Engineering Models of Stress and Strain in the ONH and Peripapillary Sclera

Closed form solutions



Attempts to mathematically model the mechanical environment of the ONH generally fall into two broad categories - closed form solutions and numerical solutions. In closed form solutions, engineering principles are used to derive equations that can be analyzed to understand the effects of selected biological parameters. Examples of closed form approaches include work by Donqui, Edwards, and Good, and a hybrid cellular solid approach by Sander. The appeal of closed form solutions is that general conclusions may be drawn from a model cast in terms of a limited number of geometric and material parameters that are felt to be of interest or might be clinically measurable. However, closed form solutions may be of limited utility because of the complexity of the ONH and peripapillary scleral tissues (e. g., the nonuniform and asymmetric geometry and material properties). The most sophisticated of these studies suggest that the structural stiffness of the sclera is the most important determinant ofmacro-level ONH biomechanics.



Numerical solutions - FE analysis



To overcome the inherent limitations of closed form solutions, researchers often utilize numerical methods to study more complex biological systems. One of the most powerful of these is FE analysis. In FE analysis, complex load-bearing structures are broken into small, regularly shaped elements (Figure 4). Stress and strain within each element is calculated and then superposed to predict the mechanical response of the entire structure.



Figure 10 Experimental results and finite element (FE) model predictions of the nonlinear, anisotropic displacement behavior of an individual monkey posterior scleral shell as IOP increases from 5 to 10 mmHg (top row), 10 to 30 mmHg (middle row), and 30 to 45 mmHg (bottom row). Experimental (gray background) and predicted (model, black background) displacements for the X-direction (left-to-right), Y-direction (top-to-bottom), and Z-direction (in-and-out) are mapped onto the outer surface of the scleral shell of a right eye (for each map superior top, inferior bottom, temporal left and nasal right). The inhomogeneity of the experimental displacement patterns are indicative of underlying tissue anisotropy, and the much greater displacements seen in the 5-10 mmHg IOP elevation as compared to the 30-45 mmHg IOP elevation reflect the highly nonlinear behavior (i. e., the sclera is stiffer and therefore more resistant to deformation at higher levels of strain). Figure, including experimental and modeling results, courtesy of Michael Girard.



The power of FE analysis lies in its ability to model structures with highly complex geometries using material properties with varying levels of complexity as warranted (e. g., inhomogeneous, anisotropic, nonlinear, or viscoelastic material descriptions). The three components necessary as input for FE models are: the 3D geometry of the tissue structure to be modeled, the material properties of the different tissues in the model, and appropriate loading and boundary conditions. These requirements have spurred the development of methodologies to isolate and describe the 3D geometry of the ONH and peripapillary sclera (Figure 8) and experimentally characterize their constituent material properties (Figure 10).



There are two basic approaches to FE modeling of the ONH: parametric and individual specific. Parametric modeling involves computing stress and strain in average, idealized geometries that do not conform to any individual’s particular anatomy. Within these models, parameters such as peripapillary scleral thickness and laminar stiffness can be varied independently to gauge the parameter’s effects on ONH biomechanics as a whole. This is a similar approach to analytical modeling, but the analyzed geometries are much more fidelic and the results more relevant and intuitive. Although parametric FE models are by nature simplified in their geometries and there are limited cases that can be modeled, these investigations yield interesting insight into the contributions of individual anatomical elements and tissue material properties to overall ONH biomechanics.



Bellezza and colleagues used parametric FE modeling to study the mechanical environment of an idealized 3D model of the posterior pole. In this study, the effects of the size and shape (aspect ratio) of an elliptical scleral canal within a spherical scleral shell of uniform thickness were studied. Idealized beamlike structures spanning the ONH were also incorporated into the model to simulate the lamina cribrosa. This study illustrated that lOP-related stress concentrations within the load-bearing connective tissues of the ONH is substantial, even at low levels of IOP. Specifically, models with larger scleral canal diameters, more elliptical canals, and thinner sclera all showed increased stresses in the ONH and peripapillary sclera for a given level of IOP. In the peripapillary sclera and ONH, stresses were as much as one and two orders of magnitude greater than IOP, respectively. While the model used in this study was idealized in terms of its material properties and geometry, it served to reinforce the concept of the peripapillary sclera and ONH as a high stress environment even at normal levels of IOP.



