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1From the Department of Mechanical and Industrial Engineering, the 2Institute for Biomaterials and Biomedical Engineering, and the 3Department of Ophthalmology and Vision Sciences, University of Toronto, Toronto, Ontario, Canada; and the 4School of Optometry, University of Waterloo, Waterloo, Ontario, Canada.
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Abstract |
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METHODS. Several models of the optic nerve head tissues (pre- and postlaminar neural tissue, lamina cribrosa, central retinal vessel, sclera, and pia mater) were constructed. Stresses, deformations, and strains were computed using finite element modeling for a range of normal and elevated intraocular pressures. Computed retinal surface deformations were compared with measured deformation patterns in enucleated human eyes. A sensitivity analysis was performed in which tissue properties and selected geometric features were varied.
RESULTS. Acute IOP-induced deformation of the vitreoretinal interface was highly dependent on optic cup shape but showed a characteristic "W-shaped" profile that did not match the deformation of the anterior surface of the lamina cribrosa. The central retinal vasculature had surprisingly little effect on optic nerve head biomechanics. At an IOP of 50 mm Hg, strains (fractional elongation) in the lamina cribrosa averaged 4% to 5.5%, dependent on model geometry, with maximum strains up to 7.7%. Strains in the lamina cribrosa were more dependent on scleral stiffness, scleral thickness, and scleral canal diameter than on lamina cribrosa stiffness and optic cup shape.
CONCLUSIONS. Computed levels of strain in the lamina cribrosa are biologically significant and capable of contributing to the development of glaucomatous optic neuropathy, even without considering the probable accentuating effect of the lamina cribrosa’s microarchitecture. Depending on optic cup shape, IOP-induced deformation of the vitreoretinal interface may not match lamina cribrosa deformation. This finding implies that scanning laser tomography has limited ability to estimate lamina cribrosa deformation when imaging the anterior topography of the optic nerve head. Biomechanical effects in the lamina cribrosa depend strongly on scleral properties.
Several attempts have been made to measure ONH biomechanics experimentally. For example, the LC has been accessed directly to measure IOP-induced deformations by histologic3 and radiographic4 methods. Such measurements, although valuable, do not give a detailed picture of the biomechanical environment within the ONH. Scanning laser tomography allows measurement of the deformation of the vitreoretinal interface in real time, but provides little direct information about the biomechanics of the underlying sclera and LC. It therefore seems that the most fruitful approach to understanding LC biomechanics is a combination of experimental measurements and modeling studies.
Recently Dongqi and Zeqin5 and Edwards and Good6 developed analytical models of LC biomechanics. These models are valuable but are limited by the complex geometry of the ONH that forced the investigators to make simplifying assumptions. For instance, they assumed a perfectly circular LC of constant thickness and uniform elastic properties. Both assumptions are reasonable in the context of an analytic model, but limit the applicability of the results.7 The models also assume that a uniform pressure is applied to the anterior LC surface. Although the vitreous humor exerts a uniform pressure on the vitreoretinal interface, its action is not directly on the LC anterior surface. The forces on the anterior surface of the LC can be different from the pressure on the retinal surface and can even have a tangential component.
Computational models have been able to overcome some of the limitations described earlier. Bellezza et al.8 used finite element modeling to consider more complex geometries and material properties. They evaluated the effect of LC geometry, mainly eccentricity, on the stress distribution within the load-bearing tissues of the ONH. Although more comprehensive than previous analytical models, these computational models still rely on some critical assumptions—namely, constant LC thickness and a uniform IOP acting directly on the anterior LC surface.
In this work, we also used finite element modeling to quantify IOP-induced stresses and strains within the ONH. However, our models expand on previous work by considering the cross section of the ONH in greater detail and allowing for imposition of more realistic boundary conditions on the LC. Our goal was to investigate the effects of peripapillary sclera and prelaminar and postlaminar neural tissue on the overall biomechanical environment within the ONH. To do so, we introduced three simplified models of the ONH and computed stresses and strains acting on ONH tissues. We also performed a preliminary geometric sensitivity analysis and a parametric study of the effects of material properties.
