Effect of Eye Globe and Optic Nerve Morphologies on Gaze-Induced Optic Nerve Head Deformations

Tingting Liu; Pham Tan Hung; Xiaofei Wang; Michaël J. A. Girard

doi:10.1167/iovs.65.8.48

Abstract

Purpose: The purpose of this study was to investigate the effect of globe and optic nerve (ON) morphologies and tissue stiffnesses on gaze-induced optic nerve head deformations using parametric finite element modeling and a design of experiment (DOE) approach.

Methods: A custom software was developed to generate finite element models of the eye using 10 morphological parameters: dural radius, scleral, choroidal, retinal, pial and peripapillary border tissue thicknesses, prelaminar tissue depth, lamina cribrosa (LC) depth, ON radius, and ON tortuosity. A central composite face-centered design (1045 models) was used to predict the effects of each morphological factor and their interactions on LC strains induced by 13 degrees of adduction. Subsequently, a further DOE analysis (1045 models) was conducted to study the effects and potential interactions between the top five morphological parameters identified from the initial DOE study and five critical tissue stiffnesses.

Results: In the DOE analysis of 10 morphological parameters, the 5 most significant factors were ON tortuosity, dural radius, ON radius, scleral thickness, and LC depth. Further DOE analysis incorporating biomechanical parameters highlighted the importance of dural and LC stiffness. A larger dural radius and stiffer dura increased LC strains but the other main factors had the opposite effects. Notably, the significant interactions were found between dural radius with dural stiffness, ON radius, and ON tortuosity.

Conclusions: This study highlights the significant impact of morphological factors on LC deformations during eye movements, with key morphological effects being more pronounced than tissue stiffnesses.

Glaucoma is one of the most common causes of blindness worldwide.¹ The biomechanical theory of glaucoma suggests that the deformations of the optic nerve (ON) head (ONH) tissues, especially the lamina cribrosa (LC), may lead to the apoptosis of retinal ganglion cells and visual field defects, either directly or indirectly.² Intraocular pressure (IOP) and cerebrospinal fluid pressure (CSFP) are the two main mechanical loads acting on the ONH that have been shown to be correlated to glaucoma pathogenesis.³^–⁵ Recent studies using finite element (FE) modeling,⁶^–⁸ optical coherence tomography,⁹^–¹³ and magnetic resonance imaging (MRI)¹⁴^–¹⁶ have highlighted that ON traction during eye movements can yield large ONH deformations, which may be as large as or significantly larger than those caused by a substantial IOP elevation to 40 or 50 mm Hg.

Previous parametric studies have also found that changes in the biomechanical and morphological parameters of the ONH significantly impact its deformations under IOP.¹⁷^–¹⁹ In vivo studies have shown that gaze-induced ONH deformations vary widely across individuals.²⁰^–²² The differences are likely due to variations in the biomechanical properties and morphologies of the eye globe, ON, and ONH. For instance, ON tortuosity varies across individuals, as shown in Figure 1. A less tortuous ON has been hypothesized to generate a larger traction force, and thus potentially larger ONH deformations.¹⁵ Therefore, to identify those who are vulnerable to ON traction during eye movements, it would be critical to identify the biomechanical and morphological factors (and their interactions) that significantly affect ONH deformations. Using FE modeling, we have previously investigated the effects of biomechanical properties on gaze-induced ONH deformations and predicted that a stiffer dura would generate a larger ONH deformation.⁶ However, the effects of eye globe and ON morphologies on gaze-induced ONH deformations and their potential interactions with biomechanical properties remain unexplored.

Figure 1.

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MRI images of the orbital region demonstrate the morphological diversity of the optic nerve (ON). These six figures show examples of ONs displaying varying degrees of curvature, ranging from straight to highly tortuous. The white lines represent the ON middle curve.

The aim of this study was to explore the effects of eye globe and ON morphologies on gaze-induced ONH deformations, and to examine any potential interactions between morphological parameters and biomechanical parameters of tissues, using parametric FE modeling and design of experiment (DOE).

