purpose. To study the relationship between intraocular pressure (IOP) and the IOP-related stress (force/cross-sectional area) it generates within the load-bearing connective tissues of the optic nerve head.

methods. Thirteen digital, three-dimensional geometries were created representing the posterior scleral shell of 13 idealized human eyes. Each three-dimensional geometry was then discretized into a finite element model consisting of 900 constituent finite elements. In five models, the scleral canal was circular (diameters of 0.50, 1.50, 1.75, 2.00, and 2.56 mm), with scleral wall thickness (0.8 mm) and inner radius (12.0 mm) held constant. In three models, the canal was elliptical (vertical-to-horizontal ratios of 2:1 [2.50 × 1.25 mm], 1.5:1 [2.1 × 1.4 mm], and 1.15:1 [1.92 × 1.67 mm]), with the same constant scleral wall thickness and inner radius. In five additional models, scleral canal size was held constant (1.92 × 1.67 mm), and either scleral wall thickness (three models, 0.5, 1.0, and 1.5 mm) or inner radius (two models, 13.0 and 14.0 mm) was varied. In all models, each finite element was assigned a single isotropic material property, either scleral (modulus of elasticity, 5500 kPa) or axonal (modulus of elasticity, 55 kPa). Maximum stresses within specific regions were calculated at an IOP of 15 mm Hg (2000 Pa).

results. Larger scleral canal diameter, elongation of the canal, and thinning of the sclera increased IOP-related stress for a given level of IOP. For all models, maximum IOP-related stress ranged from 6 × IOP (posterior sclera) to 122 × IOP (laminar trabeculae). For each model, maximum IOP-related stress was highest within the laminar trabecular region and decreased progressively through the laminar insertion, peripapillary scleral, and posterior scleral regions. Varying the inner radius had little effect on the maximum IOP-related stress within the scleral canal.

conclusions. Initial finite element models show that IOP-related stress within the load-bearing connective tissues of the optic nerve head is substantial even at low levels of IOP. Although the data suggest that scleral canal size and shape and scleral thickness are principal determinants of the magnitude of IOP-related stress within the optic nerve head, models that incorporate physiologic scleral canal and laminar geometries, a more refined finite element model meshwork, and nonisotropic material properties will be required to confirm these results.

^{ 1 }

^{ 2 }or spontaneous compression, in which differences in tissue pressure across the lamina cause axons to collapse spontaneously

^{ 3 }) but also both acute and chronic ischemia,

^{ 4 }

^{ 5 }which may be induced by the effects of IOP-related stress and strain on blood flow and diffusion within the connective tissues through which the ONH blood supply must pass.

^{ 6 }

^{ 7 }(Fig. 1) . An FEM includes the important aspects of three-dimensional (3D) geometry, material properties, boundary conditions, and mechanical loads. These models are based on a digital 3D geometry that approximates the structure being modeled, which is discretized into small, regularly shaped, “building blocks” (the finite elements) whose boundaries connect points within the geometry (called nodes). Each finite element of the model is assigned its own shape and material properties (the characteristics of its connective tissue composition that determine the element’s behavior under load). The mechanical behavior of the total structure is then calculated from the combined behavior of its constituent finite elements.

^{ 7 }

^{ 8 }

^{ 9 }

^{ 10 }

*P*is the inner pressure (IOP),

*R*is the inner radius of the sphere (approximately one half the axial length), and

*t*is the thickness of the vessel wall (scleral thickness).

^{ 11 }

*S*is the stress applied to the ends of the plate,

*a*is the length of the long axis, and

*b*is the length of the short axis.

^{ 11 }

^{ 8 }

^{ 9 }

^{ 10 }In three additional models (M9, M10, and M11) the scleral canal geometry (1.92 × 1.67 mm) and inner radius (12.0 mm) of model M8 were used, with variable scleral wall thickness (M9, 0.5 mm; M10, 1.0 mm; M11, 1.5 mm). For the final two models (M12, M13), the scleral canal geometry (1.92 × 1.67 mm) and scleral thickness (0.8 mm) of model M8 were used, and the inner radius of the posterior scleral shell was varied (M12, 13.0; M13, 14.0 mm).

