**Purpose.**:
To study the anterior–posterior lamina cribrosa deformation (LCD) and the scleral canal expansion (SCE) produced by an increase in IOP and identify the main factors and interactions that determine these responses in the monkey.

**Methods.**:
Eye-specific baseline models of the LC and sclera of both eyes of three normal monkeys were constructed. Morphing techniques were used to generate 888 models with controlled variations in LC thickness, position and modulus (stiffness), scleral thickness and modulus, and scleral canal size and eccentricity. Finite element modeling was used to simulate an increase in IOP from 10 to 15 mm Hg. A two-level, full-factorial experimental design was used to select factor combinations and to determine the sensitivity of LCD and SCE to the eight factors, independently and in interaction.

**Results.**:
LCD was between 53.6 μm (posteriorly) and −12.9 μm (anteriorly), whereas SCE was between 0.5 and 15.2 μm (all expansions). LCD was most sensitive to laminar modulus and position (24% and 21% of the variance in LCD, respectively), whereas SCE was most sensitive to scleral modulus and thickness (46% and 36% of the variance in SCE, respectively). There were also strong interactions between factors (35% and 7% of the variance in LCD and SCE, respectively).

**Conclusions.**:
IOP-related LCD and SCE result from a complex combination of factors, including geometry and material properties of the LC and sclera. This work lays the foundation for interpreting the range of individual sensitivities to IOP and illustrates that predicting individual ONH response to IOP will require the measurement of multiple factors.

^{ 1 –3 }Several studies have explored the hypothesis that an altered biomechanical environment within the optic nerve head (ONH), and the lamina cribrosa (LC) in particular, may contribute to disruption of the retinal ganglion cell axons and the subsequent loss of vision associated with glaucoma.

^{ 1,4 –15 }Hence, there has been a search for an association between changes in IOP and deformations of the LC.

^{ 5,8 –13,16 }

^{ 16,17 }Results from initial studies using radiographic

^{ 10,11 }or histologic

^{ 8,9 }techniques, or measurements of the optic disc surface

^{ 15,18 –20 }were interpreted as supportive of this paradigm. More recent numerical

^{ 6,7,12,13,21 –23 }and experimental

^{ 24,25 }studies, however, suggest that the relationship between IOP and the deformations of the LC and sclera are more complex than initially thought. It is now clear, for example, that as IOP varies, the sclera deforms, sometimes substantially, and that these deformations, when transmitted to the ONH, may play an important role in the response of the LC to IOP.

^{ 5,24 –26 }Numerical models have also predicted that IOP-related anterior–posterior displacements of the LC may be small and could be even smaller in magnitude than lateral LC displacements (i.e., in the plane of the sclera).

^{ 6,12,22,26,27 }This is consistent with recent measurements obtained using 3D histomorphometry and optical coherence tomography (Burgoyne CF, et al.

*IOVS*2008;49:ARVO E-Abstract 3655; Agoumi Y, et al.

*IOVS*2009;50:ARVO E-Abstract 4898).

^{ 25,28 }It is now generally accepted that the LC does not respond to IOP changes in isolation, but rather that the ONH and peripapillary sclera behave as a mechanical system, and that the IOP-related deformations of the lamina and sclera are linked. Still, it is not clear how the LC and sclera deform as IOP changes, or how these deformations depend on the tissues' geometry and material properties. Numerical studies suggest that proper characterization of ONH biomechanics requires considering factor interactions (i.e., that the influence of one factor depends on the level of another). Although this seems reasonable, to the best of our knowledge, the role of factor interactions on LC and sclera deformations has not been reported.

^{ 5,21 }

^{ 29,30 }This enabled us to produce new “morphed” models related to the baseline with precisely controlled variations in geometry and materials by specifying a few high-level parameters or factors. We produced six families of models, one per baseline model, spanning the physiologic ranges of the factors following a design of experiments statistical approach.