Sigal and co-workers used idealized axisymmetric FE models to pursue a more complex parametric analysis of the factors that influence the biomechanical environment within the ONH (Figure 11). In these studies, various geometric and material details of a generic model were parametrized and independently varied to assess their impact on a host of outcome measures such as strain in the lamina cribrosa and prelaminar neural tissue (Figure 11). This work identified the five most important determinants of ONH biomechanics (in rank order) as: the stiffness of the sclera, the size of the eye, IOP, the stiffness of the lamina cribrosa, and the thickness of the sclera. The finding that scleral stiffness plays a key role in ONH biomechanics is especially interesting, and was also found to be important in the analytical models of Sander and co-workers. Parametric studies such as these are important because they can be used to identify important biomechanical factors that warrant more in-depth study, thus narrowing and focusing future experimental and modeling efforts.



To address the limitations of idealized geometric and material property descriptions inherent in parametric FE models, individual-specific FE models can be created from the reconstructed geometries of particular eyes. At present, individual-specific modeling is based on high-resolution 3D reconstructions of monkey and human cadaver eyes (Figure 8), with a long-term goal to build models based on clinical imaging of living eyes so as to use them in the assignment oftarget IOP. This is especially important given that the 3D geometry of the scleral canal and peripapillary sclera largely determines the stress and strain transmitted to the contained ONH (Figure 7 Notes specifically how the 3D geometry of the scleral canal and peripapillary sclera alter the stress environment). Anatomically accurate 3D models are necessary to capture the biomechanics of anisotropic scleral material properties (varying collagen fibril orientation), and scleral canals that are noncircular and have varying optic nerve insertion angles (i. e., the optic nerve inserts from the nasal side resulting in a thinner peripapillary sclera in that quadrant). When modeling an ONH with anatomic fidelity, the tissue geometries can be constructed either by serial histologic methods or by 3D imaging, and material properties are generally determined through direct mechanical testing (Figure 10). Unfortunately, imaging of the lamina in vivo is not yet possible at the resolutions required for modeling, and no technology exists for experimental biomechanical testing of laminar beams. As a result, ONH FE models are typically constructed from eyes that are perfusion or immersion fixed at a selected IOP, and then undergo ex vivo 3D reconstruction of their connective tissues.



Bellezza and co-workers developed a histologic technique to 3D reconstruct the trabeculated structure of the lamina cribrosa from individual monkey eyes that have been perfusion fixed at varying levels of IOP (Figure 8). The resulting 3D data sets form the geometries of individual-specific FE models of the ONH at the macro-and micro-scale. Roberts, Downs, and co-workers have developed macro-scale continuum FE models of the posterior pole and ONH connective tissues from individual monkey eyes (Figure 12). In these models, the laminar microarchitecture is modeled using a continuum approach, with anisotropic material properties assigned to each FE in the ONH based on the connective tissue volume fraction and the predominant beam orientation of the contained laminar microarchitecture (Figure 13). Regional variations in connective tissue volume fraction and predominant



Figure 11 Parametric models can be used to study the influence of geometric and material property factors. To model the ONH, Sigal and colleagues created an idealized, axisymmetric (symmetric about the anterior-to-posterior axis) reference geometry and varied geometric and material property factors to assess their influence on various outcome measures of stress and strain within the model. This type of parametric sensitivity analysis is useful for identifying the tissues and anatomic structures that may be most important in the mechanical response of the ONH. Such information can serve to focus future biomechanics research and clinical device development efforts on the tissues and structures determined to be most important in ONH biomechanics. Figure courtesy of Ian Sigal.




Figure 12 Construction and results from a macro-scale continuum FE model of the posterior scleral shell and ONH of a normal monkey eye. (a) To construct the model geometry, the 3D-delineated lamina cribrosa and surrounding peripapillary sclera (see Figure 8) of an individual eye are incorporated into a generic anatomic scleral shell with regional thickness variations mapped from previous



Orientation are translated into variations in local oriented stiffness so that regions of higher and lower porosity reflect greater and lesser compliance, respectively. The inclusion of regional laminar material properties (connective tissue volume fraction and beam orientation) into FE models has a pronounced effect on the ONH’s response to IOP (Figure 14). This indicates that the regional variations in laminar geometry and structural stiffness must be represented in models to fully capture the biomechanical behavior of the ONH and suggests that the lamina is biologically optimized to withstand lOP-induced deformation.



