Individual-Specific Modeling of Rat Optic Nerve Head Biomechanics in Glaucoma

Abstract

Glaucoma is the second leading cause of blindness worldwide and is characterized by the death of retinal ganglion cells (RGCs), the cells that send vision information to the brain. Their axons exit the eye at the optic nerve head (ONH), the main site of damage in glaucoma. The importance of biomechanics in glaucoma is indicated by the fact that elevated intraocular pressure (IOP) is a causative risk factor for the disease. However, exactly how biomechanical insult leads to RGC death is not understood. Although rat models are widely used to study glaucoma, their ONH biomechanics have not been characterized in depth. Therefore, we aimed to do so through finite element (FE) modeling. Utilizing our previously described method, we constructed and analyzed ONH models with individual-specific geometry in which the sclera was modeled as a matrix reinforced with collagen fibers. We developed eight sets of scleral material parameters based on results from our previous inverse FE study and used them to simulate the effects of elevated IOP in eight model variants of each of seven rat ONHs. Within the optic nerve, highest strains were seen inferiorly, a pattern that was consistent across model geometries and model variants. In addition, changing the collagen fiber direction to be circumferential within the peripapillary sclera resulted in more pronounced decreases in strain than changing scleral stiffness. The results from this study can be used to interpret data from rat glaucoma studies to learn more about how biomechanics affects RGC pathogenesis in glaucoma.

Introduction

Glaucoma is the most prevalent cause of irreversible blindness, but current therapies are not always effective. The disease is characterized by the dysfunction of retinal ganglion cells (RGCs), which send vision information from the retina to the brain via axons that form the optic nerve. Blindness in glaucoma results from RGC apoptosis, but it is not well understood what triggers this process. A large body of evidence indicates that biomechanical insult is an important driver in glaucoma pathophysiology, including the fact that elevated intraocular pressure (IOP) is a key risk factor for the disease [1]. Further, previous modeling studies have shown that elevated IOP results in increased stress and strain in the optic nerve head (ONH), the region in the posterior eye where RGC axons exit the globe [2–4]; in fact, the ONH is also a main and early site of glaucomatous damage [5,6]. However, the links between biomechanical insult and RGC apoptosis have not been elucidated.

This incomplete understanding is due in part to the complexity of the relationship between biomechanics and cell dysfunction. Biomechanical insult to the ONH leads to tissue damage/remodeling [6], as well as alteration of the behavior of several different cell types within the ONH, and thus, disruption of RGC homeostasis [7]. In addition, different ONHs appear to have different levels of susceptibility to damage, as some people with higher-than-normal IOP never experience glaucomatous neuropathy, while others get glaucoma even at normal IOP levels [8]. However, the characteristics that predispose an ONH for damage are not well understood [6,7,9]. Therefore, detailed study of the biomechanical environment of the ONH and cell phenotypic changes occurring in glaucoma is necessary.

More precise knowledge of the sequence and interaction of events leading to RGC death would provide information for much-needed novel therapies. To better understand the role that biomechanics plays in RGC apoptosis, a model is needed which allows both precise characterization of ONH biomechanics as well as the ability to observe cellular responses at all stages of the disease. The rat ocular hypertensive models of glaucoma are good candidates because they present with important characteristics of human glaucoma when IOP is elevated, and because both chronic and acute models of IOP elevation are available [10,11]. In addition, rat models have advantages over primate models for mechanistic studies because of their lower cost, ease of care, and low genetic variability between individuals. The principle of three Rs in use of animals in research also favors the use of rodents over primates.

However, the rat ONH has substantial differences in anatomy from that of the human and the primate [12], and preliminary modeling studies on the rat ONH predicted that such differences result in patterns of mechanical stress and strain that differ from those in the human ONH [13]. Of note are five key anatomic differences (Fig. 1): (1) Rat does not have a connective tissue lamina cribrosa within its scleral canal. (2) Instead of passing through the center of the nerve, the central retinal vein (CRV) and artery (CRA) pass through the sclera on the inferior side of the nerve. (3) A vascular plexus, referred to as the perineural vascular plexus (PNVP), exists between the optic nerve and the neurovascular scleral canal wall. (4) An additional canal, the inferior arterial canal (IAC), exists in the rat sclera and is separated from the neurovascular canal by a so-called scleral sling. The CRA passes through the IAC (also referred to as the inferior arterial opening [12]). (5) On the superior side of the nerve, Bruch's membrane (BM) extends toward the nerve axis, creating a BM “overhang” around which RGC axons are forced to pass.

Fig. 1.

Histologic section of the rat (a) and schematic drawing of the human (b) ONH illustrating their anatomical differences, including 5 key differences of particular interest (numbered 1–5) as described in the text. Abbreviations: central retinal artery (A) and central retinal vein (V). Reprinted with permission from Schwaner et al. [13]. Copyright 2018 by The Journal of Biomechanical Engineering.

Thus, an in-depth characterization of rat ONH biomechanics is needed, but, so far, our two previous studies are the only existing attempts [13,14]. In the first, we presented a method for building finite element (FE) models of rat ONHs with individual-specific geometries and showed the strain patterns computed in three such models. We utilized tissue material properties similar to those used in previous human modeling studies, including the assumption that all tissues were isotropic. In the second study, we performed inverse FE modeling (iFEM) to extract material properties of the rat sclera, as scleral stiffness has been previously shown to be highly important in ONH biomechanics [4,15]. The material model chosen for the sclera in this second study incorporated nonlinear and anisotropic behavior.

