**Purpose.**:
The ability to predict the biomechanical response of the optic nerve head (ONH) to intraocular pressure (IOP) elevation holds great promise, yet remains elusive. The objective of this work was to introduce an approach to model ONH biomechanics that combines the ease of use and speed of analytical models with the flexibility and power of numerical models.

**Methods.**:
Models representing a variety of ONHs were produced, and finite element (FE) techniques used to predict the stresses (forces) and strains (relative deformations) induced on each of the models by IOP elevations (up to 10 mm Hg). Multivariate regression was used to parameterize each biomechanical response as an analytical function. These functions were encoded into a Flash-based applet. Applet utility was demonstrated by investigating hypotheses concerning ONH biomechanics posited in the literature.

**Results.**:
All responses were parameterized well by polynomials (*R* ^{2} values between 0.985 and 0.999), demonstrating the effectiveness of our fitting approach. Previously published univariate results were reproduced with the applet in seconds. A few minutes allowed for multivariate analysis, with which it was predicted that often, but not always, larger eyes experience higher levels of stress and strain than smaller ones, even at the same IOP.

**Conclusions.**:
An applet has been presented with which it is simple to make rapid estimates of IOP-related ONH biomechanics. The applet represents a step toward bringing the power of FE modeling beyond the specialized laboratory and can thus help develop more refined biomechanics-based hypotheses. The applet is available for use at www.ocularbiomechanics.com.

^{ 1,2 }Thus, it is important to understand the effects of IOP on the optic nerve head (ONH) and how this varies between individuals. Of particular interest are the effects on the lamina cribrosa (LC), a region within the ONH where insult to the retinal ganglion cell axons occurs early in the disease. Despite recent advances in ocular imaging, such as second harmonic imaging

^{ 3 }and deep scanning OCT,

^{ 4 –6 }direct measurement of the effects of IOP on the ONH remains a challenge. As a result, modeling has become a leading approach for studying ocular biomechanics.

*S*=

*PR*/2

*t*) relates the tension (

*S*) on the wall of a spherical vessel to the magnitude of the pressure (

*P*), the radius (

*R*), and the thickness of the wall (

*t*). Analytical models are attractive for their elegance and simplicity, since it is simple to enter values and compute predictions. The complexity in deriving closed-form mathematical relationships, however, has meant that analytical models are limited to highly simplified geometries, material properties and loading conditions. Laplace's law, for example, assumes a thin-walled sphere composed of a single material. These assumptions, while valid in some circumstances, are violated when there is an opening in the shell, such as the ONH. Hence Laplace's law cannot be trusted to make valid predictions involving the ONH and peripapillary sclera. In contrast, numerical models such as those analyzed using the finite element (FE) method can incorporate more realistic geometries, materials, and loadings than analytical models can and are generally easier to adapt to new conditions. Nonetheless, even relatively simple FE models can be difficult to produce and analyze, requiring particular expertise and specialized software. Consequently, the ability to predict and evaluate hypotheses of how an increase in IOP affects the biomechanics of the ONH in a simple manner that considers the complexity of the tissues continues to elude researchers.

*metamodel*, and can be used as a surrogate in lieu of the actual FE models. Not surprisingly, obtaining a close fit required relatively long polynomial functions (> 80 terms), which are inconvenient to use. Therefore in the third step, the polynomials were coded into an applet. The applet works as a black box, handling the calculations and shielding the user from the complexity of the polynomial functions. With the applet it is easy to enter a set of values for the parameters, thus defining an ONH, and almost instantly obtain predictions of the ONHs response to increases in IOP. The accuracy of the predictions made with the applet depends on the closeness of the fits and the quality of the underlying FE models. For simplicity, the models and applet in this work are based on previously reported, and thoroughly discussed, simplified models.

^{ 7 –9 }Notwithstanding the simplifications, the applet is already more comprehensive than any analytical model of the ONH and much easier to use and orders of magnitude faster to compute than even the simplest FE models.

^{ 7,8 }For this study we selected eight parameters (Table 1 and Fig. 2), out of the 21 in the original models, based on a preliminary multivariate sensitivity analysis (results not shown). In the preliminary study it was found that the eight parameters, and their interactions, accounted for between 97.7% and 99.9% of the variance in the responses. For simplicity the applet presented in this manuscript is based on these eight parameters only, acknowledging that this implies an approximation of up to 2.3% in the variance relative to a model with 21 parameters. The 13 parameters not varied here were set at their baseline levels used in our previous work.