Methods

In this study, we developed a methodology to automatically generate thousands of 3D eye models to study the effects of the eye globe and ON morphologies, as well as tissue biomechanical properties, on gaze-induced ONH deformations. Specifically, a custom-written software (C++) was designed to automatically generate FE models of the eye, each with a set of predetermined morphological and material parameters. These models were then fed into the FE solver FEBio (Musculoskeletal Research Laboratories, University of Utah, Salt Lake City, UT, USA) to predict gaze-induced ONH deformations. Configurations of all key factors were generated by a DOE approach, specifically the central composite face-centered design (CCF). The initial DOE analysis evaluated 10 morphological parameters to determine the top five, which were then combined with the five key tissue stiffnesses from previous studies.⁶ This resulted in a refined set of 10 parameters, covering both morphology and tissue stiffness, for a follow-up DOE study. Because adduction is known to induce significant ONH deformations compared to abduction,⁶^,⁷^,⁹ an adduction of 13 degrees was chosen for each model, as used in our previous work. The response of each model was characterized by the magnitude of the effective strains, the first principal and third principal strains within the LC. Below is a detailed description of the methodology.

Geometry and Biomechanical Properties of the Baseline FE Model

A whole-eye FE model was established, including the sclera, choroid, prelaminar neural tissue, LC, ON, pia mater, dura mater, orbital fat-muscle (OFM) complex, and orbital bone. The baseline geometric parameters of eye global tissues were set to averaged values reported in the literature, shown in Table 1. To maintain simplicity, we opted to combine and simulate the extraocular muscles and the orbital fat as a unified entity referred to as OFM. The eye model is symmetric up to a certain point, with the eye globe itself being symmetrical. Because ON tortuosity is introduced in the model, the ON and its pia mater becomes asymmetric starting from the posterior surface of the LC, and the dural sheath becomes asymmetric from its boundary with the sclera. Only half of the eye was reconstructed because the FE model was assumed to be symmetric about a transverse plane passing through the center of the eye globe (Fig. 2).

Table 1.

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Morphological Factors and Their Ranges

Figure 2.

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The left panel shows the reconstructed geometry and FE mesh of the eye movement model with boundary conditions and tissue connections. The right panel shows an enlarged view of the detailed ONH region (sclera, scleral fiber ring, the peripapillary border tissue, choroid, Bruch's membrane, lamina cribrosa, neural tissues, pia, and dura) illustrating the IOP and CSFP applied to each model in the primary gaze position.

The baseline biomechanical properties were the same as those used in our previous studies.⁶^,⁷ Briefly, both the sclera and LC were modeled as soft tissues reinforced with collagen fibers. We utilized the “Mooney-Rivlin with Von Mises Distributed Fibers” constitutive model in FEBio. This model assumes that the fibers are primarily located within a local 2D plane, and the parameters for fiber distribution are derived from 2D sections.²³^,²⁴ The collagen fibers in the peripapillary sclera surrounding the disc were organized into a ring, whereas those in the peripheral sclera were organized randomly (as specified by the kf parameter in Table 2) and parallel to the anterior scleral surface.²⁵ The collagen fibers in the LC exhibited lower anisotropy than that in the peripapillary sclera and were aligned radially, extending from the central vessel trunk to the LC insertion sites.²³ All other tissues were considered either hyperelastic or linear elastic, as shown in Table 2. Among them, peripapillary border tissue (PBT) is the border tissue of the choroid and sclera.²⁶ The peripapillary choroid is separated from the prelaminar neural tissue by a collagenous layer, which constitutes the border tissue of the choroid. Likewise, the scleral flange is separated from the LC by the border tissue of the sclera. Because the biomechanical behavior of the PBT has not yet been reported, we assumed that the PBT shared the same biomechanical properties as those of the pia.²⁷ All soft tissues were assumed to be incompressible. The orbital wall was considered a rigid body.

Table 2.