^{ 31,32 }Eight factors were studied: seven parameterized factors (five geometric and two material) and the baseline model. Each model was solved using finite element techniques to predict the LCD and SCE resulting from an increase in IOP. The process was repeated for all six eye-specific models. We then used factor analysis techniques to determine the sensitivity of LCD and SCE to the factors and factor interactions. Preprocessing, including morphing and meshing, and postprocessing were done using Python scripts (www.Python.org/ open-source software provided by the Python Software Foundation) and a combination of standard and custom modules in commercial software (Amira; ver. dev4.1.1; Visage Imaging, Richmond, VIC, Australia); finite element analysis and statistical design and analysis were also performed with commercial software (Abaqus, ver. 6.8.1; Dassault Systèmes, Vèlizy-Villacoublay, France; and Design-Expert, ver. 7; Stat-Ease, Minneapolis, MN, respectively).

^{ 24,33,34 }to characterize the LC microarchitecture,

^{ 35 }and in analyses of the stresses and strains within the LC induced by an acute increase in IOP (Kodiyalam S, et al.

*IOVS*2009;50:ARVO E-Abstract 4893.

^{ 12,36 }These studies provide the details of tissue preparation and 3D reconstruction of the ONH geometry in a way suitable for finite element modeling. Briefly, the eyes were perfusion fixed at an IOP of 10 mm Hg. The ONH and peripapillary sclera were trephinated (6-mm diameter) and embedded in paraffin. A microtome-based serial sectioning technique was used to acquire consecutive stained block-face images. The images were aligned and assembled into a stack (voxel resolution, 1.5 × 1.5 × 1.5 μm). The neural canal wall and the anterior and posterior surfaces of the LC and peripapillary sclera were manually delineated in custom software. The delineations were then used to construct smooth triangulated surfaces representing the eye-specific geometry of the LC and peripapillary sclera, which were then integrated into a generic shell with anatomic shape and thickness (Fig. 1).

^{ 31,32 }Scleral canal eccentricity represents the shape of the canal, similar to the ovality used for the optic disc,

^{ 37 }and was defined as the ratio of the major to minor axes of the anterior lamina insertion. The five features were parameterized using morphing techniques, which are described and discussed in detail elsewhere.

^{ 29 }Scleral canal size and eccentricity were parameterized based on the distance to the centroid of the scleral canal opening. Deformations with the same sign along the major and minor axes of the canal opening resulted in expansions and contractions of the canal that preserved canal eccentricity, whereas deformations with opposite signs varied the eccentricity and preserved canal size. LC thickness and position were parameterized based on vectors normal to the anterior and posterior lamina surface. Deformations of both anterior and posterior lamina surfaces together varied lamina position and preserved thickness, whereas deformations of only the posterior lamina surface varied lamina thickness and preserved position. Scleral thickness was parameterized as described in our previous report.

^{ 29 }In addition to the five geometric features described above, we also studied intereye variability by defining eye as a nominal categorical factor with six levels.

Factor | Factor Range | ||
---|---|---|---|

Low | High | ||

Geometry | Scleral canal size (radius), μm | 569 | 787 |

Scleral canal eccentricity | 1.24 | 1.57 | |

Sclera thickness, μm | 116 | 217 | |

Lamina cribrosa thickness, μm | 82 | 150 | |

Lamina cribrosa position, μm | 42 | 152 | |

Mechanical properties | Sclera modulus, MPa | 5.3 | 18.4 |

Lamina cribrosa modulus, MPa | 0.39 | 3.7 | |

Eye | Nominal categorical factor with six levels |

^{ 5,39 }based on the equilibrium moduli reported by Downs et al.

^{ 40 }from uniaxial tests or based on averages of C

_{1111}and C

_{2222}at 10 mm Hg for young and adult monkeys reported by Girard et al.

^{ 41,42 }from inflation tests. For the lamina modulus we considered ranges based on the literature,

^{ 5,39 }based on ratios of lamina-to-sclera modulus or based on connective tissue volume density.