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Geometry
Initially, three models were constructed, in increasing order of realism and complexity. Then variations of model 3 were constructed by independently varying scleral thickness, scleral canal diameter, and optic cup shape. All models were axisymmetric.
Model 1.
Away from the ONH region the sclera-cornea formed a 0.78-mm thick spherical shell with a 12-mm internal radius.12 In the ONH region, the sclera incorporated a scleral canal, tapered so as to have a diameter of 1.52 mm at the anterior opening. The prelaminar neural tissue included an optic cup and peripapillary rim in the ONH region. Farther from the ONH, the retina had a constant thickness of 0.24 mm before finally tapering away to zero thickness at 120° anterior of the axis of symmetry. The LC was modeled as a section of a spherical shell, concentric to the scleral–corneal shell, 0.3 mm thick with a 12.4 mm internal radius. The anterior and posterior faces of the LC had diameters of 1.80 and 2.14 mm, respectively (corresponding to an anterior LC surface area of 2.55 mm2 vs. a measured value of 2.88 ± 0.72 mm213 ). The postlaminar optic nerve extended posteriorly from the LC and filled most of the scleral canal (Fig. 1) .
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Model 3.
We extended model 1 to include more physiologically faithful details. The shape and dimensions of model 3 were based on data from literature, as well as our own estimations from histologic sections. The model did not include blood vessels but considered both the pia mater and the thinning of the sclera-cornea near the ONH region (Fig. 1) . The cup was shaped so that the cup-to-disc ratio was 0.3 when measured with respect to a level 0.05 mm below the retina surface at the rim. The LC was modeled to be 0.3 mm thick at the axis of symmetry and 0.27 mm at its periphery, with anterior and posterior diameters of 1.90 and 2.15 mm, respectively (corresponding to an anterior LC surface area of 2.83 mm2), within the ranges reported for human LC.13 16 17 18 Scleral canal diameter was 1.83 mm, again within the ranges reported in humans.19 20 We measured the pia mater and scleral thickness from sagittal histologic sections of the ONH region of human eyes fixed at an IOP of 15 mm Hg. Sections were selected to lie approximately centrally in the scleral canal, stained with picrosirius red to visualize connective tissue better, photographed, and measured with image-analysis software (Amira 3.0; TGS Inc., Bordeaux, France). We assigned the pia mater a thickness of 0.06 mm and the sclera a variable thickness, increasing from 0.46 mm at its thinnest point, adjacent to the canal, to 0.8 mm at 10° from the axis of symmetry.12 The shortest distance between the anterior surface of the LC and cerebrospinal fluid space was 0.39 mm and between the anterior surface of the LC and the pia mater was 0.28 mm. The latter value is within the ranges reported by Jonas et al.18
Sensitivity Analysis on Model 3.
Taking model 3 as the base geometry, we varied the scleral thickness away from the ONH from 0.6 to 1.0 mm,12 while keeping ONH geometry within 15° of the axis of symmetry of the ONH unchanged. Canal diameter was varied from 1.3 to 2.2 mm, representing canal cross-sectional areas of 1.33 to 3.8 mm2, within the ranges reported in human eyes.19 20 We constructed models with three different cup shapes, ranging from a steep to a gradual cup slope, but with identical internal ONH geometry, as shown in greater detail later.
Pressure Loads
A uniform IOP was applied to all interior surfaces of the eye—that is, the vitreoretinal interface and sclera-cornea anterior to the retinal termination. A range of IOPs from 0 to 50 mm Hg was considered. In model 2, constant arterial pressure of 55, 80, or 105 mm Hg was applied on the interior surface of the vessel wall. This represented a range of physiologic mean arterial pressures (
diastolic + 
systolic), corrected for the hydrostatic difference between the eye and the brachial artery, and is consistent with measurements of the mean diastolic pressure in the ophthalmic artery of 51.8 mm Hg.21 22 External surfaces of the eye were subject to a pressure of 0 mm Hg (atmospheric pressure).
In addition to the IOP, the ONH is exposed to a hydrostatic pressure from the cerebrospinal fluid (CSF). We tested the effects of CSF pressures ranging from 0 to 15 mm Hg on each of the models (1, 2, and 3). Except when indicated, all data shown are for a CSF pressure of zero.