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Tissue Biomechanical Properties

Parameterization of Morphological and Biomechanical Properties

The morphology of the FE model was parameterized using 10 factors representing the geometry of the eye globe, ONH, and that of the ON. These factors were: dural radius, scleral thickness, choroidal thickness, retinal thickness, ON radius (excluding the pia and dura), pial thickness, the thickness of PBT,²⁸ prelaminar tissue depth, central LC depth, and ON tortuosity (Fig. 3). Specifically, the thicknesses of the eye globe tissues (sclera, choroid, and retina) were modified by adjusting the distance between each tissue's boundaries and the fixed sclera-choroid interface, while maintaining the thickness of other tissue unchanged. The depth of prelaminar tissue and LC were modified by adjusting the distance from the tissue insertion point to the lowest end of the tissue center, while maintaining the position of the tissue insertion point unchanged. For the ON tissues (pia and PBT), a similar approach was used where the inner surface of each specific tissue was fixed and the outer surface were altered to vary its thickness. The radii of the dura and ON were adjusted by changing their distance from the central axis of the ON and ONH. ON tortuosity was altered by adjusting the positions of three control points along its central path. Refer to Supplementary Material SA-1 for more details on how the morphological parameters of the eye globe and ONH were varied. It should be noted that a common assumption in our study, shared among other computational parametric studies, is the presumption of independent variations among factors.

Figure 3.

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Input factors defining the parametric FE model geometry (only the ONH region of the entire eye is shown). The black points indicate insertion points when tissue depth is determined. See Table 1 for the ranges of input factors. The blue dashed line represents the symmetry axis of ONH.

The biomechanical properties of each tissue were directly modified in the input file of the FE model. The low and high levels of the biomechanical properties of LC, ON, sclera, pia, and dura were set by varying the material constants by 20% around their baseline values (see Table 3 for the exact values).

Table 3.

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Biomechanical Properties of Five Tissues and Their Ranges

Contact Definitions, Boundary, and Loading Conditions

Contacts between tissues were the same as those defined in our previous study.⁶^,⁷ Briefly, the OFM and the dura were tied together; the OFM was able to slide over the bony orbital margin with a friction factor of 0.5; the cornea-scleral shell and the OFM had a sliding contact with no friction to mimic the Tenon's capsule enveloping the eyeball. Rigid contact was assigned between the horizontal rectus muscle insertions and a “non-physiological” rigid body. This latter had a center of mass at the center of the eyeball, which was constrained with a prescribed rotation to simulate an adduction of 13 degrees. For boundary conditions, the OFM and ON were fixed at the orbital apex to mimic the fibrous adhesion of those tissues to the optic canal. The orbital bone was also held in place by fixing its outer margin. In addition to an adduction of 13 degrees, an IOP of 15 mm Hg was applied to the surface of the retinal and prelaminar tissues and a CSFP of 12.9 mm Hg was applied within the subarachnoid space of the ON. Loading was applied in two steps: first, IOP and CSFP were applied and maintained, then an adduction of 13 degrees was applied. All contact patterns, boundaries and loading conditions are illustrated in Figure 2.

FE Simulations, Post-Processing, and Output Measurement

All FE models were consistently meshed with 73,922 nodes and 62,821 elements, including 62,521 of the 8-node hexahedra and 300 of the 6-node pentahedral elements. All tissues were bonded by shared nodes at the tissue boundaries (see Fig. 2). The mesh density was numerically validated through a convergence test, which compared the average effective strain of LC computed from a model with a denser mesh (82,620 elements) to the mesh density used in this study (62,821 elements). The relative difference in LC strain was found to be 1.17% when comparing the current mesh to the denser mesh, indicating sufficient mesh density. A single model required about 20 minutes to solve on a desktop workstation (Intel Xeon Silver 4114 CPU @ 2.20 GHz, with 32 GB of memory).

The preprocessing MatLab script executed FEBio to solve the generated models and outputted the Lagrange strain tensors for each step and the volumes of all LC elements into a text file. Effective strains were calculated from the principal components of the Green Lagrange strain tensor. To isolate the specific effect of eye movement, the strains of each LC element after the first load step (with only IOP and CSFP) and those after the second load step (following eye movement) were extracted. The differential, termed “delta strain”, was then calculated for each element. This delta strain for each LC element was multiplied by its volume, and these values were summed and divided by the total LC volume, yielding the volume-weighted mean LC delta strain. For simplicity, this will be referred to as the gaze-induced LC strain in the rest of this paper, including LC effective strains, the first principal, and the third principal strains.

DOE of Morphological Factors

A CCF was chosen to analyze the effect of each factor and their interactions. In this design, there is one center point and two axial points for each factor. As this is a simulation, no repetition experiments are required for central points. Therefore, in total, there were 1045 models, including a 2-level full factorial design (1024 models) with 20 axial experiments (2 points for each of the 10 factors) and one central experiment. The independent factors were studied at 2 levels, with the low level coded as -1 and the high level coded as +1, and a central point at the baseline level coded as 0. All analyses in our study were conducted using these coded factors.