^{ 12,35 }The main differences between the ranges were that those based on the literature allowed for slightly lower limits for scleral and lamina moduli (down to 1 and 0.1 MPa, respectively), and those based on inflation tests allowed for a higher limit for the scleral modulus (up to 34.5 MPa). After analysis, the material properties turned out to be among the most influential factors. Hence, we decided to evaluate the sensitivity of the results on the assumed material properties. For this we replicated the study, repeating all runs and analyses, using different material property ranges. Although varying the ranges affected slightly the relative influence of the factors, the main results remained consistent. Therefore, for clarity, we show results obtained with laminar properties based on the studies by Roberts et al.,

^{ 12,35 }and scleral properties as an average of the values from the uniaxial

^{ 40 }and inflation

^{ 41,42 }tests, which were less than 10% different. Note that we use the term stiffness to represent the tissue mechanical properties, independent of geometry, and therefore cases with high or low Young's moduli are referred to as stiff or compliant, respectively. The concept of structural, or effective, stiffness is also useful and increasingly common,

^{ 13,21,42 }since it combines the tissue mechanical properties with aspects of its geometry, such as thickness and shape.

^{ 29,30,43 }: After the triangulated surfaces were morphed, the model volumes were meshed with four-node tetrahedra using target element sizes of 25, 50, and 125 μm for the lamina, peripapillary sclera, and scleral shell, respectively. The interior mesh was then iteratively smoothed and relaxed using Laplace's algorithm until the largest change in nodal locations was smaller than one fifth of a micrometer. For simulation and analysis the elements were converted to 10-noded tetrahedra by adding mid side nodes to the element edges. The nodal density necessary for accurate solution may depend on the parameter combinations and vary between models. Hence, the mesh refinement study had two parts. First, before morphing, we tested for sufficient nodal density in a baseline configuration. This nodal density was increased eight-fold (half target element-side length) and used as input for meshing the rest of the models. Second, after the simulations had been performed, we selected some cases with particularly large strains or deformations and tested whether these had been sufficiently refined. In every case, they were. The final models were formed by between 100,000 and 230,000 elements, depending on the geometry.

^{ 31,32 }The design was repeated six times to add the eye as a categorical (nominal) factor. Five extra runs were added to each combination of material properties, per eye, to evaluate pure error (such as drift).

^{ 31 }Pure error was 0 for both responses. This method resulted in an orthogonal, balanced design with 888 model combinations [(2

^{7}+ 5 × 4) × 6 = 888]. The order in which the models were prepared, run, and analyzed was randomized.

*P*< 0.01) and strength of factor and factor-interaction effects.

^{ 31,32 }Specifically, we determined the relative influence of the factors and their interactions by comparing their percentage contribution to the total sum of squares corrected by the mean. The sum of squares is computed as the sum of the squared differences between an observation and the mean of all observations. It is common in factorial analysis to use the sum of squares over factor levels to represent the factor contributions. (For more details of this method and the supporting rationale, please see Refs. 21, 30 –32.) The response variables were transformed to improve the normality of the responses and residuals, satisfy the requirements of ANOVA, and allow factor effects to be added in an unbiased fashion.

^{ 32 }For LCD, it was necessary to add a constant (14.1765 μm) to make the values positive and sufficiently large to avoid numerical problems.

^{ 31 }The responses were converted back to the original scale for plotting. A discussion of the rationale and consequences of the choice of 0.01 as the threshold for statistical significance has been published elsewhere.

^{ 21 }

*P*< 0.01), although their effects were relatively small.