Material Properties
The mechanical properties of the tissues that constitute the ONH and surrounding structures are not completely known. For this initial study, all materials were assumed to be linearly elastic, isotropic, and incompressible, in which case all that was needed to characterize the tissue’s biomechanical behavior was the modulus of elasticity (Young’s modulus). Table 1 summarizes relevant moduli for the tissues under consideration, from which moduli of 3 and 0.03 MPa were chosen for sclera and neural tissue, respectively. The fact that neural tissue is two orders of magnitude more compliant than the sclera is consistent with the expectation that neural tissue is unlikely to bear significant load.
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We were unable to find measurements of the mechanical properties of the central retinal vessels in the literature, and hence we assigned the central retinal artery wall a Young’s modulus of 0.3 MPa, equal to that of similar-sized vessels in other regions of the human body. There is little information about the Young’s modulus of the pia mater. Kleiven41 reported an elasticity modulus between 1.44 and 4.65 MPa, citing Zhivoderov et al.,38 whereas Brands40 used 1.86 MPa for human meninges shear modulus. Mazuchowski and Thibault39 measured the Young’s modulus of the human spinal cord, with and without pia mater present. Assuming a pia mater thickness between 0.06 and 1.6 mm and a perfectly cylindrical sample 1 cm in diameter, we derived a pia mater modulus between 2.5 and 65 MPa. We assigned the pia mater a Young’s modulus of 3 MPa, the same as for the sclera.
Parametric Study
The selection of specific mechanical properties for each of the tissues modeled required careful scrutiny. Not only could these properties affect the results obtained from each model, but the inherent physiologic variability in mechanical properties could be important in ONH biomechanics. For better understanding of the effect of our choices, we solved model 3, varying the Young’s modulus of each tissue independently over two orders of magnitude.
Numerical Details
Models were created, meshed, solved, and postprocessed using ANSYS 6.1 (Ansys, Inc., Canonsburg, PA). Finite element meshes were generated with nonuniform element density (Fig. 1) . Two-dimensional, eight-node elements (Plane82 in Ansys) were used throughout. In preliminary studies, the effect of mesh refinement was investigated in a series of meshes with element counts ranging from 217 to 41,604, and various degrees of mesh size heterogeneity. The structural percentage error in energy norm (SEPC),42 a measure of the continuity of the stress field, was plotted as a function of mesh size, and results were considered to be mesh independent when the SEPC dropped below 3% in all regions. The results discussed in this work correspond to meshes with 6,556, 20,338, and 7,123 elements (46,023, 151,025, and 44,537 degrees of freedom) for models 1, 2, and 3 respectively. Model 2 required many more elements to guarantee accurate prediction of the deformations in and near the thin blood vessel wall. The geometry variation studies required remeshing the models, and care was taken to ensure that these were still sufficiently resolved by keeping element size and nodal density roughly constant.
Numerical solution of the governing equations is difficult for incompressible materials (Poisson ratio, 
= 
), due to singular behavior. It is therefore common to use a value of close to, but not equal to . Woo et al.43 found that the use of this approximation had little effect on the results. We tested the effect of varying the Poisson ratio from 0.49 to 0.499999 and observed minor changes in the solution (e.g., <0.15% change in the average strain on the LC) and therefore used a value of = 0.49 for all simulations.
Nodes on the equator were constrained to deform radially (Fig. 1) , corresponding approximately to anterior–posterior restraint from the extraocular muscles. Nodes on the axis of symmetry were constrained to deform so that they remained on the symmetry axis. Solutions required <1 minute of CPU time on a desktop workstation (Windows XP SP1 [Microsoft Corp., Redmond, WA], and Athlon XP 2100+ CPU [AMD, Sunnyvale, CA] with 1 GB of memory).
Model Validation and Consistency Check
It is important to validate computational results; unfortunately, such validation is not trivial for the ONH. We compared acute IOP-induced deformations of the vitreoretinal interface as computed in our model with acute deformations measured in enucleated human eyes.