Our goal here was to build on these two efforts by creating rat ONH FE models incorporating individual-specific geometries and scleral material properties extracted from our previous iFEM study. We built seven ONH model geometries and analyzed eight variants of each model by varying scleral material properties, resulting in a total of 56 simulations. We analyzed strain magnitudes and patterns within the rat ONH due to elevated IOP as well as the expansion of the optic nerve and neurovascular scleral canal openings. The results from this study can be compared with patterns of damage and cell behavior in rat glaucoma studies to learn more about the role of biomechanical insult in RGC pathophysiology.

Methods

We have previously presented a detailed methodology for building FE models of rat ONHs incorporating individual-specific geometry [13]. That same method was used here, with the exception of steps to assign material properties to the sclera.

Model Geometry.

Three-dimensional (3D) digital reconstructions of eight normotensive male Brown Norway rat ONHs (ages 9.5 to 10.5 months) were built using a previously described method [12]. One of these reconstructions did not provide enough information for modeling, so it was excluded from this study. Using custom software (multiview [16]), tissue boundaries were manually delineated within radial and transverse sections through the ONH. BM, BM opening, the anterior scleral surface, the posterior scleral surface, the anterior scleral canal opening (ASCO), the posterior scleral canal opening (PSCO), the optic nerve boundary, the posterior pia mater outer surface, and the side branches of the CRV were delineated within radial sections that shared an axis of rotation passing through the center of the optic nerve. The CRA, main CRV lumen, and IAC were delineated by viewing transverse sections (perpendicular to the optic nerve axis). The resulting 3D point clouds were exported to Rhino (v5 SR 14, Robert McNeel and Associates, Seattle, WA).

A more detailed description of the geometry building methodology can be found in Ref. [13]. In brief, nonuniform rational basis spline surfaces were fit to the point clouds using several different Rhino tools. Next, through a series of Boolean operations and several other tools available in Rhino and the Rhino T-Splines plugin (Autodesk, Inc., San Rafael, CA), 3D volumes were built for the following tissues: BM, the choroid, the sclera, the PNVP, the IAC, the optic nerve, the pia mater, the CRA, and the CRV. We chose to exclude the dura mater but plan to include it in future studies that consider the effects of changes in intracranial pressure. Due to artifacts such as deformation of the sclera during tissue processing, only tissue contained within a cylinder of diameter 1.5 mm centered at BM opening were included in the model. To ensure that the resulting geometries accurately represented the true anatomy of the tissues, we used custom Rhino.Python and matlab (2017; Mathworks, Natick, MA) scripts to project the boundaries of tissue volumes onto the original digital sections through the ONH reconstructions [13]. Finally, the sclera was subdivided into several regions for assigning regional material properties. Specifically, the scleral sling (the scleral tissue between the IAC and scleral canal) was first divided from the rest of the sclera, after which the remaining scleral volume was divided into superior, inferior, temporal, and nasal quadrants (Fig. 2).

Fig. 2.

MR11OS model geometry. (a) En face view of the sclera. The scleral sling as well as the superior (S), nasal (N), inferior (I), and temporal (T) scleral quadrants are shown in different shades of blue. The neurovascular scleral canal is just superior to the sling, and the IAC is inferior to the sling. The black line indicates the division between the central region that was meshed with tetrahedral elements and the outer region that was meshed with hexahedral elements. (b) En face view of the model with only the choroid and BM hidden. The optic nerve (green), pia mater (cyan), CRV (pink), CRA (red), IAC (magenta), and PNVP (gray) are visible. (c) Superior–inferior cut plane view of the model with all tissues visible. The choroid (yellow) and BM (orange) have been added to the view. (Color version online.)

Meshing and Constraints.

Model geometries were imported into trelis (16.5; Computational Simulation Software, LLC, American Fork, UT) for meshing using a combination of four-node tetrahedral, six-node prism, and eight-node hexahedral elements (see Fig. S1 available in the Supplemental Materials on the ASME Digital Collection). The tetrahedral elements were used in the central region of each model where the complex geometry could not be meshed with hexahedral elements, with the exception of BM, which was meshed with prism elements. The peripheral region of each model containing only BM, choroid, and sclera tissue was meshed with hex elements. Each model was meshed with approximately the same mesh density of 175,000 nodes or more according to a previously conducted mesh density study [13]. Tied constraints were applied at the interface of the outer (hex elements) and inner (tet and prism elements) model regions. Due to the CRV geometry in two models, MR05OD and MR08OD, contact occurred between the inner surfaces of the CRV vessel wall, so a general contact condition without friction was applied in those models.

Material Properties.

All tissues other than the sclera were treated as isotropic neo-Hookean solids with the same elastic modulus values as in our previous publication (BM: 7 MPa; pia mater: 3 MPa; CRA wall: 0.3 MPa; CRV wall: 0.3 MPa; choroid: 0.1 MPa; PNVP: 0.1 MPa; IAC: 0.1 MPa; optic nerve: 0.03 MPa) [13]. As is standard, tissues were assumed to be nearly incompressible due to the high water content of soft biological tissues [17–19]. Although stiffness measurements have since been made on the rat optic nerve [20], unfortunately, those data were not available at the time of model construction.