^{ 7 }Thus, to determine the appropriate number of elements needed for accurate solutions, a convergence test can be performed as follows: The geometry of the structure is modeled using a series of increasingly refined finite element meshes. During the analysis of each model, the displacements at identical locations of the structure are calculated for the different meshes. The calculated solutions using the more refined meshes are more accurate, but there is also a point of diminishing returns when increasing refinement provides little improvement in the calculated solutions. When this level of accuracy is reached, the model is judged sufficiently refined. Using this methodology, element meshes made up of 900 elements provided acceptable displacement values (data not shown). As such, each of the 13 models in this report consists of 900 elements, although the 3D geometry of each element is not identical from one model to another, because of the overall variation in model geometry.

^{ 12 }The axonal material property (modulus of elasticity of 55 kPa) was chosen to be two orders of magnitude less to reflect the fact that the axons are likely to be compliant and unlikely to bear significant load.

_{1}and σ

_{2}, Fig. 2 ) of approximately 7.5 × IOP.

^{ 13 }was the first to suggest that stresses within the peripapillary sclera might be concentrated relative to the more peripheral posterior sclera because of the behavior of stress around any hole (the scleral canal) in a pressurized spherical shell (the posterior scleral shell). Greene used equation 2 to approximate the stresses near the scleral canal and showed that for a circular hole (

*a*/

*b*= 1), the stress concentration factor at the edge of the hole is 3.0. Using that equation alone to estimate the maximum level of stress around elliptical scleral canals of small and large aspect ratios suggests that the maximum stress would be approximately 35 × IOP around an elliptical canal with a small aspect ratio (model M8, Fig. 4 ) and would reach 53 × IOP around an elliptical canal with a large aspect ratio (model M6, Fig. 4 ).

^{ 14 }discussed the implications of stress within the lamina cribrosa and its relation to the viscoelastic properties of the laminar trabeculae and scleral canal. Dongqi and Zeqin

^{ 15 }recently published a mathematical model predicting the displacement of a thin circular plate representing an idealized lamina cribrosa. Yablonski and Asamoto

^{ 3 }have suggested how the interplay between IOP and radius of curvature of the lamina may affect stresses within the load-bearing tissues of the ONH. Yan et al.

^{ 16 }proposed a laminar model in which stresses (predominantly shearing stresses) are responsible for the observed posterior displacement of the lamina cribrosa at elevated IOP.

^{ 17 }

^{ 18 }

^{ 19 }and after death

^{ 20 }

^{ 21 }do not have larger disc diameters, and some have suggested that this implies that scleral canal size is not a risk factor for glaucomatous damage.

^{ 10 }Chi et al.

^{ 22 }and others

^{ 10 }have reported a larger disc area in blacks than in whites. Chi et al.

^{ 22 }have postulated that this may be one reason why blacks may have a higher susceptibility to glaucomatous damage; however, others have stressed that it is not clear that the ONH in blacks is, in fact, more susceptible to glaucomatous damage.

^{ 10 }Nesterov and Egorov

^{ 23 }have suggested that, on a theoretical basis, disc size should be one determinant for susceptibility, and Cahane and Bartov

^{ 24 }have explored the theoretical implications of both axial length and scleral thickness.

^{ 25 }and Radius

^{ 26 }have characterized the difference in laminar beam and axonal fenestration dimensions within the superior–inferior versus the nasal–temporal peripheral scleral canal. Both groups attributed early superior and inferior axonal loss in glaucoma to the relative lack of connective tissue support in these regions. The idealized geometries in our report do not address these differences in 3D laminar anatomy; however, it is hoped that ongoing studies in our laboratories involving geometries derived from 3D reconstructions of serial histologic sections eventually will do so.

^{ 27 }reported differences in eye wall stress calculations (but did not study scleral canal size and shape differences) between hypertensive glaucoma suspects in whom disease did or did not progress to glaucomatous damage. However in their study, 24-hour IOP exposure was not longitudinally assessed.

^{ 16 }

^{ 28 }

^{ 29 }

^{ 25 }

^{ 26 }Because of their thin cross-sectional area, we expect that stresses will be highest within these peripheral laminar beams. However, this basic aspect of the 3D geometry of the lamina cribrosa has not been incorporated into our initial models. In fact, within these models, the smallest beams are found within the central scleral canal, and the stress concentrations found there may thus be artifactual (Fig. 6) .

^{ 14 }and do not show this linear relationship.

^{ 30 }Future inclusion of these viscoelastic material properties along with their nonisotropic assignment will improve the accuracy of the models’ predictions and allow the study of risk factors associated with the distribution of connective tissues within the scleral canal that are separate from canal size and shape and scleral wall thickness.