The experimental study used paired human eyes obtained within 24 hours after death from the Eye Bank of Canada, Ontario Division. Eyes were obtained and managed in accordance with the provisions of the Declaration of Helsinki for research involving human tissue. Eyes were required to have optic nerves at least 5 mm long and were excluded if there was a known history of disease expected to affect connective tissue mechanical properties (e.g., scleritis, rheumatoid arthritis) or a known history of high myopia (>8.0 D). The optic nerve was tied off at a distance of 3 mm from the posterior surface of the eye to prevent blood efflux. Eyes were surgically prepared using one of two experimental approaches that were improvements over our original technique.44 In the first approach, two surgical infusion cannulae (MVS 1035; Storz Instrument Co., St. Louis, MO) were placed through the sclera and anterior pars plana 180° apart, thus allowing IOP control. The cornea was centrally trephined with an 8.5-mm trephine, and the iris and nucleus of the lens were removed. Capsular remnants and the core of the vitreous were then carefully removed with a vitrector (SITE-TXR; Site Microsurgical Systems Inc., Fort Washington, PA). The eye was rapidly refilled with isotonic saline and a specially designed optical apparatus was inserted into the hole trephined into the cornea. This device sealed against the margins of the cornea and incorporated a lens to replace the focusing power of the cornea and native lens. In the second approach, we used eyes that had been used for corneal donation. The eye was opened circumferentially at the level of the pars plana, excising the anterior chamber, iris, and lens. The vitreous and lenticular remnants were removed by vitrectomy, as just described, and a modified apparatus, consisting of a collar incorporating two cannulae and a lens, was glued to the sclera with cyanoacrylate adhesive. This provided a fluid seal around the periphery of the incision while mechanically supporting the eye. Both approaches gave comparable results, although the collar system was somewhat easier to work with. This general technique had several important advantages: It protected the posterior half of the eye from mechanical stress; it replaced the possibly opaque cornea and lens in a postmortem eye with high-quality optics, permitting excellent quality images of the ONH; and it prevented retinal detachment.
The eye was gently wrapped with gauze soaked in isotonic saline and placed in a custom holder that was mounted to the vertical rails of a standard slit-lamp headrest. IOP was sequentially set to 5, 15, 22, 30, 40, and 50 mm Hg by adjusting the height of a reservoir containing isotonic saline. This reservoir was connected to one of the cannulas, and the other cannula was clamped off. Eye temperature was kept constant at 37° by a circulating water jacket attached to a constant-temperature bath. At each pressure, the eye was allowed to equilibrate for 15 minutes and incorporate viscoelastic effects into the deformation. Then, seven topographic image series (32 aligned scanning laser images, resolution = 256 x 256 pixels) were acquired with a 10° field of view centered on the nerve head using a scanning laser tomographer (Heidelberg Retinal Tomograph, ver. 2.01a; Heidelberg Engineering, Heidelberg, Germany). This protocol ensured good estimation of image variance45 and stable ONH topography during imaging.3 The orientation of the eye was approximately determined by using insertion points of the extraocular muscles and was then adjusted by imaging and comparison of the relative locations of the macula and optic disc.
We computed the pressure-induced whole globe volume changes and compared them with the generalization of the Friedenwald ocular pressure–volume relation introduced by Silver and Geyer46 as a consistency check. We also compared our model predictions of deformation with the topographic differences between average normal subjects and average patients with glaucoma developed by Swindale et al.,47 using the HRT.
Analysis
Stresses and strains are tensors (i.e., dependent on orientation), and there is therefore no unique way to assign single values of strain and stress at a point. This makes graphical presentation of results difficult. Generally, values presented should have a reasonable physical or biological interpretation. In this work, we used the maximum principal strain and von Mises equivalent stress.48 For an incompressible material, the maximum principal strains are nonnegative, always giving a measure of maximum extension. Von Mises equivalent stresses give an indication of the forces acting throughout the tissue, while discounting the effects of hydrostatic pressure. Because of assumptions used in this model, all stresses and strains scale linearly with the applied loads. Therefore, we report stresses as multiples of the imposed IOP.