For the 10 morphological parameters (see Fig. 3), a CCF was used, resulting in a total of 1045 models. The low and high levels of all morphological parameters (excluding ON tortuosity) were set by varying them by 20% around their baseline values (see Table 1). For simplicity, we varied ON tortuosity within a narrower range, from 1.013 (low level) to 1.1 (high level), as these values have typically been observed in MRI studies. Detailed information on the morphological parameters for each model is available in the Supplementary Material SB Sheet 1.

DOE of Both Morphological and Biomechanical Factors

We conducted another DOE analysis to examine the potential interactions between morphological parameters and tissue stiffnesses. This analysis included five key morphological parameters identified from the initial DOE analysis and the stiffnesses of five tissues (LC, ON, sclera, pia, and dura) informed by our previous study.⁶^,⁷

In this DOE analysis, a CCF was used, resulting in 1045 models. The low and high levels of the biomechanical properties of LC, ON, sclera, pia, and dura were set by varying the material constants by 20% around their baseline values (see Table 3 for the exact values). The variation in morphological parameters was consistent with the initial DOE analysis. We applied a consistent 20% variation to morphological and biomechanical properties in order to better compare the effects of various factors through the coefficients of the regression modeling. Detailed information on the morphological parameters and tissue stiffnesses for each model is available in the Supplementary Material SB Sheet 2.

Statistical Analysis

All statistical analyses were carried out in Minitab (release 20; Minitab, LLC, State College, PA, USA). We applied the Bonferroni correction to our P values to account for the increased risk of type I errors due to multiple comparisons. The threshold significance level was α = 0.05/65 = 0.000769.

For each model, we reported gaze-induced LC effective strains, the first principal, and third principal strains as responses. The effects of each factor, their interactions, and second-order terms were reported, and the significance of factors was tested and ranked, as detailed below. The main effects represent the independent influences of each factor on the response variable. Interaction effects indicate that the effect of one factor depends on the level of another factor. Essentially, interactions occur when the combined effect of two factors on the response variable is not equal to the sum of their individual effects. Second-order terms represent the quadratic effects of the factors on the response variable, which capture nonlinear relationships between the independent variables and the response.

Statistical analysis of variance (ANOVA) was carried out to identify the vital factors, including each factor and interaction affecting the gaze-induced LC strains, as is standard in DOE.²⁹ After conducting a normality test, it was found that the responses meet the assumptions of ANOVA. For each response, the percentage of the total sum of squares corrected by the mean was used to represent the approximate contribution of each factor and interaction to the variance of the response, providing a measure of influence, as is usual in factor analysis.²⁹ To be influential, the contribution of all factors had to meet two criteria: statistical significance (P < 0.000769) and a contribution greater than the error. The error refers to the proportion of variance in the dependent variable that is unpredictable from the independent variables.

A regression model, evaluated for each response, was a second-order model containing the factors, their squares, and two-factor interactions. The coefficient in the regression model indicates the magnitude and direction of the factor’s effect on the response. The absolute value of the coefficient reflects the effect of the factor on the response, indicating the change in gaze-induced LC strains when the factor increases by one-unit (equivalent to a 20% change in the factor). A positive coefficient indicates that a one-unit increase in the factor leads to an increase in LC strains. Conversely, a negative coefficient indicates an inverse relationship, where a one-unit increase in the factor leads to a decrease in LC strains. The coefficient of determination R² refers to the proportion of variance explained by the regression model compared to the total variance in the dependent variable. It serves as a measure of the goodness of fit of the regression model, with higher R² values signify a better fit of the model to the data.