^{ 7,8 }All tissues were assumed linearly elastic, isotropic, and homogeneous.

^{ 7 –10 }Tissue stiffnesses were defined by Young's moduli and compressibilities by Poisson's ratios. All tissues, other than the prelaminar neural tissue (PLNT), were assumed incompressible. In this work, stiff and compliant are used to describe high and low Young's moduli, respectively. Thus, stiffness is equivalent to the tissue's mechanical property and is independent of the geometry. The parameters and their ranges have been discussed in detail elsewhere.

^{ 7 –10 }The geometric parameters were defined as described elsewhere.

^{ 7,8 }

Name | Units | Range | |
---|---|---|---|

Low | High | ||

Intraocular pressure increase* | mm Hg | 0 | 10 |

Internal radius of eye shell | mm | 9.6 | 14.4 |

Scleral shell thickness | mm | 0.64 | 0.96 |

LC anterior surface radius | mm | 0.76 | 1.14 |

Poisson ratio of prelaminar tissue | — | 0.4 | 0.49 |

Lamina cribrosa Young's modulus | MPa | 0.1 | 0.9 |

Sclera Young's modulus | MPa | 1 | 9 |

Neural tissue Young's modulus | MPa | 0.01 | 0.09 |

^{ 7 –9 }) and characterized by the 50th and 95th percentiles, the median and peak.

^{ 7 –9 }To improve regression fits the responses were transformed, with the optimal transformation for each response determined using a Box-Cox analysis.

^{ 11,12 }For all responses it was found that the optimal transformation was a (natural) logarithm.

*f*of the form where the

*x*'s are the factors, β's are the regression coefficients to be estimated, and ε is the residual. The coefficients represent the following: β

_{0}is the offset, β

*the linear factor effects, β*

_{i}*the two-factor interactions (*

_{ij}*i*≠

*j*) or the quadratic factor effects (

*i*=

*j*), and β

*the higher-order interactions and the cubic factor effects (*

_{ijk}*i*=

*j*=

*k*). We evaluated whether it was necessary to use the full function, a third-order polynomial, or if close fits could be obtained with reduced versions.

*R*

^{2}), but this coefficient is susceptible to artifacts (e.g., its value increases with the number of data points or with the range of the data). Thus we also computed the adjusted and predited

*R*

^{2}, which are less sensitive to such artifacts.

^{ 11,13 }Additionally, the signal-to-noise ratio, as the ratio of the range of the predicted values to the average prediction error was calculated.

^{ 11,12 }Statistical design and analysis were carried out using specialized software (Design-Expert 7; Stat-Ease Inc., Minneapolis, MN).

^{ 7 –9 }for example, capturing simultaneously factor interactions and cubic nonlinearities in the responses. Nevertheless, the FE models used in this work were based on our previous models. Hence, it should be possible to reproduce with the applet the previous predictions,

^{ 8 }albeit with the differences in IOP increase (up to 10 mm Hg here and 25 mm Hg previously). Thus, as a check on the applet we repeated one part of the sensitivity analysis in

^{ 8 }and compared the results.

^{ 14 –18 }

^{ 19,20 }Here we consider the following question: If all the characteristics of the ONH and sclera were known precisely, except for the mechanical properties of the LC, how much variability (uncertainty) would remain in the predicted IOP-induced stress/strain within the ONH?

^{ 7,20,21 }Hence, the effects of one parameter need to be tested for many combinations of the other parameters. Our goal with this work was to demonstrate how the applet can be used to explore these questions, not to present a comprehensive analysis on the effects of either eye size or LC properties. Hence we varied the parameters in search of interactions using an arbitrary empirical search pattern guided by our experience.

*R*

^{2}values were between 0.985 and 0.999. Other measures of the quality of fit were also excellent, demonstrating that the fits capture the responses adequately and therefore that the polynomial functions can serve as surrogates for the FE models.

