Abstract
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.
Elevated intraocular pressure (IOP) is the primary risk factor for the development of glaucoma. There is, however, a wide range of sensitivities to IOP, wherein a substantial number of individuals with normal IOP develop the disease (normotensive glaucoma), whereas other individuals with elevated IOP show no signs of the neuropathy (ocular hypertension).
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.
Traditionally there have been two approaches to model the effects of IOP on the eye: analytical and numerical. Analytical models may be written as a mathematical expression. For example, Laplace's law (S = PR/2t) 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.
The objective of this work was to introduce an approach to estimate the effects of IOP on the ONH that combines the ease of use and speed of analytical models with the flexibility and power of FE models. This approach uses surrogate models encoded in an applet. In the first part of this manuscript we describe in detail what we mean by surrogate models, demonstrate how these can be developed for the ONH, and show how encoding these surrogate models into an applet produces a tool for estimating IOP-related ONH biomechanics. We show that predictions made with the applet are virtually identical with those previously obtained with standard FE modeling. In the second part of the manuscript we demonstrate the applet's usefulness by showing how it can be used to explore some questions on ONH biomechanics posed in the literature.
Eye Size.
Uncertainty in LC Mechanical Properties.
This paper presented a methodology to produce an applet with which to estimate the biomechanical effects of an elevation in IOP on the ONH. The applet, available for use at
www.ocularbiomechanics.com, was based on producing surrogate models. This applet combines the ease of use and speed of analytical models with the power and flexibility of FE models and can be used to make predictions over a wide range of geometries and material properties. By considering these simultaneously, the resultant estimates incorporate parameter interactions and nonlinear effects, which can be substantial even in simplified models with linear materials. Predictions made with this applet corresponded well with the simpler versions in the literature. We also demonstrated how the applet can be used to explore questions about ONH biomechanics posed in the literature.
The originality of this work is twofold: To the best of my knowledge, this is the first application of surrogate models in posterior pole biomechanics and the first implementation of FE-based surrogate biomechanical models into an applet. Surrogate models are convenient because they bypass the need to explicitly compute the source models (in this case FE models) while retaining the fundamentals of the response. When surrogate models are formulated in closed form, such as the polynomials used in this work, they also allow calculation of integrals and derivatives, which are useful to identify extreme or inflection points at relatively low computational cost.
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
The ability afforded by the applet to produce rapid estimates of the effects of IOP on the ONH is useful for evaluating hypotheses of sensitivity to IOP, as was demonstrated by the two examples provided. In the first example, we have shown that the models and applet predict that often, but not always, a small increase in IOP results in higher stresses and strains within the ONH in a larger eye than in a smaller one. Higher stresses and strains in larger eyes compared with smaller eyes have been hypothesized to be one of the reasons behind the increased risk for glaucoma associated with myopic eyes, independent of IOP.
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.
In the second example it has been shown that uncertainty in LC mechanical properties translates into substantial uncertainty in the predictions of IOP-induced stress and strain within the LC, even if every other characteristic of the ONH and sclera considered by the model is known. This suggests that it is important to continue working toward characterizing LC properties, whether by measuring properties of the LC itself, or its covariations with other characteristics.
A further convenience for the applet users was the reduction in the number of parameters from 21 in the original models
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).
We recognize a potential risk with the applet introduced here, namely, that the ease of use may make it easy to dismiss the fundamental limitations of the underlying FE models and their consequences. When interpreting predictions made with the applet it is critical to consider that the physiologic relevance and accuracy of the surrogate models and applet depend on the quality of the underlying FE models. There is no a priori reason to expect that a polynomial shall provide an adequate representation of the population of FE models. Here it was found that cubic polynomials allowed accurate representation of system behavior. The polynomials used as surrogate models should not be understood to be a mechanistic relationship, but rather an approximation of the responses dependence on the parameters within the ranges studied. George Box, the famous statistician, expressed this as
13 “All models are wrong, but some are useful.” Polynomials diverge, and predictions outside the region of fit are unreliable.
For simplicity, the methodology and applet introduced and demonstrated in this work were based on simplified models of the ONH. We have discussed in depth the limitations and most salient consequences of the choices of model geometry and tissue mechanical properties,
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
The methodology can be extended to more complex FE models, although the number of models to prepare, run, and analyze increases rapidly with the number of parameters in what is often referred to as “the curse of dimensionality.”
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.
Despite the limitations the surrogate models and applet in this work are already more comprehensive than any analytical model of the ONH, and much easier and faster to use than even the simplest FE models. Also, the predictions are more directly applicable to the human ONH than Laplace's law and Friedenwald's coefficient of rigidity.
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.
We chose to analyze tensile and compressive strains and von Mises stress because studies in mechanobiology have suggested that these are potentially biologically relevant (Rogers R, et al.
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 ).
In summary, this paper has introduced an applet with which it is simple to make rapid estimates of IOP-related ONH biomechanics. Use of the applet to explore questions posed in the literature has been demonstrated. The applet represents a step toward bringing the power of FE modeling beyond the specialized laboratory, heightening appreciation of the factors influencing ONH biomechanics, and thus can help develop and refine biomechanics-based hypotheses.
Supported in part by National Institutes of Health Grant P30EY008098 (Bethesda, MD), the Eye and Ear Foundation (Pittsburgh, PA), and unrestricted grants from Research to Prevent Blindness (New York, NY).
Disclosure:
I.A. Sigal, None
I would like to acknowledge the valuable input from Paul Sanfilippo, Jonathan Grimm, Michael Girard, and Gadi Wollstein.