Results

The Effects of 10 Morphological Parameters

The average LC effective strain (i.e. delta strain after removing the effects of IOP and CSFP) across all models was 0.031, and in baseline model (when all factors are at the central point) was 0.039. In the DOE analysis, the factors and their interactions accounted for 99.69% (error = 0.31%) of the total effects in the responses. Details on the effects of 25 statistically significant factors are available in the Supplementary Material SA-2. Among these factors, the key main effect and interactions, in order, are: ON tortuosity (−0.0072, the coefficient in the regression model; and 40.07%, contribution of the total effects), dural radius (0.0063; 30.76%), ON radius (−0.0048; 17.47%), scleral thickness (−0.0025; 4.82%), LC depth (−0.0018; 2.39%), the interaction between the dural radius and the ON radius (0.0010; 0.82%), the interaction between the dural radius and the ON tortuosity (−0.0009; 0.69%), and the interaction between the ON radius and the LC depth (0.0007; 0.38%). These factors accounted for 97.40% of the total effects in the responses. In addition, the nonlinear effects of ON tortuosity and dural radius were significant.

The five most significant factors remain the same when using first and third principal strains or effective strain as the output measure. Although the rankings of these factors change slightly depending on the type of strain used, the trend of the impact of each factor on the output measure remains consistent. Details on the main and interaction effects when using the first and third principal strains as output measures can be found in Supplementary Material SA-3.

Figure 4 illustrates the influential main effects and interactions. Larger ON tortuosity, scleral thickness, ON radius, and LC depth decreased LC strains following eye movements, whereas a lager dural radius increased LC strains. As ON tortuosity increases, the rate of decrease in LC strain increased. Conversely, as dural radius increases, the rate of increase in LC strain decreased. The significant interaction occurred between dural radius with ON radius and ON tortuosity, between LC depth with ON radius. Specifically, the decrease in LC strain associated with a larger ON radius was found to be amplified when combined with a smaller dural radius. The nonlinear decrease in LC strain associated with a larger ON tortuosity was found to be amplified when combined with a larger dural radius. The decrease in LC strain associated with a larger ON radius was found to be amplified when combined with a smaller LC depth.

Figure 4.

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(A) The main effects of key factors: ON tortuosity, dural radius, ON radius, scleral thickness, and LC depth. The squared effects of ON tortuosity and dural radius were significant. (B) The interactions between dural radius and ON radius. (C) The interactions between dural radius and ON tortuosity. (D) The interactions between LC depth and ON radius. These show the fitting mean of the gaze-induced LC effective strain in low, baseline, and high levels. 0 = baseline level; −1 = low level; and 1 = high level.

Figure 5 shows the morphology of the undeformed and deformed FE models and color-coded strains (effective, first principal, and third principal strains) of ONH and LC in models with low and high levels of these five factors. Note that the color-coded strains represent the combined effects of IOP, CSFP, and eye movement.

Figure 5.

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ONH deformations induced by an adduction of 13 degrees with the 5 main factors (ON tortuosity, dural radius, ON radius, scleral thickness, and LC depth) at their low (A) and high (B–F) levels, respectively. The enlarged views of the ONH and LC show the color-coded strains, including the effective strains, and the first and third principal strains. In the enlarged views, the first value represents the LC strain induced by IOP, CSFP, and eye movement, and the second value in parentheses represents the gaze-induced LC strain after removing the effects of IOP and CSFP. Note that the LC deformations were exaggerated three times for illustration purposes.

The Effects of 5 Morphological Parameters and 5 Tissue Stiffnesses

In the DOE analysis, the factors and their interactions accounted for 99.62% (error = 0.38%) of the total effects in the responses. Details on the effects of 40 statistically significant factors are available in the Supplementary Material SA-2. Among these factors, the key main effect and interactions, in order, are: ON tortuosity (−0.0071; 36.15%), dural radius (0.0058; 24.35%), ON radius (−0.0045; 14.30%), dural stiffness (0.0036; 9.45%), scleral thickness (−0.0022;3.56%), LC stiffness (−0.0021; 3.28%), LC depth (−0.0017; 2.11%), the interaction between the dural radius and the dural stiffness (0.0012; 0.99%), the interaction between the dural radius and the ON radius (0.00093; 0.62%), and the interaction between the dural radius and the ON tortuosity (−0.0009; 0.58%). These factors accounted for 95.39% of the total effects in the responses. Similarly, only the nonlinear of ON tortuosity and dural radius were significant. Detailed main effects and interaction plots of the first principal and third principal strains can be found in the Supplementary Material SA-3.