Under the action of IOP the whole eye shell deforms. To observe only effects local to the ONH and allow for comparison with experiments, all computed anterior–posterior deformations were zeroed on the vitreoretinal interface at 5° from the center of the optic cup (the axis of symmetry), consistent with a 10° field-of-view HRT acquisition.
For each computed solution, the topography of the vitreoretinal interface at 15 mm Hg was subtracted from that at 50 mm Hg to provide a deformation map. For experimental measurements the deformation maps were obtained using the HRT’s glaucoma progression analysis routines, which normalize for movement, rotation, tilt, and magnification changes. No zeroing was necessary for the computed difference between average normal and average glaucomatous topographies.
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Figure 5 shows computed distributions of maximum principal strain (left column) and von Mises equivalent stress (right column) for models 1, 2, and 3, under an IOP of 50 mm Hg. Table 2 presents a summary of these quantities in each region for each model. The largest stresses were observed in the relatively stiff sclera and pia mater, compared with the relatively soft neural and laminar tissue. Incorporation of a simplified axisymmetric blood vessel, playing the role of the CRA, had surprisingly little effect on the strain and stress fields (top and middle rows).
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Increasing CSF pressure from 0 to 15 mm Hg at an IOP of 50 mm Hg changed the average maximum principal strain within the LC from 3.7% to 4.1% in model 1 and from 5.0% to 5.5% in model 3. These changes are small compared with the effects of geometry and material properties.
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Discussion |
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The deformation profile of the anterior surface of the LC was slightly different between models 1, 2, and 3, but not remarkably so (Fig. 4) . The main difference seemed to be in the slope at the edge, which was steeper in model 3, due to the "rotation" of the sclera (as seen in cross section), which causes the scleral angle of approach to the scleral canal to be larger in model 3 than in models 1 and 2 (Fig. 1 top vs. bottom). This difference in scleral displacement was probably due to a combination of two factors: a thinner peripapillary sclera and a larger scleral canal in model 3, as we can see from the sensitivity analysis (Fig. 6 , top and bottom rows).
As expected, stiffening of any ONH component reduced strains in the LC, whereas softening had the opposite effect (Fig. 7) . The weak dependence on pia mater modulus suggests that inclusion of the pia mater is probably not the source of the differences in strain magnitude and distribution between models 1 and 3.
Deformation of model 3 extended farther than the 5° used for zeroing and comparison with the HRT (Fig. 2 , third row) and increased in thinner or more compliant scleral tissue (Fig. 6) . This is consistent with a report by Heickell et al.,49 who suggested that the peripapillary sclera could be deformed as far as 10° away from the center of the cup.
The agreement between IOP-induced increases in intraocular volume computed on our models with Silver and Geyer’s formula46 is good at physiologic pressures (60 vs. 50 µL), but less so at elevated pressures (175 vs. 100 µL), although within the range of normal values, probably because our model does not include the stiffening of tissue known to occur at higher elongations.
Through the variations in geometry shown in Figure 6 , we see that the locations of strain concentration were independent of changes in scleral thickness, canal diameter, and cup shape. However, geometric changes, particularly scleral thickness and canal size, affected average maximum principal strain and von Mises equivalent stress in the LC. This supports the hypothesis that scleral thickness and disc size could be determinants for susceptibility to elevated IOP.8 50
The effects of CSF pressure are complex, yet average maximum principal strain within the lamina is relatively unaffected by CSF pressures up to 15 mm Hg (<0.5% increase).
Main Conclusions
Four main conclusions arise from this work. First, the deformation of the vitreoretinal interface does not necessarily correspond to the deformation of the anterior surface of the LC. Second, biologically significant levels of strain are produced within the LC as the pressure goes from physiologic to elevated, even without considering the magnifying effects of the lamina’s microarchitecture. Third, average laminar strains depend more on scleral compliance than on laminar compliance. Fourth, the simplified central retinal vasculature has surprisingly little effect on ONH biomechanics. Let us consider each of these in turn.
Lamina Deformation Need Not Match Vitreoretinal Deformation.
