3. Pairs of nucleus shear modulus data (shown as filled circles) for the twenty-two lens nuclei included in the current study. Also shown, from the main test program (Wilde et al., 2012) is the ageestiffness relationship for Model D (dashed line) and individual nucleus shear modulus data GN (shown as crosses).
C.-K. Chai et al. / Experimental Eye Research 103 (2012) 78e81
of Wilde et al., 2012) are also included in the plot. The current data correlate reasonably well with the Model D relationship and the main test data, although the quality of this correlation does appear to deteriorate with increasing age. It is noted that one of the data pairs in Fig. 3 (for a nucleus of age 48 years) lies well below the general trend implied by the rest of the data. This particular nucleus was not included in Set G and so comparisons cannot be made with the results of the main test program. One possible explanation for the anomalous behaviour of this lens is that the age of the donor may have been incorrectly transcribed in the documentation received from the eye bank. However, no evidence of any error of this sort could be found in the documentation made available to the authors and so the matter remains unresolved. In most cases, tests at 1000 rpm did not deform the nucleus sufficiently to permit a useful analysis, but data from tests conducted at high speeds (i.e. 2450 rpm, 3000 rpm and 3465 rpm) generally accord well with those collected at 2000 rpm. These higher speeds may in fact be more reliable when testing older, stiffer nuclei as the larger deformations induced are likely to lead to greater accuracy in the analysis. The nucleus stiffness data presented in this paper were obtained as an extension to the main set of spinning lens tests on the lens substance (Wilde et al., 2012). In this main test program, considerable care was taken to minimise the possibility of lens swelling prior to the test and to exclude lenses for which mechanical or swelling damage was suspected. In contrast, the data on lens nucleus presented in the current paper were obtained by mechanical tests on a lens that had been deliberately subjected to an uncertain process in which swelling was induced in the cortex, so that the cortex fibres could be removed to reveal the nuclear core. The quality of the nuclei tested in the current test program is therefore regarded as being less certain than the quality of lenses tested in the main test program. It is perhaps surprising, therefore, that reasonable agreement is achieved between the current tests and the data obtained from the main test program. It is noted that both sets of tests indicate that the nucleus stiffness increases very substantially with age (for example, the data in Fig. 3 suggest that the shear modulus increases
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by about an order of magnitude between the ages of 40 years and 60 years). The exponential nature of the relationship between age and shear modulus means that any small differences in the rate of ageing between individuals are likely to be amplified, leading to relatively large differences in nucleus shear modulus when those individuals share the same age. It seems plausible to the authors that effects of this sort will tend to swamp any minor adjustments to the stiffness of the nucleus (e.g. as a consequence of water absorption during the process of isolating the nucleus) that occur due to shortcomings in the preparation of the nucleus prior to the test. The project was supported financially by the Wellcome Trust. The assistance of Dr Val. Smith and colleagues at the Bristol Eye Bank is gratefully acknowledged. References Augusteyn, R.C., 2010. On the growth and internal structure of the human lens. Exp. Eye Res. 90 (6), 643e654. Burd, H.J., Wilde, G.S., Judge, S.J., 2011. An improved spinning lens test to determine the stiffness of the human lens. Exp. Eye Res. 92 (1), 28e39. Fisher, R.F., 1971. The elastic constants of the human lens. J. Physiol. 212 (1), 147e 180. Gullapalli, V.K., Murthy, P.R., Murthy, K.R., 1995. Colour of the nucleus as a marker of nuclear hardness, diameter and central thickness. Indian J. Ophthalmol. 43, 181e184. Heys, K.R., Cram, S.L., Truscott, R.J.W., 2004. Massive increase in the stiffness of the human lens nucleus with age: the basis for presbyopia? Mol. Vis. 10, 956e963. Heys, K.R., Friedrich, M.G., Truscott, R.J.W., 2007. Presbyopia and heat: changes associated with aging of the human lens suggest a functional role for the small heat shock protein, a-crystallin, in maintaining lens flexibility. Aging Cell 6 (6), 807e815. Hollman, K.W., O’Donnell, M., Erpelding, T.N., 2007. Mapping elasticity in human lenses using bubble-based acoustic radiation force. Exp. Eye Res. 85 (6), 890e 893. Sweeney, M.H.J., Truscott, R.J.W., 1998. An impediment to glutathione diffusion in older normal human lenses: a possible precondition for nuclear cataract. Exp. Eye Res. 67 (5), 587e595. Weeber, H.A., Eckert, G., Soergel, F., Meyer, C.H., Pechhold, W., van der Heijde, R.G.L., 2005. Dynamic mechanical properties of human lenses. Exp. Eye Res. 80 (3), 425e434. Weeber, H.A., Eckert, G., Pechhold, W., van der Heijde, R.G.L., 2007. Stiffness gradient in the crystalline lens. Graefe’s Arch. Clin. Exp. Ophthalmol. 245 (9), 1357e1366. Wilde, G.S., Burd, H.J., Judge, S.J., 2012. Shear modulus data for the human lens determined from a spinning lens test. Exp. Eye Res. 97, 36e48.