Tissue | Lamina Cribrosa | Prelaminar Neural Tissue | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Measure | Strain | Stress | Strain | Stress | ||||||||

Percentile | Tensile | Compressive | Von Mises | Tensile | Compressive | Von Mises | ||||||

50th | 95th | 50th | 95th | 50th | 95th | 50th | 95th | 50th | 95th | 50th | 95th | |

R ^{2} | 0.998 | 0.988 | 0.999 | 0.994 | 0.999 | 0.996 | 0.996 | 0.987 | 0.999 | 0.995 | 0.998 | 0.994 |

Adjusted R ^{2} | 0.998 | 0.988 | 0.999 | 0.993 | 0.999 | 0.995 | 0.996 | 0.986 | 0.999 | 0.994 | 0.998 | 0.993 |

Predicted R ^{2} | 0.998 | 0.987 | 0.999 | 0.993 | 0.999 | 0.995 | 0.996 | 0.985 | 0.999 | 0.994 | 0.998 | 0.993 |

SSq model | 903.8 | 486.8 | 1074.6 | 751.2 | 1264.3 | 1493.2 | 667.8 | 401.2 | 578.7 | 493.6 | 1104.1 | 1094.2 |

SSq residual | 2.0 | 5.7 | 1.4 | 4.7 | 1.2 | 6.6 | 2.5 | 5.4 | 0.7 | 2.7 | 2.2 | 7.0 |

Residual % | 0.2 | 1.2 | 0.1 | 0.6 | 0.1 | 0.4 | 0.4 | 1.3 | 0.1 | 0.6 | 0.2 | 0.6 |

SNR | 435 | 189 | 598 | 257 | 706 | 333 | 281 | 179 | 582 | 273 | 522 | 295 |

PRESS | 1.25 | 3.13 | 0.92 | 2.01 | 1.29 | 7.24 | 2.43 | 3.81 | 0.56 | 1.76 | 2.45 | 7.91 |

DOF model | 92 | 87 | 87 | 90 | 86 | 83 | 97 | 88 | 88 | 95 | 82 | 97 |

DOF residual | 1999 | 2004 | 2004 | 2001 | 2005 | 2008 | 1994 | 2003 | 2003 | 1996 | 2009 | 1994 |

*P*< 0.0001) contribution to the response were included in the regressions (DOF model). The rest of the DOF were grouped as a measure of the residual (DOF residual). DOF, degrees of freedom; PRESS, predicted residual sum of squares; SNR, signal-to-noise ratio; SSq, sum of squares corrected by the mean.

Example Effects on the LC of Uncertainty in Its Own Material Properties | ||
---|---|---|

Case with LC Modulus Influencing Stress More than Tensile Strain | ||

Sclera modulus: 9 MPa | Neural tissue modulus: 0.09 Mpa | Sclera thickness: 0.8 mm |

Eye radius: 14.4 mm | PLNT compressibility: 0.4 | Canal size: 0.76 mm |

Median Von Mises Stress | Median Tensile Strain | |
---|---|---|

Soft LC (modulus 0.1 MPa) | 4.6 | 0.33 |

Stiff LC (modulus 0.9 MPa) | 27.6 | 0.24 |

Ratio of largest to smallest values | 6 | 1.38 |

Case with LC Modulus Influencing Tensile Strain More than Stress | ||
---|---|---|

Sclera modulus: 5 MPa | Neural tissue modulus: 0.01 Mpa | Sclera thickness: 0.96 mm |

Eye radius: 9.6 mm | PLNT compressibility: 0.45 | Canal size: 1.14 mm |

Median Von Mises Stress | Median Tensile Strain | |
---|---|---|

Soft LC (modulus 0.1 MPa) | 9.5 | 0.94 |

Stiff LC (modulus 0.9 MPa) | 19.3 | 0.17 |

Ratio of largest to smallest values | 2 | 5.5 |

^{ 8 }but that encoded as an applet are much simpler and faster to use. Previously

^{ 8 }parameters effects were analyzed independently. The applet introduced here is much more flexible, making estimates for any combination of parameters within the ranges in Table 1.

^{ 22,23 }For these and other useful properties, surrogate models have seen application in several areas of engineering, where they are often used in optimization.

^{ 12,23,24 }

^{ 14 –18 }Our results therefore support these hypotheses but also predict a range of sensitivities due to other ocular characteristics. Specifically, it was predicted that IOP-induced stress and strain slightly decrease with increased eye size when the eyes have a thick and stiff sclera, a large canal size, and soft neural and LC tissues. It is still unknown how often these characteristics occur simultaneously. Previous nonmultivariate techniques for computing the biomechanical effects of IOP on the ONH were incapable of making a prediction such as this. Since the scleral shell was assumed spherical, eye diameter was varied rather than axial length.