Figure 6 illustrates the influential main effect and interactions. The trends for the five morphological parameters were consistent with the results from the above morphological DOE analysis. For biomechanical properties, a stiffer dura increased LC strains, whereas a stiffer LC reduced strains. The pronounced interactions occurred between dural radius with dural stiffness, ON radius, and ON tortuosity. Specifically, the increase in LC strain associated with a stiffer dura was found to be amplified when combined with a larger dural radius. Conversely, the decrease in LC strain associated with a larger ON radius was found to be amplified when combined with a smaller dural radius. Additionally, the nonlinear decrease in LC strain associated with a larger ON tortuosity was found to be amplified when combined with a larger dural radius.

Figure 6.

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(A) The main effects of key factors: ON tortuosity, dural radius, ON radius, dural stiffness, scleral thickness, LC stiffness, and LC depth. The squared effects of ON tortuosity and dural radius were significant. (B) The interactions between dural radius with dural stiffness, ON radius, and ON tortuosity. These show the fitting mean of the LC effective strain in low, baseline and high levels. 0 = baseline level; −1 = low level; and 1 = high level.

Discussion

In this study, we developed a parametric FE model and studied the effects of eye globe and ON morphologies, as well as tissue stiffnesses on ONH deformations during eye movements. Our models demonstrated that ON tortuosity, dural radius, ON radius, scleral thickness, and LC depth were the five key morphological factors that significantly affect gaze-induced ONH deformations. These parameters retained their significance in a combined analysis with tissue stiffnesses. We also observed a significant interaction between the dural radius and stiffness, proving to be a considerable factor in the ONH's response to eye movement.

Comparison of Gaze-Induced LC Strains

In this study, we reported gaze-induced LC strains as the responses, including effective, first and third principal strains. We identified the same five key factors (ON tortuosity, dural radius, ON radius, scleral thickness, and LC depth) regardless of the strain type used as the output measure. Although the rankings of these factors varied slightly with the strain type, the overall impact trend on the output measure was consistent. Similar results were found in the second DOE analysis about morphological and biomechanical factors. These findings aligned with those of a previous study,¹⁹ which also observed that the ranking of factors varies depending on the output strain types. The deformation of a point in 3D is complex and can be described by a strain tensor. From this tensor, various types of strains are derived to capture the main characteristics of deformations, including effective, first principal, and third principal strains. Effective strain is a scalar measure of the overall deformation behavior, independent of the deformation direction. However, it may not capture the effects of individual strain components, and its physical interpretation may not always be straightforward. The first principal strain quantifies the largest stretch experienced by the tissue, whereas the third principal strain represented the largest compressive strain experienced by the tissue. Although these strains can be interpreted as overall, tensile, or compressive, their biological impact remains unclear. Reporting all three strain types in this study provides a comprehensive description of deformations, which will be useful for future research.

As this study focused on the effects of geometric parameters and material properties on gaze-induced strains, we have isolated the strains induced by eye movements alone as the main output. However, it is important to note that the actual insult on neural tissue is represented by the total strain experienced by the ONH, which is induced by IOP, CSFP, and eye movement in our models. We have analyzed the impact of factors on the total strain and found that the impact of individual factors followed a consistent trend with gaze-induced strain alone, albeit with varying rankings of each factor (see Supplementary Material SA-4 for detailed information). These results suggest that ONH tissues respond to loading in a complex manner and the ranking of the importance of various parameters may change depending on the specific loading conditions.

A Larger Dural Radius Increased LC Strains During Eye Movements

Our study found that a larger dural radius (i.e. the ON sheath inner radius) leads to higher gaze-induced LC strains, similar to the effect of a stiffer dura. During eye movements, peripapillary tissues are sheared in the transverse plane by the ON sheath, resulting in significant deformations.⁷ An increased dural radius would tend to restrict eye movements by exerting a larger pulling force onto the ONH, as evidenced by the calculated traction force from the FE models (Supplementary Material SA-5). The potential for such forces to increase ONH deformations and the possibility to cause axonal death in glaucoma require further investigation.

A few studies have reported that the optic nerve sheath diameter (ONSD) is larger in normal tension glaucoma (NTG) eyes compared with healthy controls,³⁰^,³¹ suggesting that the ONH in these subjects may experience more mechanical insults during eye movement. However, the ONSD in subjects with NTG varies across different cohorts,³²^,³³ and its associations with NTG need to be further investigated. The role of dural morphology in these variations remains unexplored, and further studies are warranted.