Downs and colleagues have also used the 3D reconstruction and continuum modeling approaches to characterize and explore laminar beam biomechanics. This micro-scale modeling approach utilizes a substructuring technique based on parent macro-scale FE models to calculate the IOP-related stress and strain fields in laminar



Figure 13 Regional differences in laminar microarchitecture in a normal monkey eye. Characterization of the laminar microarchitecture utilizes the element boundaries of a continuum finite mesh to partition the lamina cribrosa connective tissue into 45 subregions. The connective tissue volume fraction (CTVF) for each region is expressed as a percentage and mapped to a grayscale value in the background. The arrows indicate the predominant orientation of the laminar beams in each region, with higher values (color-coded) indicating regions in which the beams are more highly oriented. Note that in the peripheral regions of the lamina, the beams are tethered radially into the scleral canal wall.



Beams (Figure 15). This technique reveals a complexity of lOP-related strains and stresses within the lamina cribrosa microarchitecture that is not available through macroscale FE modeling. There have been several interesting preliminary results from this work. First, stress and strain in the laminar microarchitecture are likely higher than predicted by macro-scale models of the ONH. Second, even at normal levels ofIOP, the micro-FE models predict that while the majority of laminar beams are within physiologic strain ranges, there are individual laminar beams with levels of lOP-related strain that are likely pathologic. Third, mean strain within the laminar beams of different



Figure 14 Incorporation of laminar beam orientation and connective tissue volume fraction into the material description of the lamina cribrosa affects the predictions of finite element (FE) models. Internal (vitreous) surface views of the displacement and strain in the ONH (within the heavy black outline) and peripapillary sclera in continuum models of the same eye following acute IOP elevation. In column (a), the stiffness of each laminar element is determined from both the predominant laminar beam orientation and connective tissue volume fraction (CTVF) of the contained lamina. In column (b) the orientation information (i. e., using an isotropic material stiffness), but retaining mapped CTVF produces a substantial increase in the displacement of the lamina with markedly higher strains. In column (c) all elements in the lamina are assigned the same isotropic material stiffness based on an average CTVF (with no beam orientation information). This model has slightly more central laminar displacement than case (b), but actually has lower strain in the superior lamina cribrosa owing to the fact that the superior elements in case (b) had lower CTVFs in that region (i. e., below the mean CTVF). These results suggest that representing regional laminar microarchitecture in FE models is essential to accurately predict ONH biomechanical behavior.



Histologic measurements. The segmented 3D reconstruction of the laminar connective tissue (shown) is represented in each model.



(b) A continuum FE mesh of the posterior pole is generated from the geometry. The sclera in this model is assigned uniform isotropic material properties based on previous experimental testing. The continuum elements representing the porous load-bearing laminar architecture are assigned anisotropic material properties that reflect the microstructure of the lamina enclosed by each laminar FE. This material property description is defined using a combination of the connective tissue volume fraction (CTVF) and the predominant laminar beam orientation. A visualization of the CTVF and predominant beam orientation are presented. Note that in this visualization, an anisotropy value of 1 would represent an isotropic material with no predominant orientation while larger values imply oriented laminar beams that impart higher stiffness in the direction of the plotted arrow. (c, d, e) FE results showing predicted displacement, strain, and stress distributions due to an increase in IOP from 10 to 45 mmHg. Note that in this eye, the model predicts that the ONH tilts inferiorly, the strains are highest along the superior-inferior axis of the ONH, and that the sclera bears most of the IOP-related stress.




Monkeys varies greatly, and is generally dependent on the 3D geometry of each eye’s ONH connective tissues. Finally, strain is not equally distributed though the ONH, and are concentrated in regions with less dense laminar beams. This work, while still in its early stages, holds the possibility of testing hypotheses about failure mechanisms and cellular responses at the level of the laminar beams.

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