The material model chosen to represent the sclera was an isotropic matrix reinforced by collagen fibers [21,22], the same model utilized in our previous iFEM study [14]. The strain energy density function was given by

$$W_{\mathrm{sclera}}=W_{\mathrm{gs}}+\hspace{0.17em}{W}_{\mathrm{fiber}}$$ (1)

where $W_{\mathrm{gs}}$ is the contribution of the ground substance and $W_{\mathrm{fiber}}$ is the contribution of the collagen fibers. The ground substance behavior was given by

$$W_{\mathrm{gs}}=c_1\left(I_1-3\right)+c_2\left(I_2-3\right)+\frac{K}{2}{\left(\ln J\right)}^2$$ (2)

where $c_1$ and $c_2$ are the first and second Mooney–Rivlin coefficients, $I_1$ and $I_2$ are the first and second invariants of the right Cauchy–Green deformation tensor, $K$ is the bulk modulus, and $J$ is the determinant of the deformation gradient tensor. In all simulations, we assumed that the sclera was nearly incompressible and thus set $K=$ 1 GPa to ensure that the bulk modulus was much greater than the shear modulus, as was done in our previous iFEM publication and by Girard et al. [14,19]. In addition, we always set $c_2=0$, meaning that the ground substance behaved as a neo-Hookean solid whose stiffness was dictated by $c_1$. The strain energy density function for the collagen fibers was given by

$$W_{\mathrm{fiber}}={\int }_{{\theta }_p-\frac{\mathrm{\pi }}{2}}^{{\theta }_p+\frac{\mathrm{\pi }}{2}}P(\theta )F_2\left(\lambda [\theta ]\right)\mathrm{d}\theta$$ (3)

where $P\left(\theta \right)$ is the distribution of collagen fibers, assumed in this case to be the von Mises distribution

$$P\left(\theta \right)=\frac{1}{\pi I_0\left(k_f\right)}\mathrm{e}\mathrm{xp}\left[k_f\text{cos}\left(2\left(\theta -{\theta }_p\right)\right)\right]$$ (4)

where $I_0$ is the modified Bessel function of the first kind of order 0. Further, $F_2\left(\lambda [\theta ]\right)$ is the response of the collagen fibers and is given by

$$F_2\left(\lambda \right)=\left\{\begin{array}{c}0\hspace{0.17em}\hfill \\ c_3\left(e^{c_4\left(\lambda -1\right)}\left(\mathrm{Ei}\left(c_4\lambda \right)-\mathrm{Ei}\left(c_4\right)\right)-\ln \lambda \right)\hspace{0.17em}\\ c_5\left(\lambda -1\right)+c_6\ln \lambda \hspace{0.17em}\hfill \end{array}\right.\hspace{0.17em}\genfrac{}{}{0pt}{}{\lambda \le 1\hspace{0.17em}}{\begin{array}{c}1<\lambda <{\lambda }_m\\ \lambda \ge {\lambda }_m\hspace{0.17em}\end{array}}$$ (5)

where $\lambda$ is the fiber stretch, ${\lambda }_m$ is the fiber stretch for straightened fibers, and $\mathrm{Ei}(·)$ is the exponential integral function. Note that the number “2” rather than “1” is used as the subscript on $F_2$ to be consistent with the febio theory manual. At each element in the sclera, the fibers laid within a plane locally tangent to the scleral surface. As mentioned, their orientations followed a von Mises distribution centered about a preferred direction, ${\theta }_p$ [21]. A local material coordinate system was defined at each element such that a value of ${\theta }_p=0$ deg represented a fiber axis in the circumferential direction, and a value of ${\theta }_p=90$ deg represented a fiber axis in the meridional direction. The degree of fiber alignment was specified by the fiber concentration factor, $k_f$, where a value of $k_f=0$ resulted in fibers being spread uniformly in all directions within the material plane (transverse isotropy), and a value of $k_f=\mathrm{\infty }$ resulted in all fibers being aligned in the preferred direction, ${\theta }_p$. Finally, the nonlinear stiffening behavior of the collagen fibers was dictated by the exponential fiber stress coefficient,$\hspace{0.17em}{c}_3$, and the fiber uncrimping coefficient, $c_4$. As in our previous work, we assumed that the collagen fibers never fully straightened ($\lambda <{\lambda }_m$), making it unnecessary to define values for $c_5$ and $c_6$ [14].

We utilized scleral material parameter values determined by our previous iFEM study [14], performed on normotensive eyes of male Brown Norway rats (ages 10–13 months). To avoid confusion, we would like to emphasize that the iFEM study was performed on a different set of eyes than those used for digital reconstruction [12]. We reasoned that it would have been inappropriate to combine parameters from the iFEM study into a single set of “average” parameter values for modeling studies because the scleral material model contains several parameters that interact nonlinearly with each other to influence model behavior. Thus, an “average” set of parameters could potentially result in scleral biomechanical behavior that was physiologically unrealistic. Therefore, we instead selected four eyes from the iFEM study [14], as described below. Drawing on the scleral properties from these four eyes, we created eight scleral material property sets, referred to as “model variants,” that we grouped into two sets of four.

How did we choose scleral material parameters? Since fiber stiffness likely dominates scleral mechanical behavior at high IOP, we chose four parameter sets (four eyes from the iFEM study) based on their fiber stiffnesses. Specifically, using the $c_3$ and $c_4$ values from each eye, we generated stress–stretch curves for a single fiber from the following [22]:

$$\sigma =c_3\left(e^{c_4\left(\lambda -1\right)}-1\right)$$ (6)

where $\sigma$ is the fiber stress and $\lambda$ is the fiber stretch ratio. We chose one eye with high fiber stiffness, two eyes with midrange fiber stiffnesses, and one eye with low fiber stiffness (Fig. 3). In passing we note that the range of material parameters is less than has been reported in other iFEM studies in human eyes [23,24]. This may reflect the fact that there is lower genetic variation within the inbred strain of Brown Norway rats than in the human population.