Factor | LCD | SCE |
---|---|---|

Independent | ||

Scleral canal size | 3.8 | 3.2 |

Scleral canal eccentricity | 0.0‡ | 0.0* |

Sclera thickness | 4.4 | 36.1 |

Lamina cribrosa thickness | 7.2 | 0.0* |

Lamina cribrosa position | 21.3 | 2.3 |

Sclera modulus | 3.9 | 45.8 |

Lamina cribrosa modulus | 24.4 | 3.8 |

Eye | 0.5 | 2.0 |

Interactions | ||

Canal size–lamina position | 4.0 | 0.0* |

Canal size–lamina modulus | 5.7 | 0.1 |

Sclera thickness–sclera modulus | 1.7 | 2.6 |

Sclera modulus–lamina modulus | 0.1 | 1.1 |

Lamina thickness–lamina position | 4.2 | 0.0* |

Lamina thickness–lamina modulus | 3.8 | 0.0* |

Lamina position–sclera modulus | 1.2 | 0.6 |

Lamina position–lamina modulus | 4.9 | 0.6 |

Canal size–lamina position–lamina modulus | 0.8 | 0.0* |

Sclera modulus–lamina position–lamina modulus | 0.1 | 0.2 |

All interactions combined | 34.6 | 6.8 |

Distribution of actual values | ||

Maximum, μm | 53.6 | 15.2 |

Mean, μm | 3.4 | 4.2 |

Minimum, μm | −12.9 | 0.5 |

SD, μm | 10.9 | 3.4 |

Model total sum of squares | 935.7 | 453.0 |

*P*< 0.01). All statistically significant interactions were considered for the line labeled “All interactions combined”, although, for clarity, only those with the strongest effects are shown explicitly. Note that for LCD positive values correspond to a posterior displacement. LCD was influenced most strongly by lamina cribrosa modulus and position, whereas SCE by scleral modulus and thickness. See Figure 4 for a graphic representation.

^{ 5,12,22,26,36,39,44 }suggest that the LC modulus plays an important role on the LCD. The interaction plots in Figure 9 illustrate the effects on LCD of the lamina modulus and of its interactions with other parameters. It is clear that there are strong factor interactions, including lamina modulus with canal size, lamina thickness, and position.

^{ 5,21,22,26,39 }The strong dependence of LCD on various properties of the lamina itself, even when considering simultaneous scleral deformations, had been speculated on,

^{ 1,3,12,25 }but not demonstrated quantitatively. Although LCD and SCE were each most strongly sensitive to two factors, one material and one geometric, they were also highly sensitive to other factors. This was especially clear for LCD, which was also sensitive to LC thickness, canal size, and scleral thickness and modulus.

^{ 21 }Hence, in general, ranking of factors by influence can be misleading and must be interpreted accordingly. Despite this, we chose to include Table 2 and Figure 4, because we believe that they illustrate clearly how certain factors were substantially more influential than others.

*IOVS*2008;49:ARVO E-Abstract 3655; Agoumi Y, et al.

*IOVS*2009;50:ARVO E-Abstract 4898).

^{ 8 –11,24,25,28,33,34,38,45 }It also suggests that it is important to continue development of experimental techniques to measure all the factors involved and not just laminar position and modulus.

^{ 3,13,21,23,41,46,47 }The second interaction can be interpreted as the role of laminar stiffness in the expansion of the scleral canal: increased lamina modulus (stiffer) reducing canal expansion, more so when the sclera was compliant than when it was stiff (Fig. 6). We have noted and discussed the sensitivity of canal expansion to laminar stiffness.

^{ 12,13 }

^{ 3,12,23,25,47,48 }Within this framework, a stiff sclera would deform little, with a small canal expansion (i.e., not pulling the lamina taut), resulting in the LC displacing posteriorly. Conversely, a compliant sclera would deform more, pull the lamina taut from the sides, and result in a shift of LCD anteriorly (less posteriorly or more anteriorly). The results of this work, exemplified by Figure 6, support the conceptual framework. However, we have also shown that the sensitivity of LCD to scleral and laminar properties is complex, and therefore the extent to which the conceptual framework generalizes is still to be determined. Note that even for small LCDs or SCEs, the strains and stresses within the laminar and scleral tissues could be substantial.

^{ 6 }This possibility should be considered when interpreting measurements of acute deformation and their implications on sensitivity to IOP.

^{ 10,11 }and 2D

^{ 8,9,45,49,50 }or 3D histomorphometric techniques,

^{ 24,25,27,28,33,34,38 }and with optical coherence tomography imaging (Agoumi Y, et al.

*IOVS*2009;50:ARVO E-Abstract 4898). Although valuable, the variability in the ONH's response to changes in IOP, the difficulties of accessing the interior of the ONH (as opposed to imaging the surface of the disc), and the lack of a full characterization of the geometry and material properties of each eye have prevented the experimental approaches from providing a detailed picture of LCD and SCE and of their sensitivity to the geometry and material properties of the tissues. The ability to measure LC position reliably is in development (Burgoyne CF, et al.

*IOVS*2008;49:ARVO E-Abstract 3655; Agoumi Y, et al.