A Larger ON Tortuosity Decreased LC Strains During Eye Movements

Our study suggested that increased ON tortuosity may lead to lower LC strains during eye movements. In addition, this factor exhibited a significant nonlinear shape in the scope of our study. With the increase of ON tortuosity, the reduction rate of gaze-induced LC strains increases. A relatively taut ON has the potential to exhibit rapid straightening during eye movements and thus exert more force on the ONH tissue, a phenomenon observed in our FE models (Supplementary Material SA-5). The ON tortuosity shows significant variation between healthy subjects and patients with glaucoma¹⁵^,³⁴ and high myopia.¹⁶ The effect of ON tortuosity on corresponding diseases needs to be studied.

A Thicker Sclera Decreased LC Strains During Eye Movements

Our study showed that scleral thickness significantly affects gaze-induced LC strains, with a thicker sclera associated with lower LC strains during eye movements. This finding aligns with other studies examining the effects of factors on IOP-¹⁷^,³⁵ and CSFP-induced⁵ LC strains. Because other ONH tissues are relatively compliant compared to the sclera, scleral deformation induced by eye movement can be directly transmitted to surrounding tissues, suggesting that eyes with a thinner sclera may be more sensitive to eye movements.

In case of high myopia, scleral thickness decreases significantly with increasing axial length.³⁶ In severe cases, scleral thickness can be as low as 31% of that in healthy subjects,³⁷ potentially leading to the development of staphylomas. The reduced scleral thickness³⁶ in high myopia could result in large LC deformations and increasing susceptibility to ONH damage during eye movements. Additionally, the elastically weak sclera³⁸^,³⁹ in high myopia could make it more susceptible to ONH deformations. However, as discussed above, ONH deformations in highly myopic eyes are also affected by axial length and ON tortuosity. Further studies are warranted to investigate the interactions of these factors in high myopia.

Other Factors Affecting LC Strains During Eye Movements

Our study showed that a larger ON radius (i.e. the ON parenchyma, excluding the dural and pial sheaths) had a protective effect, resulting in smaller LC strains during eye movements. There is a typical shear deformation due to ON traction, with clear temporal pulling from the dura in adduction. We speculated that a larger ON radius might possibly provide more mechanical support to the LC during ON traction, which could potentially lead to smaller gaze-induced LC strains. Direct measurement of ON size with MRI imaging⁴⁰^,⁴¹ have shown that the ON radius of subjects with glaucoma was smaller than that of healthy controls, which was significantly correlated with retinal nerve fiber layer thickness thinning and perimetric loss. A recent study suggested that myopes also tended to have smaller ONs.⁴² In view of our results, smaller ON radius leads to the increase of gaze-induced LC strains, suggesting that this factor may cause glaucoma and myopia to be more sensitive to ONH deformations caused by eye movement. The link between ON size and the gaze-induced ONH deformations in glaucoma and myopia needs further exploration.

Our study also revealed that a larger LC depth resulted in smaller gaze-induced LC strains. In this study, a greater LC depth corresponds to a more curved LC. We speculated that a larger LC depth may have greater compliance during eye movements, which could potentially lead to smaller gaze-induced LC strains. Previous studies⁴³^–⁴⁶ showed that the LC depth and LC curvature were significantly larger in POAG eyes than in healthy eyes. However, it remains unclear whether a larger LC depth/curvature is protective or detrimental in the development of glaucoma. Because LC morphology varies with race, sex, age, and axial length,⁴⁷^–⁵⁰ the relationship between the effect of LC morphology on gaze-induced ONH deformations and the development of glaucoma needs to be further studied.

Last, our study demonstrated the significance of tissue stiffness on gaze-induced LC deformations. A stiffer dura increases LC strains, which is consistent with our previous study.⁶ In addition, LC stiffness also had a strong influence on gazed-induced LC strains, where a stiffer LC resulted in a reduction of LC strains. This is straightforward as a stiffer material will deform less under the same loading condition. This observation is consistent with other studies investigating LC strains induced by IOP⁵¹^,⁵² and CSFP.¹⁸ However, the effect of tissue stiffness on gazed-induced LC strains is smaller than that of the three main morphologies (ON tortuosity, dural radius, and ON radius; see Fig. 6).