Fig. 3.

Stress–stretch curves for a single collagen fiber generated from the $c_3$ and $c_4$ values obtained in Ref. [14]. We chose to use the four parameter sets associated with the curves indicated in the figure for this study. Model variants (v) that utilized parameters from each set are also listed.

Using these parameter sets, we then created eight model variants, grouped into two sets of four. In variants 1–4, we used $c_1$, $c_3$, and $c_4$ values, determined as described above and specified uniformly throughout the sclera. We also specified eye-specific fiber orientation parameter values for the sclera, taken from the same four eyes used to specify $c_1$, $c_3$, and $c_4$ values. In variants 5–8, we used the same set of $c_1$, $c_3$, and $c_4$ values, but combined these with fiber orientation parameters averaged over several eyes (see below and Table 1). Note that the individual-specific geometries in this study did not extend past the defined peripapillary scleral region in our previous iFEM study (outer diameter of 1.5 mm) [14], so only the peripapillary ${\theta }_p$ and $k_f$ values were utilized here.

Table 1.

Scleral material properties applied in each of eight model variants (v)

	v1	v2	v3	v4	v5	v6	v7	v8
c1 (MPa)	0.043	0.04	0.045	0.031	0.043	0.04	0.045	0.031
c3 (MPa)	0.0054	0.0022	0.0001	0.0001	0.0054	0.0022	0.0001	0.0001
c4	312	243	433	349	312	243	433	349
θp_T (deg)	142	54	79	17	0	0	0	0
θp_S (deg)	137	2	19	89	0	0	0	0
θp_I (deg)	104	63	149	138	0	0	0	0
θp_N (deg)	51	173	8	129	0	0	0	0
θp_sling (deg)	Sling axis	Sling axis	Sling axis	Sling axis	Sling axis	Sling axis	Sling axis	Sling axis
kf_T	10	10	8.45	2.3	6.31	6.31	6.31	6.31
kf_S	10	4.4	1.61	9.6	6.31	6.31	6.31	6.31
kf_I	3.96	2.88	4.94	7.7	6.31	6.31	6.31	6.31
kf_N	7.41	5.45	4.48	0.86	6.31	6.31	6.31	6.31
kf_sling	6.31	6.31	6.31	6.31	6.31	6.31	6.31	6.31

In more detail, in variants 1–4, each of the scleral quadrants received its own ${\theta }_p$ (${\theta }_{p\_S},\hspace{0.17em}{\theta }_{p\_I},\hspace{0.17em}{\theta }_{p\_T},\hspace{0.17em}{\theta }_{p\_N}$) and $k_f$ ($k_{f\_S},\hspace{0.17em}{k}_{f\_I}$, $k_{f\_T}$, $k_{f\_N}$) values. In variants 5–8, we utilized fiber stiffness properties from the four selected eyes but assumed that ${\theta }_p=0\hspace{0.17em}\mathrm{deg}$ and $k_f=$ 6.31 for all scleral quadrants. This approach was motivated by observations of high variability in regional ${\theta }_p$ and $k_f$ values from the iFEM study [14], which led to concerns about noisiness in the ${\theta }_p$ and $k_f$ values for individual scleral regions. Since the overall data trends indicated that fibers were oriented circumferentially, on average, we considered it to be useful to create models that reflected this average fiber orientation.

The scleral sling was not included as a specific region in our previous iFEM study, and thus we did not have experimentally determined material parameters for this region. Based on a polarized light microscopy image kindly provided by Dr. Ian Sigal and other images from the literature [25], in all variants 1–8 we set the fiber direction within the scleral sling, ${\theta }_{\mathrm{p}\_\mathrm{sling}}$, to be parallel to the sling axis (i.e., angled slightly from the nasal-temporal direction). We also set the fiber concentration factor, $k_{\mathrm{f}\_\mathrm{sling}}$ to be 6.31, which was the average value for peripapillary scleral regions in the iFEM study.

In summary, we ran eight simulations (one for each model variant) within each of the seven rat ONH geometries, for a total of 56 simulations.

Loading Conditions.

Since the ONH geometries were perfusion-fixed at a nonzero IOP, we used the “relative displacement” method to account for the effects of prestress [26]. Therefore, the pressures in the model (IOP, CRA blood pressure (BP), and CRV BP) were ramped up in two steps, as previously described [13]. In the first step, the ONH was loaded until it reached the stress state experienced at perfusion fixation. Thus, IOP was increased from 0 to 10 mmHg, CRV BP was increased from 0 to 10 mmHg, and CRA BP was increased from 0 to 40 mmHg. In the second step, IOP was increased from 10 to 30 mmHg, CRV BP was increased from 10 to 30 mmHg, and CRA BP was increased from 40 to 110 mmHg. The resulting model configuration after the first loading step was used as the reference configuration for all displacement and strain calculations. Thus, all displacement and strain values reported reflect the change due to an IOP increase from 10 to 30 mmHg. Reported stress values reflect the change due to IOP increase from 0 to 30 mmHg (i.e., they include prestress).

Boundary Conditions.