*IOVS*2009;50:ARVO E-Abstract 4898),

^{ 51 –54 }and none of the studies cited above reported the factors identified in this work measured as the most influential on LCD and SCE. This may explain why previous studies were unable to identify good predictors for IOP-related LCD and SCE. To the best of our knowledge, this study is the first to consider factor interactions on LCD and SCE. We are working on extending the responses analyzed to include other measures of the effects of IOP, either because they are potentially biologically important, such as the stresses and strains,

^{ 6,55,56 }or because they could be measured in an experiment, such as altered blow flow.

^{ 57 }

^{ 29,30,43 }(the morphing) allowed us to evaluate the effects of interactions between geometry and material properties and to demonstrate that they are important. Our previous study on the effects of factor interactions in ONH biomechanics focused on IOP-related stress and strain

^{ 21 }and did not evaluate LCD and SCE. These responses are potentially clinically measurable. Another strength of this study is that we used updated information on the ranges of tissue geometry and material properties. As we have discussed elsewhere,

^{ 5,21,39 }unnaturally large ranges can make factors artificially influential and attenuate the influence of other factors. Previous sensitivity studies used factor information from the literature, often spanning several species, treatments, and testing procedures.

^{ 5,12,21,39,44 }Herein, we used factor ranges derived from our own measurement of the normal monkey ONH, compiled in a way that optimized their applicability to this study (e.g., all factors were measured in the same samples).

^{ 12,13,35 }and to the choice of material properties.

^{ 5,26,39 }Simulating larger increases in IOP will probably necessitate the use of nonlinear material properties. We are working on models with more realistic material properties (inhomogeneous, anisotropic, and nonlinear sclera

^{ 7,41,42,58,59 }and inhomogeneous and anisotropic lamina

^{ 7,12,13,35,36,60 }) and various loading conditions (IOP insult and cerebrospinal fluid pressure

^{ 17,61 –64 }). In light of these assumptions, the absolute magnitudes of the predictions should be interpreted carefully. We believe, however, that the relative magnitudes, which encode the factor influences, are robust. The models represent an acute deformation of the tissues due to increases in IOP and do not account for viscoelastic effects or tissue remodeling. The consequences of introducing these complexities are difficult to predict, because their effects are nonlinear and because they are further complicated by the interactions between tissue properties and other factors.

^{ 21 }

^{ 6,27,65 }and therefore cannot be used to predict the mechanics of these tissues, which may be useful in relating the model predictions with experiments and may be relevant for understanding the physiologic consequences of IOP. It is also possible that we did not morph all the key factors that determine LCD and SCE. The morphing, by design, was dependent on the six eye-specific geometries and therefore may not represent all the variations and details possible. All six eyes had similar sensitivities to the factors, and the factor “eye” (the baseline eye-specific geometry) was therefore among the least influential factors. This outcome gives us confidence that the geometries and variations considered span reasonable physiologic ranges and represent the key factors. It also suggests that, for studies focusing on factor influences over the whole factor space, it may not be necessary to analyze several eyes. Still, given the factor interactions, we expect local sensitivity to vary between individuals. The models and analysis were based on the normal monkey eye, which is smaller and has a thinner LC and sclera than the human eye.

^{ 46,66,67 }The baseline geometries and the ranges over which the geometric factors were varied were derived from 3D reconstructions and histomorphometry that may have been affected by artifacts from the embedding and fixation, such as tissue shrinkage. These artifacts and how they may affect the results from histomorphometry are discussed in detail elsewhere.

^{ 24,25,28,33,34,38 }All the geometric factors, however, were derived from the same reconstructions, and hence the factors are mutually consistent. Large variations in tissue geometry and material properties also occur normally or in pathologic conditions, such as the enlarged disc, thinned LC and sclera, and altered scleral mechanical properties associated with myopia.

^{ 68,69 }These alterations have been hypothesized to be important contributors to susceptibility to IOP,

^{ 1,5,68,69 }and in future work we will extend the parameter ranges to study them.

^{ 51 –54 }as will some of the geometric parameters used in our analysis. Combining such measurements with sensitivity analyses may allow inferences to be made regarding changes in tissue compliance over time in individual patients, a potentially powerful biomarker for assessing risk of glaucomatous neuropathy.

*The Influence of Material Properties and Geometry on Optic Nerve Head Biomechanics*. Presented at the ASME Summer Bioengineering Conference 2009. Lake Tahoe, CA, June 17–21, 2009.