Model | Scleral Canal | Scleral Shell | ||||||
---|---|---|---|---|---|---|---|---|

Shape | Dimensions (mm) | Cross-Sectional Area (mm^{2}) | Inner Radius (mm) | Wall Thickness (mm) | ||||

M1 | Circular | 0.50 × 0.50 | 0.20 | 12.0 | 0.8 | |||

M2 | Circular | 1.50 × 1.50 | 1.77 | 12.0 | 0.8 | |||

M3 | Circular | 1.75 × 1.75 | 2.41 | 12.0 | 0.8 | |||

M4 | Circular | 2.00 × 2.00 | 3.14 | 12.0 | 0.8 | |||

M5 | Circular | 2.56 × 2.56 | 5.15 | 12.0 | 0.8 | |||

M6 | Elliptical | 2.50 × 1.25 | 2.45 | 12.0 | 0.8 | |||

M7 | Elliptical | 2.10 × 1.40 | 2.31 | 12.0 | 0.8 | |||

M8 | Elliptical | 1.92 × 1.67 | 2.52 | 12.0 | 0.8 | |||

M9 | Elliptical | 1.92 × 1.67 | 2.52 | 12.0 | 0.5 | |||

M10 | Elliptical | 1.92 × 1.67 | 2.52 | 12.0 | 1.0 | |||

M11 | Elliptical | 1.92 × 1.67 | 2.52 | 12.0 | 1.5 | |||

M12 | Elliptical | 1.92 × 1.67 | 2.52 | 13.0 | 0.8 | |||

M13 | Elliptical | 1.92 × 1.67 | 2.52 | 14.0 | 0.8 |

Model | Posterior Scleral Wall Stress | Maximum PPS Stress | Maximum Laminar Insertion Stress | Maximum Laminar Trabecular Stress |
---|---|---|---|---|

Circular | ||||

M1 (0.50× 0.50) | 21.5 | 22.6 | 40.5 | 68.1 |

M2 (1.50× 1.50) | 21.5 | 29.7 | 61.0 | 107.2 |

M3 (1.75× 1.75) | 21.5 | 31.7 | 68.7 | 129.5 |

M4 (2.00× 2.00) | 21.5 | 33.8 | 76.7 | 154.1 |

M5 (2.56× 2.56) | 21.4 | 39.5 | 94.6 | 215.0 |

Elliptical | ||||

M6 (2.50× 1.25) | 21.5 | 41.1 | 79.4 | 144.4 |

M7 (2.10× 1.40) | 21.5 | 34.1 | 70.6 | 130.1 |

M8 (1.92× 1.67) | 21.5 | 32.3 | 70.4 | 133.7 |

Wall thickness | ||||

M9 (0.5) | 34.4 | 53.2 | 126.5 | 243.8 |

M8 (0.8) | 21.5 | 32.3 | 70.4 | 133.7 |

M10 (1.0) | 17.2 | 25.4 | 53.1 | 99.2 |

M11 (1.5) | 11.7 | 16.3 | 31.8 | 56.7 |

Inner radius | ||||

M8 (12.0) | 21.5 | 32.3 | 70.4 | 133.7 |

M12 (13.0) | 23.2 | 33.9 | 73.0 | 132.8 |

M13 (14.0) | 25.0 | 35.6 | 75.7 | 132.8 |

Model | Posterior Sclera | Peripapillary Sclera | Laminar Insertion Zone | Laminar Trabeculae |
---|---|---|---|---|

Circular | ||||

M1 (0.50× 0.50) | 11 | 11 | 20 | 34 |

M2 (1.50× 1.50) | 11 | 15 | 31 | 54 |

M3 (1.75× 1.75) | 11 | 16 | 34 | 65 |

M4 (2.00× 2.00) | 11 | 17 | 38 | 77 |

M5 (2.56× 2.56) | 11 | 20 | 47 | 107 |

Elliptical | ||||

M6 (2.50× 1.25) | 11 | 21 | 40 | 72 |

M7 (2.10× 1.40) | 11 | 17 | 35 | 65 |

M8 (1.92× 1.67) | 11 | 16 | 35 | 67 |

Wall thickness | ||||

M9 (0.5) | 17 | 27 | 63 | 122 |

M8 (0.8) | 11 | 16 | 35 | 67 |

M10 (1.0) | 9 | 13 | 27 | 50 |

M11 (1.5) | 6 | 8 | 16 | 28 |

Inner radius | ||||

M8 (12.0) | 11 | 16 | 35 | 67 |

M12 (13.0) | 12 | 17 | 36 | 66 |

M13 (14.0) | 13 | 18 | 38 | 66 |

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