The Interactions Affecting LC Strains During Eye Movements

This study revealed significant interactions among various factors, with the most notable being between dural radius and dural stiffness. A stiffer dura tends to increase gaze-induced LC strains, with this effect being amplified by a larger dural radius. In addition, dural radius exhibits a significant interaction with ON radius and ON tortuosity. Specifically, a larger ON radius tends to decrease gaze-induced LC strains, and this effect is amplified with a smaller dural radius. However, a larger ON tortuosity tends to nonlinearly decrease gaze-induced LC strains, with this effect being amplified by a larger dural radius. These findings underscore the importance of considering individual specific characteristics, such as eye globe and ON morphologies, as well as their biomechanical properties, in assessing the susceptibility of LC deformation during eye movements. Given the complex and multifaceted nature of morphological and biomechanical properties of the ONH, our parametric FE models provided an ideal platform for studying and quantifying the main factors and their interactions in a systematic manner to inform future experimental study design and analysis.

Limitations

In this study, several limitations warrant further discussion. First, it is important to acknowledge that variations in the dural radius were accompanied by changes in the length of the scleral flange, as the insertion point of the dura into the sclera was altered. Consequently, an increase in dural radius resulted in an enlargement of the scleral flange in the model. Although this relationship aligns with anatomic observations, where the dural radius and scleral flange size are positively correlated,⁴^,³² this confounding factor complicates the interpretation of the effect of a larger dura radius. To isolate the effects of dural radius and scleral flange size, we ran additional simulations (Supplementary Material SA-6). By increasing the dural and ON radius while keeping the scleral flange size constant, we found that a larger dural radius led to increased LC strains (despite the concurrent increase in ON radius, which typically reduces LC strain). Although the 0.2% increase is small, it reveals a clear trend. This approach confirmed that an increased dural radius contributes to higher LC strains.

Second, the morphological factors in our study included dural radius, but not dural thickness. To enhance our understandings, future studies should examine the effect of an increased dural thickness on LC deformation during eye movements.

Third, we varied these parameters by 20% from their baseline values, as the proper physiologic ranges for each parameter are unknown. Information on the precise physiological ranges and frequencies of these factors in populations would enhance our understanding of the relative importance of these factors. Although the baseline values were based on the literature with reasonable assumptions, the choice of the baseline values and ranges could affect the effects of factors and their relative importance. Thus, it is important to interpret the results within this context.

Finally, our model uses a single, idealized anatomy based on average tissue dimensions and material properties taken from various studies. Although these simplifications may not capture some of the complex behavior of the ONH, they provided a reasonable approximation, allowing us to improve our understanding of ONH biomechanics during eye movements. It will be necessary to update this work as more biomechanical information on eye and orbital tissues becomes available.

Conclusions

Our parametric finite element models demonstrated that ON tortuosity, dural radius, ON radius, scleral thickness, and LC depth were the five most important morphological factors influencing gaze-induced ONH deformations. Additionally, the stiffnesses of dura and LC were the most important biomechanical factors influencing gaze-induced ONH deformations, and the interactions between dural radius with the dural stiffness, ON radius, and ON tortuosity were significant. Our study provides an ideal platform for studying and quantifying the main factors and interactions between factors to inform experimental design and analysis. Further experimental and clinical studies are needed to explore the role of effect of individual specific characteristics on gaze-induced ONH deformations in ocular diseases, such as myopia and glaucoma.

Acknowledgments

The authors thank (1) the National Natural Science Foundation of China (12272030, 12002025), (2) the donors of the National Glaucoma Research, a program of the BrightFocus Foundation, for support of this research (G2021010S [M.G.]), (3) NMRC-LCG grant “TAckling & Reducing Glaucoma Blindness with Emerging Technologies (TARGET),” award ID: MOH-OFLCG21jun-0003 [M.G.], (4) the “Retinal Analytics through Machine learning aiding Physics (RAMP)” project that is supported by the National Research Foundation, Prime Minister's Office, Singapore under its Intra-Create Thematic Grant “Intersection Of Engineering And Health” - NRF2019-THE002-0006 awarded to the Singapore MIT Alliance for Research and Technology (SMART) Centre [M.G.].

Disclosure: T. Liu, None; P.T. Hung, None; X. Wang, None; M.J.A. Girard, None

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