To apply boundary conditions to the ONH models, we used a submodeling boundary approach, as previously described in detail [13]. However, we here solved four posterior eye model versions instead of the one model that was used previously. In more detail, to match the scleral stiffness properties of the ONH models, we generated four posterior eye model versions, one for each set of $c_1$, $c_3$, and $c_4$ values. Further, since the purpose of the posterior eye model was to provide suitable boundary conditions for the ONH model, rather than to determine strain patterns near the ONH, we assumed $k_f=0$ (transverse isotropy) throughout the sclera in all four versions, eliminating the need to specify a ${\theta }_p$ value. IOP was applied in two steps from 0 to 10 mmHg and 10 to 30 mmHg as in the ONH models. Nodes on the farthest anterior surface of the model (at the eye equator location) were not allowed to move in the $z$ or $\theta$ directions but were allowed to displace radially. Note that the $z$ axis was aligned with the central axis of the half sphere. To apply boundary conditions to the ONH models, each ONH model was semi-automatically registered with the posterior eye model [13], and displacements from nearby nodes in the posterior eye model were interpolated to nodes on the peripheral (farthest from the optic nerve) surface of the ONH model.

Outcome Measures.

All simulations were run using febio (v2.9; [22]). To assess the effect of elevated IOP on the ONH, we calculated the 50th (median) and 95th percentile first principal strain and the 50th and 5th percentile third principal strain, in the anterior nerve. The anterior nerve was defined as the nerve tissue from BM opening to approximately 150 μm posterior to the opening. The 95th and 5th percentiles were selected rather than the 100th and 0th in order to exclude the extreme values often produced by a few elements of poor shape quality. In addition, all percentiles were weighted by element volume to prevent areas of the nerve that contained large numbers of relatively small elements from biasing the results.

We also calculated the change in area of the ASCO, of the PSCO, of the nerve opening at the ASCO (nerve-ASCO), and of the nerve opening at the PSCO (nerve-PSCO) (Fig. 4). To find ASCO area, we fit a plane through the nodes lying on the ASCO edge, projected those nodes to the plane, and found the area enclosed in the resulting curve. This was repeated for the PSCO. Next, the planes fit through the ASCO and PSCO were used to select nodes outlining the nerve boundary at the level of each opening. Nodes on the nerve boundary surface within 5 μm of each plane were selected, and the areas of the nerve-ASCO and nerve-PSCO were then calculated using the same method as for the ASCO and PSCO (Fig. 4).

Fig. 4.

Methodology for calculating ASCO, PSCO, nerve-ASCO, and nerve-PSCO area. The PSCO and nerve-PSCO are shown here. (a) View of the posterior side of model MR11OS. (b) Same view as (a) but with all tissues except the sclera hidden. The nodes outlining the PSCO are shown as white dots. (c) A plane was fit through the nodes lying on the PSCO. (d) Nodes outlining the nerve-PSCO area were selected by finding nodes on the nerve surface within 5 μm of the plane. (e) Nerve-PSCO nodes were projected to the plane and the area enclosed by the resulting curve was taken as the nerve-PSCO area (shown in red). (f) Same process shown in (e) but for the PSCO nodes. (Color version online.)

As detailed by Pazos et al. [27], the eyes used in this study were obtained from rats in which one eye of each rat was normotensive (the eyes we modeled) while the contralateral eyes were hypertensive. The axonal damage within the optic nerve of each hypertensive eye was assessed using a grading system [28], which assigned scores from 1 (no damage) to 5 (degeneration involving the entire nerve area). We tested each of our biomechanical modeling outcome measures for association with the damage score in the contralateral eye using Pearson correlation to determine whether there were animal-specific anatomic features that would lead to an association between biomechanical factors and nerve damage. Note that this approach assumes a high degree of contralateral concordance in anatomy and other physiological properties of the rat ONH.

Results

As shown in Fig. 5, within model variants 1–4, anterior nerve principal strains depended on both geometry and scleral material properties. Within variant 1, there was especially large differences in strain between geometries, with the range in 50th percentile third principal strain being −2.4% to −4.1%. Also within variants 1–4, the ordering of geometries with respect to strain level was not consistent. For example, MR11OD had one of the highest 95th percentile 3rd principal strain values in variant 1, yet also had the lowest value among models in variant 2. In variants 5–8, within which only stiffness values were changed, the variation in strain between geometries and between variants was much smaller. For example, the maximum difference in 50th percentile 3rd principal strain between geometries was only 0.58% and the maximum difference produced by changes in material properties was only 0.26%. In addition, the ordering of geometry with respect to strain value was consistent across variants 5–8. Finally, all strain values within variants 5–8 were smaller in magnitude than those in variants 1–4.

Fig. 5.

Principal strains in the anterior nerve from all model simulations. The groupings of lines from top to bottom are the 95th percentile 1st principal strain, 50th percentile 1st principal strain, 50th percentile 3rd principal strain, and 5th percentile third principal strain. The different colors indicate different ONH model geometries.

Like the principal strain values, principal stress values had smaller magnitude in variants 5–8 compared to variants 1–4 (Fig. 6). However, changes in stress due to geometry were on the same order as those due to differences in material properties. Overall, the models demonstrated more compressive stress in the nerve than tensile stress, with some models even having negative values for the 50th percentile first principal stress.

Fig. 6.

First and third principal stresses in the anterior nerve for all models. Asterisks indicate first principal stress and open circles indicate third principal stress. Solid lines indicate 50th percentile values and dashed lines indicate either 95th percentile first principal stress or 5th percentile third principal stress.

The magnitudes of ASCO, PSCO, nerve-ASCO, and nerve-PSCO expansion were all low, with values ranging from approximately 0.5% to 3.1% increases in area (Fig. 7). Trends in the data were similar to those seen in the strains. More variation occurred within variants 1–4 compared to 5–8, and variants 5–8 had lower change in area than variants 1–4, in general.

Fig. 7.