Pressure is a normal stress—that is, a force that, by definition, acts at 90° to any surface to which it is exposed. Therefore, when the cup walls are steep, IOP on the vitreoretinal interface induces lateral displacement, in addition to posterior displacement. This lateral displacement stretches tissue away from the center of the cup, resulting in cup enlargement, as well as deepening. This complex pattern of displacement leads to a situation in which the magnitude of optic cup deepening is not maximum at the center of the cup floor, but is offset from the center. This deformation profile was also observed experimentally, at least in some eyes (Figs. 3 4) .
The anterior surface of the lamina did not experience the same magnitude of lateral displacement forces and did not necessarily move in the same way as the vitreoretinal interface. This means that measurements of acute ONH topographic change (e.g., using the HRT) can be a poor surrogate for the displacement of the LC. The magnitude of this discrepancy is expected to be largest in cups with steep walls.
The LC Is Exposed to Biologically Significant Levels of Strain.
The mechanisms by which stress and strain affect cellular functions, leading to changes in astrocyte phenotype and glaucomatous optic neuropathy, are not well understood. The disruption of axonal transport observed in glaucoma could be a consequence of cytoskeletal alterations and microtubule loss that result from transient stretch injury,51 but other cellular functions have also been shown to be affected by stress and strain, including cell adhesion, transmembrane transport, and RNA processing.52 53 To elucidate the possible relevance of a given level of stress or strain requires the determination of normal physiologic levels. Downs et al.28 considered a physiologic level of strain in the sclera to be between 0% and 1% and suggested that levels above 3.5% could be pathophysiologic, because they result in detectable changes in scleral material properties (stiffening). Slightly higher strain levels (5%–6%) were sufficient to induce a wide range of biological effects in neuronal cells in in vitro models.6 54 The same threshold for axonal injury was observed by Margulies and Thibault.55 Using adult guinea pig optic nerves Bain and Meaney56 estimated a Lagrangian strain threshold of 21% for morphologic axonal injury and 18% for deterioration of nerve function. These values are lower than the 27.1% measured in infant guinea pigs.57 Using a mixture of experiments and computational models of a guinea pig brain Anderson58 developed a probabilistic model of injury based on the peak von Mises stress in a region. Stresses of 27 to 43 kPa (202–322 mm Hg) resulted in 50% chance of injury in up to 20% of the affected region. Variable shear stress, such as may occur in tissue during diurnal variations in IOP, leads to injury in three times more cells than does steady shear stress.59
The peak levels of strain and stress predicted by our models exceed most of the thresholds just quoted. Some of the extreme strains and stresses predicted by the models are due to nonphysiologic geometric details (e.g., nonrounded "corners"); however, strain levels in the LC in particular are nontrivial, consistent with the hypothesis that mechanical effects can contribute to damage to nerve fibers and loss of visual function observed in glaucoma.
Average Laminar Strain Is Most Affected by Scleral Compliance, Less Affected by Laminar Compliance, and Least Affected by Neural and Pia Mater Compliance.
The natural variability and inherent inaccuracy in measurements of the mechanical properties of the tissues of the ONH compel us to test the effect that our choice of material properties has on our calculations. Through the parametric studies, we found that average laminar maximum principal strain depends more on scleral than laminar compliance, with a relatively weak dependence on neural and pia mater compliances. Further, the thickness of the sclera and the size of the scleral canal have an appreciable influence on LC strains. This suggests that for proper understanding and modeling of ONH biomechanics, it is essential to know the mechanical properties and geometry of the sclera, at least as accurately as those of the LC.
A Simplified Central Retinal Vasculature Seems to Have Little Effect on ONH Biomechanics.
Except for some local stress concentrations, ONH biomechanics were only weakly affected by the simplified central retinal vasculature. This result is counter to our original expectations. Direct measurements of the mechanical properties (e.g., stiffness) of the central vessel walls and incorporation of the central retinal vasculature in a fully three-dimensional model is necessary to confirm this result. In particular, because the central retinal vascular tree branches into the prelaminar neural region, it is still possible that it plays an important role in the support of the nerve fiber layer, especially in the slowly progressing disease where tissue remodeling is important.