^{ 7 –9 }to the eight most influential ones. This was only discussed briefly here for simplicity and because it was done using statistical techniques similar to those we have applied elsewhere.

^{ 7,21,22 }Although we acknowledge that not accounting explicitly for 13 parameters implies an approximation of up to 2.3% in the variance, we believe that reducing the number of parameters by 61% was worthwhile, especially when considering that this reduces the number of two- and three-factor interactions dramatically (by 98.78% and 99.99%, respectively).

^{ 13 }“All models are wrong, but some are useful.” Polynomials diverge, and predictions outside the region of fit are unreliable.

^{ 9,25 }of the parameters and their ranges,

^{ 7,8,10 }and of the responses analyzed.

^{ 7,8,10,26 }Hence, these will not be discussed at length again. Instead, we summarize earlier discussions, with a focus on the limitations and considerations most relevant to this work. The models represent only an acute deformation of the tissues due to increases in IOP and do not account for the long-term remodeling processes that are known to take place as glaucoma develops.

^{ 27 –32 }The models do not account for LC microarchitecture, which may amplify the levels of strain (Kodiyalam S, et al.

*IOVS*2009;50:ARVO E-Abstract 4893), and do not consider the stresses at the baseline IOP. The models were based on a simplified axisymmetric geometry and therefore do not completely reflect the complex architecture of the ONH region or the corneoscleral shell (which is not of constant thickness).

^{ 33 }In addition, the ONH geometry differs between individuals in more complex ways than can be captured by the factors considered.

^{ 34,35 }

^{ 23 }In recent years there have been substantial advances in imaging and other experimental techniques, which have been applied to the posterior pole and ONH.

^{ 3,6,33,36,37 }We are working to integrate these advances into improved FE models that incorporate more realistic anatomies (like the variations in scleral shell thickness

^{ 33,38,39 }), material properties (anisotropic and nonlinear scleral properties,

^{ 36,40,41 }lamina cribrosa anisotropy, and inhomogeneity

^{ 3,19,35 }), and loading (larger IOP insult and cerebrospinal fluid pressure

^{ 42 –46 }). More complex models will require even more effort to produce and parameterize and have higher computational requirements. The time savings of surrogate models will be even greater in such models.

^{ 20 }This study differs from most of the numerical studies of ONH biomechanics in that we analyzed relatively low levels of IOP (from 5 to 15 mm Hg). We did this for several reasons: First, normal IOP is much more common than elevated IOP,

^{ 1,2 }and therefore the analysis is relevant to a larger group. Second, there is better information on which to base the parameters and their ranges for normal eyes.

^{ 9,18,31 }Third, small IOP elevations may be particularly informative in understanding the pathogenesis of low-tension glaucoma. Further, as we have demonstrated before, ONH biomechanics are complex, even with simplified geometries and material properties.

^{ 8,9,21,26 }Simulating a relatively small IOP increase allowed us to use linear materials, whose stiffness can be specified by a single parameter for each tissue—the Young's modulus. Studies of ocular tissue properties have shown that while the assumption of linear scleral properties is reasonably adequate at low levels of IOP (under 10 mm Hg), it becomes increasingly problematic at elevated IOP (above 20 mm Hg), because as the tissue stretches it stiffens.

^{ 36,40,41,47 –50 }We believe that a solid understanding of ONH biomechanics at low pressures helps build up for understanding larger pressure increases.

*IOVS*2009;50:ARVO E-Abstract 888).

^{ 51 –55 }We have previously discussed the need to differentiate between tensile and compressive strains, as well as the value of computing peak and median levels of strain.

^{ 26 }The LC is where insult to the retinal ganglion cell axons is believed to initiate in glaucoma,

^{ 2,56 }whereas the PLNT is also of interest since it changes so dramatically during the development of glaucomatous neuropathy.

^{ 34,57,58 }Work is underway on extending the responses analyzed to include other potentially biologically important measures of the effects of IOP (like the shearing strains

^{ 26,59 –61 }) and those measurable in the experiment (such as LC displacement and canal expansion

^{ 6,31,34,37,62,63 }).

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