Changes in area of the ASCO, PSCO, nerve-ASCO, and nerve-PSCO for different ONH models and variants. Different ONH models are indicated by color.

All outcome measures were tested for possible association with contralateral eye optic nerve damage grade. The r² values and p values of the resulting linear regressions were highly variable between model variants for all outcome measures, and there were no significant correlations between any outcome measure and damage grade. We plotted 95th percentile strain versus damage grade from model variants 1, 4, 5, and 8 as an example (Fig. 8).

Fig. 8.

First principal 95th percentile strain versus contralateral eye damage for model variants 1, 4, 5, and 8. Similar to the situation with all other outcome measures and for all model variants, r² and p values were highly variable. No correlations achieved significance. Y-axis limits are different in each plot to better show data spread.

To examine the patterns of strain within the nerve due to changes in material properties, we plotted heat maps showing first and third principal strains in all model variants of MR11OS (Fig. 9). As seen in the en face views and superior–inferior cut views, first and third principal strain concentrations occurred primarily in the inferior nerve. This effect was less pronounced in variants 5–8 compared to variants 1–4, but the regions of highest strain were still found in the inferior nerve. This trend was also apparent across variants of the other six ONH models (data not shown). As seen in the nasal-temporal cut of MR11OS, it appears that there was a slight tendency for higher strains to occur toward the nasal side of the nerve, but this was not as pronounced as the difference between the inferior and superior nerve.

Fig. 9.

First (top subrows) and third (bottom subrows) principal strains for different model variants (columns) of MR11OS. The highest strains are concentrated in the inferior nerve regardless of model variant, although the pattern is less pronounced in variants 5–8. Tensile strain is shown in red and compressive strain is shown in blue. The en face view (top), superior (S)–inferior (I) cut plane view (middle) and nasal (N)-temporal (T) cut plane view (bottom) are shown. Scale bars shown in the far-left column are 100 μm. Dashed lines in the en face view indicate S–I and N–T cut planes. Tissue colors are: nerve (green), CRV (pink), CRA (red), BM (orange), choroid (yellow), sclera (blue), pia mater (cyan). The undeformed configuration is shown in all images to ensure viewing of a consistent cut plane across columns. (Color version online.)

To examine how the patterns of strain varied between geometries, we plotted heat maps showing first and third principal strain in four eyes for variant 1 (Fig. 10). Again, the highest strains and largest regions of high strain were consistently found in the inferior nerve. This effect was seen in all four eyes but was most pronounced in MR08OD. Compared to MR10OS and MR09OS, MR05OD and MR08OD had lower strains in the superior nerve, particularly near the BM overhang on the superior side of the anterior nerve.

Fig. 10.

First (middle subcolumns) and third (right subcolumns) principal strain in variant 1 of ONH models MR01OS (top left), MR05OD (top right), MR08OD (bottom left), and MR09OS (bottom right). The en face (top subrows), superior–inferior (middle subrows) cut plane, and nasal-temporal (bottom subrows) cut plane views are shown. Scale bars are 100 μm. All colors and abbreviations are as in Fig. 9. The undeformed configuration is shown in all images to ensure viewing of consistent cut planes.

Discussion

This study is the first to perform individual-specific computational modeling of rat ONH biomechanics while incorporating a fiber-reinforced model for the sclera, an improvement on past models that modeled the sclera as a simple neo-Hookean solid. We built seven rat ONH models with individual-specific geometries and incorporated material properties that were informed by our previous iFEM study. To assess the effects of elevated IOP on the rat ONH, eight variants of each ONH model were solved, each with a different set of material properties for the sclera. Outcome measures included 50th and 95th percentile first principal stress and strain, 50th and 5th percentile third principal stress and strain, ASCO expansion, PSCO expansion, nerve-ASCO expansion, and nerve-PSCO expansion.

In our previous publication [13], we presented strain magnitudes and patterns for three ONH models that were also used in this study (MR05OD, MR04OD, and MR10OS) in which neo-Hookean behavior was assumed for the sclera. In this study, the strain magnitudes of these ONH models were lower in all variants (Fig. 5) compared to our previous study. However, the strain patterns were qualitatively similar between the simulations in this study and the previous publication [13], in spite of the different scleral material model used here. Notably, as in the previous study, highest strains occurred primarily in the inferior nerve. In addition, this trend was present across variants of all ONH models. We hypothesize that this pattern is due to the presence of the blood vessels and the IAC directly inferior to the rat optic nerve, creating a weak point in the sclera. This sets the rat apart from the human ONH, which experiences strain patterns that are largely axisymmetric [3]. Interestingly, there are two published patterns of glaucomatous damage that involve the superior and inferior regions of the rat nerve. Namely, RGC axonal degeneration occurs preferentially in the superior nerve [11], while astrocytes (known to be mechanosensitive) are activated and reorient their processes in the inferior nerve before those in the superior nerve [29]. Although it is confusing why RGC axons would preferentially degenerate in the superior nerve, we hypothesize that the inferior astrocytes withdraw processes extending into the superior nerve, leaving the superior axons more vulnerable to apoptosis. Our data support this possibility, but further study will have to be carried out to determine its validity.