Limitations
The results in this study were obtained from simplified models, and naturally there are aspects of the biomechanics of the ONH that are impossible to capture. For example, with axial symmetry, it is impossible to consider any geometry other than a perfectly circular scleral canal or the oblique insertion of the optic nerve into the sclera, which could affect strain and stress levels.19 20 We developed the models in an attempt to identify the fundamental elements involved in ONH biomechanics, and they cannot, by definition, tell us about individual-specific aspects. They are therefore, by construction, preliminary. Still, they provide the opportunity to perform analysis in a controlled situation that incorporates potentially interesting biomechanical features. For example, they allow us to model the IOP-induced deformation of the vitreoretinal interface. Also, whereas previous models assumed that the peripheral LC is pulled by lateral forces coplanar to the undeformed LC, our models allow for effects of peripapillary scleral thickness variations, resulting in rotation of the scleral canal in the anterior–posterior direction, akin to a twisting of the LC periphery.
Despite the idealized nature of the models, this work represents, to our knowledge, the first attempt to model in detail the depth of the human ONH from vitreoretinal interface to optic nerve. Ongoing work seeks to incorporate more physiologically accurate geometries (including detailed blood vessels), complex material properties (including anisotropy, nonlinearity, and large deformation effects), and a full evaluation of the effect of CSF pressure. Use of more advanced material properties will also depend on the availability of experimental data that provide the necessary parameters and allow validation of the results.
It is important to interpret the results presented in this work in their proper context. First, we modeled the LC as a homogenous tissue, where in reality it is composed of relatively rigid connective tissue and more compliant neural tissue. The stresses and strains we computed represent averages over tissue regions larger than several LC pores; however, individual components can experience stresses or strains much larger than the average (Downs J, et al. IOVS 2004;45:ARVO E-Abstract 2157). This work helps establish a lower bound for the stresses and strains experienced by individual cells in the LC, as well as guiding boundary condition selection for more sophisticated models of individual laminar pores. Second, the assumed incompressibility of the tissues could also be inaccurate—for example, if elevated IOP induced axoplasmic flow from the prelaminar neural region into the LC or the postlaminar neural region.
Despite these restrictions, it is encouraging that the pattern of computed topographic change is similar to that measured experimentally, at least in some eyes (Figs. 3 4) , with maximal deflection of prelaminar neural tissues occurring in a ring at the bottom of the optic cup. Nevertheless, there is significant inter-eye variability that cannot be fully captured by generic simplified models, and the process of validation discussed herein should not be considered definitive. We intend to show only that the predicted deformations fall within the range of experimental measurements on real eyes. In addition, the use of postmortem tissue may induce artifacts, including the development of edema that could affect ONH deformation under elevated IOP. Future work will explore the development of individual-specific models, with geometries based on histologic samples and a comprehensive sensitivity analysis on geometry.
Notwithstanding the difference in scale, there is a surprising similarity in the shape of the deformation curves, shown in Figure 4 , of computed IOP-induced deformations and differences between average normal and average glaucomatous eyes. Even the bend approximately 0.8 mm from the cup center is repeated. Still, our models represent only acute (passive) deformation of the tissue in response to a step change in IOP. This ignores long-term remodeling processes that distinguish glaucomatous from normal eyes, and this similarity could be a coincidence.
In closing, a consistent morphologic finding was that the sclera thins as it approaches the scleral canal, as shown in model 3. Because the sclera is the stiffest ONH tissue, it supports much of the overall load. In particular, the peripapillary sclera must support relatively high stresses, which are compounded by peripapillary scleral thinning. Our calculations suggest that peripapillary scleral thinning confers a biomechanical disadvantage, because it produces higher strains in the LC. Downs et al.60 have identified a zone of thickened sclera surrounding the peripapillary sclera in the monkey eye. This ring would partially "shield" the scleral canal from large deformation. Evidence of such a thickened zone in the human eyes is equivocal12 and should be further investigated.
Studies have shown that a thinner central cornea is a significant risk factor for development61 and progression62 of glaucomatous visual field loss. It is not yet clear whether this finding is explainable solely by artifactually low IOP readings through thin corneas, or whether there are other factors that are responsible.63 64 If there were a correlation between corneal thickness and scleral thickness, thinner corneas would be associated with more LC deformation at a given IOP level. This speculation remains to be tested experimentally.