Comparing the strain magnitudes across model variants emphasizes the importance of scleral collagen fiber direction on optic nerve strains. In variants 5–8, the circumferentially aligned fibers resulted in decreased strain in every ONH model compared to variants 1–4. Also, there was low variability between ONH models within variants 5–8. In addition to the high variability seen in variants 1–4, some interesting patterns involving the relative strains between ONH models emerged. Note that strain magnitudes of the OD (right) and OS (left) eyes tended to be grouped together in variants 1 and 2. For example, in variant 1, the OS eyes were the four eyes with highest third principal strain magnitude and the OD eyes were the three eyes with lowest third principal strain magnitude. In addition, from variant 1 to 2, the third principal strains in the OS eyes decreased, while the third principal strains in the OD eyes increased. These observations indicate that there was interaction between the regional fiber alignment (scleral anisotropic properties) and geometry, but perhaps not between scleral stiffness and geometry. In other words, the changes in material properties between variants 1 and 2 produced different effects on strain in different ONH geometries, but the same effect did not occur between variants 5 and 6, where only scleral stiffness was changed. The similar patterns seen in canal and nerve expansion are consistent with this point. However, the apparent lack of interaction between geometry and scleral stiffness may be due to the fact that scleral stiffness did not actually vary between models as much as Fig. 3 would initially seem to suggest (see below).

This observation also brings to light an important point for future fiber-reinforced models of the sclera. In our previous iFEM study and in this study, we defined the material axes at each element such that the i vector (along which ${\theta }_p=0\hspace{0.17em}\mathrm{deg}$) pointed in the counterclockwise direction (when looking at the posterior sclera surface), and the j vector (along which ${\theta }_p=90\hspace{0.17em}\mathrm{deg}$) pointed at the optic nerve. This convention was used for both OD and OS eyes. In the future, it would be better to switch the direction of the i component for OS eyes to be clockwise, so that a ${\theta }_p$ angle of 45 deg or 135 deg would result in fiber orientations that had the same effect on both OS eyes and OD eyes. In other words, this change would result in the i vector always pointing from inferior to temporal, temporal to superior, superior to nasal, and nasal to inferior quadrants.

One might ask why there seemed to be so little change in strain with different fiber stiffnesses, and why decreasing the fiber stiffness did not always result in decreased strain, as seen when comparing variants 5 and 7 (Fig. 5). This is likely due to the way in which we calculated strain. Recall that the model was prestressed by raising IOP from 0 mmHg to 10 mmHg and that all strains were calculated with respect to the resulting state. This was done to account for the effects of preload (perfusion fixation) in the histomorphometric method. By comparing the curves in Fig. 3, one can see that even though there was a wide range of stretch values between curves at a given stress, the slopes of the curves were fairly similar once a certain stress threshold was reached. For example, at a stress of 0.05 MPa, the slopes of the fiber stress–stretch curves for model variants 5–8 were approximately 17.4, 12.7, 21.8, and 17.6 MPa, respectively. The stiffness of the fibers in each model will depend on their stress level at 10 mmHg and higher, and the overall scleral stiffness will also depend on $c_1$. In other words, the tangent moduli of fibers within different material variants varied more at low stress than high stress. Thus, larger differences in strain would likely have been observed if we did not have to include such a large preloading step (10 mmHg) in each simulation. Therefore, the changes in strain observed between model variants are unintuitive, but not unreasonable. In future experiments obtaining rat ONH anatomical data, efforts should be made to either eliminate or reduce the amount of preload necessary.

Due to the difficulty and insufficiency of current imaging methods (discussed below), we did not perform experimental validation of our results. Further, little experimental data are available in the literature. To the best of our knowledge, the only publications detailing strains in the rodent ONH are several studies done on mouse eyes [30–32]. Laser-scanning microscopy was used on genetically modified mice (not rats) to image astrocytes in the ONH during whole globe inflation from 10 mmHg to 30 mmHg. Average first principal strain of 7.7% was documented within the astrocytic lamina of the mouse ONH [30], which is similar to the region within which we calculated 50th percentile strain. The 50th percentile strain ranges in our simulations were 1.5% to 3.2% for first principal strain and −2.0% to −4.1% for third principal strain. The lower values in our simulations may be due to several factors, including the possibility that material properties in our simulations were too stiff. However, the experimental studies on the mouse ONH did not report strain values by clock-hour region of the nerve (only peripheral versus central [32]), so we cannot compare our observations of strain patterns with their results. Experimental validation is an important goal of future work.

Finally, there were no significant correlations between any outcome measures and contralateral eye optic nerve damage grade. This result was disappointing, but not entirely unexpected. Although the models incorporate individual-specific geometry, they do not incorporate individual-specific material properties, nor do they account for the highly variable IOP burden in the ocular hypertensive eye [28]. As discussed above, there was obvious interaction between model geometry and material properties, and both are likely essential for computing deformation at the accuracy needed to predict damage levels of a given eye. Other limitations also prevented our models from computing the true strain values experienced by a given eye, and they are discussed below.

Limitations.

The largest limitations of this study have to do with material properties. Although we made significant improvements from our previous rat ONH modeling study by treating the sclera as a fiber-reinforced matrix and by incorporating parameter values that were derived from experiments on rat sclera, even these improvements had drawbacks. First, the material model that we selected is still only an approximation of scleral structure. It is likely that the rat sclera contains multiple layers of fibers through its thickness, each with unique organization [33]. von Mises distributions cannot always accurately describe the fiber directions of multiple fiber layers [34]. In addition, the organization of fibers changes gradually with distance from the nerve and location around the nerve [35], as opposed to being organized in quadrants. Better representation of local collagen fiber organization is an important point of improvement to focus on for future studies. Further, we utilized reduced sets of material parameters in variants 1–4 (12 parameters were varied) and 5–8 (five parameters were varied) compared to the iFEM study (19 parameters). This decision decreased the heterogeneity of material properties in the simulations of this study, potentially resulting in an underestimation of optic nerve strains. Next, as discussed in detail elsewhere [14], there were limitations with our iFEM study that affected the accuracy of retrieved parameter values. Unfortunately, we were unable to effectively account for preload which likely resulted in an overestimation of scleral stiffness [26]. Also, as mentioned above, we were concerned about noisiness of the $k_f$ and ${\theta }_p$ estimates for individual regions. Finally, the eyes tested in the iFEM study [14] were not the same as the eyes used to inform the model geometries [12] of this study.

The choice of material properties for all other tissues was an additional limitation. All other tissues were modeled as neo-Hookean solids, even though they probably are anisotropic. The pia mater in particular is more likely to behave like a transversely isotropic solid due to the collagen fibers in its structure. We also do not have experimental measurements of the stiffnesses of the other tissues in the rat ONH. These tissues are difficult to access and extremely small, making them difficult to mechanically test. However, advances in inverse methods paired with noninvasive imaging may provide solutions to this problem in the future [36].

Constraints should be added to the anterior faces of the CRA and CRV. In most models, the CRA (and adjacent CRV wall) “bent” around the scleral sling toward the superior CRV wall. In some model variants, this resulted in contact between the inferior and superior CRV walls that was likely not physiologic. The bending of the CRV also resulted in small areas of high strain in the anterior–inferior nerve (not the main strain concentration seen in the superior–inferior cut plane view) that surround the CRA on either side and can be seen in the en face view (Figs. 9 and 10). Constraints to enforce more realistic deformation should be added.

Although the histomorphometric method used to build the rat ONH digital reconstructions accurately preserves important anatomic and morphological relationships, there is a known issue with shrinkage associated with the embedding and fixation steps [12]. Unfortunately, the extent of shrinkage is not well characterized and may vary from tissue to tissue, so we could not correct for it. Ideally, improvements to noninvasive imaging technologies such as optical coherence tomography will replace methods such as histomorphometry and thus eliminate the shrinkage problem. Until such methods can accurately image all tissues in the rat ONH, based on our experience here and with a recently published sensitivity study [37], we suggest that experimentalists focus first on obtaining accurate information on the boundaries (including the canals, scleral sling, etc.) and local thickness of the sclera. We also intuit that the precise shape of the optic nerve and thickness of the pia mater are important, but more work is needed to confirm these points.

Another limitation of the histomorphometry method is the need for a preload. As discussed above, accounting for the effects of preload in our models likely decreased the differences that we saw between models. Future work should also focus on eliminating or reducing the amount of preload necessary for obtaining anatomical data.

Finally, the computed results from these models have not been validated. This would be extremely difficult and would require highly accurate, noninvasive imaging of the rat ONH to be carried out while controlling IOP. It may be possible to do this in the future using optical coherence tomography [38]. However, complex corrections would have to be made for differences in optical properties and sizes between rat eyes in order to obtain accurate measurements of deformation [39,40]. In addition, recent publications have described optical imaging of ONH connective tissues and of green fluorescent protein-expressing cells in the mouse [30–32,41]. Although this approach relies on a transgenic mouse line, extending such ideas to rats is possible in principle.

Conclusion

We built seven FE models with individual-specific rat ONH geometries to evaluate the effects of elevating IOP from 10 mmHg to 30 mmHg on the ONH. Improving on a previous study, we modeled the sclera as a fiber-reinforced matrix to account for its nonlinear and anisotropic behavior. We solved eight variants of each ONH model by incorporating different sets of experimentally derived scleral material properties for a total of 56 simulations. Regardless of scleral property differences, strains were highest in the inferior optic nerve. In model variants with highly aligned circumferential fibers in the sclera, optic nerve strains were substantially reduced. No significant correlations were found between contralateral eye optic nerve damage grade and any of the outcome measures, which included strain, stress, scleral canal expansion, and optic nerve expansion. The strain patterns and effects of material property changes reported here can be used to inform future modeling studies on the rat ONH and interpret results from rat glaucoma studies to better understand how biomechanical insult affects RGC pathogenesis in glaucoma.

Supplementary Material

Supplementary Material

Supplementary Figure

Funding Data

• •National Institutes of Health (NIH). NEI:T32EY007092 (SAS), NEI:F31EY028832 (SAS) NEI:R01EY025286 (CRE), NEI:R01EY010145 (JCM), P30 EY010572 (JCM); NEI:R01EY011610 (CFB) (Funder ID: 10.13039/100000002).
• •Georgia Research Alliance (CRE) (Funder ID: 10.13039/100008065).
• •Research to Prevent Blindness, unrestricted departmental funding grant (Casey Eye Institute) (Funder ID: 10.13039/100006668).
• •Alcon Research Institute (CFB), Good Samaritan (Devers Eye Institute) Foundation (CFB), Sears Medical Trust (CFB) (Funder ID: 10.13039/100007817).

Nomenclature

ASCO =

anterior scleral canal opening

BM =

Bruch's membrane

BP =

blood pressure

CRA =

central retinal artery

CRV =

central retinal vein

FE =

finite element

iFEM =

inverse finite element modeling

IAC =

inferior arterial canal

IOP =

intraocular pressure

nerve-ASCO =

nerve opening at the anterior scleral canal opening

nerve-PSCO =

nerve opening at the posterior scleral canal opening

ONH =

optic nerve head

PNVP =

perineural vascular plexus

PSCO =

posterior scleral canal opening

RGC =

retinal ganglion cell

3D =

three dimensional