In summary, despite the relative simplicity of the models presented herein, important conclusions can be made that are likely to hold true for most eyes. The deformation of the vitreoretinal interface does not necessarily correspond to that of the anterior surface of the LC; biologically significant levels of strain are produced within the LC; and average strains within the LC depend more strongly on scleral and laminar compliance than on neural and pia mater compliance.
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Acknowledgements |
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Footnotes |
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Supported by the Canadian Institutes of Health Research (CRE, JGF), Glaucoma Research Society of Canada (JGF), and Consejo Nacional de Ciencia y Tecnologia de Mexico (CONACYT; IAS).
Submitted for publication February 10, 2004; revised July 8, 2004; accepted August 5, 2004.
Disclosure: I.A. Sigal, None; J.G. Flanagan, None; I. Tertinegg, None; C.R. Ethier, None
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked "advertisement" in accordance with 18 U.S.C. 
1734 solely to indicate this fact.
Corresponding author: C. Ross Ethier, Department of Mechanical and Industrial Engineering, 5 King’s College Road, University of Toronto, Toronto, Ontario M5S 3G8, Canada; ethier{at}mie.utoronto.ca.
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A. Hommer, G. Fuchsjager-Mayrl, H. Resch, C. Vass, G. Garhofer, and L. Schmetterer Estimation of Ocular Rigidity Based on Measurement of Pulse Amplitude Using Pneumotonometry and Fundus Pulse Using Laser Interferometry in Glaucoma Invest. Ophthalmol. Vis. Sci., September 1, 2008; 49(9): 4046 - 4050. [Abstract] [Full Text] [PDF]
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A. P. Wells, D. F. Garway-Heath, A. Poostchi, T. Wong, K. C. Y. Chan, and N. Sachdev Corneal Hysteresis but Not Corneal Thickness Correlates with Optic Nerve Surface Compliance in Glaucoma Patients Invest. Ophthalmol. Vis. Sci., August 1, 2008; 49(8): 3262 - 3268. [Abstract] [Full Text] [PDF]
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H. Yang, J. C. Downs, C. Girkin, L. Sakata, A. Bellezza, H. Thompson, and C. F. Burgoyne 3-D Histomorphometry of the Normal and Early Glaucomatous Monkey Optic Nerve Head: Lamina Cribrosa and Peripapillary Scleral Position and Thickness Invest. Ophthalmol. Vis. Sci., October 1, 2007; 48(10): 4597 - 4607. [Abstract] [Full Text] [PDF]
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J. C. Downs, H. Yang, C. Girkin, L. Sakata, A. Bellezza, H. Thompson, and C. F. Burgoyne Three-Dimensional Histomorphometry of the Normal and Early Glaucomatous Monkey Optic Nerve Head: Neural Canal and Subarachnoid Space Architecture Invest. Ophthalmol. Vis. Sci., July 1, 2007; 48(7): 3195 - 3208. [Abstract] [Full Text] [PDF]
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 S Cirovic, R M Bhola, D R Hose, I C Howard, P V Lawford, J E Marr, and M A Parsons Computer modelling study of the mechanism of optic nerve injury in blunt trauma Br. J. Ophthalmol., June 1, 2006; 90(6): 778 - 783. [Abstract] [Full Text] [PDF]
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I. A. Sigal, J. G. Flanagan, and C. R. Ethier Factors Influencing Optic Nerve Head Biomechanics Invest. Ophthalmol. Vis. Sci., November 1, 2005; 46(11): 4189 - 4199. [Abstract] [Full Text] [PDF]
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E. Spoerl, A. G. Boehm, and L. E. Pillunat The Influence of Various Substances on the Biomechanical Behavior of Lamina Cribrosa and Peripapillary Sclera Invest. Ophthalmol. Vis. Sci., April 1, 2005; 46(4): 1286 - 1290. [Abstract] [Full Text] [PDF]
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) and von Mises equivalent stress (
) within the LC.
